I.N. Bronshtein · K.A. Semendyayev · G. Musiol · H. Muehlig
Handbook of Mathematics
I.N. Bronshtein · K.A. Semendyay...
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I.N. Bronshtein · K.A. Semendyayev · G. Musiol · H. Muehlig
Handbook of Mathematics
I.N. Bronshtein · K.A. Semendyayev · G. Musiol · H. Muehlig
Handbook of Mathematics 5th Ed.
With 745 Figures and 142 Tables
123
Ilja N. Bronshtein † Konstantin A. Semendyayev † Prof. Dr. Gerhard Musiol Prof. Dr. Heiner Muehlig
Based on the 6th edition of Bronshtein/Semendyayev/Musiol/Muehlig “Taschenbuch der Mathematik”, 2005. Published by Wissenschaftlicher Verlag Harri Deutsch GmbH, Frankfurt am Main.
Library of Congress Control Number: 2007930331
ISBN 978-3-540-72121-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Typesetting by the authors Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover: WMXDesign GmbH, Heidelberg Translation: Gabriela Sz´ ep, Budapest Printed on acid-free paper
62/3180/YL - 5 4 3 2 1 0
Preface to the Fifth English Edition This fth edition is based on the fourth English edition (2003) and corresponds to the improved sixth German edition (2005). It contains all the chapters of the both mentioned editions, but in a renewed revised and extendet form. So in the work at hand, the classical areas of Engineering Mathematics required for current practice are presented, such as \Arithmetic", \Functions", \Geometry", \Linear Algebra", \Algebra and Discrete Mathematics", (including \Logic", \Set Theory", \Classical Algebraic Structures", \Finite Fields", \Elementary Number Theory", "Cryptology", \Universal Algebra", \Boolean Algebra and Swich Algebra", \Algorithms of Graph Theory", \Fuzzy Logic"), \Di erentiation", \Integral Calculus", \Differential Equations", \Calculus of Variations", \Linear Integral Equations", \Functional Analysis", \Vector Analysis and Vector Fields", \Function Theory", \Integral Transformations", \Probability Theory and Mathematical Statistics". Fields of mathematics that have gained importance with regards to the increasing mathematical modeling and penetration of technical and scienti c processes also receive special attention. Included amongst these chapters are \Stochastic Processes and Stochastic Chains" as well as \Calculus of Errors", \Dynamical Systems and Chaos", \Optimization", \Numerical Analysis", \Using the Computer" and \Computer Algebra Systems". The chapter 21 containing a large number of useful tables for practical work has been completed by adding tables with the physical units of the International System of Units (SI). Dresden, February 2007 Prof. Dr. Gerhard Musiol Prof. Dr. Heiner Muhlig
From the Preface to the Fourth English Edition The \Handbook of Mathematics" by the mathematician, I. N. Bronshtein and the engineer, K. A. Semendyayev was designed for engineers and students of technical universities. It appeared for the
rst time in Russian and was widely distributed both as a reference book and as a text book for colleges and universities. It was later translated into German and the many editions have made it a permanent xture in German-speaking countries, where generations of engineers, natural scientists and others in technical training or already working with applications of mathematics have used it. On behalf of the publishing house Harri Deutsch, a revision and a substantially enlarged edition was prepared in 1992 by Gerhard Musiol and Heiner Muhlig, with the goal of giving "Bronshtein" the modern practical coverage requested by numerous students, university teachers and practitioners. The original style successfully used by the authors has been maintained. It can be characterized as \short, easily understandable, comfortable to use, but featuring mathematical accuracy (at a level of detail consistent with the needs of engineers)" . Since 2000, the revised and extended fth German edition of the revision has been on the market. Acknowledging the success that \Bronstein" has experienced in the German-speaking countries, Springer-Verlag Heidelberg/Germany is publishing a fourth English edition, which corresponds to the improved and extended fth German edition. The book is enhanced with over a thousand complementary illustrations and many tables. Special functions, series expansions, inde nite, de nite and elliptic integrals as well as integral transformations See Preface to the First Russian Edition
VI and statistical distributions are supplied in an extensive appendix of tables. In order to make the reference book more e ective, clarity and fast access through a clear structure were the goals, especially through visual clues as well as by a detailed technical index and colored tabs. An extended bibliography also directs users to further resources. We would like to cordially thank all readers and professional colleagues who helped us with their valuable statements, remarks and suggestions on the German edition of the book during the revision process. Special thanks go to Mrs. Professor Dr. Gabriela Szep (Budapest), who made this English debut version possible. Furthermore our thanks go to the co-authors for the critical treatment of their chapters. Dresden, June 2003 Prof. Dr. Gerhard Musiol Prof. Dr. Heiner Muhlig
Co-authors Some chapters and sections originated through a cooperation with the co-authors.
Chapter or Section
Spherical Trigonometry (3.4.1{3.4.3) Spherical Curves (3.4.3.4) Logic (5.1), Set Theory (5.2), Classic Algebraic Structures (5.3), Applications of Groups, Rings and Fields, Vektor Spaces (besides 5.3.4, 5.3.7.6) Universal Algebra (5.6), Boolean Algebra and Switch Algebra (5.7) Groups Representations, Applications of Groups (5.3.4, 5.3.5.4{5.3.5.6) Elementary Number Theory (5.4), Cryptology (5.5), Graphs (5.8) Fuzzy{Logic (5.9) Non-Linear Partial Di erential Equations, Solitonen (9.2.4) Linear Integral Equations (11.) Optimization (18.) Functional Analyzis (12.) Elliptic Functions (14.6) Dynamical Systems and Chaos (17.) Computer Algebra Systems (19.8.4, 20.)
Co-author
Dr. H. Nickel , Dresden Prof. L. Marsolek, Berlin
Dr. J. Brunner, Dresden Prof. Dr. R. Reif, Dresden Prof. Dr. U. Baumann, Dresden Prof. Dr. A. Grauel, Soest Prof. Dr. P. Ziesche, Dresden Dr. I. Steinert, Dusseldorf Prof. Dr. M. Weber, Dresden Dr. N. M. Fleischer , Moscow Prof. Dr. V. Reitmann, Dresden, St. Petersburg Prof. Dr. G. Flach, Dresden
Contents VII
Contents List of Tables 1 Arithmetic 1.1
1.2
Elementary Rules for Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1.1 Natural, Integer, and Rational Numbers . . . . . . . . . . . . . . . 1.1.1.2 Irrational and Transcendental Numbers . . . . . . . . . . . . . . . 1.1.1.3 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1.4 Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1.5 Commensurability . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Methods for Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2.1 Direct Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2.2 Indirect Proof or Proof by Contradiction . . . . . . . . . . . . . . 1.1.2.3 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . 1.1.2.4 Constructive Proof . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Sums and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3.1 Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Powers, Roots, and Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4.1 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4.2 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4.3 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4.4 Special Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5.2 Algebraic Expressions in Detail . . . . . . . . . . . . . . . . . . . 1.1.6 Integral Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6.1 Representation in Polynomial Form . . . . . . . . . . . . . . . . . 1.1.6.2 Factorizing a Polynomial . . . . . . . . . . . . . . . . . . . . . . . 1.1.6.3 Special Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6.4 Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6.5 Determination of the Greatest Common Divisor of Two Polynomials 1.1.7 Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.7.1 Reducing to the Simplest Form . . . . . . . . . . . . . . . . . . . . 1.1.7.2 Determination of the Integral Rational Part . . . . . . . . . . . . . 1.1.7.3 Decomposition into Partial Fractions . . . . . . . . . . . . . . . . 1.1.7.4 Transformations of Proportions . . . . . . . . . . . . . . . . . . . 1.1.8 Irrational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 De nition of a Finite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Special Finite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Mean Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5.1 Arithmetic Mean or Arithmetic Average . . . . . . . . . . . . . . 1.2.5.2 Geometric Mean or Geometric Average . . . . . . . . . . . . . . . 1.2.5.3 Harmonic Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5.4 Quadratic Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . .
XL 1 1 1 1 2 2 3 4 4 5 5 5 6 6 6 7 7 7 8 9 9 10 10 11 11 11 11 12 12 14 14 14 15 15 17 17 18 18 18 19 19 19 19 20 20 20
VIII Contents 1.3
1.4
1.5
1.2.5.5 Relations Between the Means of Two Positive Values . . . . . . . . Business Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Calculation of Interest or Percentage . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Calculation of Compound Interest . . . . . . . . . . . . . . . . . . . . . . . 1.3.2.1 Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2.2 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Amortization Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3.1 Amortization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3.2 Equal Principal Repayments . . . . . . . . . . . . . . . . . . . . . 1.3.3.3 Equal Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Annuity Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4.1 Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4.2 Future Amount of an Ordinary Annuity . . . . . . . . . . . . . . . 1.3.4.3 Balance after n Annuity Payments . . . . . . . . . . . . . . . . . 1.3.5 Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Pure Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1.2 Properties of Inequalities of Type I and II . . . . . . . . . . . . . . 1.4.2 Special Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.1 Triangle Inequality for Real Numbers . . . . . . . . . . . . . . . . 1.4.2.2 Triangle Inequality for Complex Numbers . . . . . . . . . . . . . . 1.4.2.3 Inequalities for Absolute Values of Di erences of Real and Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.4 Inequality for Arithmetic and Geometric Means . . . . . . . . . . 1.4.2.5 Inequality for Arithmetic and Quadratic Means . . . . . . . . . . . 1.4.2.6 Inequalities for Di erent Means of Real Numbers . . . . . . . . . . 1.4.2.7 Bernoulli's Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.8 Binomial Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.9 Cauchy{Schwarz Inequality . . . . . . . . . . . . . . . . . . . . . 1.4.2.10 Chebyshev Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.11 Generalized Chebyshev Inequality . . . . . . . . . . . . . . . . . . 1.4.2.12 Holder Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.13 Minkowski Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Solution of Linear and Quadratic Inequalities . . . . . . . . . . . . . . . . . 1.4.3.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3.2 Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3.3 Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3.4 General Case for Inequalities of Second Degree . . . . . . . . . . . Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Imaginary and Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . 1.5.1.1 Imaginary Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1.2 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Geometric Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2.1 Vector Representation . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2.2 Equality of Complex Numbers . . . . . . . . . . . . . . . . . . . . 1.5.2.3 Trigonometric Form of Complex Numbers . . . . . . . . . . . . . . 1.5.2.4 Exponential Form of a Complex Number . . . . . . . . . . . . . . 1.5.2.5 Conjugate Complex Numbers . . . . . . . . . . . . . . . . . . . . 1.5.3 Calculation with Complex Numbers . . . . . . . . . . . . . . . . . . . . . . 1.5.3.1 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . 1.5.3.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20 21 21 22 22 22 23 23 23 24 25 25 25 25 26 28 28 28 29 30 30 30 30 30 30 30 30 31 31 31 32 32 32 33 33 33 33 33 34 34 34 34 34 34 34 35 35 36 36 36 36
Contents IX
1.6
1.5.3.3 Division . . . . . . . . . . . . . . . . . . . . . . . 1.5.3.4 General Rules for the Basic Operations . . . . . . 1.5.3.5 Taking Powers of Complex Numbers . . . . . . . . 1.5.3.6 Taking of the n-th Root of a Complex Number . . Algebraic and Transcendental Equations . . . . . . . . . . . . . . . 1.6.1 Transforming Algebraic Equations to Normal Form . . . . . 1.6.1.1 De nition . . . . . . . . . . . . . . . . . . . . . . 1.6.1.2 Systems of n Algebraic Equations . . . . . . . . . 1.6.1.3 Superuous Roots . . . . . . . . . . . . . . . . . . 1.6.2 Equations of Degree at Most Four . . . . . . . . . . . . . . 1.6.2.1 Equations of Degree One (Linear Equations) . . . 1.6.2.2 Equations of Degree Two (Quadratic Equations) . 1.6.2.3 Equations of Degree Three (Cubic Equations) . . 1.6.2.4 Equations of Degree Four . . . . . . . . . . . . . . 1.6.2.5 Equations of Higher Degree . . . . . . . . . . . . 1.6.3 Equations of Degree n . . . . . . . . . . . . . . . . . . . . . 1.6.3.1 General Properties of Algebraic Equations . . . . 1.6.3.2 Equations with Real Coecients . . . . . . . . . . 1.6.4 Reducing Transcendental Equations to Algebraic Equations 1.6.4.1 De nition . . . . . . . . . . . . . . . . . . . . . . 1.6.4.2 Exponential Equations . . . . . . . . . . . . . . . 1.6.4.3 Logarithmic Equations . . . . . . . . . . . . . . . 1.6.4.4 Trigonometric Equations . . . . . . . . . . . . . . 1.6.4.5 Equations with Hyperbolic Functions . . . . . . .
2 Functions 2.1
Notion of Functions . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 De nition of a Function . . . . . . . . . . . . . . . . . . 2.1.1.1 Function . . . . . . . . . . . . . . . . . . . . . 2.1.1.2 Real Functions . . . . . . . . . . . . . . . . . 2.1.1.3 Functions of Several Variables . . . . . . . . . 2.1.1.4 Complex Functions . . . . . . . . . . . . . . . 2.1.1.5 Further Functions . . . . . . . . . . . . . . . . 2.1.1.6 Functionals . . . . . . . . . . . . . . . . . . . 2.1.1.7 Functions and Mappings . . . . . . . . . . . . 2.1.2 Methods for De ning a Real Function . . . . . . . . . . 2.1.2.1 De ning a Function . . . . . . . . . . . . . . . 2.1.2.2 Analytic Representation of a Function . . . . . 2.1.3 Certain Types of Functions . . . . . . . . . . . . . . . . 2.1.3.1 Monotone Functions . . . . . . . . . . . . . . 2.1.3.2 Bounded Functions . . . . . . . . . . . . . . . 2.1.3.3 Even Functions . . . . . . . . . . . . . . . . . 2.1.3.4 Odd Functions . . . . . . . . . . . . . . . . . 2.1.3.5 Representation with Even and Odd Functions . 2.1.3.6 Periodic Functions . . . . . . . . . . . . . . . 2.1.3.7 Inverse Functions . . . . . . . . . . . . . . . . 2.1.4 Limits of Functions . . . . . . . . . . . . . . . . . . . . 2.1.4.1 De nition of the Limit of a Function . . . . . . 2.1.4.2 De nition by Limit of Sequences . . . . . . . . 2.1.4.3 Cauchy Condition for Convergence . . . . . . 2.1.4.4 In nity as a Limit of a Function . . . . . . . . 2.1.4.5 Left-Hand and Right-Hand Limit of a Function
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X Contents
2.2
2.3
2.4
2.5 2.6
2.7
2.1.4.6 Limit of a Function as x Tends to In nity . . . . . . . . . . . 2.1.4.7 Theorems About Limits of Functions . . . . . . . . . . . . . 2.1.4.8 Calculation of Limits . . . . . . . . . . . . . . . . . . . . . . 2.1.4.9 Order of Magnitude of Functions and Landau Order Symbols 2.1.5 Continuity of a Function . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5.1 Notion of Continuity and Discontinuity . . . . . . . . . . . . 2.1.5.2 De nition of Continuity . . . . . . . . . . . . . . . . . . . . 2.1.5.3 Most Frequent Types of Discontinuities . . . . . . . . . . . . 2.1.5.4 Continuity and Discontinuity of Elementary Functions . . . . 2.1.5.5 Properties of Continuous Functions . . . . . . . . . . . . . . Elementary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.2 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.3 Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Transcendental Functions . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.1 Exponential Functions . . . . . . . . . . . . . . . . . . . . . 2.2.2.2 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 2.2.2.3 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 2.2.2.4 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . 2.2.2.5 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . 2.2.2.6 Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . 2.2.3 Composite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Linear Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Quadratic Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Cubic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Polynomials of n-th Degree . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Parabola of n-th Degree . . . . . . . . . . . . . . . . . . . . . . . . . Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Special Fractional Linear Function (Inverse Proportionality) . . . . . . 2.4.2 Linear Fractional Function . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Curves of Third Degree, Type I . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Curves of Third Degree, Type II . . . . . . . . . . . . . . . . . . . . . 2.4.5 Curves of Third Degree, Type III . . . . . . . . . . . . . . . . . . . . . 2.4.6 Reciprocal Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Square Root of a Linear Binomial . . . . . . . . . . . . . . . . . . . . 2.5.2 Square Root of a Quadratic Polynomial . . . . . . . . . . . . . . . . . 2.5.3 Power Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exponential Functions and Logarithmic Functions . . . . . . . . . . . . . . . 2.6.1 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Error Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Exponential Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5 Generalized Error Function . . . . . . . . . . . . . . . . . . . . . . . . 2.6.6 Product of Power and Exponential Functions . . . . . . . . . . . . . . Trigonometric Functions (Functions of Angles) . . . . . . . . . . . . . . . . . 2.7.1 Basic Notion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1.1 De nition and Representation . . . . . . . . . . . . . . . . . 2.7.1.2 Range and Behavior of the Functions . . . . . . . . . . . . . 2.7.2 Important Formulas for Trigonometric Functions . . . . . . . . . . . .
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Contents XI
2.7.2.1 Relations Between the Trigonometric Functions of the Same Angle (Addition Theorems) . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2.2 Trigonometric Functions of the Sum and Di erence of Two Angles 2.7.2.3 Trigonometric Functions of an Integer Multiple of an Angle . . . . 2.7.2.4 Trigonometric Functions of Half-Angles . . . . . . . . . . . . . . . 2.7.2.5 Sum and Di erence of Two Trigonometric Functions . . . . . . . . 2.7.2.6 Products of Trigonometric Functions . . . . . . . . . . . . . . . . 2.7.2.7 Powers of Trigonometric Functions . . . . . . . . . . . . . . . . . 2.7.3 Description of Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . 2.7.3.2 Superposition of Oscillations . . . . . . . . . . . . . . . . . . . . . 2.7.3.3 Vector Diagram for Oscillations . . . . . . . . . . . . . . . . . . . 2.7.3.4 Damping of Oscillations . . . . . . . . . . . . . . . . . . . . . . . 2.8 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 De nition of the Inverse Trigonometric Functions . . . . . . . . . . . . . . . 2.8.2 Reduction to the Principal Value . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3 Relations Between the Principal Values . . . . . . . . . . . . . . . . . . . . 2.8.4 Formulas for Negative Arguments . . . . . . . . . . . . . . . . . . . . . . . 2.8.5 Sum and Di erence of arcsin x and arcsin y . . . . . . . . . . . . . . . . . . 2.8.6 Sum and Di erence of arccos x and arccos y . . . . . . . . . . . . . . . . . . 2.8.7 Sum and Di erence of arctan x and arctan y . . . . . . . . . . . . . . . . . . 2.8.8 Special Relations for arcsin x arccos x arctan x . . . . . . . . . . . . . . . . 2.9 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 De nition of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . 2.9.2 Graphical Representation of the Hyperbolic Functions . . . . . . . . . . . . 2.9.2.1 Hyperbolic Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2.2 Hyperbolic Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2.3 Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2.4 Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . . 2.9.3 Important Formulas for the Hyperbolic Functions . . . . . . . . . . . . . . . 2.9.3.1 Hyperbolic Functions of One Variable . . . . . . . . . . . . . . . . 2.9.3.2 Expressing a Hyperbolic Function by Another One with the Same Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.3.3 Formulas for Negative Arguments . . . . . . . . . . . . . . . . . . 2.9.3.4 Hyperbolic Functions of the Sum and Di erence of Two Arguments (Addition Theorems) . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.3.5 Hyperbolic Functions of Double Arguments . . . . . . . . . . . . . 2.9.3.6 De Moivre Formula for Hyperbolic Functions . . . . . . . . . . . . 2.9.3.7 Hyperbolic Functions of Half-Argument . . . . . . . . . . . . . . . 2.9.3.8 Sum and Di erence of Hyperbolic Functions . . . . . . . . . . . . 2.9.3.9 Relation Between Hyperbolic and Trigonometric Functions with Complex Arguments z . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Area Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1.1 Area Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1.2 Area Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1.3 Area Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1.4 Area Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.2 Determination of Area Functions Using Natural Logarithm . . . . . . . . . 2.10.3 Relations Between Di erent Area Functions . . . . . . . . . . . . . . . . . . 2.10.4 Sum and Di erence of Area Functions . . . . . . . . . . . . . . . . . . . . . 2.10.5 Formulas for Negative Arguments . . . . . . . . . . . . . . . . . . . . . . .
79 79 79 80 81 81 82 82 82 82 83 83 84 84 84 85 86 86 86 86 87 87 87 88 88 88 88 89 89 89 89 89 89 90 90 90 90 91 91 91 91 91 92 92 92 93 93 93
XII Contents 2.11 Curves of Order Three (Cubic Curves) . . . . . . . . . . . . . 2.11.1 Semicubic Parabola . . . . . . . . . . . . . . . . . . . 2.11.2 Witch of Agnesi . . . . . . . . . . . . . . . . . . . . . 2.11.3 Cartesian Folium (Folium of Descartes) . . . . . . . . 2.11.4 Cissoid . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.5 Strophoide . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Curves of Order Four (Quartics) . . . . . . . . . . . . . . . . 2.12.1 Conchoid of Nicomedes . . . . . . . . . . . . . . . . . 2.12.2 General Conchoid . . . . . . . . . . . . . . . . . . . . 2.12.3 Pascal's Limacon . . . . . . . . . . . . . . . . . . . . 2.12.4 Cardioid . . . . . . . . . . . . . . . . . . . . . . . . . 2.12.5 Cassinian Curve . . . . . . . . . . . . . . . . . . . . . 2.12.6 Lemniscate . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Cycloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.1 Common (Standard) Cycloid . . . . . . . . . . . . . . 2.13.2 Prolate and Curtate Cycloids or Trochoids . . . . . . 2.13.3 Epicycloid . . . . . . . . . . . . . . . . . . . . . . . . 2.13.4 Hypocycloid and Astroid . . . . . . . . . . . . . . . . 2.13.5 Prolate and Curtate Epicycloid and Hypocycloid . . . 2.14 Spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14.1 Archimedean Spiral . . . . . . . . . . . . . . . . . . . 2.14.2 Hyperbolic Spiral . . . . . . . . . . . . . . . . . . . . 2.14.3 Logarithmic Spiral . . . . . . . . . . . . . . . . . . . 2.14.4 Evolvent of the Circle . . . . . . . . . . . . . . . . . . 2.14.5 Clothoid . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 Various Other Curves . . . . . . . . . . . . . . . . . . . . . . 2.15.1 Catenary Curve . . . . . . . . . . . . . . . . . . . . . 2.15.2 Tractrix . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Determination of Empirical Curves . . . . . . . . . . . . . . . 2.16.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . 2.16.1.1 Curve-Shape Comparison . . . . . . . . . . 2.16.1.2 Recti cation . . . . . . . . . . . . . . . . . . 2.16.1.3 Determination of Parameters . . . . . . . . . 2.16.2 Useful Empirical Formulas . . . . . . . . . . . . . . . 2.16.2.1 Power Functions . . . . . . . . . . . . . . . 2.16.2.2 Exponential Functions . . . . . . . . . . . . 2.16.2.3 Quadratic Polynomial . . . . . . . . . . . . 2.16.2.4 Rational Linear Function . . . . . . . . . . . 2.16.2.5 Square Root of a Quadratic Polynomial . . . 2.16.2.6 General Error Curve . . . . . . . . . . . . . 2.16.2.7 Curve of Order Three, Type II . . . . . . . . 2.16.2.8 Curve of Order Three, Type III . . . . . . . 2.16.2.9 Curve of Order Three, Type I . . . . . . . . 2.16.2.10 Product of Power and Exponential Functions 2.16.2.11 Exponential Sum . . . . . . . . . . . . . . . 2.16.2.12 Numerical Example . . . . . . . . . . . . . . 2.17 Scales and Graph Paper . . . . . . . . . . . . . . . . . . . . . 2.17.1 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . 2.17.2 Graph Paper . . . . . . . . . . . . . . . . . . . . . . . 2.17.2.1 Semilogarithmic Paper . . . . . . . . . . . . 2.17.2.2 Double Logarithmic Paper . . . . . . . . . . 2.17.2.3 Graph Paper with a Reciprocal Scale . . . .
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2.17.2.4 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.1 De nition and Representation . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.1.1 Representation of Functions of Several Variables . . . . . . . . . . 2.18.1.2 Geometric Representation of Functions of Several Variables . . . . 2.18.2 Di erent Domains in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.2.1 Domain of a Function . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.2.2 Two-Dimensional Domains . . . . . . . . . . . . . . . . . . . . . . 2.18.2.3 Three or Multidimensional Domains . . . . . . . . . . . . . . . . . 2.18.2.4 Methods to Determine a Function . . . . . . . . . . . . . . . . . . 2.18.2.5 Various Ways to De ne a Function . . . . . . . . . . . . . . . . . . 2.18.2.6 Dependence of Functions . . . . . . . . . . . . . . . . . . . . . . . 2.18.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.3.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.3.2 Exact De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.3.3 Generalization for Several Variables . . . . . . . . . . . . . . . . . 2.18.3.4 Iterated Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.5 Properties of Continuous Functions . . . . . . . . . . . . . . . . . . . . . . 2.18.5.1 Theorem on Zeros of Bolzano . . . . . . . . . . . . . . . . . . . . . 2.18.5.2 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . . 2.18.5.3 Theorem About the Boundedness of a Function . . . . . . . . . . . 2.18.5.4 Weierstrass Theorem (About the Existence of Maximum and Minimum) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19 Nomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.1 Nomograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.2 Net Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.3 Alignment Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.3.1 Alignment Charts with Three Straight-Line Scales Through a Point 2.19.3.2 Alignment Charts with Two Parallel and One Inclined Straight-Line Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.3.3 Alignment Charts with Two Parallel Straight Lines and a Curved Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.4 Net Charts for More Than Three Variables . . . . . . . . . . . . . . . . . .
3 Geometry 3.1
Plane Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.1 Point, Line, Ray, Segment . . . . . . . . . . . . . . 3.1.1.2 Angle . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.3 Angle Between Two Intersecting Lines . . . . . . . . 3.1.1.4 Pairs of Angles with Intersecting Parallels . . . . . . 3.1.1.5 Angles Measured in Degrees and in Radians . . . . . 3.1.2 Geometrical De nition of Circular and Hyperbolic Functions . 3.1.2.1 De nition of Circular or Trigonometric Functions . . 3.1.2.2 De nitions of the Hyperbolic Functions . . . . . . . 3.1.3 Plane Triangles . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3.1 Statements about Plane Triangles . . . . . . . . . . 3.1.3.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Plane Quadrangles . . . . . . . . . . . . . . . . . . . . . . . 3.1.4.1 Parallelogram . . . . . . . . . . . . . . . . . . . . . 3.1.4.2 Rectangle and Square . . . . . . . . . . . . . . . . .
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3.2
3.3
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3.5
3.1.4.3 Rhombus . . . . . . . . . . . . . . . . . . . . . . . 3.1.4.4 Trapezoid . . . . . . . . . . . . . . . . . . . . . . . 3.1.4.5 General Quadrangle . . . . . . . . . . . . . . . . . 3.1.4.6 Inscribed Quadrangle . . . . . . . . . . . . . . . . . 3.1.4.7 Circumscribing Quadrangle . . . . . . . . . . . . . 3.1.5 Polygons in the Plane . . . . . . . . . . . . . . . . . . . . . . 3.1.5.1 General Polygon . . . . . . . . . . . . . . . . . . . 3.1.5.2 Regular Convex Polygons . . . . . . . . . . . . . . . 3.1.5.3 Some Regular Convex Polygons . . . . . . . . . . . 3.1.6 The Circle and Related Shapes . . . . . . . . . . . . . . . . . 3.1.6.1 Circle . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6.2 Circular Segment and Circular Sector . . . . . . . . 3.1.6.3 Annulus . . . . . . . . . . . . . . . . . . . . . . . . Plane Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.1 Calculations in Right-Angled Triangles in the Plane 3.2.1.2 Calculations in General Triangles in the Plane . . . 3.2.2 Geodesic Applications . . . . . . . . . . . . . . . . . . . . . . 3.2.2.1 Geodetic Coordinates . . . . . . . . . . . . . . . . . 3.2.2.2 Angles in Geodesy . . . . . . . . . . . . . . . . . . 3.2.2.3 Applications in Surveying . . . . . . . . . . . . . . Stereometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Lines and Planes in Space . . . . . . . . . . . . . . . . . . . . 3.3.2 Edge, Corner, Solid Angle . . . . . . . . . . . . . . . . . . . . 3.3.3 Polyeder or Polyhedron . . . . . . . . . . . . . . . . . . . . . 3.3.4 Solids Bounded by Curved Surfaces . . . . . . . . . . . . . . Spherical Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Basic Concepts of Geometry on the Sphere . . . . . . . . . . 3.4.1.1 Curve, Arc, and Angle on the Sphere . . . . . . . . 3.4.1.2 Special Coordinate Systems . . . . . . . . . . . . . 3.4.1.3 Spherical Lune or Biangle . . . . . . . . . . . . . . 3.4.1.4 Spherical Triangle . . . . . . . . . . . . . . . . . . . 3.4.1.5 Polar Triangle . . . . . . . . . . . . . . . . . . . . . 3.4.1.6 Euler Triangles and Non-Euler Triangles . . . . . . 3.4.1.7 Trihedral Angle . . . . . . . . . . . . . . . . . . . . 3.4.2 Basic Properties of Spherical Triangles . . . . . . . . . . . . . 3.4.2.1 General Statements . . . . . . . . . . . . . . . . . . 3.4.2.2 Fundamental Formulas and Applications . . . . . . 3.4.2.3 Further Formulas . . . . . . . . . . . . . . . . . . . 3.4.3 Calculation of Spherical Triangles . . . . . . . . . . . . . . . 3.4.3.1 Basic Problems, Accuracy Observations . . . . . . . 3.4.3.2 Right-Angled Spherical Triangles . . . . . . . . . . 3.4.3.3 Spherical Triangles with Oblique Angles . . . . . . . 3.4.3.4 Spherical Curves . . . . . . . . . . . . . . . . . . . Vector Algebra and Analytical Geometry . . . . . . . . . . . . . . . 3.5.1 Vector Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1.1 De nition of Vectors . . . . . . . . . . . . . . . . . 3.5.1.2 Calculation Rules for Vectors . . . . . . . . . . . . . 3.5.1.3 Coordinates of a Vector . . . . . . . . . . . . . . . . 3.5.1.4 Directional Coecient . . . . . . . . . . . . . . . . 3.5.1.5 Scalar Product and Vector Product . . . . . . . . . 3.5.1.6 Combination of Vector Products . . . . . . . . . . .
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3.6
3.5.1.7 Vector Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1.8 Covariant and Contravariant Coordinates of a Vector . . . . . . 3.5.1.9 Geometric Applications of Vector Algebra . . . . . . . . . . . . . 3.5.2 Analytical Geometry of the Plane . . . . . . . . . . . . . . . . . . . . . . 3.5.2.1 Basic Concepts, Coordinate Systems in the Plane . . . . . . . . . 3.5.2.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . 3.5.2.3 Special Notation in the Plane . . . . . . . . . . . . . . . . . . . 3.5.2.4 Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2.5 Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2.6 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2.7 Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2.8 Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2.9 Quadratic Curves (Curves of Second Order or Conic Sections) . . 3.5.3 Analytical Geometry of Space . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3.1 Basic Concepts, Spatial Coordinate Systems . . . . . . . . . . . 3.5.3.2 Transformation of Orthogonal Coordinates . . . . . . . . . . . . 3.5.3.3 Special Quantities in Space . . . . . . . . . . . . . . . . . . . . . 3.5.3.4 Line and Plane in Space . . . . . . . . . . . . . . . . . . . . . . 3.5.3.5 Surfaces of Second Order, Equations in Normal Form . . . . . . . 3.5.3.6 Surfaces of Second Order or Quadratic Surfaces, General Theory Di erential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Plane Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.1 Ways to De ne a Plane Curve . . . . . . . . . . . . . . . . . . . 3.6.1.2 Local Elements of a Curve . . . . . . . . . . . . . . . . . . . . . 3.6.1.3 Special Points of a Curve . . . . . . . . . . . . . . . . . . . . . . 3.6.1.4 Asymptotes of Curves . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.5 General Discussion of a Curve Given by an Equation . . . . . . . 3.6.1.6 Evolutes and Evolvents . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.7 Envelope of a Family of Curves . . . . . . . . . . . . . . . . . . . 3.6.2 Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2.1 Ways to De ne a Space Curve . . . . . . . . . . . . . . . . . . . 3.6.2.2 Moving Trihedral . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2.3 Curvature and Torsion . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3.1 Ways to De ne a Surface . . . . . . . . . . . . . . . . . . . . . . 3.6.3.2 Tangent Plane and Surface Normal . . . . . . . . . . . . . . . . 3.6.3.3 Line Elements of a Surface . . . . . . . . . . . . . . . . . . . . . 3.6.3.4 Curvature of a Surface . . . . . . . . . . . . . . . . . . . . . . . 3.6.3.5 Ruled Surfaces and Developable Surfaces . . . . . . . . . . . . . 3.6.3.6 Geodesic Lines on a Surface . . . . . . . . . . . . . . . . . . . .
4 Linear Algebra 4.1
4.2
Matrices . . . . . . . . . . . . . . . . . . . . . 4.1.1 Notion of Matrix . . . . . . . . . . . . 4.1.2 Square Matrices . . . . . . . . . . . . . 4.1.3 Vectors . . . . . . . . . . . . . . . . . . 4.1.4 Arithmetical Operations with Matrices 4.1.5 Rules of Calculation for Matrices . . . . 4.1.6 Vector and Matrix Norms . . . . . . . . 4.1.6.1 Vector Norms . . . . . . . . . 4.1.6.2 Matrix Norms . . . . . . . . . Determinants . . . . . . . . . . . . . . . . . .
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4.4
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4.2.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Rules of Calculation for Determinants . . . . . . . . . . . . . . . . . . . . . 4.2.3 Evaluation of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Transformation of Coordinate Systems . . . . . . . . . . . . . . . . . . . . . 4.3.2 Tensors in Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Tensors with Special Properties . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3.1 Tensors of Rank 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3.2 Invariant Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Tensors in Curvilinear Coordinate Systems . . . . . . . . . . . . . . . . . . 4.3.4.1 Covariant and Contravariant Basis Vectors . . . . . . . . . . . . . 4.3.4.2 Covariant and Contravariant Coordinates of Tensors of Rank 1 . . 4.3.4.3 Covariant, Contravariant and Mixed Coordinates of Tensors of Rank 2 .................................... 4.3.4.4 Rules of Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Pseudotensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5.1 Symmetry with Respect to the Origin . . . . . . . . . . . . . . . . 4.3.5.2 Introduction to the Notion of Pseudotensors . . . . . . . . . . . . Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Linear Systems, Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1.2 Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1.3 Linear Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1.4 Calculation of the Inverse of a Matrix . . . . . . . . . . . . . . . . 4.4.2 Solution of Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . 4.4.2.1 De nition and Solvability . . . . . . . . . . . . . . . . . . . . . . . 4.4.2.2 Application of Pivoting . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2.3 Cramer's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2.4 Gauss's Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Overdetermined Linear Equation Systems . . . . . . . . . . . . . . . . . . . 4.4.3.1 Overdetermined Linear Systems of Equations and Linear Mean Square Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3.2 Suggestions for Numerical Solutions of Mean Square Value Problems Eigenvalue Problems for Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 General Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Special Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2.1 Characteristic Polynomial . . . . . . . . . . . . . . . . . . . . . . 4.5.2.2 Real Symmetric Matrices, Similarity Transformations . . . . . . . 4.5.2.3 Transformation of Principal Axes of Quadratic Forms . . . . . . . 4.5.2.4 Suggestions for the Numerical Calculations of Eigenvalues . . . . . 4.5.3 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . .
5 Algebra and Discrete Mathematics 5.1 5.2
Logic . . . . . . . . . . . . . . . . . . . . 5.1.1 Propositional Calculus . . . . . . 5.1.2 Formulas in Predicate Calculus . . Set Theory . . . . . . . . . . . . . . . . . 5.2.1 Concept of Set, Special Sets . . . . 5.2.2 Operations with Sets . . . . . . . 5.2.3 Relations and Mappings . . . . . 5.2.4 Equivalence and Order Relations . 5.2.5 Cardinality of Sets . . . . . . . . .
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5.3
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5.5
Classical Algebraic Structures . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3.1 De nition and Basic Properties . . . . . . . . . . . . . . 5.3.3.2 Subgroups and Direct Products . . . . . . . . . . . . . . 5.3.3.3 Mappings Between Groups . . . . . . . . . . . . . . . . . 5.3.4 Group Representations . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4.2 Particular Representations . . . . . . . . . . . . . . . . . 5.3.4.3 Direct Sum of Representations . . . . . . . . . . . . . . . 5.3.4.4 Direct Product of Representations . . . . . . . . . . . . . 5.3.4.5 Reducible and Irreducible Representations . . . . . . . . 5.3.4.6 Schur's Lemma 1 . . . . . . . . . . . . . . . . . . . . . . 5.3.4.7 Clebsch{Gordan Series . . . . . . . . . . . . . . . . . . . 5.3.4.8 Irreducible Representations of the Symmetric Group SM . 5.3.5 Applications of Groups . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5.1 Symmetry Operations, Symmetry Elements . . . . . . . . 5.3.5.2 Symmetry Groups or Point Groups . . . . . . . . . . . . 5.3.5.3 Symmetry Operations with Molecules . . . . . . . . . . . 5.3.5.4 Symmetry Groups in Crystallography . . . . . . . . . . . 5.3.5.5 Symmetry Groups in Quantum Mechanics . . . . . . . . 5.3.5.6 Further Applications of Group Theory in Physics . . . . . 5.3.6 Rings and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6.2 Subrings, Ideals . . . . . . . . . . . . . . . . . . . . . . . 5.3.6.3 Homomorphism, Isomorphism, Homomorphism Theorem 5.3.6.4 Finite Fields and Shift Registers . . . . . . . . . . . . . . 5.3.7 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.7.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.7.2 Linear Dependence . . . . . . . . . . . . . . . . . . . . . 5.3.7.3 Linear Mappings . . . . . . . . . . . . . . . . . . . . . . 5.3.7.4 Subspaces, Dimension Formula . . . . . . . . . . . . . . . 5.3.7.5 Euclidean Vector Spaces, Euclidean Norm . . . . . . . . . 5.3.7.6 Linear Operators in Vector Spaces . . . . . . . . . . . . . Elementary Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1.1 Divisibility and Elementary Divisibility Rules . . . . . . . 5.4.1.2 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . 5.4.1.3 Criteria for Divisibility . . . . . . . . . . . . . . . . . . . 5.4.1.4 Greatest Common Divisor and Least Common Multiple . 5.4.1.5 Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . . 5.4.2 Linear Diophantine Equations . . . . . . . . . . . . . . . . . . . . 5.4.3 Congruences and Residue Classes . . . . . . . . . . . . . . . . . . 5.4.4 Theorems of Fermat, Euler, and Wilson . . . . . . . . . . . . . . . 5.4.5 Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cryptology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Problem of Cryptology . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Cryptosystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Mathematical Foundation . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Security of Cryptosystems . . . . . . . . . . . . . . . . . . . . . . 5.5.4.1 Methods of Conventional Cryptography . . . . . . . . . .
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5.8
5.9
5.5.4.2 Linear Substitution Ciphers . . . . . . . . . . . . . 5.5.4.3 Vigenere Cipher . . . . . . . . . . . . . . . . . . . . 5.5.4.4 Matrix Substitution . . . . . . . . . . . . . . . . . . 5.5.5 Methods of Classical Cryptanalysis . . . . . . . . . . . . . . 5.5.5.1 Statistical Analysis . . . . . . . . . . . . . . . . . . 5.5.5.2 Kasiski{Friedman Test . . . . . . . . . . . . . . . . 5.5.6 One-Time Pad . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.7 Public Key Methods . . . . . . . . . . . . . . . . . . . . . . . 5.5.7.1 Die{Hellman Key Exchange . . . . . . . . . . . . 5.5.7.2 One-Way Function . . . . . . . . . . . . . . . . . . 5.5.7.3 RSA Method . . . . . . . . . . . . . . . . . . . . . 5.5.8 AES Algorithm (Advanced Encryption Standard) . . . . . . . 5.5.9 IDEA Algorithm (International Data Encryption Algorithm) Universal Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Congruence Relations, Factor Algebras . . . . . . . . . . . . 5.6.3 Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 Homomorphism Theorem . . . . . . . . . . . . . . . . . . . . 5.6.5 Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.6 Term Algebras, Free Algebras . . . . . . . . . . . . . . . . . Boolean Algebras and Switch Algebra . . . . . . . . . . . . . . . . . 5.7.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Duality Principle . . . . . . . . . . . . . . . . . . . . . . . . 5.7.3 Finite Boolean Algebras . . . . . . . . . . . . . . . . . . . . 5.7.4 Boolean Algebras as Orderings . . . . . . . . . . . . . . . . . 5.7.5 Boolean Functions, Boolean Expressions . . . . . . . . . . . . 5.7.6 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.7 Switch Algebra . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms of Graph Theory . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Basic Notions and Notation . . . . . . . . . . . . . . . . . . . 5.8.2 Traverse of Undirected Graphs . . . . . . . . . . . . . . . . . 5.8.2.1 Edge Sequences or Paths . . . . . . . . . . . . . . . 5.8.2.2 Euler Trails . . . . . . . . . . . . . . . . . . . . . . 5.8.2.3 Hamiltonian Cycles . . . . . . . . . . . . . . . . . . 5.8.3 Trees and Spanning Trees . . . . . . . . . . . . . . . . . . . . 5.8.3.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3.2 Spanning Trees . . . . . . . . . . . . . . . . . . . . 5.8.4 Matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.5 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.6 Paths in Directed Graphs . . . . . . . . . . . . . . . . . . . . 5.8.7 Transport Networks . . . . . . . . . . . . . . . . . . . . . . . Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.1 Basic Notions of Fuzzy Logic . . . . . . . . . . . . . . . . . . 5.9.1.1 Interpretation of Fuzzy Sets . . . . . . . . . . . . . 5.9.1.2 Membership Functions on the Real Line . . . . . . . 5.9.1.3 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . 5.9.2 Aggregation of Fuzzy Sets . . . . . . . . . . . . . . . . . . . 5.9.2.1 Concepts for Aggregation of Fuzzy Sets . . . . . . . 5.9.2.2 Practical Aggregator Operations of Fuzzy Sets . . . 5.9.2.3 Compensatory Operators . . . . . . . . . . . . . . . 5.9.2.4 Extension Principle . . . . . . . . . . . . . . . . . . 5.9.2.5 Fuzzy Complement . . . . . . . . . . . . . . . . . .
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336 336 336 337 337 337 338 338 338 339 339 339 340 340 340 340 341 341 341 341 342 342 343 343 343 344 345 346 348 348 351 351 352 353 354 354 355 356 357 357 358 360 360 360 361 363 365 365 366 368 368 368
Contents XIX
5.9.3 Fuzzy-Valued Relations . . . . . . . . . . . . . . . 5.9.3.1 Fuzzy Relations . . . . . . . . . . . . . . 5.9.3.2 Fuzzy Product Relation R S . . . . . . 5.9.4 Fuzzy Inference (Approximate Reasoning) . . . . . 5.9.5 Defuzzi cation Methods . . . . . . . . . . . . . . 5.9.6 Knowledge-Based Fuzzy Systems . . . . . . . . . . 5.9.6.1 Method of Mamdani . . . . . . . . . . . 5.9.6.2 Method of Sugeno . . . . . . . . . . . . . 5.9.6.3 Cognitive Systems . . . . . . . . . . . . 5.9.6.4 Knowledge-Based Interpolation Systems
6 Di erentiation 6.1
6.2
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Di erentiation of Functions of One Variable . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Di erential Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Rules of Di erentiation for Functions of One Variable . . . . . . . . . . . . . 6.1.2.1 Derivatives of the Elementary Functions . . . . . . . . . . . . . . . 6.1.2.2 Basic Rules of Di erentiation . . . . . . . . . . . . . . . . . . . . 6.1.3 Derivatives of Higher Order . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3.1 De nition of Derivatives of Higher Order . . . . . . . . . . . . . . 6.1.3.2 Derivatives of Higher Order of some Elementary Functions . . . . . 6.1.3.3 Leibniz's Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3.4 Higher Derivatives of Functions Given in Parametric Form . . . . . 6.1.3.5 Derivatives of Higher Order of the Inverse Function . . . . . . . . . 6.1.4 Fundamental Theorems of Di erential Calculus . . . . . . . . . . . . . . . . 6.1.4.1 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4.2 Fermat's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4.3 Rolle's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4.4 Mean Value Theorem of Di erential Calculus . . . . . . . . . . . . 6.1.4.5 Taylor's Theorem of Functions of One Variable . . . . . . . . . . . 6.1.4.6 Generalized Mean Value Theorem of Di erential Calculus (Cauchy's Theorem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5 Determination of the Extreme Values and Inection Points . . . . . . . . . . 6.1.5.1 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5.2 Necessary Conditions for the Existence of a Relative Extreme Value 6.1.5.3 Relative Extreme Values of a Di erentiable, Explicit Function . . . 6.1.5.4 Determination of Absolute Extrema . . . . . . . . . . . . . . . . . 6.1.5.5 Determination of the Extrema of Implicit Functions . . . . . . . . Di erentiation of Functions of Several Variables . . . . . . . . . . . . . . . . . . . . 6.2.1 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1.1 Partial Derivative of a Function . . . . . . . . . . . . . . . . . . . 6.2.1.2 Geometrical Meaning for Functions of Two Variables . . . . . . . . 6.2.1.3 Di erentials of x and f (x) . . . . . . . . . . . . . . . . . . . . . . 6.2.1.4 Basic Properties of the Di erential . . . . . . . . . . . . . . . . . . 6.2.1.5 Partial Di erential . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Total Di erential and Di erentials of Higher Order . . . . . . . . . . . . . . 6.2.2.1 Notion of Total Di erential of a Function of Several Variables (Complete Di erential) . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2.2 Derivatives and Di erentials of Higher Order . . . . . . . . . . . . 6.2.2.3 Taylor's Theorem for Functions of Several Variables . . . . . . . . 6.2.3 Rules of Di erentiation for Functions of Several Variables . . . . . . . . . . 6.2.3.1 Di erentiation of Composite Functions . . . . . . . . . . . . . . . 6.2.3.2 Di erentiation of Implicit Functions . . . . . . . . . . . . . . . . .
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XX Contents 6.2.4 Substitution of Variables in Di erential Expressions and Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4.1 Function of One Variable . . . . . . . . . . . . . . . . . . . . . . . 6.2.4.2 Function of Two Variables . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Extreme Values of Functions of Several Variables . . . . . . . . . . . . . . . 6.2.5.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5.2 Geometric Representation . . . . . . . . . . . . . . . . . . . . . . 6.2.5.3 Determination of Extreme Values of Functions of Two Variables . . 6.2.5.4 Determination of the Extreme Values of a Function of n Variables . 6.2.5.5 Solution of Approximation Problems . . . . . . . . . . . . . . . . 6.2.5.6 Extreme Value Problem with Side Conditions . . . . . . . . . . . .
7 In nite Series 7.1
7.2
7.3
Sequences of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Properties of Sequences of Numbers . . . . . . . . . . . . . . 7.1.1.1 De nition of Sequence of Numbers . . . . . . . . . . 7.1.1.2 Monotone Sequences of Numbers . . . . . . . . . . 7.1.1.3 Bounded Sequences . . . . . . . . . . . . . . . . . . 7.1.2 Limits of Sequences of Numbers . . . . . . . . . . . . . . . . Number Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 General Convergence Theorems . . . . . . . . . . . . . . . . 7.2.1.1 Convergence and Divergence of In nite Series . . . . 7.2.1.2 General Theorems about the Convergence of Series . 7.2.2 Convergence Criteria for Series with Positive Terms . . . . . . 7.2.2.1 Comparison Criterion . . . . . . . . . . . . . . . . . 7.2.2.2 D'Alembert's Ratio Test . . . . . . . . . . . . . . . 7.2.2.3 Root Test of Cauchy . . . . . . . . . . . . . . . . . 7.2.2.4 Integral Test of Cauchy . . . . . . . . . . . . . . . . 7.2.3 Absolute and Conditional Convergence . . . . . . . . . . . . 7.2.3.1 De nition . . . . . . . . . . . . . . . . . . . . . . . 7.2.3.2 Properties of Absolutely Convergent Series . . . . . 7.2.3.3 Alternating Series . . . . . . . . . . . . . . . . . . . 7.2.4 Some Special Series . . . . . . . . . . . . . . . . . . . . . . . 7.2.4.1 The Values of Some Important Number Series . . . 7.2.4.2 Bernoulli and Euler Numbers . . . . . . . . . . . . 7.2.5 Estimation of the Remainder . . . . . . . . . . . . . . . . . . 7.2.5.1 Estimation with Majorant . . . . . . . . . . . . . . 7.2.5.2 Alternating Convergent Series . . . . . . . . . . . . 7.2.5.3 Special Series . . . . . . . . . . . . . . . . . . . . . Function Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . 7.3.2.1 De nition, Weierstrass Theorem . . . . . . . . . . . 7.3.2.2 Properties of Uniformly Convergent Series . . . . . 7.3.3 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3.1 De nition, Convergence . . . . . . . . . . . . . . . 7.3.3.2 Calculations with Power Series . . . . . . . . . . . . 7.3.3.3 Taylor Series Expansion, Maclaurin Series . . . . . . 7.3.4 Approximation Formulas . . . . . . . . . . . . . . . . . . . . 7.3.5 Asymptotic Power Series . . . . . . . . . . . . . . . . . . . . 7.3.5.1 Asymptotic Behavior . . . . . . . . . . . . . . . . . 7.3.5.2 Asymptotic Power Series . . . . . . . . . . . . . . .
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Contents XXI
7.4
Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Trigonometric Sum and Fourier Series . . . . . . . . . . . . . . . . 7.4.1.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1.2 Most Important Properties of the Fourier Series . . . . . 7.4.2 Determination of Coecients for Symmetric Functions . . . . . . . 7.4.2.1 Di erent Kinds of Symmetries . . . . . . . . . . . . . . . 7.4.2.2 Forms of the Expansion into a Fourier Series . . . . . . . 7.4.3 Determination of the Fourier Coecients with Numerical Methods 7.4.4 Fourier Series and Fourier Integrals . . . . . . . . . . . . . . . . . 7.4.5 Remarks on the Table of Some Fourier Expansions . . . . . . . . .
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Inde nite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Primitive Function or Antiderivative . . . . . . . . . . . . . . . . . . 8.1.1.1 Inde nite Integrals . . . . . . . . . . . . . . . . . . . . . . 8.1.1.2 Integrals of Elementary Functions . . . . . . . . . . . . . . 8.1.2 Rules of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Integration of Rational Functions . . . . . . . . . . . . . . . . . . . 8.1.3.1 Integrals of Integer Rational Functions (Polynomials) . . . 8.1.3.2 Integrals of Fractional Rational Functions . . . . . . . . . . 8.1.3.3 Four Cases of Partial Fraction Decomposition . . . . . . . . 8.1.4 Integration of Irrational Functions . . . . . . . . . . . . . . . . . . . 8.1.4.1 Substitution to Reduce to Integration of Rational Functions 8.1.4.2 Integration of Binomial Integrands . . . . . . . . . . . . . . 8.1.4.3 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . 8.1.5 Integration of Trigonometric Functions . . . . . . . . . . . . . . . . 8.1.5.1 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.5.2 Simpli ed Methods . . . . . . . . . . . . . . . . . . . . . . 8.1.6 Integration of Further Transcendental Functions . . . . . . . . . . . 8.1.6.1 Integrals with Exponential Functions . . . . . . . . . . . . 8.1.6.2 Integrals with Hyperbolic Functions . . . . . . . . . . . . . 8.1.6.3 Application of Integration by Parts . . . . . . . . . . . . . 8.1.6.4 Integrals of Transcendental Functions . . . . . . . . . . . . De nite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Basic Notions, Rules and Theorems . . . . . . . . . . . . . . . . . . 8.2.1.1 De nition and Existence of the De nite Integral . . . . . . 8.2.1.2 Properties of De nite Integrals . . . . . . . . . . . . . . . . 8.2.1.3 Further Theorems about the Limits of Integration . . . . . 8.2.1.4 Evaluation of the De nite Integral . . . . . . . . . . . . . . 8.2.2 Application of De nite Integrals . . . . . . . . . . . . . . . . . . . . 8.2.2.1 General Principles for Application of the De nite Integral . 8.2.2.2 Applications in Geometry . . . . . . . . . . . . . . . . . . 8.2.2.3 Applications in Mechanics and Physics . . . . . . . . . . . 8.2.3 Improper Integrals, Stieltjes and Lebesgue Integrals . . . . . . . . . 8.2.3.1 Generalization of the Notion of the Integral . . . . . . . . . 8.2.3.2 Integrals with In nite Integration Limits . . . . . . . . . . 8.2.3.3 Integrals with Unbounded Integrand . . . . . . . . . . . . . 8.2.4 Parametric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4.1 De nition of Parametric Integrals . . . . . . . . . . . . . . 8.2.4.2 Di erentiation Under the Symbol of Integration . . . . . . 8.2.4.3 Integration Under the Symbol of Integration . . . . . . . . 8.2.5 Integration by Series Expansion, Special Non-Elementary Functions .
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8 Integral Calculus 8.1
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XXII Contents 8.3
8.4
8.5
Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Line Integrals of the First Type . . . . . . . . . . . . . . . . . . 8.3.1.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1.2 Existence Theorem . . . . . . . . . . . . . . . . . . . 8.3.1.3 Evaluation of the Line Integral of the First Type . . . 8.3.1.4 Application of the Line Integral of the First Type . . . 8.3.2 Line Integrals of the Second Type . . . . . . . . . . . . . . . . 8.3.2.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2.2 Existence Theorem . . . . . . . . . . . . . . . . . . . 8.3.2.3 Calculation of the Line Integral of the Second Type . . 8.3.3 Line Integrals of General Type . . . . . . . . . . . . . . . . . . 8.3.3.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3.2 Properties of the Line Integral of General Type . . . . 8.3.3.3 Integral Along a Closed Curve . . . . . . . . . . . . . 8.3.4 Independence of the Line Integral of the Path of Integration . . 8.3.4.1 Two-Dimensional Case . . . . . . . . . . . . . . . . . 8.3.4.2 Existence of a Primitive Function . . . . . . . . . . . 8.3.4.3 Three-Dimensional Case . . . . . . . . . . . . . . . . 8.3.4.4 Determination of the Primitive Function . . . . . . . 8.3.4.5 Zero-Valued Integral Along a Closed Curve . . . . . . Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1.1 Notion of the Double Integral . . . . . . . . . . . . . 8.4.1.2 Evaluation of the Double Integral . . . . . . . . . . . 8.4.1.3 Applications of the Double Integral . . . . . . . . . . 8.4.2 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2.1 Notion of the Triple Integral . . . . . . . . . . . . . . 8.4.2.2 Evaluation of the Triple Integral . . . . . . . . . . . . 8.4.2.3 Applications of the Triple Integral . . . . . . . . . . . Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Surface Integral of the First Type . . . . . . . . . . . . . . . . 8.5.1.1 Notion of the Surface Integral of the First Type . . . . 8.5.1.2 Evaluation of the Surface Integral of the First Type . 8.5.1.3 Applications of the Surface Integral of the First Type 8.5.2 Surface Integral of the Second Type . . . . . . . . . . . . . . . 8.5.2.1 Notion of the Surface Integral of the Second Type . . 8.5.2.2 Evaluation of Surface Integrals of the Second Type . . 8.5.3 Surface Integral in General Form . . . . . . . . . . . . . . . . . 8.5.3.1 Notion of the Surface Integral in General Form . . . . 8.5.3.2 Properties of the Surface Integrals . . . . . . . . . . . 8.5.3.3 An Application of the Surface Integral . . . . . . . . .
9 Di erential Equations 9.1
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Ordinary Di erential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 First-Order Di erential Equations . . . . . . . . . . . . . . . . . . . . . . . 9.1.1.1 Existence Theorems, Direction Field . . . . . . . . . . . . . . . . . 9.1.1.2 Important Solution Methods . . . . . . . . . . . . . . . . . . . . . 9.1.1.3 Implicit Di erential Equations . . . . . . . . . . . . . . . . . . . . 9.1.1.4 Singular Integrals and Singular Points . . . . . . . . . . . . . . . . 9.1.1.5 Approximation Methods for Solution of First-Order Di erential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Di erential Equations of Higher Order and Systems of Di erential Equations
462 463 463 463 463 464 464 464 466 466 467 467 467 468 468 468 469 469 469 470 471 471 471 472 474 476 476 476 479 479 479 480 481 482 483 483 484 485 485 485 486
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Contents XXIII
9.2
9.1.2.1 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 9.1.2.2 Lowering the Order . . . . . . . . . . . . . . . . . . . . . . . . . . 499 9.1.2.3 Linear n-th Order Di erential Equations . . . . . . . . . . . . . . 500 9.1.2.4 Solution of Linear Di erential Equations with Constant Coecients 502 9.1.2.5 Systems of Linear Di erential Equations with Constant Coecients 505 9.1.2.6 Linear Second-Order Di erential Equations . . . . . . . . . . . . . 507 9.1.3 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 9.1.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 514 9.1.3.2 Fundamental Properties of Eigenfunctions and Eigenvalues . . . . 515 9.1.3.3 Expansion in Eigenfunctions . . . . . . . . . . . . . . . . . . . . . 516 9.1.3.4 Singular Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 Partial Di erential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 9.2.1 First-Order Partial Di erential Equations . . . . . . . . . . . . . . . . . . . 517 9.2.1.1 Linear First-Order Partial Di erential Equations . . . . . . . . . . 517 9.2.1.2 Non-Linear First-Order Partial Di erential Equations . . . . . . . 519 9.2.2 Linear Second-Order Partial Di erential Equations . . . . . . . . . . . . . . 522 9.2.2.1 Classi cation and Properties of Second-Order Di erential Equations with Two Independent Variables . . . . . . . . . . . . . . . . . . . 522 9.2.2.2 Classi cation and Properties of Linear Second-Order Di erential Equations with more than two Independent Variables . . . . . . . . . . 523 9.2.2.3 Integration Methods for Linear Second-Order Partial Di erential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 9.2.3 Some further Partial Di erential Equations From Natural Sciences and Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 9.2.3.1 Formulation of the Problem and the Boundary Conditions . . . . . 534 9.2.3.2 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 9.2.3.3 Heat Conduction and Di usion Equation for Homogeneous Media . 537 9.2.3.4 Potential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 538 9.2.3.5 Schrodinger's Equation . . . . . . . . . . . . . . . . . . . . . . . . 538 9.2.4 Non-Linear Partial Di erential Equations: Solitons, Periodic Patterns and Chaos 546 9.2.4.1 Formulation of the Physical-Mathematical Problem . . . . . . . . 546 9.2.4.2 Korteweg de Vries Equation (KdV) . . . . . . . . . . . . . . . . . 548 9.2.4.3 Non-Linear Schrodinger Equation (NLS) . . . . . . . . . . . . . . 549 9.2.4.4 Sine{Gordon Equation (SG) . . . . . . . . . . . . . . . . . . . . . 549 9.2.4.5 Further Non-linear Evolution Equations with Soliton Solutions . . 551
10 Calculus of Variations
10.1 De ning the Problem . . . . . . . . . . . . . . . . . . . . . . 10.2 Historical Problems . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Isoperimetric Problem . . . . . . . . . . . . . . . . . 10.2.2 Brachistochrone Problem . . . . . . . . . . . . . . . . 10.3 Variational Problems of One Variable . . . . . . . . . . . . . 10.3.1 Simple Variational Problems and Extremal Curves . . 10.3.2 Euler Di erential Equation of the Variational Calculus 10.3.3 Variational Problems with Side Conditions . . . . . . 10.3.4 Variational Problems with Higher-Order Derivatives . 10.3.5 Variational Problem with Several Unknown Functions 10.3.6 Variational Problems using Parametric Representation 10.4 Variational Problems with Functions of Several Variables . . . 10.4.1 Simple Variational Problem . . . . . . . . . . . . . . . 10.4.2 More General Variational Problems . . . . . . . . . . 10.5 Numerical Solution of Variational Problems . . . . . . . . . .
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XXIV Contents 10.6 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 10.6.1 First and Second Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 10.6.2 Application in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
11 Linear Integral Equations
11.1 Introduction and Classi cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Fredholm Integral Equations of the Second Kind . . . . . . . . . . . . . . . . . . . 11.2.1 Integral Equations with Degenerate Kernel . . . . . . . . . . . . . . . . . . 11.2.2 Successive Approximation Method, Neumann Series . . . . . . . . . . . . . 11.2.3 Fredholm Solution Method, Fredholm Theorems . . . . . . . . . . . . . . . 11.2.3.1 Fredholm Solution Method . . . . . . . . . . . . . . . . . . . . . . 11.2.3.2 Fredholm Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Numerical Methods for Fredholm Integral Equations of the Second Kind . . 11.2.4.1 Approximation of the Integral . . . . . . . . . . . . . . . . . . . . 11.2.4.2 Kernel Approximation . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4.3 Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Fredholm Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . 11.3.1 Integral Equations with Degenerate Kernels . . . . . . . . . . . . . . . . . . 11.3.2 Analytic Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Reduction of an Integral Equation into a Linear System of Equations . . . . 11.3.4 Solution of the Homogeneous Integral Equation of the First Kind . . . . . . 11.3.5 Construction of Two Special Orthonormal Systems for a Given Kernel . . . 11.3.6 Iteration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Theoretical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Solution by Di erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.3 Solution of the Volterra Integral Equation of the Second Kind by Neumann Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.4 Convolution Type Volterra Integral Equations . . . . . . . . . . . . . . . . 11.4.5 Numerical Methods for Volterra Integral Equations of the Second Kind . . . 11.5 Singular Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Abel Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Singular Integral Equation with Cauchy Kernel . . . . . . . . . . . . . . . . 11.5.2.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . 11.5.2.2 Existence of a Solution . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2.3 Properties of Cauchy Type Integrals . . . . . . . . . . . . . . . . . 11.5.2.4 The Hilbert Boundary Value Problem . . . . . . . . . . . . . . . . 11.5.2.5 Solution of the Hilbert Boundary Value Problem (in short: Hilbert Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2.6 Solution of the Characteristic Integral Equation . . . . . . . . . .
12 Functional Analysis
12.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Notion of a Vector Space . . . . . . . . . . . . 12.1.2 Linear and Ane Linear Subsets . . . . . . . . 12.1.3 Linearly Independent Elements . . . . . . . . . 12.1.4 Convex Subsets and the Convex Hull . . . . . . 12.1.4.1 Convex Sets . . . . . . . . . . . . . . 12.1.4.2 Cones . . . . . . . . . . . . . . . . . 12.1.5 Linear Operators and Functionals . . . . . . . 12.1.5.1 Mappings . . . . . . . . . . . . . . . 12.1.5.2 Homomorphism and Endomorphism .
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12.2
12.3
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12.1.5.3 Isomorphic Vector Spaces . . . . . . . . . . . . . . . . . 12.1.6 Complexi cation of Real Vector Spaces . . . . . . . . . . . . . . . 12.1.7 Ordered Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 12.1.7.1 Cone and Partial Ordering . . . . . . . . . . . . . . . . . 12.1.7.2 Order Bounded Sets . . . . . . . . . . . . . . . . . . . . 12.1.7.3 Positive Operators . . . . . . . . . . . . . . . . . . . . . 12.1.7.4 Vector Lattices . . . . . . . . . . . . . . . . . . . . . . . Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Notion of a Metric Space . . . . . . . . . . . . . . . . . . . . . . . 12.2.1.1 Balls, Neighborhoods and Open Sets . . . . . . . . . . . . 12.2.1.2 Convergence of Sequences in Metric Spaces . . . . . . . . 12.2.1.3 Closed Sets and Closure . . . . . . . . . . . . . . . . . . 12.2.1.4 Dense Subsets and Separable Metric Spaces . . . . . . . . 12.2.2 Complete Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . 12.2.2.1 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . 12.2.2.2 Complete Metric Spaces . . . . . . . . . . . . . . . . . . 12.2.2.3 Some Fundamental Theorems in Complete Metric Spaces 12.2.2.4 Some Applications of the Contraction Mapping Principle 12.2.2.5 Completion of a Metric Space . . . . . . . . . . . . . . . 12.2.3 Continuous Operators . . . . . . . . . . . . . . . . . . . . . . . . . Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Notion of a Normed Space . . . . . . . . . . . . . . . . . . . . . . 12.3.1.1 Axioms of a Normed Space . . . . . . . . . . . . . . . . . 12.3.1.2 Some Properties of Normed Spaces . . . . . . . . . . . . 12.3.2 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2.1 Series in Normed Spaces . . . . . . . . . . . . . . . . . . 12.3.2.2 Examples of Banach Spaces . . . . . . . . . . . . . . . . 12.3.2.3 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Ordered Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . 12.3.4 Normed Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Notion of a Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . 12.4.1.1 Scalar Product . . . . . . . . . . . . . . . . . . . . . . . 12.4.1.2 Unitary Spaces and Some of their Properties . . . . . . . 12.4.1.3 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2.1 Properties of Orthogonality . . . . . . . . . . . . . . . . 12.4.2.2 Orthogonal Systems . . . . . . . . . . . . . . . . . . . . 12.4.3 Fourier Series in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . 12.4.3.1 Best Approximation . . . . . . . . . . . . . . . . . . . . 12.4.3.2 Parseval Equation, Riesz{Fischer Theorem . . . . . . . . 12.4.4 Existence of a Basis, Isomorphic Hilbert Spaces . . . . . . . . . . . Continuous Linear Operators and Functionals . . . . . . . . . . . . . . . . 12.5.1 Boundedness, Norm and Continuity of Linear Operators . . . . . . 12.5.1.1 Boundedness and the Norm of Linear Operators . . . . . 12.5.1.2 The Space of Linear Continuous Operators . . . . . . . . 12.5.1.3 Convergence of Operator Sequences . . . . . . . . . . . . 12.5.2 Linear Continuous Operators in Banach Spaces . . . . . . . . . . . 12.5.3 Elements of the Spectral Theory of Linear Operators . . . . . . . . 12.5.3.1 Resolvent Set and the Resolvent of an Operator . . . . . . 12.5.3.2 Spectrum of an Operator . . . . . . . . . . . . . . . . . . 12.5.4 Continuous Linear Functionals . . . . . . . . . . . . . . . . . . . .
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12.6
12.7
12.8
12.9
12.5.4.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.4.2 Continuous Linear Functionals in Hilbert Spaces, Riesz Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.4.3 Continuous Linear Functionals in L p . . . . . . . . . . . . . . . . 12.5.5 Extension of a Linear Functional . . . . . . . . . . . . . . . . . . . . . . . . 12.5.6 Separation of Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.7 Second Adjoint Space and Reexive Spaces . . . . . . . . . . . . . . . . . . Adjoint Operators in Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.1 Adjoint of a Bounded Operator . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.2 Adjoint Operator of an Unbounded Operator . . . . . . . . . . . . . . . . . 12.6.3 Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.3.1 Positive De nite Operators . . . . . . . . . . . . . . . . . . . . . . 12.6.3.2 Projectors in a Hilbert Space . . . . . . . . . . . . . . . . . . . . . Compact Sets and Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.1 Compact Subsets of a Normed Space . . . . . . . . . . . . . . . . . . . . . . 12.7.2 Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.2.1 De nition of Compact Operator . . . . . . . . . . . . . . . . . . . 12.7.2.2 Properties of Linear Compact Operators . . . . . . . . . . . . . . 12.7.2.3 Weak Convergence of Elements . . . . . . . . . . . . . . . . . . . 12.7.3 Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.4 Compact Operators in Hilbert Space . . . . . . . . . . . . . . . . . . . . . . 12.7.5 Compact Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . Non-Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8.1 Examples of Non-Linear Operators . . . . . . . . . . . . . . . . . . . . . . . 12.8.2 Di erentiability of Non-Linear Operators . . . . . . . . . . . . . . . . . . . 12.8.3 Newton's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8.4 Schauder's Fixed-Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . 12.8.5 Leray{Schauder Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8.6 Positive Non-Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . 12.8.7 Monotone Operators in Banach Spaces . . . . . . . . . . . . . . . . . . . . Measure and Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.1 Sigma Algebra and Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.2 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.2.1 Measurable Function . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.2.2 Properties of the Class of Measurable Functions . . . . . . . . . . 12.9.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.3.1 De nition of the Integral . . . . . . . . . . . . . . . . . . . . . . . 12.9.3.2 Some Properties of the Integral . . . . . . . . . . . . . . . . . . . 12.9.3.3 Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . . . 12.9.4 Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.5 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.5.1 Formula of Partial Integration . . . . . . . . . . . . . . . . . . . . 12.9.5.2 Generalized Derivative . . . . . . . . . . . . . . . . . . . . . . . . 12.9.5.3 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.5.4 Derivative of a Distribution . . . . . . . . . . . . . . . . . . . . .
13 Vector Analysis and Vector Fields
13.1 Basic Notions of the Theory of Vector Fields . . . . . 13.1.1 Vector Functions of a Scalar Variable . . . . 13.1.1.1 De nitions . . . . . . . . . . . . . . 13.1.1.2 Derivative of a Vector Function . . 13.1.1.3 Rules of Di erentiation for Vectors
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13.1.1.4 Taylor Expansion for Vector Functions . . . . . . . . . . . . . . . 13.1.2 Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2.1 Scalar Field or Scalar Point Function . . . . . . . . . . . . . . . . 13.1.2.2 Important Special Cases of Scalar Fields . . . . . . . . . . . . . . 13.1.2.3 Coordinate De nition of a Field . . . . . . . . . . . . . . . . . . . 13.1.2.4 Level Surfaces and Level Lines of a Field . . . . . . . . . . . . . . 13.1.3 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.3.1 Vector Field or Vector Point Function . . . . . . . . . . . . . . . . 13.1.3.2 Important Cases of Vector Fields . . . . . . . . . . . . . . . . . . 13.1.3.3 Coordinate Representation of Vector Fields . . . . . . . . . . . . . 13.1.3.4 Transformation of Coordinate Systems . . . . . . . . . . . . . . . 13.1.3.5 Vector Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Di erential Operators of Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Directional and Space Derivatives . . . . . . . . . . . . . . . . . . . . . . . 13.2.1.1 Directional Derivative of a Scalar Field . . . . . . . . . . . . . . . 13.2.1.2 Directional Derivative of a Vector Field . . . . . . . . . . . . . . . 13.2.1.3 Volume Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Gradient of a Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2.1 De nition of the Gradient . . . . . . . . . . . . . . . . . . . . . . 13.2.2.2 Gradient and Volume Derivative . . . . . . . . . . . . . . . . . . . 13.2.2.3 Gradient and Directional Derivative . . . . . . . . . . . . . . . . . 13.2.2.4 Further Properties of the Gradient . . . . . . . . . . . . . . . . . . 13.2.2.5 Gradient of the Scalar Field in Di erent Coordinates . . . . . . . . 13.2.2.6 Rules of Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.3 Vector Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.4 Divergence of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.4.1 De nition of Divergence . . . . . . . . . . . . . . . . . . . . . . . 13.2.4.2 Divergence in Di erent Coordinates . . . . . . . . . . . . . . . . . 13.2.4.3 Rules for Evaluation of the Divergence . . . . . . . . . . . . . . . 13.2.4.4 Divergence of a Central Field . . . . . . . . . . . . . . . . . . . . . 13.2.5 Rotation of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.5.1 De nitions of the Rotation . . . . . . . . . . . . . . . . . . . . . . 13.2.5.2 Rotation in Di erent Coordinates . . . . . . . . . . . . . . . . . . 13.2.5.3 Rules for Evaluating the Rotation . . . . . . . . . . . . . . . . . . 13.2.5.4 Rotation of a Potential Field . . . . . . . . . . . . . . . . . . . . . 13.2.6 Nabla Operator, Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . 13.2.6.1 Nabla Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.6.2 Rules for Calculations with the Nabla Operator . . . . . . . . . . . 13.2.6.3 Vector Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.6.4 Nabla Operator Applied Twice . . . . . . . . . . . . . . . . . . . . 13.2.6.5 Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.7 Review of Spatial Di erential Operations . . . . . . . . . . . . . . . . . . . 13.2.7.1 Fundamental Relations and Results (see Table 13.2) . . . . . . . . 13.2.7.2 Rules of Calculation for Spatial Di erential Operators . . . . . . . 13.2.7.3 Expressions of Vector Analysis in Cartesian, Cylindrical, and Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Integration in Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Line Integral and Potential in Vector Fields . . . . . . . . . . . . . . . . . . 13.3.1.1 Line Integral in Vector Fields . . . . . . . . . . . . . . . . . . . . . 13.3.1.2 Interpretation of the Line Integral in Mechanics . . . . . . . . . . . 13.3.1.3 Properties of the Line Integral . . . . . . . . . . . . . . . . . . . . 13.3.1.4 Line Integral in Cartesian Coordinates . . . . . . . . . . . . . . .
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XXVIII Contents 13.3.1.5 Integral Along a Closed Curve in a Vector Field . . . . . . . . . . . 13.3.1.6 Conservative or Potential Field . . . . . . . . . . . . . . . . . . . 13.3.2 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2.1 Vector of a Plane Sheet . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2.2 Evaluation of the Surface Integral . . . . . . . . . . . . . . . . . . 13.3.2.3 Surface Integrals and Flow of Fields . . . . . . . . . . . . . . . . . 13.3.2.4 Surface Integrals in Cartesian Coordinates as Surface Integral of Second Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 Integral Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.3.1 Integral Theorem and Integral Formula of Gauss . . . . . . . . . . 13.3.3.2 Integral Theorem of Stokes . . . . . . . . . . . . . . . . . . . . . . 13.3.3.3 Integral Theorems of Green . . . . . . . . . . . . . . . . . . . . . 13.4 Evaluation of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Pure Source Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Pure Rotation Field or Zero-Divergence Field . . . . . . . . . . . . . . . . . 13.4.3 Vector Fields with Point-Like Sources . . . . . . . . . . . . . . . . . . . . . 13.4.3.1 Coulomb Field of a Point-Like Charge . . . . . . . . . . . . . . . . 13.4.3.2 Gravitational Field of a Point Mass . . . . . . . . . . . . . . . . . 13.4.4 Superposition of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.4.1 Discrete Source Distribution . . . . . . . . . . . . . . . . . . . . . 13.4.4.2 Continuous Source Distribution . . . . . . . . . . . . . . . . . . . 13.4.4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Di erential Equations of Vector Field Theory . . . . . . . . . . . . . . . . . . . . . 13.5.1 Laplace Di erential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 Poisson Di erential Equation . . . . . . . . . . . . . . . . . . . . . . . . . .
14 Function Theory
14.1 Functions of Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Continuity, Di erentiability . . . . . . . . . . . . . . . . . . . . . . . 14.1.1.1 De nition of a Complex Function . . . . . . . . . . . . . . 14.1.1.2 Limit of a Complex Function . . . . . . . . . . . . . . . . . 14.1.1.3 Continuous Complex Functions . . . . . . . . . . . . . . . 14.1.1.4 Di erentiability of a Complex Function . . . . . . . . . . . 14.1.2 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.2.1 De nition of Analytic Functions . . . . . . . . . . . . . . . 14.1.2.2 Examples of Analytic Functions . . . . . . . . . . . . . . . 14.1.2.3 Properties of Analytic Functions . . . . . . . . . . . . . . . 14.1.2.4 Singular Points . . . . . . . . . . . . . . . . . . . . . . . . 14.1.3 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.3.1 Notion and Properties of Conformal Mappings . . . . . . . 14.1.3.2 Simplest Conformal Mappings . . . . . . . . . . . . . . . . 14.1.3.3 The Schwarz Reection Principle . . . . . . . . . . . . . . 14.1.3.4 Complex Potential . . . . . . . . . . . . . . . . . . . . . . 14.1.3.5 Superposition Principle . . . . . . . . . . . . . . . . . . . . 14.1.3.6 Arbitrary Mappings of the Complex Plane . . . . . . . . . 14.2 Integration in the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 De nite and Inde nite Integral . . . . . . . . . . . . . . . . . . . . . 14.2.1.1 De nition of the Integral in the Complex Plane . . . . . . . 14.2.1.2 Properties and Evaluation of Complex Integrals . . . . . . 14.2.2 Cauchy Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2.1 Cauchy Integral Theorem for Simply Connected Domains . 14.2.2.2 Cauchy Integral Theorem for Multiply Connected Domains
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14.3
14.4
14.5 14.6
14.2.3 Cauchy Integral Formulas . . . . . . . . . . . . . . . . . . . . . . . 14.2.3.1 Analytic Function on the Interior of a Domain . . . . . . 14.2.3.2 Analytic Function on the Exterior of a Domain . . . . . . Power Series Expansion of Analytic Functions . . . . . . . . . . . . . . . . 14.3.1 Convergence of Series with Complex Terms . . . . . . . . . . . . . 14.3.1.1 Convergence of a Number Sequence with Complex Terms 14.3.1.2 Convergence of an In nite Series with Complex Terms . . 14.3.1.3 Power Series with Complex Terms . . . . . . . . . . . . . 14.3.2 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 Principle of Analytic Continuation . . . . . . . . . . . . . . . . . . 14.3.4 Laurent Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.5 Isolated Singular Points and the Residue Theorem . . . . . . . . . 14.3.5.1 Isolated Singular Points . . . . . . . . . . . . . . . . . . 14.3.5.2 Meromorphic Functions . . . . . . . . . . . . . . . . . . 14.3.5.3 Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . 14.3.5.4 Residue . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.5.5 Residue Theorem . . . . . . . . . . . . . . . . . . . . . . Evaluation of Real Integrals by Complex Integrals . . . . . . . . . . . . . 14.4.1 Application of Cauchy Integral Formulas . . . . . . . . . . . . . . 14.4.2 Application of the Residue Theorem . . . . . . . . . . . . . . . . . 14.4.3 Application of the Jordan Lemma . . . . . . . . . . . . . . . . . . 14.4.3.1 Jordan Lemma . . . . . . . . . . . . . . . . . . . . . . . 14.4.3.2 Examples of the Jordan Lemma . . . . . . . . . . . . . . Algebraic and Elementary Transcendental Functions . . . . . . . . . . . . 14.5.1 Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.2 Elementary Transcendental Functions . . . . . . . . . . . . . . . . 14.5.3 Description of Curves in Complex Form . . . . . . . . . . . . . . . Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.1 Relation to Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . 14.6.2 Jacobian Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.3 Theta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.4 Weierstrass Functions . . . . . . . . . . . . . . . . . . . . . . . . .
15 Integral Transformations
15.1 Notion of Integral Transformation . . . . . . . . . . . . . . . . . . . . . 15.1.1 General De nition of Integral Transformations . . . . . . . . . . 15.1.2 Special Integral Transformations . . . . . . . . . . . . . . . . . . 15.1.3 Inverse Transformations . . . . . . . . . . . . . . . . . . . . . . 15.1.4 Linearity of Integral Transformations . . . . . . . . . . . . . . . 15.1.5 Integral Transformations for Functions of Several Variables . . . 15.1.6 Applications of Integral Transformations . . . . . . . . . . . . . 15.2 Laplace Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Properties of the Laplace Transformation . . . . . . . . . . . . . 15.2.1.1 Laplace Transformation, Original and Image Space . . . 15.2.1.2 Rules for the Evaluation of the Laplace Transformation 15.2.1.3 Transforms of Special Functions . . . . . . . . . . . . . 15.2.1.4 Dirac Function and Distributions . . . . . . . . . . . 15.2.2 Inverse Transformation into the Original Space . . . . . . . . . . 15.2.2.1 Inverse Transformation with the Help of Tables . . . . . 15.2.2.2 Partial Fraction Decomposition . . . . . . . . . . . . . 15.2.2.3 Series Expansion . . . . . . . . . . . . . . . . . . . . . 15.2.2.4 Inverse Integral . . . . . . . . . . . . . . . . . . . . . .
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15.3
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15.6
15.2.3 Solution of Di erential Equations using Laplace Transformation . . . . . . . 15.2.3.1 Ordinary Linear Di erential Equations with Constant Coecients 15.2.3.2 Ordinary Linear Di erential Equations with Coecients Depending on the Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3.3 Partial Di erential Equations . . . . . . . . . . . . . . . . . . . . Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Properties of the Fourier Transformation . . . . . . . . . . . . . . . . . . . 15.3.1.1 Fourier Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1.2 Fourier Transformation and Inverse Transformation . . . . . . . . 15.3.1.3 Rules of Calculation with the Fourier Transformation . . . . . . . 15.3.1.4 Transforms of Special Functions . . . . . . . . . . . . . . . . . . . 15.3.2 Solution of Di erential Equations using the Fourier Transformation . . . . . 15.3.2.1 Ordinary Linear Di erential Equations . . . . . . . . . . . . . . . 15.3.2.2 Partial Di erential Equations . . . . . . . . . . . . . . . . . . . . Z-Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 Properties of the Z-Transformation . . . . . . . . . . . . . . . . . . . . . . . 15.4.1.1 Discrete Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1.2 De nition of the Z-Transformation . . . . . . . . . . . . . . . . . . 15.4.1.3 Rules of Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1.4 Relation to the Laplace Transformation . . . . . . . . . . . . . . . 15.4.1.5 Inverse of the Z-Transformation . . . . . . . . . . . . . . . . . . . 15.4.2 Applications of the Z-Transformation . . . . . . . . . . . . . . . . . . . . . 15.4.2.1 General Solution of Linear Di erence Equations . . . . . . . . . . 15.4.2.2 Second-Order Di erence Equations (Initial Value Problem) . . . . 15.4.2.3 Second-Order Di erence Equations (Boundary Value Problem) . . Wavelet Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.1 Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.2 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.3 Wavelet Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.4 Discrete Wavelet Transformation . . . . . . . . . . . . . . . . . . . . . . . . 15.5.4.1 Fast Wavelet Transformation . . . . . . . . . . . . . . . . . . . . . 15.5.4.2 Discrete Haar Wavelet Transformation . . . . . . . . . . . . . . . 15.5.5 Gabor Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Walsh Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.1 Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.2 Walsh Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16 Probability Theory and Mathematical Statistics
16.1 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.1 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.2 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.3 Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.4 Collection of the Formulas of Combinatorics (see Table 16.1) . 16.2 Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Event, Frequency and Probability . . . . . . . . . . . . . . . 16.2.1.1 Events . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1.2 Frequencies and Probabilities . . . . . . . . . . . . 16.2.1.3 Conditional Probability, Bayes Theorem . . . . . . 16.2.2 Random Variables, Distribution Functions . . . . . . . . . . 16.2.2.1 Random Variable . . . . . . . . . . . . . . . . . . . 16.2.2.2 Distribution Function . . . . . . . . . . . . . . . . . 16.2.2.3 Expected Value and Variance, Chebyshev Inequality
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Contents XXXI
16.2.2.4 Multidimensional Random Variable . . . . . . . . . . . . . . . . . 16.2.3 Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.3.1 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . 16.2.3.2 Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . . . 16.2.3.3 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.4 Continuous Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.4.1 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.4.2 Standard Normal Distribution, Gaussian Error Function . . . . . . 16.2.4.3 Logarithmic Normal Distribution . . . . . . . . . . . . . . . . . . 16.2.4.4 Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . 16.2.4.5 Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.4.6 2 (Chi-Square) Distribution . . . . . . . . . . . . . . . . . . . . . 16.2.4.7 Fisher F Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.4.8 Student t Distribution . . . . . . . . . . . . . . . . . . . . . . . . 16.2.5 Law of Large Numbers, Limit Theorems . . . . . . . . . . . . . . . . . . . . 16.2.6 Stochastic Processes and Stochastic Chains . . . . . . . . . . . . . . . . . . 16.2.6.1 Basic Notions, Markov Chains . . . . . . . . . . . . . . . . . . . . 16.2.6.2 Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Mathematical Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 Statistic Function or Sample Function . . . . . . . . . . . . . . . . . . . . . 16.3.1.1 Population, Sample, Random Vector . . . . . . . . . . . . . . . . . 16.3.1.2 Statistic Function or Sample Function . . . . . . . . . . . . . . . . 16.3.2 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.2.1 Statistical Summarization and Analysis of Given Data . . . . . . . 16.3.2.2 Statistical Parameters . . . . . . . . . . . . . . . . . . . . . . . . 16.3.3 Important Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.3.1 Goodness of Fit Test for a Normal Distribution . . . . . . . . . . . 16.3.3.2 Distribution of the Sample Mean . . . . . . . . . . . . . . . . . . . 16.3.3.3 Con dence Limits for the Mean . . . . . . . . . . . . . . . . . . . 16.3.3.4 Con dence Interval for the Variance . . . . . . . . . . . . . . . . . 16.3.3.5 Structure of Hypothesis Testing . . . . . . . . . . . . . . . . . . . 16.3.4 Correlation and Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.4.1 Linear Correlation of two Measurable Characters . . . . . . . . . . 16.3.4.2 Linear Regression for two Measurable Characters . . . . . . . . . . 16.3.4.3 Multidimensional Regression . . . . . . . . . . . . . . . . . . . . . 16.3.5 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.5.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.5.2 Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.5.3 Example of a Monte Carlo Simulation . . . . . . . . . . . . . . . . 16.3.5.4 Application of the Monte Carlo Method in Numerical Mathematics 16.3.5.5 Further Applications of the Monte Carlo Method . . . . . . . . . . 16.4 Calculus of Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 Measurement Error and its Distribution . . . . . . . . . . . . . . . . . . . . 16.4.1.1 Qualitative Characterization of Measurement Errors . . . . . . . . 16.4.1.2 Density Function of the Measurement Error . . . . . . . . . . . . . 16.4.1.3 Quantitative Characterization of the Measurement Error . . . . . 16.4.1.4 Determining the Result of a Measurement with Bounds on the Error 16.4.1.5 Error Estimation for Direct Measurements with the Same Accuracy 16.4.1.6 Error Estimation for Direct Measurements with Di erent Accuracy 16.4.2 Error Propagation and Error Analysis . . . . . . . . . . . . . . . . . . . . . 16.4.2.1 Gauss Error Propagation Law . . . . . . . . . . . . . . . . . . . . 16.4.2.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
754 754 755 756 757 758 758 759 759 760 761 762 763 763 764 765 765 768 769 769 769 770 772 772 773 774 774 776 777 778 779 779 779 780 781 783 783 783 784 785 787 787 788 788 788 790 792 793 793 794 794 796
XXXII Contents 17 Dynamical Systems and Chaos
17.1 Ordinary Di erential Equations and Mappings . . . . . . . . . . . . . . . . . . 17.1.1 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1.2 Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2 Qualitative Theory of Ordinary Di erential Equations . . . . . . . . . . 17.1.2.1 Existence of Flows, Phase Space Structure . . . . . . . . . . . 17.1.2.2 Linear Di erential Equations . . . . . . . . . . . . . . . . . . . 17.1.2.3 Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2.4 Invariant Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2.5 Poincare Mapping . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2.6 Topological Equivalence of Di erential Equations . . . . . . . . 17.1.3 Discrete Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . 17.1.3.1 Steady States, Periodic Orbits and Limit Sets . . . . . . . . . . 17.1.3.2 Invariant Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 17.1.3.3 Topological Conjugacy of Discrete Systems . . . . . . . . . . . 17.1.4 Structural Stability (Robustness) . . . . . . . . . . . . . . . . . . . . . 17.1.4.1 Structurally Stable Di erential Equations . . . . . . . . . . . . 17.1.4.2 Structurally Stable Discrete Systems . . . . . . . . . . . . . . 17.1.4.3 Generic Properties . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Quantitative Description of Attractors . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 Probability Measures on Attractors . . . . . . . . . . . . . . . . . . . . 17.2.1.1 Invariant Measure . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1.2 Elements of Ergodic Theory . . . . . . . . . . . . . . . . . . . 17.2.2 Entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.2.1 Topological Entropy . . . . . . . . . . . . . . . . . . . . . . . 17.2.2.2 Metric Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.3 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.4 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.4.1 Metric Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 17.2.4.2 Dimensions De ned by Invariant Measures . . . . . . . . . . . 17.2.4.3 Local Hausdor Dimension According to Douady and Oesterle 17.2.4.4 Examples of Attractors . . . . . . . . . . . . . . . . . . . . . . 17.2.5 Strange Attractors and Chaos . . . . . . . . . . . . . . . . . . . . . . . 17.2.6 Chaos in One-Dimensional Mappings . . . . . . . . . . . . . . . . . . . 17.3 Bifurcation Theory and Routes to Chaos . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Bifurcations in Morse{Smale Systems . . . . . . . . . . . . . . . . . . . 17.3.1.1 Local Bifurcations in Neighborhoods of Steady States . . . . . 17.3.1.2 Local Bifurcations in a Neighborhood of a Periodic Orbit . . . 17.3.1.3 Global Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 Transitions to Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2.1 Cascade of Period Doublings . . . . . . . . . . . . . . . . . . . 17.3.2.2 Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2.3 Global Homoclinic Bifurcations . . . . . . . . . . . . . . . . . 17.3.2.4 Destruction of a Torus . . . . . . . . . . . . . . . . . . . . . .
18 Optimization
18.1 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.1 Formulation of the Problem and Geometrical Representation 18.1.1.1 The Form of a Linear Programming Problem . . . . 18.1.1.2 Examples and Graphical Solutions . . . . . . . . . . 18.1.2 Basic Notions of Linear Programming, Normal Form . . . . .
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Contents XXXIII
18.1.2.1 Extreme Points and Basis . . . . . . . . . . . . . . . . . . 18.1.2.2 Normal Form of the Linear Programming Problem . . . . . 18.1.3 Simplex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.3.1 Simplex Tableau . . . . . . . . . . . . . . . . . . . . . . . 18.1.3.2 Transition to the New Simplex Tableau . . . . . . . . . . . 18.1.3.3 Determination of an Initial Simplex Tableau . . . . . . . . 18.1.3.4 Revised Simplex Method . . . . . . . . . . . . . . . . . . . 18.1.3.5 Duality in Linear Programming . . . . . . . . . . . . . . . 18.1.4 Special Linear Programming Problems . . . . . . . . . . . . . . . . . 18.1.4.1 Transportation Problem . . . . . . . . . . . . . . . . . . . 18.1.4.2 Assignment Problem . . . . . . . . . . . . . . . . . . . . . 18.1.4.3 Distribution Problem . . . . . . . . . . . . . . . . . . . . . 18.1.4.4 Travelling Salesman . . . . . . . . . . . . . . . . . . . . . . 18.1.4.5 Scheduling Problem . . . . . . . . . . . . . . . . . . . . . . 18.2 Non-linear Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1 Formulation of the Problem, Theoretical Basis . . . . . . . . . . . . 18.2.1.1 Formulation of the Problem . . . . . . . . . . . . . . . . . 18.2.1.2 Optimality Conditions . . . . . . . . . . . . . . . . . . . . 18.2.1.3 Duality in Optimization . . . . . . . . . . . . . . . . . . . 18.2.2 Special Non-linear Optimization Problems . . . . . . . . . . . . . . . 18.2.2.1 Convex Optimization . . . . . . . . . . . . . . . . . . . . . 18.2.2.2 Quadratic Optimization . . . . . . . . . . . . . . . . . . . 18.2.3 Solution Methods for Quadratic Optimization Problems . . . . . . . 18.2.3.1 Wolfe's Method . . . . . . . . . . . . . . . . . . . . . . . . 18.2.3.2 Hildreth{d'Esopo Method . . . . . . . . . . . . . . . . . . 18.2.4 Numerical Search Procedures . . . . . . . . . . . . . . . . . . . . . . 18.2.4.1 One-Dimensional Search . . . . . . . . . . . . . . . . . . . 18.2.4.2 Minimum Search in n-Dimensional Euclidean Vector Space 18.2.5 Methods for Unconstrained Problems . . . . . . . . . . . . . . . . . 18.2.5.1 Method of Steepest Descent (Gradient Method) . . . . . . 18.2.5.2 Application of the Newton Method . . . . . . . . . . . . . 18.2.5.3 Conjugate Gradient Methods . . . . . . . . . . . . . . . . 18.2.5.4 Method of Davidon, Fletcher and Powell (DFP) . . . . . . 18.2.6 Evolution Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.6.1 Mutation{Selection{Strategy . . . . . . . . . . . . . . . . 18.2.6.2 Recombination . . . . . . . . . . . . . . . . . . . . . . . . 18.2.7 Gradient Methods for Problems with Inequality Type Constraints) . 18.2.7.1 Method of Feasible Directions . . . . . . . . . . . . . . . . 18.2.7.2 Gradient Projection Method . . . . . . . . . . . . . . . . . 18.2.8 Penalty Function and Barrier Methods . . . . . . . . . . . . . . . . . 18.2.8.1 Penalty Function Method . . . . . . . . . . . . . . . . . . 18.2.8.2 Barrier Method . . . . . . . . . . . . . . . . . . . . . . . . 18.2.9 Cutting Plane Methods . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Discrete Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . 18.3.1 Discrete Dynamic Decision Models . . . . . . . . . . . . . . . . . . . 18.3.1.1 n-Stage Decision Processes . . . . . . . . . . . . . . . . . . 18.3.1.2 Dynamic Programming Problem . . . . . . . . . . . . . . . 18.3.2 Examples of Discrete Decision Models . . . . . . . . . . . . . . . . . 18.3.2.1 Purchasing Problem . . . . . . . . . . . . . . . . . . . . . 18.3.2.2 Knapsack Problem . . . . . . . . . . . . . . . . . . . . . . 18.3.3 Bellman Functional Equations . . . . . . . . . . . . . . . . . . . . . 18.3.3.1 Properties of the Cost Function . . . . . . . . . . . . . . .
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XXXIV Contents 18.3.3.2 Formulation of the Functional Equations . . . . . . 18.3.4 Bellman Optimality Principle . . . . . . . . . . . . . . . . . 18.3.5 Bellman Functional Equation Method . . . . . . . . . . . . . 18.3.5.1 Determination of Minimal Costs . . . . . . . . . . . 18.3.5.2 Determination of the Optimal Policy . . . . . . . . 18.3.6 Examples of Applications of the Functional Equation Method 18.3.6.1 Optimal Purchasing Policy . . . . . . . . . . . . . . 18.3.6.2 Knapsack Problem . . . . . . . . . . . . . . . . . .
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19.1 Numerical Solution of Non-Linear Equations in a Single Unknown . . . . 19.1.1 Iteration Method . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.1.1 Ordinary Iteration Method . . . . . . . . . . . . . . . . 19.1.1.2 Newton's Method . . . . . . . . . . . . . . . . . . . . . 19.1.1.3 Regula Falsi . . . . . . . . . . . . . . . . . . . . . . . . 19.1.2 Solution of Polynomial Equations . . . . . . . . . . . . . . . . . 19.1.2.1 Horner's Scheme . . . . . . . . . . . . . . . . . . . . . 19.1.2.2 Positions of the Roots . . . . . . . . . . . . . . . . . . . 19.1.2.3 Numerical Methods . . . . . . . . . . . . . . . . . . . . 19.2 Numerical Solution of Equation Systems . . . . . . . . . . . . . . . . . . 19.2.1 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . 19.2.1.1 Triangular Decomposition of a Matrix . . . . . . . . . . 19.2.1.2 Cholesky's Method for a Symmetric Coecient Matrix . 19.2.1.3 Orthogonalization Method . . . . . . . . . . . . . . . . 19.2.1.4 Iteration Methods . . . . . . . . . . . . . . . . . . . . . 19.2.2 Non-Linear Equation Systems . . . . . . . . . . . . . . . . . . . 19.2.2.1 Ordinary Iteration Method . . . . . . . . . . . . . . . . 19.2.2.2 Newton's Method . . . . . . . . . . . . . . . . . . . . . 19.2.2.3 Derivative-Free Gauss{Newton Method . . . . . . . . . 19.3 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.1 General Quadrature Formulas . . . . . . . . . . . . . . . . . . . 19.3.2 Interpolation Quadratures . . . . . . . . . . . . . . . . . . . . . 19.3.2.1 Rectangular Formula . . . . . . . . . . . . . . . . . . . 19.3.2.2 Trapezoidal Formula . . . . . . . . . . . . . . . . . . . 19.3.2.3 Simpson's Formula . . . . . . . . . . . . . . . . . . . . 19.3.2.4 Hermite's Trapezoidal Formula . . . . . . . . . . . . . 19.3.3 Quadrature Formulas of Gauss . . . . . . . . . . . . . . . . . . . 19.3.3.1 Gauss Quadrature Formulas . . . . . . . . . . . . . . . 19.3.3.2 Lobatto's Quadrature Formulas . . . . . . . . . . . . . 19.3.4 Method of Romberg . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.4.1 Algorithm of the Romberg Method . . . . . . . . . . . 19.3.4.2 Extrapolation Principle . . . . . . . . . . . . . . . . . . 19.4 Approximate Integration of Ordinary Di erential Equations . . . . . . . 19.4.1 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . 19.4.1.1 Euler Polygonal Method . . . . . . . . . . . . . . . . . 19.4.1.2 Runge{Kutta Methods . . . . . . . . . . . . . . . . . . 19.4.1.3 Multi-Step Methods . . . . . . . . . . . . . . . . . . . 19.4.1.4 Predictor{Corrector Method . . . . . . . . . . . . . . . 19.4.1.5 Convergence, Consistency, Stability . . . . . . . . . . . 19.4.2 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . 19.4.2.1 Di erence Method . . . . . . . . . . . . . . . . . . . . 19.4.2.2 Approximation by Using Given Functions . . . . . . . .
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19 Numerical Analysis
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Contents XXXV
19.4.2.3 Shooting Method . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Approximate Integration of Partial Di erential Equations . . . . . . . . . . . 19.5.1 Di erence Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5.2 Approximation by Given Functions . . . . . . . . . . . . . . . . . . . 19.5.3 Finite Element Method (FEM) . . . . . . . . . . . . . . . . . . . . . . 19.6 Approximation, Computation of Adjustment, Harmonic Analysis . . . . . . . 19.6.1 Polynomial Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 19.6.1.1 Newton's Interpolation Formula . . . . . . . . . . . . . . . . 19.6.1.2 Lagrange's Interpolation Formula . . . . . . . . . . . . . . . 19.6.1.3 Aitken{Neville Interpolation . . . . . . . . . . . . . . . . . . 19.6.2 Approximation in Mean . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6.2.1 Continuous Problems, Normal Equations . . . . . . . . . . . 19.6.2.2 Discrete Problems, Normal Equations, Householder's Method 19.6.2.3 Multidimensional Problems . . . . . . . . . . . . . . . . . . 19.6.2.4 Non-Linear Least Squares Problems . . . . . . . . . . . . . . 19.6.3 Chebyshev Approximation . . . . . . . . . . . . . . . . . . . . . . . . 19.6.3.1 Problem De nition and the Alternating Point Theorem . . . 19.6.3.2 Properties of the Chebyshev Polynomials . . . . . . . . . . . 19.6.3.3 Remes Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 19.6.3.4 Discrete Chebyshev Approximation and Optimization . . . . 19.6.4 Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6.4.1 Formulas for Trigonometric Interpolation . . . . . . . . . . . 19.6.4.2 Fast Fourier Transformation (FFT) . . . . . . . . . . . . . . 19.7 Representation of Curves and Surfaces with Splines . . . . . . . . . . . . . . . 19.7.1 Cubic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.7.1.1 Interpolation Splines . . . . . . . . . . . . . . . . . . . . . . 19.7.1.2 Smoothing Splines . . . . . . . . . . . . . . . . . . . . . . . 19.7.2 Bicubic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.7.2.1 Use of Bicubic Splines . . . . . . . . . . . . . . . . . . . . . 19.7.2.2 Bicubic Interpolation Splines . . . . . . . . . . . . . . . . . . 19.7.2.3 Bicubic Smoothing Splines . . . . . . . . . . . . . . . . . . . 19.7.3 Bernstein{Bezier Representation of Curves and Surfaces . . . . . . . . 19.7.3.1 Principle of the B{B Curve Representation . . . . . . . . . . 19.7.3.2 B{B Surface Representation . . . . . . . . . . . . . . . . . . 19.8 Using the Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.8.1 Internal Symbol Representation . . . . . . . . . . . . . . . . . . . . . 19.8.1.1 Number Systems . . . . . . . . . . . . . . . . . . . . . . . . 19.8.1.2 Internal Number Representation . . . . . . . . . . . . . . . . 19.8.2 Numerical Problems in Calculations with Computers . . . . . . . . . . 19.8.2.1 Introduction, Error Types . . . . . . . . . . . . . . . . . . . 19.8.2.2 Normalized Decimal Numbers and Round-O . . . . . . . . . 19.8.2.3 Accuracy in Numerical Calculations . . . . . . . . . . . . . . 19.8.3 Libraries of Numerical Methods . . . . . . . . . . . . . . . . . . . . . 19.8.3.1 NAG Library . . . . . . . . . . . . . . . . . . . . . . . . . . 19.8.3.2 IMSL Library . . . . . . . . . . . . . . . . . . . . . . . . . . 19.8.3.3 Aachen Library . . . . . . . . . . . . . . . . . . . . . . . . . 19.8.4 Application of Computer Algebra Systems . . . . . . . . . . . . . . . . 19.8.4.1 Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . 19.8.4.2 Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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910 911 911 912 913 917 917 917 918 918 919 919 921 922 922 923 923 924 925 926 927 927 928 931 931 931 932 933 933 933 935 935 935 936 936 936 936 938 939 939 939 941 944 944 945 946 946 946 949
XXXVI Contents 20 Computer Algebra Systems
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1.1 Brief Characterization of Computer Algebra Systems . 20.1.2 Examples of Basic Application Fields . . . . . . . . . 20.1.2.1 Manipulation of Formulas . . . . . . . . . . 20.1.2.2 Numerical Calculations . . . . . . . . . . . . 20.1.2.3 Graphical Representations . . . . . . . . . . 20.1.2.4 Programming in Computer Algebra Systems 20.1.3 Structure of Computer Algebra Systems . . . . . . . . 20.1.3.1 Basic Structure Elements . . . . . . . . . . . 20.2 Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.1 Basic Structure Elements . . . . . . . . . . . . . . . . 20.2.2 Types of Numbers in Mathematica . . . . . . . . . . . 20.2.2.1 Basic Types of Numbers in Mathematica . . 20.2.2.2 Special Numbers . . . . . . . . . . . . . . . 20.2.2.3 Representation and Conversion of Numbers . 20.2.3 Important Operators . . . . . . . . . . . . . . . . . . 20.2.4 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.4.1 Notions . . . . . . . . . . . . . . . . . . . . 20.2.4.2 Nested Lists, Arrays or Tables . . . . . . . . 20.2.4.3 Operations with Lists . . . . . . . . . . . . . 20.2.4.4 Special Lists . . . . . . . . . . . . . . . . . . 20.2.5 Vectors and Matrices as Lists . . . . . . . . . . . . . . 20.2.5.1 Creating Appropriate Lists . . . . . . . . . . 20.2.5.2 Operations with Matrices and Vectors . . . . 20.2.6 Functions . . . . . . . . . . . . . . . . . . . . . . . . 20.2.6.1 Standard Functions . . . . . . . . . . . . . . 20.2.6.2 Special Functions . . . . . . . . . . . . . . . 20.2.6.3 Pure Functions . . . . . . . . . . . . . . . . 20.2.7 Patterns . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.8 Functional Operations . . . . . . . . . . . . . . . . . 20.2.9 Programming . . . . . . . . . . . . . . . . . . . . . . 20.2.10Supplement about Syntax, Information, Messages . . . 20.2.10.1 Contexts, Attributes . . . . . . . . . . . . . 20.2.10.2 Information . . . . . . . . . . . . . . . . . . 20.2.10.3 Messages . . . . . . . . . . . . . . . . . . . 20.3 Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.1 Basic Structure Elements . . . . . . . . . . . . . . . . 20.3.1.1 Types and Objects . . . . . . . . . . . . . . 20.3.1.2 Input and Output . . . . . . . . . . . . . . . 20.3.2 Types of Numbers in Maple . . . . . . . . . . . . . . . 20.3.2.1 Basic Types of Numbers in Maple . . . . . . 20.3.2.2 Special Numbers . . . . . . . . . . . . . . . 20.3.2.3 Representation and Conversion of Numbers . 20.3.3 Important Operators in Maple . . . . . . . . . . . . . 20.3.4 Algebraic Expressions . . . . . . . . . . . . . . . . . . 20.3.5 Sequences and Lists . . . . . . . . . . . . . . . . . . . 20.3.6 Tables, Arrays, Vectors and Matrices . . . . . . . . . . 20.3.6.1 Tables and Arrays . . . . . . . . . . . . . . . 20.3.6.2 One-Dimensional Arrays . . . . . . . . . . . 20.3.6.3 Two-Dimensional Arrays . . . . . . . . . . . 20.3.6.4 Special Commands for Vectors and Matrices
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Contents XXXVII
20.3.7 Procedures, Functions and Operators . . . . . . . . . 20.3.7.1 Procedures . . . . . . . . . . . . . . . . . . 20.3.7.2 Functions . . . . . . . . . . . . . . . . . . . 20.3.7.3 Functional Operators . . . . . . . . . . . . . 20.3.7.4 Di erential Operators . . . . . . . . . . . . 20.3.7.5 The Functional Operator map . . . . . . . . 20.3.8 Programming in Maple . . . . . . . . . . . . . . . . . 20.3.9 Supplement about Syntax, Information and Help . . . 20.3.9.1 Using the Maple Library . . . . . . . . . . . 20.3.9.2 Environment Variable . . . . . . . . . . . . 20.3.9.3 Information and Help . . . . . . . . . . . . . 20.4 Applications of Computer Algebra Systems . . . . . . . . . . 20.4.1 Manipulation of Algebraic Expressions . . . . . . . . . 20.4.1.1 Mathematica . . . . . . . . . . . . . . . . . 20.4.1.2 Maple . . . . . . . . . . . . . . . . . . . . . 20.4.2 Solution of Equations and Systems of Equations . . . 20.4.2.1 Mathematica . . . . . . . . . . . . . . . . . 20.4.2.2 Maple . . . . . . . . . . . . . . . . . . . . . 20.4.3 Elements of Linear Algebra . . . . . . . . . . . . . . . 20.4.3.1 Mathematica . . . . . . . . . . . . . . . . . 20.4.3.2 Maple . . . . . . . . . . . . . . . . . . . . . 20.4.4 Di erential and Integral Calculus . . . . . . . . . . . 20.4.4.1 Mathematica . . . . . . . . . . . . . . . . . 20.4.4.2 Maple . . . . . . . . . . . . . . . . . . . . . 20.5 Graphics in Computer Algebra Systems . . . . . . . . . . . . 20.5.1 Graphics with Mathematica . . . . . . . . . . . . . . 20.5.1.1 Basic Elements of Graphics . . . . . . . . . . 20.5.1.2 Graphics Primitives . . . . . . . . . . . . . . 20.5.1.3 Syntax of Graphical Representation . . . . . 20.5.1.4 Graphical Options . . . . . . . . . . . . . . 20.5.1.5 Two-Dimensional Curves . . . . . . . . . . . 20.5.1.6 Parametric Representation of Curves . . . . 20.5.1.7 Representation of Surfaces and Space Curves 20.5.2 Graphics with Maple . . . . . . . . . . . . . . . . . . 20.5.2.1 Two-Dimensional Graphics . . . . . . . . . . 20.5.2.2 Three-Dimensional Graphics . . . . . . . . .
21 Tables 21.1 21.2 21.3 21.4 21.5 21.6 21.7
Frequently Used Mathematical Constants . . . . Natural Constants . . . . . . . . . . . . . . . . . Metric Pre xes . . . . . . . . . . . . . . . . . . International System of Physical Units (SI-Units) Important Series Expansions . . . . . . . . . . . Fourier Series . . . . . . . . . . . . . . . . . . . Inde nite Integrals . . . . . . . . . . . . . . . . 21.7.1 Integral Rational Functions . . . . . . . . 21.7.1.1 Integrals with X = ax + b . . . 21.7.1.2 Integrals with X = ax2 + bx + c 21.7.1.3 Integrals with X = a2 x2 . . . 21.7.1.4 Integrals with X = a3 x3 . . . 21.7.1.5 Integrals with X = a4 + x4 . . . 21.7.1.6 Integrals with X = a4 ; x4 . . .
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21.8
21.9 21.10 21.11 21.12 21.13 21.14
21.15 21.16 21.17 21.18 21.19 21.20 21.21
21.7.1.7 Some Cases of Partial Fraction Decomposition . . . . . . . . 21.7.2 Integrals of Irrational Functions ..................... p 21.7.2.1 Integrals with x andpa2 b2 x . . . . . . . . . . . . . . . . . 21.7.2.2 Other Integralspwith x . . . . . . . . . . . . . . . . . . . . 21.7.2.3 Integrals with pax + b . . p ................... 21.7.2.4 Integrals with pax + b and fx + g . . . . . . . . . . . . . . 21.7.2.5 Integrals with pa2 ; x2 . . . . . . . . . . . . . . . . . . . . . 21.7.2.6 Integrals with px2 + a2 . . . . . . . . . . . . . . . . . . . . . 21.7.2.7 Integrals with px2 ; a2 . . . . . . . . . . . . . . . . . . . . . 21.7.2.8 Integrals with ax2 + bx + c . . . . . . . . . . . . . . . . . . 21.7.2.9 Integrals with other Irrational Expressions . . . . . . . . . . 21.7.2.10 Recursion Formulas for an Integral with Binomial Di erential 21.7.3 Integrals of Trigonometric Functions . . . . . . . . . . . . . . . . . . . 21.7.3.1 Integrals with Sine Function . . . . . . . . . . . . . . . . . . 21.7.3.2 Integrals with Cosine Function . . . . . . . . . . . . . . . . . 21.7.3.3 Integrals with Sine and Cosine Function . . . . . . . . . . . . 21.7.3.4 Integrals with Tangent Function . . . . . . . . . . . . . . . . 21.7.3.5 Integrals with Cotangent Function . . . . . . . . . . . . . . . 21.7.4 Integrals of other Transcendental Functions . . . . . . . . . . . . . . . 21.7.4.1 Integrals with Hyperbolic Functions . . . . . . . . . . . . . . 21.7.4.2 Integrals with Exponential Functions . . . . . . . . . . . . . 21.7.4.3 Integrals with Logarithmic Functions . . . . . . . . . . . . . 21.7.4.4 Integrals with Inverse Trigonometric Functions . . . . . . . . 21.7.4.5 Integrals with Inverse Hyperbolic Functions . . . . . . . . . . De nite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.8.1 De nite Integrals of Trigonometric Functions . . . . . . . . . . . . . . 21.8.2 De nite Integrals of Exponential Functions . . . . . . . . . . . . . . . 21.8.3 De nite Integrals of Logarithmic Functions . . . . . . . . . . . . . . . 21.8.4 De nite Integrals of Algebraic Functions . . . . . . . . . . . . . . . . . Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.9.1 Elliptic Integral of the First Kind F (' k) k = sin . . . . . . . . . . 21.9.2 Elliptic Integral of the Second Kind E (' k) k = sin . . . . . . . . . 21.9.3 Complete Elliptic Integral, k = sin . . . . . . . . . . . . . . . . . . . Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bessel Functions (Cylindrical Functions) . . . . . . . . . . . . . . . . . . . . . Legendre Polynomials of the First Kind . . . . . . . . . . . . . . . . . . . . . Laplace Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.14.1Fourier Cosine Transformation . . . . . . . . . . . . . . . . . . . . . . 21.14.2Fourier Sine Transformation . . . . . . . . . . . . . . . . . . . . . . . 21.14.3Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 21.14.4Exponential Fourier Transformation . . . . . . . . . . . . . . . . . . . Z Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 21.17.1Standard Normal Distribution for 0.00 x 1.99 . . . . . . . . . . . 21.17.2Standard Normal Distribution for 2.00 x 3.90 . . . . . . . . . . . 2 Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fisher F Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Student t Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1029 1030 1030 1030 1031 1032 1033 1035 1036 1038 1040 1040 1041 1041 1043 1045 1049 1049 1050 1050 1051 1053 1054 1055 1056 1056 1057 1058 1059 1061 1061 1061 1062 1063 1064 1066 1067 1072 1072 1078 1083 1085 1086 1089 1091 1091 1092 1093 1094 1096 1097
Contents XXXIX
22 Bibliography Index Mathematic Symbols
1098 1109 A
XL List of Tables
ListofTables 1.1 1.2 1.3 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 5.1 5.2 5.3
De nition of powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pascal's triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Auxiliary values for the solution of equations of degree three . . . . . . . . . . . . . Domain and range of trigonometric functions . . . . . . . . . . . . . . . . . . . . . Signs of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Values of trigonometric functions for 0 30 45 60 and 90: . . . . . . . . . . . . Reduction formulas and quadrant relations of trigonometric functions . . . . . . . . Relations between the trigonometric functions of the same argument in the interval 0 < < =2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Domains and ranges of the inverses of trigonometric functions . . . . . . . . . . . . Relations between two hyperbolic functions with the same arguments for x > 0 . . . Domains and ranges of the area functions . . . . . . . . . . . . . . . . . . . . . . . For the approximate determination of an empirically given function relation . . . . Names of angles in degree and radian measure . . . . . . . . . . . . . . . . . . . . . Properties of some regular polygons . . . . . . . . . . . . . . . . . . . . . . . . . . De ning quantities of a right angled-triangle in the plane . . . . . . . . . . . . . . . De ning quantities of a general triangle, basic problems . . . . . . . . . . . . . . . Conversion between Degrees and Gons . . . . . . . . . . . . . . . . . . . . . . . . . Directional angle in a segment with correct sign for arctan . . . . . . . . . . . . . . Regular polyeders with edge length a . . . . . . . . . . . . . . . . . . . . . . . . . . De ning quantities of a spherical right-angled triangle . . . . . . . . . . . . . . . . First and second basic problems for spherical oblique triangles . . . . . . . . . . . . Third basic problem for spherical oblique triangles . . . . . . . . . . . . . . . . . . Fourth basic problem for spherical oblique triangles . . . . . . . . . . . . . . . . . . Fifth and sixth basic problemes for a spherical oblique triangle . . . . . . . . . . . . Scalar product of basis vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vector product of basis vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scalar product of reciprocal basis vectors . . . . . . . . . . . . . . . . . . . . . . . Vector product of reciprocal basis vectors . . . . . . . . . . . . . . . . . . . . . . . Vector equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric application of vector algebra . . . . . . . . . . . . . . . . . . . . . . . . Equation of curves of second order. Central curves ( 6= 0) . . . . . . . . . . . . . . Equations of curves of second order. Parabolic curves ( = 0) . . . . . . . . . . . . Coordinate signs in the octants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connections between Cartesian, cylindrical, and spherical polar coordinates . . . . Notation for the direction cosines under coordinate transformation . . . . . . . . . Type of surfaces of second order with 6= 0 (central surfaces) . . . . . . . . . . . . Type of surfaces of second order with = 0 (paraboloid, cylinder and two planes) . Tangent and normal equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vector and coordinate equations of accompanying con gurations of a space curve . . Vector and coordinate equations of accompanying con gurations as functions of the arclength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equations of the tangent plane and the surface normal . . . . . . . . . . . . . . . . Truth table of propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . NAND function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NOR function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 13 42 77 78 78 78 80 85 90 92 112 129 139 141 144 145 145 154 168 170 171 172 173 186 186 186 186 188 189 205 206 208 210 211 224 224 226 241 241 246 286 288 288
List of Tables XLI
5.4 5.5 5.6 5.7 5.8 5.9 6.1 6.2 6.3 7.1 7.2 7.3 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 11.1 13.1 13.2 13.3 13.4 14.1 14.2 14.3 15.1 15.2 16.1 16.2 16.3 16.4 16.5 16.6 17.1 19.1 19.2
Primitive Bravais lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bravais lattice, crystal systems, and crystallographic classes . . . . . . . . . . . . . Some Boolean functions with two variables . . . . . . . . . . . . . . . . . . . . . . Tabular representation of a fuzzy set . . . . . . . . . . . . . . . . . . . . . . . . . . t- and s-norms, p 2 IR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of operations in Boolean logic and in fuzzy logic . . . . . . . . . . . . . Derivatives of elementary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Di erentiation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivatives of higher order of some elementary functions . . . . . . . . . . . . . . . The rst Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximation formulas for some frequently used functions . . . . . . . . . . . . . Basic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Important rules of calculation of inde nite integrals . . . . . . . . . . . . . . . . . . Substitutions for integration of irrational functions I . . . . . . . . . . . . . . . . . Substitutions for integration of irrational functions II . . . . . . . . . . . . . . . . . Important properties of de nite integrals . . . . . . . . . . . . . . . . . . . . . . . Line integrals of the rst type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curve elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plane elements of area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of the double integral . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of the triple integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary regions of curved surfaces . . . . . . . . . . . . . . . . . . . . . . . . . Roots of the Legendre polynomial of the rst kind . . . . . . . . . . . . . . . . . . . Relations between the components of a vector in Cartesian, cylindrical, and spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental relations for spatial di erential operators . . . . . . . . . . . . . . . . Expressions of vector analysis in Cartesian, cylindrical, and spherical coordinates . . Line, surface, and volume elements in Cartesian, cylindrical, and spherical coordinates Real and imaginary parts of the trigonometric and hyperbolic functions . . . . . . . Absolute values and arguments of the trigonometric and hyperbolic functions . . . . Periods, roots and poles of Jacobian functions . . . . . . . . . . . . . . . . . . . . . Overview of integral transformations of functions of one variable . . . . . . . . . . . Comparison of the properties of the Fourier and the Laplace transformation . . . . Collection of the formulas of combinatorics . . . . . . . . . . . . . . . . . . . . . . Relations between events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Con dence level for the sample mean . . . . . . . . . . . . . . . . . . . . . . . . . . Error description of a measurement sequence . . . . . . . . . . . . . . . . . . . . . Steady state types in three-dimensional phase space . . . . . . . . . . . . . . . . . Helping table for FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
310 311 344 360 367 369 381 386 387 412 413 419 428 430 435 436 443 465 465 474 475 479 480 482 574 648 658 659 660 700 700 704 708 730 747 748 773 776 777 794 809 916 920
XLII List of Tables 19.3 19.4 19.5 19.6 19.7 19.8 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 20.10 20.11 20.12 20.13 20.14 20.15 20.16 20.17 20.18 20.19 20.20 20.21 20.22 20.23 20.24 20.25 20.26 20.27 20.28 20.29 21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8 21.9 21.10 21.11 21.12 21.13 21.14 21.15 21.16
Number systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters for the basic forms . . . . . . . . . . . . . . . . . . . . . Mathematica, numerical operations . . . . . . . . . . . . . . . . . . Mathematica, commands for interpolation . . . . . . . . . . . . . . . Mathematica, numerical solution of di erential equations . . . . . . Maple, options for the command fsolve . . . . . . . . . . . . . . . . . Mathematica, Types of numbers . . . . . . . . . . . . . . . . . . . . Mathematica, Important operators . . . . . . . . . . . . . . . . . . Mathematica, Commands for the choice of list elements . . . . . . . Mathematica, Operations with lists . . . . . . . . . . . . . . . . . . Mathematica, Operation Table . . . . . . . . . . . . . . . . . . . . Mathematica, Operations with matrices . . . . . . . . . . . . . . . . Mathematica, Standard functions . . . . . . . . . . . . . . . . . . . Mathematica, Special functions . . . . . . . . . . . . . . . . . . . . Maple, Basic types . . . . . . . . . . . . . . . . . . . . . . . . . . . Maple, Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maple, Types of numbers . . . . . . . . . . . . . . . . . . . . . . . . Maple, Arguments of function convert . . . . . . . . . . . . . . . . Maple, Standard functions . . . . . . . . . . . . . . . . . . . . . . . Maple, Special functions . . . . . . . . . . . . . . . . . . . . . . . . Mathematica, Commands for manipulation of algebraic expressions . Mathematica, Algebraic polynomial operations . . . . . . . . . . . . Maple, Operations to manipulate algebraic expressions . . . . . . . Mathematica, Operations to solve systems of equations . . . . . . . Maple, Matrix operations . . . . . . . . . . . . . . . . . . . . . . . Maple, Operations of the Gaussian algorithm . . . . . . . . . . . . . Mathematica, Operations of di erentiation . . . . . . . . . . . . . . Mathematica, Commands to solve di erential equations . . . . . . . Maple, Options of operation dsolve . . . . . . . . . . . . . . . . . . Mathematica, Two-dimensional graphic objects . . . . . . . . . . . Mathematica, Graphics commands . . . . . . . . . . . . . . . . . . Mathematica, Some graphical options . . . . . . . . . . . . . . . . . Mathematica, Options for 3D graphics . . . . . . . . . . . . . . . . Maple, Options for Plot command . . . . . . . . . . . . . . . . . . Maple, Options of command plot3d . . . . . . . . . . . . . . . . . . Frequently Used Constants . . . . . . . . . . . . . . . . . . . . . . . Natural Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . Metric Pre xes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . International System of Physical Units (SI-Units) . . . . . . . . . . Important Series Expansions . . . . . . . . . . . . . . . . . . . . . . Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inde nite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . De nite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bessel Functions (Cylindrical Functions) . . . . . . . . . . . . . . . Legendre Polynomials of the First Kind . . . . . . . . . . . . . . . . Laplace Transformation . . . . . . . . . . . . . . . . . . . . . . . . Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . Z -Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . .
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937 939 946 947 948 950 957 959 960 960 961 962 963 963 968 968 970 971 975 976 979 980 981 986 990 991 992 994 997 999 999 1000 1005 1006 1008 1010 1010 1012 1012 1015 1020 1023 1056 1061 1063 1064 1066 1067 1072 1086 1089
List of Tables XLIII
21.17 21.18 21.19 21.20 21.21
Standard Normal Distribution 2 Distribution . . . . . . . . Fisher F Distribution . . . . . Student t Distribution . . . . Random Numbers . . . . . .
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1091 1093 1094 1096 1097
1
1 Arithmetic
1.1 Elementary Rules for Calculations 1.1.1 Numbers
1.1.1.1 Natural, Integer, and Rational Numbers 1. Denitions and Notation
The positive and negative integers, fractions, and zero are together called the rational numbers. In relation to these we use the following notation (see 5.2.1, 1., p. 290): Set of natural numbers: IN=f0 1 2 3 : : :g Set of integers: Z= f: : : ;2 ;1 0 1 2 : : :g Set of rational numbers: Q= fxjx = pq with p 2 Z q 2 Z and q 6= 0g . The notion of natural numbers arose from enumeration and ordering. The natural numbers are also called the non-negative integers.
2. Properties of the Set of Rational Numbers
The set of rational numbers is in nite. The set is ordered, i.e., for any two di erent given numbers a and b we can tell which is the smaller one. The set is dense everywhere, i.e., between any two di erent rational numbers a and b (a < b) there
is at least one rational number c (a < c < b). Consequently, there is an in nite number of other rational numbers between any two di erent rational numbers.
3. Arithmetical Operations
The arithmetical operations (addition, subtraction, multiplication and division) can be performed with any two rational numbers, and the result is a rational number. The only exception is division by zero, which is not possible: The operation written in the form a : 0 is meaningless because it does not have any result: If a 6= 0, then there is no rational number b such that b 0 = a could be ful lled, and if a = 0 then b can be any of the rational numbers. The frequently occurring formula a : 0 = 1 (in nity) does not mean that the division is possible it is only the notation for the statement: If the denominator approaches zero and, e.g., the numerator does not, then the absolute value (magnitude) of the quotient exceeds any nite limit.
4. Decimal Fractions, Continued Fractions
Every rational number a can be represented as a terminating or periodically in nite decimal fraction or as a nite continued fraction (see 1.1.1.4, p. 3).
5. Geometric Representation
If we x an origin (the zeropoint) 0, a positive direction (orientation), and the unit of length l (measuring rule, see also 2.17.1, p. 114) and (Fig. 1.1), then every rational number a corresponds to a certain point on this line. This point has the coordinate a, and it is a so-called rational point. The line is called the numerical axis. Because the set of rational numbers is dense everywhere, between two rational points there are in nitely many further rational points. B −2 −3 −11 − 4
−1
1 3 2 2
0 l=1
Figure 1.1
8 3 3 x
A 0
K 1
2
Figure 1.2
3
2 1. Arithmetic
1.1.1.2 Irrational and Transcendental Numbers
The set of rational numbers is not satisfactory for calculus. Even though it is dense everywhere, it does not cover the whole numerical axis. For example if we rotate the diagonal AB of the unit square around A so that B goes into the point K , then K does not have any rational coordinate (Fig. 1.2). The introduction of irrational numbers allows us to assign a number to every point of the numerical axis. In textbooks we can nd exact de nitions for irrational numbers, e.g., by nests of intervals. For this survey it is enough to note that the irrational numbers take all the non-rational points of the numerical axis and every irrational number corresponds to a point of the axis, and that every irrational number can be represented as a non-periodic in nite decimal fraction. First of all, the non-integer real roots of the algebraic equation xn + an;1 xn;1 + + a1x + a0 = 0 (n > 1 integer integer coecients) (1.1a) belong to the irrational numbers. These roots are called algebraic irrationals. A: The simplest examples of algebraic irrationals are the real roots of the equation xn ; a = 0, as p numbers of the form n a , if they are not rational. p p B: 2 2 = 1:414 : : : 3 10 = 2:154 : : : are algebraic irrationals. The irrational numbers which are not algebraic irrationals are called transcendental. A: = 3:141592 : : : e = 2:718281 : : : are transcendental numbers. B: The decimal logarithm of the integers, except the numbers of the form 10n, are transcendental. The non-integer roots of the quadratic equation x2 + a1x + a0 = 0 (a1 a0 integers) (1.1b) p are called quadratic irrationals. They have the form (a + b D)=c (a b c integers, c 6= 0 D > 0, square-free number). The division of a line segment a in the ratio of the golden section x=a = (a ; x)=x (see p 3.5.2.3, 3., p. 193) leads to the quadratic equation x2 + x ; 1 = p 0, if a = 1. The solution x = ( 5 ; 1)=2 is a quadratic irrational. It contains the irrational number 5 .
1.1.1.3 Real Numbers
Rational and irrational numbers together form the set of real numbers, which is denoted by IR.
1. Most Important Properties
The set of real numbers has the following important properties (see also 1.1.1.1, 2., p. 1). It is: Finite. Ordered. Dense everywhere. Closed, i.e., every point of the numerical axis corresponds to a real number. This statement does not hold for the rational numbers.
2. Arithmetical Operations
Arithmetical operations can be performed with any two real numbers and the result is a real number, too. The only exception is division by zero (see 1.1.1.1, 3., p. 1). Raising to a power and also its inverse operation can be performed among real numbers so it is possible to take an arbitrary root of any positive number every positive real number has a logarithm for an arbitrary positive basis, except that 1 cannot be a basis. A further generalization of the notion of numbers leads us to the concept of complex numbers (see 1.5, p. 34).
3. Interval of Numbers
A connected set of real numbers with endpoints a and b is called an interval of numbers with endpoints a and b, where a < b and a is allowed to be ;1 and b is allowed to be +1. If the endpoint itself does
1.1 Elementary Rules for Calculations 3
not belong to the interval, then this end of the interval is open, in the opposite case it is closed. We de ne an interval by its endpoints a and b, putting them in braces. We use a bracket for a closed end of the interval and a parenthesis for an open one. We distinguish between open intervals (a b), half-open (half-closed) intervals a b) or (a b] and closed intervals a b], according to whether none of the endpoints, one of the endpoints or both endpoints belong to it, respectively. We frequently meet the notation ]a b instead of (a b) for open intervals, and analogously a b instead of a b). In the case of graphical representations, we denote the open end of the interval by a round arrow head, the closed one by a lled point.
1.1.1.4 Continued Fractions
Continued fractions are nested fractions, by which rational and irrational numbers can be represented and approximated even better than by decimal representation (see 19.8.1.1, p. 937 and A and B on p. 4). 1. Rational Numbers p =a + 1 : (1.2) The continued fraction of a rational number is 1 q 0 a1 + nite. For a positive rational number which is 1 a2 + . greater than 1 it has the form (1.2). We abbrevi1 .. + p ate it by the symbol = a0 a1 a2 : : : an] with an;1 + a1 q n ak 1 (k = 1 2 : : : n). The numbers ak are calculated with the help of the Euclidean algorithm:
! p = a + r1 0 < r1 < 1 q 0 q q q = a + r2 0 < r2 < 1 r1 1 r1 r1 r1 = a + r3 0 < r3 < 1 r2 2 r2 r2 ... ... ... ! rn;2 = a + rn 0 < rn < 1 n ;1 rn;1 rn;1 rn;1 rn;1 = a (r = 0) : n n+1 rn 61 = 2 + 7 = 2 + 1 = 2 + 1 = 2 3 1 6] . 6 27 27 3+ 7 3+ 1 1 1+ 6
2. Irrational Numbers
(1.3a) (1.3b) (1.3c) (1.3d) (1.3e)
Continued fractions of irrational numbers do not break o . We call them in nite continued fractions with a0 a1 a2 : : :]. If some numbers ak are repeated in an in nite continued fraction, then this fraction is called a periodic continued fraction or recurring chain fraction. Every periodic continued fraction represents a quadratic irrationality, and conversely, every quadratic irrationality has a representation in the form of a periodic continued fraction. p The numberp 2 = 1:4142135 : : : is a quadratic irrationality and it has the periodic continued fraction representation 2 = 1 2 2 2 : : :].
4 1. Arithmetic
3. Aproximation of Real Numbers
If = a0 a1 a2 : : :] is an arbitrary real number, then every nite continued fraction k = a0 a1 a2 : : : ak ] = pq
(1.4)
represents an approximation of . The continued fraction k is called the k-th approximant of . It can be calculated by the recursive formula pk;2 (k 1 p = 1 p = a q = 0 q = 1): (1.5) k = pqk = aak pqk;1 + ;1 0 0 ;1 0 +q k
k k;1
k ;2
According to the Liouville approximation theorem, the following estimat holds: j ; k j = j ; pqk j < q12 : (1.6) k k Furthermore, it can be shown that the approximants approach the real number with increasing accuracy alternatively from above and from below. The approximants converge to especially fast if the numbers ai (i = 1 2 : : : k) in (1.4) have large values. Consequently, the convergence is worst for the numbers 1 1 1 : : :]. A: From the decimal presentation of the continued fraction representation = 3 7 15 1 292 : : :] follows with the help of (1.3a){(1.3e). The corresponding approximants (1.5) with the estimat according to (1.6) are: 1 = 22 with j ; 1 j < 12 2 10;2, 2 = 333 with j ; 2 j < 1 2 9 10;5, 7 7 106 106 1 8 10;5. The actual errors are much smaller. They are less than 3 = 355 with j ; 3j < 113 1132 1:3 10;3 for 1, 8:4 10;5 for 2 and 2:7 10;7 for 3 . The approximants 1 2 and 3 represent better approximations for than the decimal representation with the corresponding number of digits. B: The formula of the golden section x=a = (a ; x)=x (see 1.1.1.2, p. 2, 3.5.2.3, 3., p. 193 and 17.3.2.4, 4., p. 845) can be represented by the following two continued fractions: x = a 1 1 1 : : :] and p x = a2 (1 + 5) = a2 (1 + 2 4 4 4 : : :]). The approximant 4 delivers in the rst case an accuracy of 0:018 a, in the second case of 0:000 001 a.
1.1.1.5 Commensurability
We call two numbers a and b commensurable, i.e., measurable by the same number, if both are an integer multiple of a third number c. From a = mc b = nc (m n 2 Z) it follows that a = x (x rational): (1.7) b Otherwise a and b are incommensurable. A: In a pentagon the sides and diagonals are incommensurable segments (see A in 3.1.5.3, p. 138). Today we consider that it was Hippasos from Metapontum (450 BC) who discovered irrational numbers by this example. B: The length of p a side and a diagonal of a square are incommensurable because their ratio is the irrational number 2 . C: The lengths of the golden section (see 1.1.1.2,p p. 2 and 3.5.2.3, 3.,p. 193) are incommensurable, because their ratio contains the irrational number 5 .
1.1.2 Methods for Proof Mostly we use three types of proofs: direct proof, indirect proof,
1.1 Elementary Rules for Calculations 5
proof by (mathematical or arithmetical) induction. We also talk about constructive proof.
1.1.2.1 Direct Proof
We start with a theorem which is already proved (premise p) and we derive the truth of the theorem we want to prove (conclusion q). The logical steps we mostly use for our conclusions are implication and equivalence.
1. Direct Proof by Implication
The implication p ) q means that the truth of the conclusion follows from the truth of the premise (see \ Implication " in the truth table, 5.1.1, p. 286). p Prove the inequality a +2 b ab for a > 0 b > 0. The premise is the well-known binomial formula (a + b)2 = a2 + 2ab + b2 . It follows by subtracting 4ab that (a + b)2 ; 4ab = (a ; b)2 0 we certainly obtain the statement from this inequality if we restrict our investigations only to the positive square roots because of a > 0 and b > 0.
2. Direct Proof by Equivalence
The proof will be delivered by verifying an equivalent statement. In practice it means that all the arithmetical operations which we use for changing p into q must be uniquely invertible. Prove the inequality 1 + a + a2 + + an < 1 ;1 a for 0 < a < 1. Multiplying by 1 ; a we obtain: 1 ; a + a ; a2 + a2 ; a3 + an ; an+1 = 1 ; an+1 < 1. This last inequality is true because of the assumption 0 < an+1 < 1, and the inequality we started from also holds because all the arithmetical operations we used are uniquely invertible.
1.1.2.2 Indirect Proof or Proof by Contradiction
To prove the statement q we start from its negation q", and from q" we arrive at a false statement r, i.e., q" ) r (see also 5.1.1, 7., p. 288). In this case q" must be false, because using the implication a false assumption can result only in a false conclusion (see truth table 5.1.1, p. 286). If q" is false q must be true. p p p Prove that the number 2 is irrational. Suppose, 2 is rational. Then the equality 2 = ab holds for some integers a b and b 6= 0. We can assume that the numbers a b are coprime numbers, i.e., they 2 p do not have any common divisor. We get ( 2)2 = 2 = ab2 or a2 = 2b2 , therefore, a2 is an even number, and this is possible only if a = 2n is an even number. We deduce that a2 = 4n2 = 2b2 holds, and hence b must be an even number, too. It is obviously a contradiction to the assumption that a and b are coprime.
1.1.2.3 Mathematical Induction
With this method, we prove theorems or formulas depending on natural numbers n. The principle of mathematical induction is the following: If the statement is valid for a natural number n0 , and if from the validity of the statement for a natural number n n0 the validity of the statement follows for n +1, then the statement is valid for every natural number n n0 . According to these, the steps of the proof are: 1. Basis of the Induction: We show that the statement is valid for n = n0. Mostly we can choose n0 = 1. 2. Induction Hypothesis: We suppose n is an integer such that the statement is valid (premise p). 3. Induction Conclusion: We formulate the proposition for n + 1 (conclusion q). 4. Proof of the Implication: p ) q. We call steps 3. and 4. together the induction step or logical deduction from n to n + 1.
6 1. Arithmetic Prove the formula sn = 1 + 1 + 1 + + 1 = n . 12 23 34 n(n + 1) n + 1 The steps of the proof by induction are: 1. n = 1 : s1 = 1 1 2 = 1 +1 1 is obviously true. 2. Suppose sn = 1 1 2 + 2 1 3 + 3 1 4 + + n(n1+ 1) = n +n 1 holds for an n 1. 3. Supposing 2. we have to show: sn+1 = nn ++ 12 . 4. The proof: sn+1 = 1 1 2 + 2 1 3 + 3 1 4 + + n(n1+ 1) + (n + 1)(1 n + 2) = sn + (n + 1)(1 n + 2) = n + 1 n2 + 2n + 1 = (n + 1)2 = n + 1 . = n + 1 (n + 1)(n + 2) (n + 1)(n + 2) (n + 1)(n + 2) n + 2
1.1.2.4 Constructive Proof
In approximation theory, for instance, the proof of an existence theorem usually follows a constructive process, i.e., the steps of the proof give a method of calculation for a result which satis es the propositions of the existence theorem. The existence of a third-degree interpolation-spline function (see 19.7.1.1, 1., p. 931) can be proved in the following way: We show that the calculation of the coecients of a spline satisfying the requirements of the existence theorem results in a tridiagonal linear equation system, which has a unique solution (see 19.7.1.1, 2., p. 932).
1.1.3 Sums and Products 1.1.3.1 Sums 1. Denition
To briey denote a sum we use the summation sign P:
a1 + a2 + : : : + an =
n X
k=1
ak :
(1.8)
With this notation we denote the sum of n summands ak (k = 1 2 : : : n). We call k the running index or summation variable.
2. Rules of Calculation
1. Sum of Summands Equal to Each Other , i.e., ak = a for k = 1 2 : : : n: n X
ak = na:
k=1
(1.9a)
2. Multiplication by a Constant Factor n X
cak = c
k=1
n X
k=1
ak :
3. Separating a Sum n X
k=1
ak =
m X
k=1
ak +
(1.9b) n X
k=m+1
ak (1 < m < n):
(1.9c)
4. Addition of Sums with the Same Length n X
k=1
(ak + bk + ck + : : :) =
n X
k=1
ak +
n X
k=1
bk +
n X
k=1
ck + : : : :
(1.9d)
1.1 Elementary Rules for Calculations 7
5. Renumbering n X
k=1
ak =
m+ Xn;1 k=m
n X
ak;m+1
k=m
ak =
n;X m+l k=l
ak+m;l :
(1.9e)
6. Exchange the Order of Summation in Double Sums n X m X
i=1 k=1
m X n X
aik =
aik :
k=1 i=1
(1.9f)
1.1.3.2 Products 1. Denition
The abbreviated notation for a product is the product sign Q: n Y
a1a2 : : : an =
k=1
ak :
(1.10)
With this notation we denote a product of n factors ak (k = 1 2 : : : n), where k is called the running index.
2. Rules of Calculation
1. Product of Coincident Factors , i.e., ak = a for k = 1 2 : : : n: n Y
ak = an:
k=1
(1.11a)
2. Factor out a Constant Factor n Y
k=1
(cak ) = cn
n Y
k=1
ak :
(1.11b)
3. Separating into Partial Products n Y
k=1
ak =
m Y
k=1
ak
n Y
k=m+1
ak (1 < m < n):
4. Product of Products n Y
k=1
ak bk ck : : : =
5. Renumbering n Y
k=1
ak =
m+Y n;1 k=m
n Y
k=1
ak
(1.11c)
n n Y Y bk ck : : : :
k=1
ak;m+1
(1.11d)
k=1
n Y k=m
ak =
n;Y m+l k=l
ak+m;l :
(1.11e)
6. Exchange the Order of Multiplication in Double Products n Y m Y
i=1 k=1
aik =
m Y n Y
k=1 i=1
aik :
(1.11f)
1.1.4 Powers, Roots, and Logarithms 1.1.4.1 Powers
The notation ax is used for the algebraic operation of raising to a power. The number a is called the base, x is called the exponent or power, and ax is called the power. Powers are de ned as in Table 1.1.
8 1. Arithmetic For the allowed values of bases and exponents we have the following
Rules of Calculation:
x ax : ay = aay = ax;y (1.12) x x ax bx = (a b)x ax : bx = abx = ab (1.13) (ax)y = (ay )x = ax y (1.14) x x ln a a =e (a > 0): (1.15) Here ln a is the natural logarithm of a where e = 2:718281828459 : : : is the base. Special powers are +1 if n even (;1)n = ; (1.16a) a0 = 1 for any a 6= 0 : (1.16b) 1 if n odd Table 1.1 De nition of powers
ax ay = ax+y
base a
exponent x 0 arbitrary real, 6= 0 n = 1 2 3 : : :
positive real
0
1.1.4.2 Roots
power ax 1 an = a| a a{z : : : a} (a to the power n) n factors n = ;1 ;2 ;3 : : : an = ;1n
a p p p rational: q a q = q ap (p, q integer, q > 0) (q-th root of a to the power p) irrational: pk lim pk lim a qk k!1 q k!1 k
positive
0
According 1.1 the n-th root of a positive number a is the positive number denoted by pn a (toa >Table 0 real n > 0 integer): (1.17a) We call this operation taking of the root or extraction of the root, and a is called the radicand, n is called the radical or index. The solution of the equation xn = a (a real or complex (1.17b) p n > 0 integer) is often denoted by x = n a. But we must not be confused: In this relation this notation denotes all the solutions of the equation, i.e., it represents n di erent values xk (k = 1 2 : : : n). In the cace of negative or complex values they are to be determined by (1.140b) (see 1.5.3.6, p. 38). The equation x2 = 4 has two real solutions, namely 2. p 3 The p equation x = ;8 has three roots among the complex numbers: x1 = 1+i 3 x2 = ;2 and x3 = 1 ; i 3 , but only one among the reals.
1.1 Elementary Rules for Calculations 9
1.1.4.3 Logarithms 1. Denition
The logarithm u of a positive number x > 0 to the base b > 0, b 6= 1, is the exponent of the power which has the value x with b in the base. We denote it by u = logb x. Consequently the equation bu = x (1.18a) yields logb x = u (1.18b) and conversely the second one yields the rst one. In particular we have for b > 1 logb 1 = 0 logb b = 1 logb 0 = ;1 (1.18c) +1 for b < 1: The logarithm of negative numbers can be de ned only among the complex numbers. To take the logarithm of a given number means to nd its logarithm. We take the logarithm of an expression when we transform it like (1.19a, 1.19b). The determination of a number or an expression from its logarithm is called raising to a power.
2. Some Properties of the Logarithm
a) Every positive number has a logarithm to any positive base, except the base b = 1. b) For x > 0 and y > 0 the following Rules of Calculation are valid for any b (which is allowed to be
a base):
!
log xy = log x ; log y (1.19a) p log xn = n log x in particular log n x = n1 log x : (1.19b) With (1.19a, 1.19b) we can calculate the logarithm of products and fractions as sums or di erences of logarithms. 3x2 p3 y 3x2 p3 y Take the logarithm of the expression 2zu3 : log 2zu3 = log 3x2 p3 y ; log (2zu3) = log 3 + 2 log x + 13 log y ; log 2 ; log z ; 3 log u: Often the reverse transformation is required, i.e., we have to rewrite an expression containing logarithms of di erent amounts into one, which is the logarithm of one expression. 3x2 p3 y log 3 + 2 log x + 1 log y ; log 2 ; log z ; 3 log u = log . 3 2zu3 c) Logarithms to di erent bases are proportional, i.e., the logarithm to a base a can be change into a logarithm to the base b by multiplication: loga x = M logb x where M = loga b = log1 a : (1.20) log (xy) = log x + log y
We call M the modulus of the transformation.
b
1.1.4.4 Special Logarithms
1. The logarithm to the base 10 is called the decimal or Briggsian logarithm. We write log10 x = lg x and log (x10 ) = + log x is valid:
2. The logarithm to the base e is called the natural or Neperian logarithm. We write
loge x = ln x: The modulus of transformation to change from the natural logarithm into the decimal is M = log e = ln110 = 0:4342944819
(1.21) (1.22) (1.23)
10 1. Arithmetic and to change from the decimal into the natural one it is (1.24) M1 = M1 = ln 10 = 2:3025850930 : 3. The logarithm to base 2 is called the binary logarithm. We write log2 x = ld x or log2 x = lb x: (1.25) 4. We can nd the values of the decimal and natural logarithm in logarithm tables. Some time ago the logarithm was used for numerical calculation of powers, and it often made numerical multiplication and division easier. Mostly the decimal logarithm was used. Today pocket calculators and personal computers make these calculations. Every number given in decimal form (so every real number), which is called in this relation the antilog, can be written in the form x = x^10k with 1 x^ < 10 (1.26a) by factoring out an appropriate power of ten: 10k with integer k. This form is called the half-logarithmic representation. Here x^ is given by the sequence of gures of x, and 10k is the order of magnitude of x. Then for the logarithm we have log x = k + log x^ with 0 log x^ < 1 i.e., log x^ = 0 : : : : (1.26b) Here k is the so-called characteristic and the sequence of gures behind the decimal point of log x^ is called the the mantissa. The mantissa can be found in logarithm tables. lg 324 = 2:5105, the characteristic is 2, the mantissa is 5105. If we multiply or divide this number by 10n, for example 324000 3240 3:24 0:0324, its logarithms have the same mantissa, here 5105, but di erent characteristics. That is why the mantissas are given in logarithm tables. In order to get the mantissa of a number x rst we have to move the decimal point right or left to get a number between 1 and 10, and the characteristic of the antilog x is determined by how many digits k the decimal point was moved. 5. Slide rule Beside the logarithm, the slide rule was of important practical help in numerical calculations. The slide rule works by the principle of the form (1.19a), so we multiply and divide by adding and subtracting numbers. On the slide rule the scale-segments are denoted according to the logarithm values, so multiplication and division can be performed as addition or subtraction (see Scale and Graph Papers 2.17.1, p. 114).
1.1.5 Algebraic Expressions 1.1.5.1 Denitions
1. Algebraic Expression
One or more algebraic quantities, such as numbers or p symbols, are called an algebraic expression or term if they are connected by the symbols, + ; : , etc., as well as by di erent types of braces for xing the order of operations.
2. Identity
is an equality relation between two algebraic expressions if for arbitrary values of the symbols in them the equality holds.
3. Equation
is an equality relation between two algebraic expressions if the equality holds only for a few values of the symbols. For instance an equality relation F (x) = f (x) (1.27) between two functions with the same independent variable is considered as an equation with one variable if it holds only for certain values of the variable. If the equality is valid for every value of x, we call it an identity, or we say that the equality holds identically, and we write F (x) f (x).
1.1 Elementary Rules for Calculations 11
4. Identical Transformations
are performed in order to change an algebraic expression into another one if the two expression are identically equal. Our goal is to have another form, e.g., to get a shorter form or a more convenient form for further calculations. We often want to have the expression in a form which is especially good for solving an equation, or taking the logarithm, or for calculating the derivative or integral of it, etc.
1.1.5.2 Algebraic Expressions in Detail 1. Principal Quantities
We call principal quantities the literal symbols occurring in algebraic expressions, according to which the expression is classi ed. They must be xed in any single case. In the case of functions, the independent variables are the principal quantities. The other quantities not given by numbers are the parameters of the expression. In some expressions the parameters are called coecients. We talk about coecients, e.g., in the cases of polynomials, Fourier series, and linear di erential equations, etc. An expression belongs to a certain class, depending on which kind of operations are performed on the principal quantities. Usually, we use the last letters of the alphabet x, y, z, u, v, : : : to denote the principal quantities and the rst letters a, b, c, : : : for parameters. The letters m, n, p, : : : are usually used for positive integer parameter values, for instance for indices in summations or in iterations.
2. Integral Rational Expressions
are expressions which contain only addition, subtraction, and multiplication of the principal quantities. The term also means powers of them with non-negative integer exponents.
3. Rational Expressions
contain also division by principal quantities, i.e., division by integral rational expressions, so principal quantities can have negative integers in the exponent.
4. Irrational Expressions
contain roots, i.e., non-integer rational powers of integral rational or rational expressions with respect to their principal quantities, of course.
5. Transcendental Expressions
contain exponential, logarithmic or trigonometric expressions of the principal quantities, i.e., there can be irrational numbers in the exponent of an expression of principal quantities, or an expression of principal quantities can be in the exponent, or in the argument of a trigonometric or logarithmic expression.
1.1.6 Integral Rational Expressions
1.1.6.1 Representation in Polynomial Form
Every integral rational expression can be changed into polynomial form by elementary transformations, as in addition, subtraction, and multiplication of monomials and polynomials. (;a3 + 2a2x ; x3 )(4a2 + 8ax) + (a3 x2 + 2a2x3 ; 4ax4 ) ; (a5 + 4a3x2 ; 4ax4 ) = ;4a5 + 8a4 x ; 4a2x3 ; 8a4x + 16a3x2 ; 8ax4 + a3x2 + 2a2 x3 ; 4ax4 ; a5 ; 4a3 x2 + 4ax4 = ;5a5 + 13a3x2 ; 2a2x3 ; 8ax4 .
1.1.6.2 Factorizing a Polynomial
Often we can decompose a polynomial into a product of monomials and polynomials. To do so, we can use factoring out, grouping, special formulas, and special properties of equations. A: Factoring out: 8ax2y ; 6bx3 y2 + 4cx5 = 2x2 (4ay ; 3bxy2 + 2cx3): B: Grouping: 6x2 + xy ; y2 ; 10xz ; 5yz = 6x2 + 3xy ; 2xy ; y2 ; 10xz ; 5yz = 3x(2x + y) ; y(2x + y) ; 5z(2x + y) = (2x + y)(3x ; y ; 5z): C: Using of properties of equations (see also 1.6.3.1, p. 43): P (x) = x6 ; 2x5 + 4x4 + 2x3 ; 5x2 . a) Factoring out x2 . b) Realizing that 1 = 1 and 2 = ;1 are the roots of the equation P (x) = 0
12 1. Arithmetic and dividing P (x) by x2 (x ; 1)(x + 1) = x4 ; x2 we get the quotient x2 ; 2x + 5. We can no longer decompose this expression into real factors because p = ;2, q = 5, p2=4 ; q < 0, so nally we have the decomposition: x6 ; 2x5 + 4x4 + 2x3 ; 5x2 = x2 (x ; 1)(x + 1)(x2 ; 2x + 5).
1.1.6.3 Special Formulas
(x y)2 = x2 2xy + y2 (x + y + z)2 = x2 + y2 + z2 + 2xy + 2xz + 2yz (x + y + z + + t + u)2 = x2 + y2 + z2 + + t2 + u2 + +2xy + 2xz + + 2xu + 2yz + + 2yu + + 2tu (x y)3 = x3 3x2 y + 3xy2 y3: We calculate the expression (x y)n by the binomial formula (see (1.36a){(1.37a)). (x + y)(x ; y) = x2 ; y2 xn ; yn = xn;1 + xn;2y + + xyn;2 + yn;1 (for integer n, and n > 1), x;y xn + yn = xn;1 ; xn;2 y + ; xyn;2 + yn;1 (for odd n, and n > 1) x+y xn ; yn = xn;1 ; xn;2 y + + xyn;2 ; yn;1 (for even n, and n > 1): x+y
(1.28) (1.29) (1.30) (1.31) (1.32) (1.33) (1.34) (1.35)
1.1.6.4 Binomial Theorem
1. Power of an Algebraic Sum of Two Summands (First Binomial Formula)
The formula (a + b)n = an + nan;1 b + n(n ; 1) an;2b2 + n(n ; 1)(n ; 2) an;3b3 2! 3! n ( n ; 1) : : : ( n ; m + 1) n ;m m ++ a b + + nabn;1 + bn (1.36a) m! is called the binomial theorem, where a and b are real or complex values and n = 1 2 : : : . Using the binomial coecients delivers a shorter and more convenient notation: ! ! ! ! ! ! n abn;1 + n bn(1.36b) (a + b)n = n0 an + n1 an;1b + n2 an;2 b2 + n3 an;3b3 + + n ; 1 n or ! n n X (a + b)n = an;k bk : (1.36c) k=0 k
2. Power of an Algebraic Di erence (Second Binomial Formula)
or
n ; 2) an;3b3 (a ; b)n = an ; nan;1 b + n(n2!; 1) an;2 b2 ; n(n ; 1)( 3! : (n ; m + 1) an;mbm + + (;1)nbn + + (;1)m n(n ; 1) : :m ! (a ; b)n =
n n! X (;1)k an;k bk : k=0 k
(1.37a) (1.37b)
1.1 Elementary Rules for Calculations 13
3. Binomial Coe cients
The de nition is for non-negative and integer n and k: ! n = n! (1.38a) k (n ; k)!k! (0 k n) where n! is the product of the positive integers from 1 to n, and it is called n factorial: n! = 1 2 3 : : : n and by de nition 0! = 1: (1.38b) We can easily see the binomial coecients from the Pascal triangle in Table 1.2. The rst and the last number is equal to one in every row every other coecient is the sum of the numbers standing on left and on right in the row above it. Simple calculations verify the following formulas: ! ! ! ! ! n = 1 n = n n = 1: (1.39b) n = n = n! (1.39a) 0 1 n k n ; k k!(n ; k)!
! ! ! ! ! n+1 = n + n ;1 + n; 2 ++ k : k+1 k k k k ! ! ! ! n+1 = n+1 n : n = n;k n : (1.39d) k n;k+1 k k+1 k+1 k ! ! ! n+1 = n + n : k+1 k+1 k
(1.39c) (1.39e) (1.39f)
Table 1.2 Pascal's triangle
n 0 1 2 3 4 5 6 6 ...
1 "! 6 0
1
1 6 "! 6 1
1 5
Coecients 1 1 1 1 2 3 3 4 6 10 10 15 20 "! "! 6 6 2 3
1 4 15 "! 6 4
1 5
1 6 "! 6 5
1
1 "! 6 6
For an arbitrary real value ( 2 IR) and a non-negative integer k one can de ne the binomial coe! cient k : ! ! = ( ; 1)( ; 2) ( ; k + 1) for integer k and k 1 = 1: (1.40) k! 0 k ! ; 12 = ; 12 (; 12 ; 1)(; 12 ; 2) = ; 5 : 3 3! 16
14 1. Arithmetic
4. Properties of the Binomial Coe cients
The binomial coecients increase until the middle of the binomial formula (1.36b), then decrease. The binomial coecients are equal for the terms standing in symmetric positions with respect to the start and the end of the expression. The sum of the binomial coecients in the binomial formula of degree n is equal to 2n . The sum of the coecients at the odd positions is equal to the sum of the coecients at the even
positions.
5. Binomial Series
The formula (1.36a) of the binomial theorem can also be extended for negative and fraction exponents. If jbj < a, then (a + b)n has a convergent innite series (see also 21.5, p. 1015): n ; 2) an;3b3 + : (1.41) (a + b)n = an + nan;1 b + n(n2!; 1) an;2b2 + n(n ; 1)( 3!
1.1.6.5 Determination of the Greatest Common Divisor of Two Polynomials
It is possible that two polynomials P (x) of degree n and Q(x) of degree m with n m have a common polynomial factor, which contains x. The least common multiple of these factors is the greatest common divisor of the polynomials. P (x) = (x ; 1)2(x ; 2)(x ; 4) Q(x) = (x ; 1)(x ; 2)(x ; 3) the greates common devisor is (x ; 1)(x ; 2). If P (x) and Q(x) do not have any common polynomial factor, we call them relatively prime or coprime. In this case, their greatest common divisor is a constant. The greatest common divisor of two polynomials P (x) and Q(x) can be determined by the Euclidean algorithm without decomposing them into factors: 1. Division of P (x) by Q(x) = R0 (x) results in the quotient T1(x) and the remainder R1 (x): P (x) = Q(x)T1 (x) + R1 (x) : (1.42a) 2. Division of Q(x) by R1 (x) results in the quotient T2(x) and the remainder R2 (x): Q(x) = R1(x)T2 (x) + R2(x): (1.42b) 3. Division of R1 (x) by R2 (x) results in T3 (x) and R3(x), etc. The greatest common divisor of the two polynomials is the last non-zero remainder Rk (x). This method is known from the arithmetic of natural numbers (see 1.1.1.4, p. 3). We determine the greatest common divisor, for instance, when we solve equations, and we want to separate the roots with higher multiplicity, and when we apply the Sturm method (see 1.6.3.2, 2., p. 44).
1.1.7 Rational Expressions
1.1.7.1 Reducing to the Simplest Form
Every rational expression can be written in the form of a quotient of two coprime polynomials. To do this, we need only elementary transformations such as addition, subtraction, multiplication and division of polynomials and fractions and simpli cation of fractions. 3x + 2x z+ y 2 x + z ;y + z : Find the most simple form of x x2 + z12 (3xz + 2x + y)z2 + ;y2z + x + z = 3xz3 + 2xz2 + yz2 + (x3 z2 + x)(;y2z + x + z) = (x3z2 + x)z z x3z3 + xz
1.1 Elementary Rules for Calculations 15
3xz3 + 2xz2 + yz2 ; x3 y2z3 ; xy2z + x4 z2 + x2 + x3 z3 + xz . x3 z3 + xz
1.1.7.2 Determination of the Integral Rational Part
A quotient of two polynomials with the same variable x is a proper fraction if the degree of the numerator is less than the degree of the denominator. In the opposite case, we call it an improper fraction. Every improper fraction can be decomposed into a sum of a proper fraction and a polynomial by dividing the numerator by the denominator, i.e., separating the integral rational part. 4 3 22a2 x2 ; 24a3 x + 10a4 Determine the integral rational part of R(x) = 3x ; 10ax x+ : 2 ; 2ax + 3a2 3 x ; 5a4 ; 2 a (3x4 ;10ax3 +22a2x2 ;24a3x +10a4) : (x2 ; 2ax + 3a2) = 3x2 ; 4ax + 5a2 + x2 ; 2ax ; 3a2 3x4 ; 6ax3 + 9a2 x2 ; 4ax33 +13a22x22 ;24a33x ; 4ax + 8a x ;12a x 5a2 x2 ;12a3x +10a4 5a2 x2 ;10a3x +15a4 2a3 x ; 5a4 ; 2a3x; 5a4: R(x) = 3x2 ; 4ax + 5a2 + x; 2 ; 2ax + 3a2 : The integral rational part of a rational function R(x) is considered to be as an asymptotic approximation for R(x) because for large values of jxj, the value of the proper fraction part tends to zero, and R(x) behaves as its polynomial part.
1.1.7.3 Decomposition into Partial Fractions
Every proper fraction P (x) = anxn + an;1xn;1 + + a1 x + a0 R(x) = Q (n < m) (1.43) (x) bm xm + bm;1 xm;1 + + b1 x + b0 can be decomposed uniquely into a sum of partial fractions. In (1.43) the coecients a0 a1 : : : an b0 b1 : : : bm are arbitrary real or complex numbers we can suppose that bm = 1 , otherwise we can divide the numerator and the denominator by it. The partial fractions have the form p 2 A Dx + E (1.44a) with ; q < 0 : (1.44b) k 2 l (x ; ) (x + px + q) 2 If we restrict our investigation to real numbers, the following four cases, 1, 2, 3 and 4 can occur. If we consider complex numbers we have only two cases: cases 1 and 2. In the complex case, every fraction R(x) can be decomposed into a sum of fractions having the form (1.44a), where A and are complex numbers. We will use it when we solve linear di erential equations.
1. Decomposition into Partial Fractions, Case 1
Suppose the equation Q(x) = 0 for the polynomial Q(x) in the denominator has m di erent simple roots 1 ,. . . m . Then the decomposition has the form P (x) an xn + + a 0 A1 A2 Am (1.45a) Q(x) = (x ; 1)(x ; 2 ) : : : (x ; m ) = x ; 1 + x ; 2 + + x ; m with coecients (1.45b) A1 = QP0((1)) A2 = QP0((2)) : : : Am = QP0((m)) 1
2
m
where in the denominator we have the substitution values of the derivative dQ dx for x = 1 x = 2 : : : .
16 1. Arithmetic A: 6xx3;;x x+ 1 = Ax + x B; 1 + x C+ 1 , 1 = 0 , 2 = +1 and 3 = ;1 2
P (1) P (;1) P (x) = 6x2 ; x + 1 , Q0(x) = 3x2 ; 1 , A = QP0(0) (0) = ;1, B = Q0 (1) = 3 and C = Q0(;1) = 4, P (x) = ; 1 + 3 + 4 . Q(x) x x;1 x+1 Another possibility to determine the coecients A1 A2 : : : Am is called the method of comparing coecients or method of undetermined coecients. In the following cases we have to use this method. 2 2 (x + 1) + Cx(x ; 1) . B: 6x x3;;x x+ 1 = Ax + x B; 1 + x C+ 1 = A(x ; 1) + Bx x(x2 ; 1) The coecients of the corresponding powers of x must be equal on the two sides of the equality, so we obtain the equations 6 = A + B + C , ;1 = B ; C , 1 = ;A, which have the same solution for A, B and C as in example A.
2. Decomposition into Partial Fractions, Case 2
Suppose the polynomial Q(x) in the denominator can be decomposed into a product of powers of linear factors in the real case this means that the equation Q(x) = 0 has m real roots counted by multiplicity. Then the decomposition has the form P (x) = anxn + an;1xn;1 + + a0 = A1 + A2 + + Ak1 Q(x) (x ; 1)k1 (x ; 2)k2 : : : (x ; i )ki x ; 1 (x ; 1)2 (x ; 1 )k1 + B1 + B2 2 + + Bk2 k2 + + Lki ki : (1.46) x ; 2 (x ; 2 ) (x ; 2) (x ; i ) x+1 A1 B1 B2 B3 x(x ; 1)3 = x + x ; 1 + (x ; 1)2 + (x ; 1)3 . The coecients A1, B1 , B2, B3 can be determined by the method of comparing coecients.
3. Decomposition into Partial Fractions, Case 3
If the equation Q(x) = 0 for a polynomial with real coecients in the denominator also has complex roots but only with multiplicity one, then the decomposition has the form P (x) = anxn + an;1xn;1 + + a0 k 1 Q(x) (x ; 1) (x ; 2)k2 : : : (x2 + p1 x + q1 )(x2 + p2x + q2 ) : : : A1 + A2 + + Dx + E + Fx + G + : = x; (1.47) 1 (x ; 1 )2 x2 + p1x + q1 x2 + p2x + q2 The quadratic denominator x2 + px + q arises from the fact that if a polynomial with real coecients has a complex root, the complex conjugate of this root is also a root, and they have the same multiplicity. 3x2 ; 2 A Dx + E 2 (x + x + 1)(x + 1) = x + 1 + x2 + x + 1 . The coecients A, D, E can be determined by the method of comparing coecients.
4. Decomposition into Partial Fractions, Case 4
Analogously, if the real equation Q(x) = 0 for the polynomial of the denominator has complex roots with a higher multiplicity, we decompose the fraction into the form P (x) = anxn + an;1xn;1 + + a0 Q(x) (x ; 1)k1 (x ; 2)k2 : : : (x2 + p1 x + q1 )l1 (x2 + p2x + q2)l2 : : : A1 + A2 + + D1x + E1 + D2 x + E2 + = x; 1 (x ; 1 )2 x2 + p1x + q1 (x2 + p1 x + q1 )2
1.1 Elementary Rules for Calculations 17
Dl1 x + El1 + F1x + G1 + + Fl2 x + Gl2 + : (1.48) (x2 + p1x + q1)l1 x2 + p2x + q2 (x2 + p2 x + q2 )l2 5x2 ; 4x + 16 = A + D1x + E1 + D2x + E2 . The coecients A, D , E , D , E will 1 1 2 2 (x ; 3)(x2 ; x + 1)2 x ; 3 x2 ; x + 1 (x2 ; x + 1)2 be determined by the method of comparing coecients. +
1.1.7.4 Transformations of Proportions
The equality a=c (1.49a) yields ad = bc ac = db db = ac ab = dc b d and furthermore a b = c d a b = c d a c = b d a + b = c + d: b d a c c d a;b c;d From the equalities of the proportions a1 = a2 = = an (1.50a) a1 + a2 + + an = a1 : it follows that b1 b2 bn b1 + b2 + + bn b1
1.1.8 Irrational Expressions
(1.49b) (1.49c) (1.50b)
Every irrational expression can be written in a simpler form by 1. simplifying the exponent, 2. taking out terms from the radical sign and 3. moving the irrationality into the numerator. 1. Simplify the Exponent We can simplify the exponent if the radicand can be factorized and the index of the radical and the exponents in the radicand have a common factor. We divide the index of the radical and the exponents by their greatest common divisor. q6 q q 16(x12 ; 2x11 + x10 ) = 6 42 x52 (x ; 1)2 = 3 4x5 (x ; 1) . 2. Moving the Irrationality There are di erent ways to move the irrationality into the numerator. s 2 p3 2 s s s p2xy x 2 xy x 2 xy z A: 2y = 4y2 = 2y . B: 3 4yz2 = 3 8y3zz3 = 22xy yz . p p C: x +1py = px ; y p = xx2;; yy . x+ y x; y p3 p3 2 p3 p3 2 2 2 D: x +1p3 y = p3x ;x2 y +p3 y p3 2 = x ; xx3 +y +y y . x+ y x ;x y+ y 3. Simplest Forms of Powers and Radicals Also powers and radicals can be transformed into the simplest form. v v px 3xpx(p2 + px) 3xp2x + 3x2 u 81x6 u 9x3 t4 p p = u t p p = p3x p A: u = = 2;x . 2;x ( 2 ; x)4 ( 2 ; x)2 2; x p p p p B: px + 3 x2 + 4 x3 + 12 x7 px ; p3 x + p4 x ; 12 x5 = (x1=2 +x2=3 +x3=4 +x7=12 )(x1=2 ;x1=3 + =4 + x11=12 + x + x5=6 ; x11=12 ; x13=12 ; x1=4 ; x5=12 ) = x + x7=6 + x5=4 + x13=12 ; x5=6p; x ; x13p=12 ; x11p=12 + x3p 12 13 12 11 4 3 4 5 3=4 1=6 1=3 1=2 7 = 6 5 = 4 13 = 12 11 = 12 3 = 4 px4 3 ; x =p6 x ;p3 x p; x + x = x ; x ; x + x = x (1 ; x ; x + x ) = x (1 ; x ; x + x).
18 1. Arithmetic
1.2 Finite Series
1.2.1 De nition of a Finite Series The sum
sn = a0 + a1 + a2 + + an =
n X
ai
i=0
(1.51)
is called a nite series. The summands ai (i = 0 1 2 : : : n) are given by certain formulas, they are numbers, and they are the terms of the series.
1.2.2 Arithmetic Series
1. Arithmetic Series of First Order
is a nite series where the terms form an arithmetic sequence, i.e., the di erence of two terms standing after each other is a constant: %ai = ai+1 ; ai = d = const holds, so ai = a0 + id: (1.52a) With these we have: sn = a0 + (a0 + d) + (a0 + 2d) + + (a0 + nd) (1.52b) sn = a0 +2 an (n + 1) = n +2 1 (2a0 + nd): (1.52c)
2. Arithmetic Series of k-th Order
is a nite series, where the k-th di erences %k ai of the sequence a0 , a1, a2, . . . ,an are constants. The di erences of higher order are calculated by the formula % ai = %;1 ai+1 ; %;1 ai ( = 2 3 : : : k): (1.53a) It is convenient to calculate them from the dierence schema (also di erence table or triangle schema):
a0 a1 a2 a3 ...
%a0 %a1 %a2 ...
%2a0 %2a1 %2a2 ... %2an;2
%3 a0 %3 a
1
... %3 a
n;3
. . . %k a
0
. . . %k a 1 ... %k an;k ...
... ...
%n a0
%an;1 an The following formulas hold for the terms and the sum: ! ! ! ai = a0 + 1i %a0 + 2i %2a0 + + ki %k a0 (i = 1 2 : : : n) ! ! ! ! + 1 %k a : sn = n +1 1 a0 + n +2 1 %a0 + n +3 1 %2a0 + + nk + 0 1
(1.53b)
(1.53c) (1.53d)
1.2 Finite Series 19
1.2.3 Geometric Series
The sum (1.51) is called a geometric series, if the terms form a geometric sequence, i.e., the ratio of two successive terms is a constant: ai+1 = q = const holds so a = a qi: (1.54a) i 0 ai With these we have n+1 sn = a0 + a0q + a0 q2 + + a0qn = a0 q q ;;1 1 for q 6= 1 (1.54b) sn = (n + 1)a0 for q = 1: (1.54c) If n ! 1 (see 7.2.1.1, 2., p. 406), then we get an innite geometric series, which has a limit if jqj < 1, and it is s = 1 a;0 q : (1.54d)
1.2.4 Special Finite Series
1 + 2 + 3 + + (n ; 1) + n = n(n + 1) 2 ( n + 1)(2p + n) p + (p + 1) + (p + 2) + + (p + n) = 2 1 + 3 + 5 + + (2n ; 3) + (2n ; 1) = n2 2 + 4 + 6 + + (2n ; 2) + 2n = n(n + 1) n + 1) 2 1 + 22 + 32 + + (n ; 1)2 + n2 = n(n + 1)(2 6 2 2 13 + 23 + 33 + + (n ; 1)3 + n3 = n (n + 1) 4 2 ; 1) n (4 n 2 2 2 2 1 + 3 + 5 + + (2n ; 1) = 3 13 + 33 + 53 + + (2n ; 1)3 = n2 (2n2 ; 1) 1)(3n2 + 3n ; 1) 14 + 24 + 34 + + n4 = n(n + 1)(2n + 30 n nxn+1 1 + 2x + 3x2 + + nxn;1 = 1 ; (n + 1)x + (x 6= 1) : (1 ; x)2
(1.55) (1.56) (1.57) (1.58) (1.59) (1.60) (1.61) (1.62) (1.63) (1.64)
1.2.5 Mean Values
(See also 16.3.4.1, 1., p. 779 and 16.4, p. 787)
1.2.5.1 Arithmetic Mean or Arithmetic Average
The arithmetic mean of the n quantities a1 a2 : : : an is the expression n X xA = a1 + a2 +n + an = n1 ak : k=1 For two values a and b we have: xA = a +2 b :
(1.65a) (1.65b)
20 1. Arithmetic The values a , xA and b form an arithmetic sequence.
1.2.5.2 Geometric Mean or Geometric Average
The geometric mean of n positive quantities a1 a2 : : : an is the expression
xG = pn a1 a2 : : : an =
n ! n1 Y ak :
For two values p a and b we have xG = ab :
xG
xG a
a)
.
b
(1.66a)
k=1
b) Figure 1.3
b a
.
(1.66b) The values a xG and b form a geometric sequence. If a and b are given line segments, p then we can get a segment with length xG = ab with the help of one of the constructions shown in Fig. 1.3a or in Fig. 1.3b. A special case of the geometric mean is when we want to divide a line segment according to the golden section (see 3.5.2.3, 3., p. 193).
1.2.5.3 Harmonic Mean
The harmonic mean of n quantities a1 a2 : : : an is the expression
;1
;1 " X n xH = n1 ( a1 + a1 + + a1 ) = n1 a1 : 1 2 n k=1 k For two values a and b we have
;1 xH = 21 a1 + 1b xH = a2+abb :
(1.67a) (1.67b)
1.2.5.4 Quadratic Mean
The quadratic mean of n quantities a1 , a2 ,. . . , an is the expression v s u n u X xQ = n1 (a1 2 + a2 2 + + an2 ) = t n1 a2k : k=1 For two values a and b, we have s 2 2 xQ = a +2 b : The quadratic mean is important in the theory of observational error (see 16.4, p. 787).
(1.68a) (1.68b)
1.2.5.5 Relations Between the Means s of Two Positive Values 2 2 p For xA = a +2 b xG = ab xH = a2+abb xQ = a +2 b we have 1. if a < b, then a < xH < xG < xA < xQ < b 2. if a = b, then a = x A = xG = x H = x Q = b :
(1.69a) (1.69b)
1.3 Business Mathematics 21
1.3 Business Mathematics
Business calculations are based on the use of arithmetic and geometric series, on formulas (1.52a){ (1.52c) and (1.54a){(1.54d). However these applications in banking are so varied and special that a special discipline has developed using speci c terminology. So business arithmetic is not con ned only to the calculation of the principal by compound interest or the calculation of annuities. It also includes the calculation of interest, repayments, amortization, calculation of instalment payments, annuities, depreciation, e ective interest yield and the yield on investment. Basic concepts and formulas for calculations are discussed below. For studying business mathematics in detail, you will have to consult the specialist literature on the subject. Actuarial mathematics and risk theory use the methods of probability theory and mathematical statistics, and they represent a separate discipline, so we do not discuss them here.
1.3.1 Calculation of Interest or Percentage 1. Percentage or Interest
p K , where K denotes the principal in business mathematics. The expression p percent of K means 100 The symbol for percent is %, i.e., we have the equalities p p% = 100 or 1% = 0:01: (1.70)
2. Increment
If K is raised by p%, we get the increased value p : K~ = K 1 + 100 (1.71) Relating the increment K p to the new value K~ , then the proportion K p : K~ = p~ : 100, K~ contains 100 100 p 100 p~ = 100 + p (1.72) percent of increment. If an article has a value of 200 and a 15% extra charge is added, the nal value is 230. This price contains p~ = 15 100 = 13:04 percent increment for the user. 115
3. Discount or Reduction
If we reduce the value K by p% rebate, we get the reduced value p : K~ = K 1 ; 100 p to the new value K~ , then we realize If we compare the reduction K 100
p 100 p~ = 100 ;p
(1.73)
(1.74)
percent of rebate. If an article has a value 300, and they give a 10% discount, it will be sold for 270. This price 100 = 11:11 percent rebate for the buyer. contains p~ = 10 90
22 1. Arithmetic
1.3.2 Calculation of Compound Interest 1.3.2.1 Interest
Interest is either payment for the use of a loan or it is a revenue realized from a receivable. For a principal K , placed for a whole period of interest (usually one year),
p K 100 (1.75) interest is paid at the end of the period of interest. Here p is the rate of interest for the period of interest, and we say that p% interest is paid for the principal K .
1.3.2.2 Compound Interest
Compound interest is computed on the principal and on any interest earned that has not been paid or withdrawn. It is the return on the principal for two or more time periods. The interest of the principal increased by interest is called compound interest. We discuss di erent cases depending on how the principal is changing.
1. Single Deposit
Compounded annually the principal K increases after n years up to the nal value Kn. At the end of the n-th year this value is: p n : Kn = K 1 + 100 (1.76) p = q and we call q the accumulation factor or growth factor. For briefer notation we substitute 1 + 100 Interest may be compounded for any period of time: annually, half-annually, monthly, daily, and so on. If we divide the year into m equal interest periods the interest will be added to the principal K at the end of every period. Then the interest is K 100p m for one interest period, and the principal increases after n years with m interest period up to the value mn Kmn = K 1 + 100p m : (1.77) m The quantity 1 + p is known as the nominal rate, and 1 + p as the eective rate. 100 100m A principal of 5000, with a nominal interest 7:2% annually, increases within 6 years a) compounded annually to K6 = 5000(1 + 0:072)6 = 7588:20, b) compounded monthly to K72 = 5000(1 + 0 072=12)72 = 7691:74.
2. Regular Deposits
Suppose we deposit the same amount E in equal intervals. Such an interval must be equal to an interest period. We can deposit at the beginning of the interval, or at the end of the interval. At the end of the n-th interest period we have the balance Kn:
a) Depositing at the Beginning: n Kn = Eq qq ;;11 :
(1.78a)
3. Depositing in the Course of the Year
b) Depositing at the End: n Kn = E qq ;;11 :
(1.78b)
A year or an interest period is divided into m equal parts. At the beginning or at the end of each of these time periods the same amount E is deposited and bears interest until the end of the year. In this way, after one year we have the balance K1:
1.3 Business Mathematics 23
a) Depositing" at the Beginning:
b) Depositing" at the End:
+ 1)p : ; 1)p : (1.79b) K1 = E m + (m200 (1.79a) K1 = E m + (m200 In the second year the total K1 bears interest, and further deposits and interests are added like in the rst year, so after n years the balance Kn for midterm deposits and yearly interest payment is:
a) Depositing" at the Beginning:
+ 1)p qn ; 1 : (1.80a) Kn = E m + (m200 q;1
b) Depositing at the End: " ( m ; 1)p qn ; 1 : Kn = E m + 200 q;1
(1.80b)
At a yearly rate of interest p = 5:2% a depositor deposits 1000 at the end of every month. After how many years will it reach the balance 500 000?
5:2 1:052n ; 1 , follows the answer, n = From (1.80b), for instance, from 500 000 = 1000 12 + 11200 0:052 22:42 years.
1.3.3 Amortization Calculus 1.3.3.1 Amortization
Amortization is the repayment of credits. Our assumptions: 1. For a debt S the debtor is charged at p% interest at the end of an interest period. 2. After N interest period the debt is completely repaid. The charge of the debtor consists of interest and principal repayment for every interest period. If the interest period is one year, the amount to be paid during the whole year is called an annuity. There are di erent possibilities for a debtor. For instance, the repayments can be made at the interest date, or meanwhile the amount of repayment can be di erent time by time, or it can be constant during the whole term.
1.3.3.2 Equal Principal Repayments
The amortization instalments are paid during the year, but no midterm compound interest is calculated. We use the following notation: S debt (interest payment at the end of a period with p%), S principal repayment (T = const), T = mN m number of repayments during one interest period, N number of interest periods until the debt is fully repaid. Besides the principal repayments the debtor also has to pay the interest charges:
a) Interest Zn for the n-th Interest Period: p S 1 ; 1 n ; m + 1 : (1.81a) Zn = 100 N 2m
b) Total Interest Z to be Paid for a Debt S mN Times, During N Interest Periods with an Interest Rate p% :
N X Z = Z = p S N ; 1 + m + 1 : (1.81b) n=1
n
A debt of 60 000 has a yearly interest rate of 8%. The principal repayment of 1000 for 60 months should be paid at the end of the months. How much is the actual interest at the end of each year? The interest for every year is calculated by (1.81a) with S = 60000 p = 8 N = 5 and m = 12. They are enumerated in the annexed table.
100
2 1. year: 2. year: 3. year: 4. year: 5. year:
2m Z1 = 4360 Z2 = 3400 Z3 = 2440 Z4 = 1480 Z5 = 520 Z = 12200
24 1. Arithmetic
The total interest can be calculated also by (1.81b) as Z = 8 60000 5 ; 1 + 13 = 12 200. 100 2 24
1.3.3.3 Equal Annuities
S the interest payable decreases over the course of time (see For equal principal repayments T = mN the previous example). In contrast to this, in the case of equal annuities the same amount is repaid for every interest period. A constant annuity A containing the principal repayment and the interest is repaid, i.e., the charge of the debtor is constant during the whole period of repayment. With the notation S debt (interest payment of p% at the end of a period), A annuity for every interest period (A const), a one instalment paid m times per interest period (a const), p the accumulation factor, q = 1 + 100 after n interest periods the remaining outstanding debt Sn is: " ; 1)p qn ; 1 : (1.82) Sn = S qn ; a m + (m200 q;1 n Here the term Sq denotes the value of the debt S after n interest periods with compound interest (see (1.76)). The second term in (1.82) gives the value of the midterm repayments a with compound interest (see (1.80b) with E = a). For the annuity: " ; 1)p : A = a m + (m200 (1.83) Here paying A once means the same as paying a m times. From (1.83) it follows that A ma. Because after N interest periods the debt must be completely repaid, from (1.82) for SN = 0 considering (1.83) we get: A = S qN qqN;;11 = S 1 q;;q;1N : (1.84)
To solve a problem of business mathematics we can express, from (1.84), any of the quantities A S q or N if the others are known. A: A loan of 60 000 bears 8% interest per year, and is to be repaid over 5 years in equal instalments. How much is the yearly annuity A and the monthly instalment a? From (1.84) and (1.83) we get: A = 60 000 0:081 = 15 027:39, a = 1502711:39 8 = 1207:99. 1 ; 1:085 12 + 200 B: A loan of S = 100 000 is to be repaid during N = 8 years in equal annuities with an interest rate of 7:5%. At the end of every year 5000 extra repayment must be made. How much will the monthly instalment be? For the annuity A per year according to (1.84) we get A = 100 000 0:0751 = 1; 1:0758 17 072:70. Because A consists of 12 monthly instalments a , and because of the 5000 extra payment
7:5 + 5000 = 17 072:70 follows, so the monthly at the end of the year, from (1.83) A = a 12 + 11200 charge is a = 972:62.
1.3 Business Mathematics 25
1.3.4 Annuity Calculations 1.3.4.1 Annuities
If a series of payments is made regularly at the same time intervals, in equal or varying amounts, at the beginning or at the end of the interval, we call it annuity payments. We distinguish: a) Payments on an Account The periodic payments, called rents, are paid on an account and bear compound interest. We use the formulas of 1.3.2. b) Receipt of Payments The payments of rent are made from capital bearing compound interest. We use the formulas of the annuity calculations in 1.3.3, where the annuities are called rents. If no more than the actual interest is paid as a rent, we call it a perpetual annuity. Rent payments (deposits and payo s) can be made at the interest terms, or at shorter intervals during the period of interest, i.e. in the course of the year.
1.3.4.2 Future Amount of an Ordinary Annuity
The date of the interest calculations and the payments should coincide. The interest is calculated at p% compound interest, and the payments (rents) on the account are always the same, R. The future value of the ordinary annuity Rn , i.e., the amount to which the regular deposits increase after n periods amounts to: n p : Rn = R qq ;;11 with q = 1 + 100 (1.85) The present value of an ordinary annuity R0 is the amount which should be paid at the beginning of the rst interest period (one time) to reach the nal value Rn with compound interest during n periods: R0 = Rqnn with q = 1 + p : (1.86) 100 A man claims 5000 at the end of every year for 10 years from a rm. Before the rst payment the rm declares bankruptcy. Only the present value of the ordinary annuity R0 can be asked from the administration of the bankrupt's estate. With an interest of 4% per year the man gets: n ;n 04;10 = 40 554:48. R0 = q1n R qq ;;11 = R 1q;;q 1 = 5000 1 ;01::04
1.3.4.3 Balance after n Annuity Payments
For ordinary annuity payments capital K is at our disposal bearing p% interest. After every interest period an amount r is paid. The balance Kn after n interest periods, i.e., after n rent payments, is: n p : Kn = Kqn ; Rn = Kqn ; r qq ;;11 with q = 1 + 100 (1.87a) Conclusions from (1.87a): p r = K 100 (1.87b) p r > K 100 (1.87c)
Consequently Kn = K holds, so the capital does not change. This is the case of perpetual annuity.
The capital will be completely used up after N rent payments. From (1.87a) it follows for KN = 0: N (1.87d) K = qrN qq ;;11 : If midterm interest is calculated and midterm rents are paid, and the original interest period is divided into m equal intervals, then in the formulas (1.85){(1.87a) n is replaced by mn and accordingly q = p by q = 1 + p . 1 + 100 100m
26 1. Arithmetic What amount must be deposited monthly at the end of the month for 20 years, from which a rent of 2000 should be paid monthly for 20 years, and the interest period is one month with an interest rate of 0:5%. From (1.87d) we get for n = 20 12 = 240 the sum K which is necessary for the required payments: 1:005240 ; 1 K = 1:2000 005240 0:005 =240 279 161:54. The necessary monthly deposits R are given by (1.85): ;1 R240 = 279 161:54 = R 1:005 0:005 , i.e., R = 604:19.
1.3.5 Depreciation
1. Methods of Depreciation
Depreciation is the term most often used to indicate that assets have declined in service potential in a given year either due to obsolescence or physical factors. Depreciation is a method whereby the original (cost) value at the beginning of the reporting year is reduced to the residual value at year-end. We use the following concepts: A depreciation base, N useful life (given in years), Rn residual value after n years (n N ), an (n = 1 2 : : : N ) depreciation rate in the n-th year. The methods of depreciation di er from each other depending on the amortization rate: straight-line method, i.e., equal yearly rates, decreasing-charge method, i.e., decreasing yearly rates.
2. Straight-Line Method
The yearly depreciations are constant, i.e., for amortization rates an and the remaining value Rn after n years we have: (1.88) Rn = A ; n A ;NRN (n = 1 2 : : : N ): (1.89) an = A ;NRN = a If we substitute RN = 0, then the value of the given thing is reduced to zero after N years, i.e., it is totally depreciated. The purchase price of a machine is A = 50 000. In 5 years it should be depreciated to a value R5 = 10 000. With linear depreciation acYear Depreciation Depreciation Residual Cumulated depr. in % cording to (1.88) and (1.89) base expense value of the depr. base we have the annexed amor1 50 000 8000 42 000 16.0 tization schedule: 2 42 000 8000 34 000 19.0 It shows that the percent3 34 000 8000 26 000 23.5 age of accumulated depreci4 26 000 8000 18 000 30.8 ation with respect to the ac5 18 000 8000 10 000 44.4 tual initial value is increasing.
3. Arithmetically Declining Balance Depreciation
In this case the depreciation is not constant. It is decreasing yearly by the same amount d, by the so-called multiple. For depreciation in the n-th year we have: an = a1 ; (n ; 1)d (n = 2 3 : : : N + 1 a1 and d are given): (1.90) N Considering the equality A ; RN = P an from the previous equation it follows that:
d = 2 NaN1 ;(N(A;;1)RN )] :
n=1
(1.91)
1.3 Business Mathematics 27
For d = 0 we get the special case of straight-line depreciation. If d > 0, it follows from (1.91) that (1.92) a1 > A ;NRN = a where a is the depreciation rate for straight-line depreciation. The rst depreciation rate a1 of the arithmetically-declining balance depreciation must satisfy the following inequality: A ; R N < a < 2 A ; RN : (1.93) 1 N N A machine of 50 000 purchase price is to be depreciated to the value 10 000 within 5 years by arithmetically declining depreciation. In the rst year 15 000 should be depreciated. The annexed depreciYear Depretiation Depreciation Residual Depreciation in % ation schedule is calbase expense value of depr. base culated by the given 1 50 000 15 000 35 000 30.0 formulas, and it shows 2 35 000 11 500 23 500 32.9 that with the excep3 23 500 8 000 15 500 34.0 tion of the last rate the 4 15 500 4 500 11 000 29.0 percentage of depreci5 11 000 1 000 10 000 9.1 ation is fairly equal.
4. Digital Declining Balance Depreciation
Digital depreciation is a special case of arithmetically declining depreciation. Here it is required that the last depreciation rate aN should be equal to the multiple d. From aN = d it follows that N) d = 2(NA(N;+R1) (1.94a) a1 = Nd a2 = (N ; 1)d : : : aN = d: (1.94b) The purchase price of a machine is A = 50 000. This machine is to be depreciated in 5 years to the value R5 = 10 000 by digital depreciation. Year Depreciation Depreciation Residual Depreciation in % base expense value of the depr. base The annexed depreciation 1 50 000 a1 = 5d = 13 335 36 665 26:7 schedule, calculated by the 2 36 665 a2 = 4d = 10 668 25 997 29:1 given formulas, shows that 3 25 997 a3 = 3d = 8 001 17 996 30:8 the percentage of the de4 17 996 a4 = 2d = 5 334 12 662 29:6 preciation is fairly equal. 5 12 662 a5 = d = 2 667 9 995 21:1
5. Geometrically Declining Balance Depreciation
Consider geometrically declining depreciation where p% of the actual value is depreciated every year. For the residual value Rn after n years we have: p n (n = 1 2 : : :) : Rn = A 1 ; 100 (1.95) Usually A (the acquisition cost) is given. The useful life of the asset is N years long. If from the quantities RN p and N , two is given, the third one can be calculated by the formula (1.95). A: A machine with a purchase value 50 000 is to be geometrically depreciated yearly by 10%. After how many years will its value drop below 10 000 for the rst time? Based on (1.95), we get that 000=50 000) N = ln(10 ln(1 ; 0:1) = 15:27 years.
B: For a purchase price of A = 1000 the residual value Rn should be represented for n = 1 2 : : : 10 years by a) straight-line, b) arithmetically declining, c) geometrically declining depreciation. The results are shown in Fig. 1.4.
28 1. Arithmetic
6. Depreciation with Di erent Types of Deprecation
Rn
A=1000
Since in the case of geometrically declining depreciation the residual value cannot become equal to zero for a nite n, it is reasonable after a certain time, e.g., after m years, to switch over to straight-line depreciation. 600 We determine m so that from this time on the geometrically declining depreciation rate is smaller than the 400 straight-line depreciation rate. From this requirement it follows that: 200 m > N ; 100 (1.96) p : 0 6 8 10 n 2 4 Here m is the last year of geometrically declining depreciation and N is the last year of linear depreciation Figure 1.4 when the residual value becomes zero. A machine with a purchase value of 50 000 is to be depreciated to zero within 15 years, for m years by geometrically declining depreciation with 14% of the residual value, then with the straightline method. From (1.96) we get m > 15 ; 100 = 7:76, i.e., after m = 8 years it is reasonable to switch 14 over to straight-line depreciation. 800
arithmetically declining geometrically declining linear
1.4 Inequalities
1.4.1 Pure Inequalities 1.4.1.1 Denitions 1. Inequalities
Inequalities are comparisons of two real algebraic expressions represented by one of the following signs: Type I > (\greater") Type II < (\smaller") Type III 6= (\not equal") Type IIIa (\greater or smaller") Type IV (\greater or equal") Type IVa 6< (\not smaller") Type V (\smaller or equal") Type Va 6> (\not greater") The notation III and IIIa, IV and IVa, and V and Va have the same meaning, so they can be replaced by each other. The notation III can also be used for those types of quantities for which the notions of \greater" or \smaller" cannot be de ned, for instance for complex numbers or vectors, but in this case it cannot be replaced by IIIa.
2. Identical Inequalities, Inequalities of the Same and of the Opposite Sense, Equivalent Inequalities
1. Identical Inequalities are valid for arbitrary values of the letters contained in them. 2. Inequalities of the Same Sense belong to the same type from the rst two, i.e., both belong to type I or both belong to type II. 3. Inequalities of the Opposite Sense belong to di erent types of the rst two, i.e., one to type I, the other to type II. 4. Equivalent Inequalities are inequalities if they are valid exactly for the same values of the unknowns contained in them.
3. Solution of Inequalities
Similarly to equalities, inequalities can contain unknown quantities which are usually denoted by the last letters of the alphabet. The solution of an inequality or a system of inequalities means the determination of the limits for the unknowns between which they can change, keeping the inequality or system
1.4 Inequalities 29
of inequalities true. We can look for the solutions of any kind of inequality mostly we have to solve pure inequalities of type I and II.
1.4.1.2 Properties of Inequalities of Type I and II
1. Change the Sense of the Inequality
If a > b holds, then b < a is valid, if a < b holds, then b > a is valid.
(1.97a) (1.97b)
If a > b and b > c hold, then a > c is valid if a < b and b < c hold, then a < c is valid.
(1.98a) (1.98b)
2. Transitivity
3. Addition and Subtraction of a Quantity If a > b holds, then a c > b c is valid if a < b holds, then a c < b c is valid.
(1.99a) (1.99b) By adding or subtracting the same amount to the both sides of inequality, the sense of the inequality does not change.
4. Addition of Inequalities
If a > b and c > d hold, then a + c > b + d is valid if a < b and c < d hold, then a + c < b + d is valid. Two inequalities of the same sense can be added.
(1.100a) (1.100b)
5. Subtraction of Inequalities
If a > b and c < d hold, then a ; c > b ; d is valid (1.101a) if a < b and c > d hold, then a ; c < b ; d is valid. (1.101b) Inequalities of the opposite sense can be subtracted the result keeps the sense of the rst inequality. Subtracting inequalities of the same sense is not allowed.
6. Multiplication and Division of an Inequality by a Quantity
(1.102a) If a > b and c > 0 hold, then ac > bc and ac > cb are valid, if a < b and c > 0 hold, then ac < bc and a < b are valid, (1.102b) c c if a > b and c < 0 hold, then ac < bc and a < b are valid, (1.102c) c c if a < b and c < 0 hold, then ac > bc and ac > cb are valid. (1.102d) Multiplication or division of both sides of an inequality by a positive value does not change the sense of the inequality. Multiplication or division by a negative value changes the sense of the inequality.
7. Inequalities and Reciprocal Values
If 0 < a < b or a < b < 0 hold, then a1 > 1b is valid.
(1.103)
30 1. Arithmetic
1.4.2 Special Inequalities
1.4.2.1 Triangle Inequality for Real Numbers
For arbitrary real numbers a b a1 a2 : : : an, there are the inequalities ja + bj jaj + jbj ja1 + a2 + + an j ja1j + ja2 j + + janj : (1.104) The absolute value of the sum of two or more real numbers is less than or equal to the sum of their absolute values. The equality holds only if the summands have the same sign.
1.4.2.2 Triangle Inequality for Complex Numbers For n complex numbers z1 z2 : : : zn 2 C
X n n zk = jz1 + z2 + + znj jz1 j + jz2j + + jznj = X jzk j: k=1 k=1
(1.105)
1.4.2.3 Inequalities for Absolute Values of Di erences of Real and Complex Numbers
For arbitrary real numbers a b 2 IR, there are the inequalities jaj ; jbj ja ; bj jaj + jbj: (1.106) The absolute value of the di erence of two real numbers is less than or equal to the sum of their absolute values, but greater than or equal to the di erence of their absolute values. For two arbitrary complex numbers z1 z2 2 C jjz1j ; jz2 jj jz1 ; z2 j jz1 j + jz2j : (1.107)
1.4.2.4 Inequality for Arithmetic and Geometric Means
a1 + a2 + + an pn a a a for a > 0 : (1.108) 1 2 n i n The arithmetic mean of n positive numbers is greater than or equal to their geometric mean. Equality holds only if all the n numbers are equal.
1.4.2.5 Inequality for Arithmetic and Quadratic Means a1 + a2 + + an s a1 2 + a2 2 + + an2 : n n
(1.109)
The absolute value of the arithmetic mean of numbers is less than or equal to their quadratic mean.
1.4.2.6 Inequalities for Di erent Means of Real Numbers
For the harmonic, geometric, arithmetic, and quadratic means of two positive real numbers a and b with a < b the following inequalities hold (see also 1.2.5.5, p. 20): a < xH < xG < xA < xQ < b: (1.110a) Here s 2 2 p (1.110b) xA = a +2 b xG = ab xH = a2+abb xQ = a +2 b :
1.4.2.7 Bernoulli's Inequality
For every real number a ;1 and integer n 1 holds (1 + a)n 1 + n a :
(1.111)
1.4 Inequalities 31
The equality holds only for n = 1 , or a = 0.
1.4.2.8 Binomial Inequality
For arbitrary real numbers a b 2 IR, we have ja bj 21 (a2 + b2 ) :
(1.112)
1.4.2.9 Cauchy{Schwarz Inequality
The Cauchy{Schwarz inequality holds for arbitrary real numbers ai bj 2 IR :
q
q
or
ja1b1 + a2 b2 + + anbn j a1 2 + a22 + + an2 b1 2 + b2 2 + + bn 2
(1.113a)
(a1b1 + a2 b2 + + anbn)2 (a1 2 + a22 + + an2 )(b1 2 + b2 2 + + bn2 ): (1.113b) For two nite sequences of n real numbers, the sum of the pairwise products is less than or equal to the product of the square roots of the sums of the squares of these numbers. Equality holds only if a1 : b1 = a2 : b2 = = an : bn . If n = 3 and fa1 a2 a3g and fb1 b2 b3 g are considered as vectors in a Cartesian coordinate system, then the Cauchy{Schwarz inequality means that the absolute value of the scalar product of two vectors is less than or equal to the product of absolute values of these vectors. If n > 3, then this statement can be extended for vectors in n-dimensional Euclidean space. Considering that for complex numbers jzj2 = z z (z is the complex conjugate of z), the inequality (1.113b) is valid also for arbitrary complex numbers zi wj 2 C: (z1 w1 + z2 w2 + + zn wn) (z1w1 + z2 w2 + + zn wn) (z1 z1 + z2 z2 + + zn zn )(w1 w1 + w2 w2 + + wn wn). An analogous statement is the Cauchy{Schwarz inequality for convergent in nite series and for certain integrals: 1 X
n=1
"Z b a
!2 X ! X ! 1 1 anbn an 2 bn 2 n=1
(1.114)
n=1
2 Z b ! Zb ! f (x) '(x) dx f (x)]2 dx '(x)]2 dx : a
1.4.2.10 Chebyshev Inequality
a
(1.115)
If a1 a2 : : : an, b1 b2 : : : bn are real positive numbers, then we have the following inequalities: a1 + a2 + + an b1 + b2 + + bn ! a1b1 + a2 b2 + + an bn (1.116a) n n n for a1 a2 : : : an and b1 b2 : : : bn or a1 a2 : : : an and b1 b2 : : : bn and a1 + a2 + + an b1 + b2 + + bn ! a1b1 + a2 b2 + + an bn (1.116b) n n n for a1 a2 : : : an and b1 b2 : : : bn : For two nite sequences with n positive numbers, the product of the arithmetic means of these sequences is less than or equal to the aritmetic mean of the pairwise products if both sequences are increasing or
32 1. Arithmetic both are decreasing but the inequality is valid in the opposite sense if one of the sequences is increasing and the other one is decreasing.
1.4.2.11 Generalized Chebyshev Inequality
If a1 a2 : : : an, b1 b2 : : : bn are real positive numbers, then we have the inequalities s k k s k k s k k a1 + a2 + + an k k b1 + b1 + + bn k (a1 b1 )k + (a2 b2 )k + + (an bn )k (1.117a) n n n for a1 a2 : : : an and b1 b2 : : : bn or a1 a2 : : : an and b1 b2 : : : bn and s s k k s k k k k k a1 k + a2 k + + an k k b1 + b1 + + bn k (a1b1 ) + (a2 b2 )n + + (anbn) (1.117b) n n for a1 a2 : : : an and b1 b2 : : : bn :
1.4.2.12 Holder Inequality
1. Holder Inequality for Series If p and q are two real numbers such that p1 + 1q = 1 holds, and
x1 x2 : : : xn and y1 y2 : : : yn are arbitrary 2n complex numbers, then we have: "X
p1 "X
1q n n n X jxk yk j jxk jp jyk jq : k=1
k=1
k=1
(1.118a)
This inequality is also valid for countable in nite pairs of numbers: 1 X
k=1
jxk yk j
"X 1
k=1
jxk jp
p1 "X 1
k=1
jyk jq
1q
(1.118b)
where from the convergence of the series on the right-hand side the convergence of the left-hand side follows. 2. Holder Inequality for Integrals If f (x) and g(x) are two measurable functions on the measure space (X A ) (see 12.9.2, p. 636), then we have:
Z
X
2Z 3 p1 2Z 3 q1 p q 4 5 4 jf (x)g(x)jd jf (x)j d jg(x)j d 5 : X
X
(1.118c)
1.4.2.13 Minkowski Inequality
1. Minkowski Inequality for Series If p 1 holds, and fxk gkk==11 and fyk g1k=1 with xk yk 2 C are two sequences of numbers, then we have:
"X 1
k=1
p1
jxk + yk jp
"X 1
k=1
1p " X 1
jxk jp +
k=1
p1
jyk jp :
(1.119a)
2. Minkowski Inequality for Integrals If f (x) and g(x) are two measurable functions on the measure space (X A ) (see 12.9.2, p. 636), then we have: 3 p1 2Z 2Z 3 p1 2Z 3 p1 4 jf (x) + g(x)jpd 5 4 jf (x)jpd 5 + 4 jg(x)jpd 5 : X
X
X
(1.119b)
1.4 Inequalities 33
1.4.3 Solution of Linear and Quadratic Inequalities 1.4.3.1 General Remarks
During the solution of an inequality we transform it into an equivalent inequality. Similarly to the solution of an equation we can add the same expression to both sides formally, it may seem that we bring a summand from one side to the other, changing its sign. Furthermore one can multiply or divide both sides of an inequality by a non-zero expression, where the inequality keeps its sense if this expression has a positive value, and changes its sense if this expression has a negative value. An inequality of rst degree can always be transformed into the form ax > b: (1.120) The simplest form of an inequality of second degree is x2 > m (1.121a) or x2 < m (1.121b) and in the general case it has the form ax2 + bx + c > 0 (1.122a) or ax2 + bx + c < 0: (1.122b)
1.4.3.2 Linear Inequalities
Inequalities of rst degree have the solution x > ab for a > 0 (1.123a)
and x < ab for a < 0:
(1.123b)
and x2 < m
(1.124b)
5x + 3 < 8x + 1, 5x ; 8x < 1 ; 3, ;3x < ;2, x > 2 . 3
1.4.3.3 Quadratic Inequalities
Inequalities of second degree in the form x2 > m (1.124a) have solutions
a) x2 > m : For m 0 the solution is x > pm and x < ;pm (jxj > pm) for m < 0 the inequality holds identically.
b) x2 < m : For m > 0 the solution is ; pm < x < +pm (jxj < pm) for m 0 there is no solution.
1.4.3.4 General Case for Inequalities of Second Degree
(1.125a) (1.125b) (1.126a) (1.126b)
ax2 + bx + c > 0 (1.127a) or ax2 + bx + c < 0: (1.127b) We divide the inequality by a. If a < 0 then the sense of the inequality changes, but in any case it will have the form x2 + px + q < 0 (1.127c) or x2 + px + q > 0: (1.127d) By completing the square it follows that p 2 p 2 2 2 x+ 2 < 2 ;q (1.127e) or x + p2 > p2 ; q: (1.127f) 2 Denoting x + 2p by z and 2p ; q by m, we obtain the inequality
34 1. Arithmetic z2 < m (1.128a) or z2 > m: Solving these inequalities, we get the values for x. 2 A: ;2x2 + 14x ; 20 > 0, x2 ; 7x + 10 < 0, x ; 27 < 94 , ; 32 < x ; 27 < 32 , ; 32 + 72 < x < 23 + 72 . The solution is 2 < x < 5. B: x2 + 6x + 15 > 0, (x + 3)2 > ;6. The inequality holds identically. 2 C: ;2x2 + 14x ; 20 < 0, x ; 27 > 94 , x ; 72 > 32 and x ; 72 < ; 32 . The solution intervals are x > 5 and x < 2.
(1.128b)
1.5 Complex Numbers
1.5.1 Imaginary and Complex Numbers 1.5.1.1 Imaginary Unit
The imaginary unit is denoted by i, which represents a number di erent from any real number, and whose square is equal to ;1. In electronics, instead of i the letter j is usually used to avoid accidently confusing it with the intensity of current, also denoted by i. The introduction of the imaginary unit leads to the generalization of the notion of numbers to the complex numbers, which play a very important role in algebra and analysis. The complex numbers have several interpretations in geometry and physics.
1.5.1.2 Complex Numbers
The algebraic form of a complex number is z = a + i b: (1.129a) When a and b take all possible real values, we get all possible complex numbers z. The number a is the real part, the number b is the imaginary part of the number z : a = Re(z) b = Im(z): (1.129b) For b = 0 we have z = a, so the real numbers form a subset of the complex numbers. For a = 0 we have z = i b, which is a \pure imaginary number". The set of complex numbers is denoted by C . Remark: Functions w = f (z) with complex variable z = x + i y will be discussed in function theory (see 14.1, p. 671 ).
1.5.2 Geometric Representation 1.5.2.1 Vector Representation
Similarly to the representation of the real numbers on the numerical axis, the complex numbers can be represented as points in the plane, the so-called Gaussian number plane: A number z = a + i b is represented by the point whose abscissa is a and ordinate is b (Fig. 1.5). The real numbers are on the axis of abscissae which is also called the real axis, the pure imaginary numbers are on the axis of ordinates which is also called the imaginary axis. On this plane every point is given uniquely by its position vector or radius vector (see 3.5.1.1, 6., p. 180), so every complex number corresponds to a vector which starts at the origin and is directed to the point de ned by the complex number. So, complex numbers can be represented as points or as vectors (Fig. 1.6).
1.5.2.2 Equality of Complex Numbers
Two complex numbers are equal by de nition if their real parts and imaginary parts are equal to each other. From a geometric viewpoint, two complex numbers are equal if the position vectors correspond-
1.5 Complex Numbers 35
ing to them are equal. In the opposite case the complex numbers are not equal. The notions \greater" and \smaller" are meaningless for complex numbers. y imag. axis
y imag. axis z=a+b i b
0 real axis a
Figure 1.5
y imag. axis b
z
x
0
real axis
x
Figure 1.6
z ρ
ϕ 0 real axis a
x
Figure 1.7
1.5.2.3 Trigonometric Form of Complex Numbers
The form z = a+ib (1.130a) is called the algebraic form of the complex number. When polar coordinates are used, we get the trigonometric form of the complex numbers (Fig. 1.7): z = (cos ' + i sin '): (1.130b) The length of the position vector of a point = jzj is called the absolute value or the magnitude of the complex number , the angle ', given in radian measure, is called the argument of the complex number and is denoted by arg z: = jzj ' = arg z = ! + 2k with 0 < 1 ; < ! + k = 0 1 2 : : : : (1.130c) We call ' the principal value of the complex number. The relations between , ' and a , b for a point are the same as between the Cartesian and polar coordinates of a point (see 3.5.2.2, p. 191): p a = cos ' (1.131a) b = sin ' (1.131b) = a2 + b2 (1.131c)
8 > arctan b > > > 8 a > > > arccos a for b 0 > 0 + > > < > < 2 ' = > ; arccos a for b < 0 > 0 ;2 ' = > > > : > unde ned for = 0 > arctan b + > a > > (1.131d) > : arctan b ; a
for a > 0 for a = 0, b > 0, for a = 0, b < 0, for a < 0, b 0, for a < 0, b < 0.
The complex number z = 0 has absolute value equal to zero its argument arg 0 is unde ned.
1.5.2.4 Exponential Form of a Complex Number
(1.131e)
We call the representation z = ei ' (1.132a) the exponential form of the complex number, where is the magnitude and ' is the argument. The Euler relation is the formula ei ' = cos ' + i sin ' : (1.132b)
36 1. Arithmetic We represent a complex number in three forms: p a) z = 1 + i 3 (algebraic form), b) z = 2 cos 3 + i sin 3 (trigonometric form), c) z = 2 ei 3 (exponential form), considering the principal value of it. If we do not restrict ourselves
only to the principal
value, we have p d) z = 1+i 3 = 2 exp i 3 + 2k = 2 cos 3 + 2k + i sin 3 + 2k (k = 0 1 2 : : :) .
1.5.2.5 Conjugate Complex Numbers
Two complex numbers z and z are called conjugate complex numbers if their real parts are equal and their imaginary parts di er only in sign: Re(z ) = Re(z) Im(z ) = ;Im(z) : (1.133a) The geometric interpretation of points corresponding to the conjugate complex numbers are points symmetric with respect to the real axis. Conjugate complex numbers have the same absolute value, their arguments di er only in sign: z = a + i b = (cos ' + i sin ') = ei ' (1.133b) ;i ' z = a ; i b = (cos ' ; i sin ') = e : (1.133c) Instead of z one often uses the notation z for the conjugate of z.
1.5.3 Calculation with Complex Numbers 1.5.3.1 Addition and Subtraction
Addition and subtraction of two or more complex numbers given in algebraic form is de ned by the formula z1 + z2 ; z3 + = (a1 + i b1) + (a2 + i b2) ; (a3 + i b3) + = (a1 + a2 ; a3 + ) + i (b1 + b2 ; b3 + ) : (1.134) We make the calculations in the same way as we do by the usual binomials. As a geometric interpretation of addition and subtraction we consider the addition and subtraction of the corresponding vectors (Fig. 1.8). For these we use the usual rules for vector calculations (see 3.5.1.1, p. 180). For z and z , z + z is always real, and z ; z is pure imaginary. y imag. axis z1+z2-z3 z1+z2
z1
-z3 0
y imag. axis z z 1 2
z2 real axis x z3
Figure 1.8
1.5.3.2 Multiplication
z2 z1 0
1 real axis x
Figure 1.9
y imag. axis zi
z . 0 real axis x
Figure 1.10
The multiplication of two complex numbers z1 and z2 given in algebraic form is de ned by the following formula z1 z2 = (a1 + i b1)(a2 + i b2) = (a1 a2 ; b1 b2 ) + i (a1b2 + b1a2 ) : (1.135a) For numbers given in trigonometric form we have z1 z2 = 1(cos '1 + i sin '1)] 2 (cos '2 + i sin '2 )] (1.135b) = 1 2 cos('1 + '2 ) + i sin('1 + '2 )]
1.5 Complex Numbers 37
i.e., the absolute value of the product is equal to the product of the absolute values of the factors, and the argument of the product is equal to the sum of the arguments of the factors. The exponential form of the product is z1 z2 = 1 2 ei('1 +'2) : (1.135c) The geometric interpretation of the product of two complex numbers z1 and z2 is a vector such that we rotate the vector corresponding to z1 by the argument of z2 clockwise or counterclockwise according to the sign of this argument, and the length of the vector will be stretched by jz2j. The product z1 z2 can also be represented with similar triangles (Fig. 1.9). The multiplication of a complex number z by i means a rotation by =2 and the absolute value does not change (Fig. 1.10). For z and z : zz = 2 = jzj2 = a2 + b2: (1.136)
1.5.3.3 Division
Division is de ned as the inverse operation of multiplication. For complex numbers given in algebraic form, we have z1 = a1 + i b1 = a1 a2 + b1 b2 + i a2 b1 ; a1b2 : (1.137a) z2 a2 + i b2 a22 + b2 2 a2 2 + b2 2 For complex numbers given in trigonometric form we have z1 = 1(cos '1 + i sin '1) = 1 cos(' ; ' ) + i sin(' ; ' )] (1.137b) 1 2 1 2 z2 2(cos '2 + i sin '2) 2 i.e., the absolute value of the quotient is equal to the ratio of the absolute values of the dividend and the divisor the argument of the quotient is equal to the di erence of the arguments. For the exponential form we get z1 = 1 ei('1 ;'2 ): (1.137c) z 2
2
In the geometric representation we get the vector corresponding to z1 =z2 if we rotate the vector representing z1 by ; arg z2, then we make a contraction by jz2 j. Remark: Division by zero is impossible.
1.5.3.4 General Rules for the Basic Operations
As we can observe we can make our calculations with complex numbers z = a + i b in the same way as we do with binomials, only we have to consider that i 2 = ;1. We know how to divide binomials by a real number. So, on division of a complex number by a complex number, rst we clear the denominator from the imaginary part of the divisor, and we multiply the numerator and the denominator of the fraction by the complex conjugate of the divisor. This is possible because (a + i b)(a ; i b) = a2 + b2 (1.138) is a real number. (3 ; 4i )(;1 + 5i )2 + 10 + 7i = (3 ; 4i )(1 ; 10i ; 25) + (10 + 7i )i = ;2(3 ; 4i )(12 + 5i ) + 1 + 3i 5i 1 + 3i 5i i 1 + 3i 7 ; 10i = ;2(56 ; 33i )(1 ; 3i ) + 7 ; 10i = ;2(;43 ; 201i ) + 7 ; 10i = 1 (50+191i ) = 10+38:2i : 5 (1 + 3i )(1 ; 3i ) 5 10 5 5
1.5.3.5 Taking Powers of Complex Numbers
The n-th power of a complex number could be calculated using the binomial formula, but it would be very inconvenient. For practical reasons we use the trigonometric form and the so-called de Moivre formula: (cos ' + i sin ')]n = n (cos n' + i sin n') (1.139a)
38 1. Arithmetic i.e., the absolute value is raised to the n-th power, and the argument is multiplied by n. In particular, we have: i 2 = ;1 i 3 = ;i i 4 = +1 (1.139b) in general i 4n+k = i k : (1.139c)
1.5.3.6 Taking of the n-th Root of a Complex Number
Taking of the n-th root is the inverse operation of taking powers. For z = (cos ' +i sin ') the notation
p
(1.140a) z1=n = n z with n > 0 integer is the shorthand notation for the n di erent values ! !k = pn cos ' +n2k + i sin ' +n2k
(k = 0 1 2 : : : n ; 1): (1.140b) While addition, subtraction, multiplication, division, and taking a power with integer exponent have unique results, taking of the n-th root has n di erent solutions !k . The geometric interpretations of the points !k are the vertices of a regular n-gonp whose center is at the origin. In Fig. 1.11 the six values of 6 z are represented.
z
y imag. axis
ω1
ω2
ω0 ω3
0
real axis
x
ω5 ω4
Figure 1.11
1.6 Algebraic and Transcendental Equations
1.6.1 Transforming Algebraic Equations to Normal Form 1.6.1.1 Denition
The variable x in the equality F (x) = f (x) (1.141) is called the unknown if the equality is valid only for certain values x1 x2 : : : xn of the variable, and these values are called the solutions or the roots of the equation. Two equations are considered equivalent if they have exactly the same roots. An equation is called an algebraic equation if the functions F (x) and f (x) are algebraic, i.e., they are rational or irrational expressions of course one of them can be constant. Every algebraic equation can be transformed into normal form P (x) = anxn + an;1xn;1 + + a1 x + a0 = 0 (1.142) by algebraic transformations. The roots of the original equation occur among the roots of the normal form, but under certain circumstances some are superuous. The leading coecient an is frequently transformed to the value 1. The exponent n is called the degree of the equation. p 1 + x2 ; 6 = 1 + x ; 3 . The transformations Determine the normal form of the equation x ; 3( x ; 2) x step by steppare: p x(x ; 1p+ x2 ; 6) = 3x(x ; 2) + 3(x ; 2)(x ; 3) x2 ; x + x x2 ; 6 = 3x2 ; 6x + 3x2 ; 15x + 18 x x2 ; 6 = 5x2 ; 20x +18 x2 (x2 ; 6) = 25x4 ; 200x3 +580x2 ; 720x +324 24x4 ; 200x3 + 586x2 ; 720x + 324 = 0: The result is an equation of fourth degree in normal form.
1.6.1.2 System of n Algebraic Equations
Every system of algebraic equations can be transformed to normal form, i.e., into a system of polynomial equations:
1.6 Algebraic and Transcendental Equations 39
P1(x y z : : :) = 0 P2(x y z : : :) = 0 : : : Pn(x y z : : :) = 0 : (1.143) The Pi (i = 1 2 : : : n) are polynomials in x y z : : : . ; 1 = pz , 3. xy = z . Determine the normal form of the equation system: 1. pxy = z1 , 2. xy ; 1 The normal form is: 1. x2 z2 ; y = 0 , 2. x2 ; 2x + 1 ; y2z + 2yz ; z = 0, 3. xy ; z = 0.
1.6.1.3 Superuous Roots
It can happen that after transformation of an algebraic equation into normal form P (x) = 0, it has solutions which are not solutions of the original equation. This happens in two cases: 1. Disappearing Denominator If an equation has the form of a fraction P (x) = 0 (1.144a) Q(x) with polynomials P (x) and Q(x), then we get the normal form by multiplying by the denominator: P (x) = 0: (1.144b) The roots of (1.144b) are the same as those of the original equation (1.144a), except if a root x = of the equation P (x) = 0 is also a root of the equation Q(x) = 0. In this case we should simplify rst by the term x ; , actually by (x ; )k . Anyway if we perform a so-called non-identical transformation, we still have to substitute the roots we get into the original equation (see also 1.6.3.1, p. 43). 3 3 A: x x; 1 = x ;1 1 or xx ;;11 = 0 (1). If we do not simplify by x ; 1, then the root x1 = 1 satis es the equation x3 ; 1 = 0, but it does not satisfy (1), because it makes the denominator zero. 3 2 B: x ;x23;x 2+x +3x1; 1 = 0 (2). If we do not simplify by the term (x ; 1)2, the equation does not have any root, because (x ; 1)3 = 0 has the root x1 = 1, but the denominator is also zero here. After simpli cation, (2) has a simple root x = 1. 2. Irrational Equations If in the equation we also have an unknown in the radicand, then it is possible that the normal form of the equation has roots which do not satisfy the original equation. Consequently every solution we get from the normal form must be substituted into the original equation in order whetherpit satis es it or not. px +to7 +check 1 = 2x or x + 7 = 2x ; 1 (1) x + 7 = (2x ; 1)2 or 4x2 ; 5x ; 6 = 0 (2). The solutions of (2) are x1 = 2 x2 = ;3=4. The solution x1 satis es (1), but the solution x2 does not.
1.6.2 Equations of Degree at Most Four
1.6.2.1 Equations of Degree One (Linear Equations) 1. Normal Form ax + b = 0 :
2. Number of Solutions There is a unique solution x1 = ; ab :
(1.145) (1.146)
1.6.2.2 Equations of Degree Two (Quadratic Equations)
1. Normal Form
ax2 + bx + c = 0 or divided by a: x2 + px + q = 0 :
(1.147a) (1.147b)
40 1. Arithmetic 2. Number of Real Solutions of a Real Equation Depending on the sign of the discriminant 2 D = 4ac ; b2 or D = q ; p4 (1.148)
we have: for D < 0, there are two real solutions (two real roots), for D = 0, there is one real solution (two coincident roots), for D > 0, there is no real solution (two complex roots). 3. Properties of the Roots of a Quadratic Equation If x1 and x2 are the roots of the quadratic equation (1.147a) or (1.147b), then the following equalities hold: x1 + x2 = ; ab = ;p x1 x2 = ac = q : (1.149)
4. Solution of Quadratic Equations Method 1: Factorization of ax2 + bx + c = a(x ; )(x ; ) (1.150a)
or x2 + px + q = (x ; )(x ; ) (1.150b)
if it is successful, immediately gives the roots x1 = x2 = : x2 + x ; 6 = 0, x2 + x ; 6 = (x + 3)(x ; 2) , x1 = ;3 x2 = 2. Method 2: Using the solution formula a) for (1.147a) the solutions are
p
2 x12 = ;b 2ba ; 4ac (1.152a) where we use the second formula if b is an even integer b) for (1.147b) the solutions are s 2 p x12 = ; 2 p4 ; q :
v u u !2 b ; 2 t 2b ; ac or x12 = a
(1.151)
(1.152b)
(1.153)
1.6.2.3 Equations of Degree Three (Cubic Equations) 1. Normal Form
ax3 + bx2 + cx + d = 0
(1.154a) b or after dividing by a and substituting y = x + we have 3a 3 y + 3py + 2q = 0 or in reduced form y3 + p y + q = 0 (1.154b) where 3 2 q = 2q = 272ba3 ; 3bca2 + ad and p = 3p = 3ac3a;2 b : (1.154c) 2. Number of Real Solutions Depending on the sign of the discriminant D = q2 + p3 (1.155) we have : for D > 0, one real solution (one real and two complex roots), for D < 0, three real solutions (three di erent real roots), for D = 0, one real solution (one real root with multiplicity three) in the case p = q = 0 or two real solutions (a single and a double real root) in the case p3 = ;q2 6= 0.
1.6 Algebraic and Transcendental Equations 41
3. Properties of the Roots of a Cubic Equation If x1 x2, and x3 are the roots of the cubic equation (1.154a), then the following equalities hold: (1.156) x1 + x2 + x3 = ; ab x1 + x1 + x1 = ; dc x1 x2 x3 = ; da : 1 2 3 4. Solution of a Cubic Equation Method 1: If it is possible to decompose the left-hand side into a product of linear terms ax3 + bx2 + cx + d = a(x ; )(x ; )(x ; ) (1.157a)
we immediately get the roots x1 = x2 = x3 = : (1.157b) x3 + x2 ; 6x = 0, x3 + x2 ; 6x = x(x + 3)(x ; 2) x1 = 0, x2 = ;3, x3 = 2. Method 2: Using the Formula of Cardano. By substituting y = u + v the equation (1.154b) has the form u3 + v3 + (u + v)(3uv + 3p) + 2q = 0: (1.158a) This equation is obviously satis ed if u3 + v3 = ;2q and uv = ;p (1.158b) hold. If we write (1.158b) in the form u3 + v3 = ;2q u3v3 = ;p3 (1.158c) then we have two unknowns u3 and v3, and we know their sum and product. Therefore using Vieta's root theorem (see 1.6.3.1, 3., p. 43) the solutions of the quadratic equation w2 ; (u3 + v3 )w + u3v3 = w2 + 2qw ; p3 = 0 (1.158d) can be calculated. We get q q w1 = u3 = ;q + q2 + p3 w2 = v3 = ;q ; q2 + p3 (1.158e) so for the solution y of (1.154b) the Cardano formula results in
r r q q y = u + v = 3 ;q + q2 + p3 + 3 ;q ; q2 + p3 : (1.158f) Because the third root of a complex number means three di erent numbers (see (1.140b) on p. 38) we could have nine di erent cases, but because of uv = ;p, the solutions are reduced to the following three: y1 = u1 + v1 (if possible, consider the real third roots u1 and v1 such that u1v1 = ;p) (1.158g) p p y2 = u1 ; 12 + 2i 3 + v1 ; 12 ; 2i 3 (1.158h) p p y3 = u1 ; 12 ; 2i 3 + v1 ; 21 + 2i 3 : (1.158i) p p y3 + 6y + 2 = 0 with p = 2 q = 1 and q2 + p3 = 9 and u = 3 ;1 + 3 = 3 2 = 1:2599, p p v = 3 ;1 ; 3 = 3 ;p4 = ;1:5874. The real root is y1 = u + v = ;0:3275, the complex roots are y23 = ; 12 (u + v) i 23 (u ; v) = 0:1638 i 2:4659. Method 3: If we have a real equation, we can use the auxiliary values given in Table 1.3. With p from (1.154b) we substitute q r = jpj (1.159)
42 1. Arithmetic where the sign of r is the same as the sign of q. With this, using Table 1.3, we can determine the value of the auxiliary variable ' and with it we can tell the roots y1, y2 and y3 depending on the signs of p and D = q2 + p3 . Table 1.3 Auxiliary values for the solution of equations of degree three
p 0 cosh ' = rq3 y1 = ;2r cosh '3 p y2 = r cosh '3 + i 3 r sinh '3 p y = r cosh ' ; i 3 r sinh ' 3
3
3
p>0 sinh ' = rq3
y1 = ;2r sinh '3 p y2 = r sinh '3 + i 3 r cosh '3 p y3 = r sinh '3 ; i 3 r cosh '3
p y3 ; 9y + 4 = 0. p = ;3 q = 2 q2 + p3 < 0, r = 3 cos ' = p2 = 0:3849 ' = 67220: 3 3 p p p y1 = ;2 3 cos 22270 = ;3:201 y2 = 2 3 cos(60 ; 22270) = 2:747, y3 = 2 3 cos(60 + 22270) = 0:455. Checking: y1 + y2 + y3 = 0:001 which can be considered 0 for the accuracy of our calculations. Method 4: Numeric approximate solution, see 19.1.2, p. 887 numeric approximate solution by the help of a nomogram, see 2.19, p. 127.
1.6.2.4 Equations of Degree Four
1. Normal Form
ax4 + bx3 + cx2 + dx + e = 0: If all the coecients are real, this equation has 0 or 2 or 4 real solutions. 2. Special Forms If b = d = 0 hold, then we can calculate the roots of ax4 + cx2 + e = 0 (biquadratic equation) by the formulas p2 x1234 = py y = ;c 2ca ; 4ae : For a = e and b = d, the roots of the equation ax4 + bx3 + cx2 + bx + a = 0 can be calculated by the formulas p2 p2 2 x1234 = y 2y ; 4 y = ;b b 2;a 4ac + 8a :
3. Solution of a General Equation of Degree Four Method 1: If we can somehow factorize the left-hand side of the equation ax4 + bx3 + cx2 + dx + e = 0 = a(x ; )(x ; )(x ; )(x ; ) then the roots can be immediately determined: x1 = x2 = x3 = x4 = : x4 ; 2x3 ; x2 + 2x = 0, x(x2 ; 1)(x ; 2) = x(x ; 1)(x + 1)(x ; 2) x1 = 0, x2 = 1, x3 = ;1, x4 = 2.
(1.160) (1.161a) (1.161b) (1.161c) (1.161d) (1.162a) (1.162b)
1.6 Algebraic and Transcendental Equations 43
Method 2: The roots of the equation (1.162a) for a = 1 coincide with the roots of the equation ! x2 + (b + A) x2 + y + by A; d = 0 (1.163a) p where A = 8y + b2 ; 4c and y is one of the real roots of the equation of third degree 8y3 ; 4cy2 + (2bd ; 8e)y + e(4c ; b2 ) ; d2 = 0 (1.163b) 3 bc b b with B = 8 ; 2 + d = 6 0. The case B = 0 gives by the help of the substitution x = u ; 4 a biquadratic equation of the form (1.161a). Method 3: Approximate solution, see 19.1.2, p. 887.
1.6.2.5 Equations of Higher Degree
It is impossible to give a formula or a nite sequence of formulas which produce the roots of an equation of degree ve or higher.
1.6.3 Equations of Degree
n
1.6.3.1 General Properties of Algebraic Equations 1. Roots
The left-hand side of the equation xn + an;1 xn;1 + : : : + a0 = 0 (1.164a) is a polynomial Pn(x) of degree n, and a solution of (1.164a) is a root of the polynomial Pn(x). If is a root of the polynomial, then Pn(x) is divisible by (x ; ). Generally Pn(x) = (x ; )Pn;1(x) + Pn(): (1.164b) Here Pn;1(x) is a polynomial of degree n ; 1. If Pn(x) is divisible by (x ; )k , but it is not divisible by (x ; )k+1 then is called a root of order k of the equation Pn(x) = 0. In this case is a common root of the polynomial Pn(x) and its derivatives to order (k ; 1).
2. Fundamental Theorem of Algebra
Every equation of degree n whose coecients are real or complex numbers has n real or complex roots, where the roots of higher order are counted by their multiplicity. If we denote the roots of P (x) by : : : and they have multiplicity k l m : : :, then the product representation of the polynomial is P (x) = (x ; )k (x ; )l (x ; )m : : : : (1.165a) The solution of the equation P (x) = 0 can be simpli ed if we can reduce the equation to another one, which has the same roots, but only with multiplicity one. In order to get this, we decompose the polynomial into a product of two factors P (x) = Q(x)T (x) (1.165b) such that T (x) = (x ; )k;1(x ; )l;1 : : : Q(x) = (x ; )(x ; ) : : : : (1.165c) Because the roots of the polynomial P (x) with higher multiplicity are the roots of its derivative P 0(x), too, T (x) is the greatest common denominator of the polynomial P (x) and its derivative P 0(x) (see 1.1.6.5, p.14). If we divide P (x) by T (x) we get the polynomial Q(x) which has all the roots of P (x) , and each root occurs with multiplicity one.
3. Theorem of Vieta About Roots
The relations between the n roots x1 , x2 , . . . , xn and the coecients of the equation (1.164a) are:
x1 + x2 + : : : + xn =
n X i=1
xi = ;an;1
44 1. Arithmetic x1 x2 + x1 x3 + : : : + xn;1xn = x1 x2 x3 + x1x2 x4 + : : : + xn;2 xn;1xn =
n X ij =1 i<j
xixj = an;2
n X
ijk=1 i<j 0 (Fig. 2.2b). By sign(x) (read \signum x"), we denote the sign function .
Figure 2.2
2.1.3 Certain Types of Functions 2.1.3.1 Monotone Functions
y
y
If a function satis es the relations x1 x2 x x1 x2 x 0 0 f (x2) f (x1) or f (x2 ) f (x1 ) (2.7a) a) b) for arbitrary arguments x1 and x2 with x2 > x1 in its domain, then it is called monotonically increasing or monotonically decreasing (Fig. 2.3a,b). Figure 2.3 If one of the above relations (2.7a) does not hold for every x in the domain of the function, but it is valid, e.g., in an interval or on a half-axis, then the function is said to be monotonic in this domain. Functions satisfying the relations f (x2) > f (x1) or f (x2) < f (x1 ) (2.2)
50 2. Functions i.e., when the equality never holds in (2.7a), are called strictly monotonically increasing or strictly monotonically decreasing. In Fig. 2.3a there is a representation of a strictly monotonically increasing function in Fig. 2.3b there is the graph of a monotonically decreasing function being constant between x1 and x2 . y = e;x is strictly monotonically decreasing, y = ln x is strictly monotonically increasing.
2.1.3.2 Bounded Functions
A function is said to be bounded above if there is a number (called an upper bound) such that the values of the function never exceed it. A function is called bounded below if there is a number (called a lower bound) such that the values of the function are never less than this number. If a function is bounded above and below, we simply call it bounded. (If a function has one upper bound, obviously it has an in nite number of upper bounds, all numbers greater than this one. It can be proven that among the upper bounds there is always a smallest, the so-called least upper bound. Similar statements are valid for lower bounds.) A: y = 1 ; x2 is bounded above (y 1). B: y = ex is bounded below (y > 0). C: y = sin x is bounded (;1 y +1). D: y = 1 +4 x2 is bounded (0 < y 4).
2.1.3.3 Even Functions
Even functions (Fig. 2.4a) satisfy the relation f (;x) = f (x): (2.8a) If D is the domain of f , then (x 2 D) ) (;x 2 D) (2.8b) should hold. A: y = cos x, B: y = x4 ; 3x2 + 1.
y
a)
2.1.3.4 Odd Functions
Odd functions (Fig. 2.4b) satisfy the relation f (;x) = ;f (x): If D is the domain of f , then (x 2 D) ) (;x 2 D) should hold. A: y = sin x B: y = x3 ; x:
y
0
x
0
b)
x
Figure 2.4 (2.9a) (2.9b)
2.1.3.5 Representation with Even and Odd Functions
If for the domain D of a function f the condition \from x 2 D it follows that ;x 2 D" holds, then f can be written as a sum of an even function g and an odd function u: f (x) = g(x) + u(x) with g(x) = 12 f (x) + f (;x)] u(x) = 21 f (x) ; f (;x)] : (2.10) f (x) = ex = 12 ex + e;x + 12 ex ; e;x = cosh x + sinh x (see 2.9.1, p. 87).
2.1.3.6 Periodic Functions
Periodic functions satisfy the relation f (x + T ) = f (x) T const T 6= 0: (2.11) Obviously, if the above equality holds for some T , it holds for any integer multiple of T . The smallest positive number T satisfying the relation is called the period (Fig. 2.5).
y
0
T
T
Figure 2.5
x
2.1 Notion of Functions 51
2.1.3.7 Inverse Functions
If the function y = f (x) is a one-to-one function, i.e., if x1 x2 2 D and x1 6= x2 , then f (x1 ) 6= f (x2 ), then there is a function y = '(x) such that for every pair of values (a b) satisfying the equality b = f (a), the equality a = '(b) will be valid, and for every pair of values for which a = '(b) holds, b = f (a) will be valid. The functions y = '(x) and y = f (x) (2.12) are inverse functions of each other. Obviously, every strictly monotonic function has an inverse function. The graph of an inverse function y = '(x) is obtained by reection of the graph of y = f (x) with respect to the line y = x (Fig. 2.6). y
y=e x
y y=x 2
y
y= x
0
a)
0
x
p _ y=arcsin x _ 21 −p y=sin x 2 −1 p x _ −1 1 2 _ −p 2
nx y=l
x
b)
c) Figure 2.6
Examples of Inverse Functions: A: y = f (x) = xp2 with D : x 0, with D : x 0, y = '(x) = x B: y = f (x) = ex with D : ;1 < x < 1,
W : y 0 W : y 0. W : y > 0 y = '(x) = ln x with D : x > 0, W : ;1 < y < 1. C: y = f (x) = sin x with D : ;=2 x =2, W : ;1 y 1 y = '(x) = arcsin x with D : ;1 x 1, W : ;=2 y =2. In order to get the explicit form of the inverse function of y = f (x) we exchange x and y in the expression, then from the equation x = f (y) we express y, so we have y = '(x). The representations y = f (x) and x = '(y) are equivalent. Therefore, we have two important formulas f ('(y)) = y and '(f (x)) = x: (2.13)
2.1.4 Limits of Functions
2.1.4.1 Denition of the Limit of a Function
The function y = f (x) has the limit A at x = a f (x) = A or f (x) ! A for x ! a (2.14) xlim !a if as x approaches the value a in nitely closely, the value of f (x) approaches the value A in nitely closely. The function f (x) does not have to be de ned at a, and even if de ned, it does not matter whether f (a) is equal to A. Precise De nition: The limit (2.14) exists, if for any given positive number " there is a positive number such that for every x 6= a belonging to the domain and satisfying the inequality jx ; aj < (2.15a)
52 2. Functions the inequality jf (x) ; Aj < " (2.15b) holds eventually with the expection of the point a (Fig. 2.7). If a is an endpoint of a connected region, then the inequality jx ; aj < is reduced either to a ; < x or to x < a + .
2.1.4.2 Denition by Limit of Sequences (see
y A+e A A−e 0
a−h a a+h
x
7.1.2, p. 405)
Figure 2.7 A function f (x) has the limit A at x = a if for every sequence x1 x2 : : : xn : : : of the values of x from the domain and converging to a (but being not equal to a), the sequence of the corresponding values of the function f (x1 ) f (x2) : : : f (xn) : : : converges to A.
2.1.4.3 Cauchy Condition for Convergence
A necessary and sucient condition for a function f (x) to have a limit at x = a is that for any two values x1 6= a and x2 6= a belonging to the domain and being close enough to a, the values f (x1 ) and f (x2) are also close enough to each other. Precise De nition: A necessary and sucient condition for a function f (x) to have a limit at x = a is that for any given positive number " there is a positive number such that for arbitrary values x1 and x2 belonging to the domain and satisfying the inequalities 0 < jx1 ; aj < and 0 < jx2 ; aj < (2.16a) the inequality jf (x1) ; f (x2)j < " (2.16b) holds.
2.1.4.4 Innity as a Limit of a Function
The symbol jf (x)j = 1 (2.17) xlim !a means that as x approaches a, the absolute value jf (x)j does not have an upper bound, and the closer we are to a, the larger is its greatest lower bound. Precise De nition: The equality (2.17) holds if for any given positive number K there is a positive number such that for any x 6= a from the interval a; <x K : (2.18b) If all the values of f (x) in the interval a; <x 0 such that for any x < ;N the corresponding value of f (x) is in the interval A ; " < f (x) < A + ". A: x!lim+1 x +x 1 = 1, B: x!;1 lim x + 1 = 1, C: x!;1 lim ex = 0. x
Case b) Assume that for any positive number K , there is a positive number N such that if x > N or x < ;N then the absolute value of the function is larger then K . In this case we write lim jf (x)j = 1 or x!;1 lim jf (x)j = 1: (2.20c) x!+1 3 3 A: lim x ; 1 = +1, B: lim x ; 1 = ;1, x2 3 1 ; C: x!lim+1 x2x = ;1,
x2 3 1 ; D: x!;1 lim x2x = +1.
x!+1
x!;1
2.1.4.7 Theorems About Limits of Functions
1. Limit of a Constant Function The limit of a constant function is the constant itself: A = A: xlim !a
(2.21) 2. Limit of a Sum or a Di erence If among a nite number of functions each has a limit, then the limit of their sum or di erence is equal to the sum or di erence of their limits (if this last expression does not contain 1 ; 1): f (x) + '(x) ; (x)] = xlim f (x) + xlim '(x) ; xlim (x) : (2.22) xlim !a !a !a !a 3. Limit of Products If among a nite number of functions each has a limit, then the limit of their product is equal to the product of their limits (if this last expression does not contain a 0 1 type):
f (x) '(x) (x)] = xlim f (x) xlim '(x) xlim (x) : xlim !a !a !a !a
(2.23)
54 2. Functions 4. Limit of a Quotient The limit of the quotient of two functions is equal to the quotient of their limits, in the case when both limits exist and the limit of the denominator is not equal to zero (and this last expression is not an 1=1 type):
f (x) f (x) = xlim !a : (2.24) lim ' x!a (x) Also if the denominator is equal to zero, we can usually tell if the limit exists or not, checking the sign of the denominator (the indeterminate form is 0=0). Similarly, we can calculate the limit of a power by taking a suitable power of the limit (if it is not a 00, 11, or 10 type). 5. Pinching If the values of a function f (x) lie between the values of the functions '(x) and (x), i.e., '(x) < f (x) < (x), and if xlim '(x) = A and xlim (x) = A hold, then f (x) has a limit, too, and !a !a f (x) = A: (2.25) xlim !a xlim !a '(x)
2.1.4.8 Calculation of Limits
The calculation of the value of a limit can be made by using the following transformations and the theorems of 2.1.4.7:
1. Suitable Transformations
We transform the expression into a form such that we can tell the limit. There are several types of recommended transformations in di erent cases we show three of them. x3 ; 1 = lim (x2 + x + 1) = 3. A: xlim !1 x ; 1 p1 + x ; x1!1 (p1 + x ; 1)(p1 + x + 1) p p1 +1x + 1 = 21 . B: xlim = xlim = xlim !0 !0 !0 x x( 1 + x + 1) sin 2x = lim 2(sin 2x) = 2 lim sin 2x = 2 . Here we refer to the well-known theorem C: xlim !0 x!0 2x!0 2x x 2x sin = 1. lim !0
2. Bernoulli{l'Hospital Rule
0 0 1 In the case of indeterminate forms like 00 , 1 1 , 0 1 , 1 ; 1 , 0 , 1 , 1 , one often applies the Bernoulli{l'Hospital rule (usually called l'Hospital rule for short): Suppose xlim '(x) = 0 and xlim (x) = 0 or xlim '(x) = 1 and xlim (x) = 1, and suppose that there !a !a !a !a is an interval containing a such that the functions '(x) and (x) are de ned and di erentiable in this '0(x) exists. Then interval except perhaps at a, and 0(x) 6= 0 in this interval, and xlim !a 0 (x)
'(x) = lim '0(x) : (2.26) x!a 0 (x) Remark: If the limit of the ratio of the derivatives does not exist, it does not mean that the original limit does not exist. Maybe it does, but we cannot tell this using l'Hospital's rule. '0(x) is still an indeterminate form, and the numerator and denominator satisfy the assumptions If xlim !a 0 (x) of the above theorem, we can again use l'Hospital's rule. Case a) Indeterminate Forms 00 or : We use the theorem after checking if the conditions are ful lled: f (x) = xlim xlim !a !a (x)
1 1
2.1 Notion of Functions 55
2 cos 2x 2 ln sin 2x = lim sin 2x = lim 2 tan x = lim cos2 x = lim cos2 2x = 1. lim x!0 cos x x!0 tan 2x x!0 2 x!0 ln sin x x!0 cos2 x sin x cos22x Case b) Indeterminate Form 0 : If we have f (x) = '(x) (x) and xlim '(x) = 0 and !a lim ( x ) = 1 , than in order to use l'Hospital's rule for lim f ( x ) we transform it into one of the forms x! a x!a '(x) or lim (x) , so we reduce it to an indeterminate form 0 or 1 like in case a). xlim !a x!a 1 1 0 1 (x) '(x) ; 2x = lim ;2 = 2 . lim ( ; 2x) tan x = x!lim x!=2 =2 cot x x!=2 ; sin12x Case c) Indeterminate Form : If f (x) = '(x) ; (x) and xlim '(x) = 1 and xlim (x) = 1 , !a !a 0 1 then we can transform this expression into the form or usually in several di erent ways for instance 0 1 !, 1 1 1 as ' ; = ; ' ' . Then we proceed as in case a).
1
1;1
!
x ln x ; x + 1 = 0 . Applying l'Hospital rule twice we get lim x ; 1 = xlim x!1 x ; 1 ln x !1 0 0 x ln x ; ln 1x 0 1 1 ! B C B x ln x ; x + 1 ln x 1 B C B x C C lim = xlim A = xlim A= 2. 1 1 x!1 x ln x ; ln x !1 @ !1 @ 1 ln x + 1 ; x x + x2 Case d) Indeterminate Forms 00 0 11 : If f (x) = '(x)(x) and xlim '(x) = 0 and xlim (x) = !a !a 0, then we rst nd the limit A of ln f (x) = (x) ln '(x), which has the form 0 1 (case b)), then we can nd the value eA. The procedures in the cases 10 and 11 are similar. ln x = lim(;x) = 0, i.e., A = ln X = 0, lim xx = X ln xx = x ln x xlim x ln x = xlim x!0 !0 !0 x;1 x!0 x = 1. so X = 1 and nally xlim x !0
1
3. Taylor Expansion
Besides l'Hospital's rule the expansion of functions of indeterminate form into Taylor series can be applied (see 6.1.4.5, p. 389). ! x3 + x5 ; ! x ; x ; x ; sin x = lim 3! 5! 1 ; x2 + = 1 . lim = lim x!0 x!0 x!0 3! 5! x3 x3 6
2.1.4.9 Order of Magnitude of Functions and Landau Order Symbols
Comparing two functions, we often consider their mutual behavior with respect to a certain argument x = a. It is also convenient to compare the order of magnitude of the functions. 1. A function f (x) tends to in nity with a higher order than a function g(x) at a if the quotient fg((xx)) and the absolute values of f (x) exceed any limit as x tends to a.
56 2. Functions 2. A function f (x) tends to zero with a higher order than a function g(x) at a if the absolute values of
f (x) g(x) and the quotient fg((xx)) tends to zero as x tends to a. 3. Two functions f (x) and g(x) tend to zero or to in nity by the same order (or order of magnitude) at a if 0 < m < fg((xx)) < M holds for the absolute value of their quotient as x tends to a, where M is a nite number. 4. Landau Order Symbols The mutual behavior of two functions at a point x = a can be described by the Landau order symbols O (\big O"), or o (\small o") as follows: If x ! a then f (x) = A 6= 0 A = const f (x) = O(g(x)) means that xlim (2.27a) !a g (x) and f (x) = 0 f (x) = o(g(x)) means that xlim (2.27b) !a g (x) where a = 1 is also possible. The Landau order symbols have meaning only if we assume the x tends to a given a. sin x = 1 6= 0, A: sin x = O(x) for x ! 0 , because with f (x) = sin x and g(x) = x we have: xlim !0 x i.e., sin x behaves like x in the neighborhood of x = 0. B: For f (x) = 1 ; cos x and g(x) = sin x the function f (x) vanishes with a higher order than g(x): f (x) lim = lim 1 ; cos x = 0, i.e., 1 ; cos x = o(sin x) for x ! 0. x!0 g (x) x!0 sin x C: f (x) and g(x) vanish by the same order for f (x) = 1 ; cos x g(x) = x2 : lim f (x) = lim 1 ; cos x = 1 , i.e., 1 ; cos x = O(x2) for x ! 0. x!0 g (x) x!0 x2 2
5. Polynomial The order of magnitude of polynomials at 1 can be expressed by their degree. So
the function f (x) = x has order 1, a polynomial of degree n + 1 has an order higher by one than a polynomial of degree n. 6. Exponential Function The exponential function tends to in nity more quickly than any high power xn (n is a xed positive number): ex = 1: (2.28a) xlim !1 xn The proof follows by applying l'Hospital's rule for a natural number n: ex = lim ex = : : : = lim ex = 1: (2.28b) xlim !1 xn x!1 nxn;1 x!1 n! 7. Logarithmic Function The logarithm tends to in nity more slowly than any small positive power x ( is a xed positive number): log x lim (2.29) x!1 x = 0: The proof is with the help of l'Hospital's rule.
2.1 Notion of Functions 57
2.1.5 Continuity of a Function 2.1.5.1 Notion of Continuity and Discontinuity
Most functions occurring in practice are continuous, i.e., for small changes of the argument x a continuous function y(x) changes also only a little. The graphical representation of such a function results in a continuous curve. If the curve is broken at some points, the corresponding function is discontinuous, and the values of the arguments where the breaks are, are the points of discontinuity. Fig. 2.9 shows the curve of a function, which is piecewise continuous. The points of discontinuity are A, B , C , D, E , F and G . The arrow-heads show that the endpoints do not belong to the curve.
y
B 0
A
C DE FG x
2.1.5.2 Denition of Continuity
Figure 2.9 A function y = f (x) is called continuous at the point x = a if 1. f (x) is de ned at a 2. the limit xlim f (x) exists and is equal to f (a). !a This is exactly the case if for an arbitrary " > 0 there is a (") > 0 such that jf (x) ; f (a)j < " for every x with jx ; aj < (2.30) holds. We also talk about one-sided (left- or right-hand sided) continuity, if instead of xlim f (x) = f (a) we !a consider only the one-sided limit x!lim f ( x ) or lim f ( x ) and this is equal to the substitution value a;0 x!a+0 f (a). If a function is continuous for every x in a given interval from a to b, then the function is called continuous in this interval, which can be open, half-open, or closed (see 1.1.1.3, 3., p. 2). If a function is de ned and continuous at every point of the numerical axis, it is said to be continuous everywhere. A function has a point of discontinuity at x = a, which is an interior point or an endpoint of its domain, if the function is not de ned here, or f (a) is not equal to the limit xlim f (x), or the limit does not exist. !a p If the function is de ned only on one side of x = a, e.g., + x for x = 0 and arccos x for x = 1 , then it is not a point of discontinuity but it is a termination. A function f (x) is called piecewise continuous, if it is continuous at every point of an interval except at a nite number of points, and at these points it has nite jumps.
2.1.5.3 Most Frequent Types of Discontinuities 1. Values of the Function Tend to Innity
The most frequent discontinuity is if the function tends to 1 (points B , C , and E in Fig. 2.9). A: f (x) = tan x f 2 ; 0 = +1 f 2 + 0 = ;1. The type of discontinuity (see Fig. 2.34, p. 76) is the same as at E in Fig. 2.9. For the meaning of the symbols f (a ; 0) f (a + 0) see 2.1.4.5, p. 52. B: f (x) = (x ;1 1)2 f (1 ; 0) = +1 f (1 + 0) = +1. The type of discontinuity is the same as at the point B in Fig. 2.9. 1 C: f (x) = e x;1 f (1 ; 0) = 0 f (1 + 0) = 1. The type of discontinuity is the same as at C in Fig. 2.9, with the di erence that this function f (x) is not de ned at x = 1.
58 2. Functions
2. Finite Jump
Passing through x = a the function f (x) jumps from a nite value to another nite value (like at the points A, F , G in Fig. 2.9, p. 57): The value of the function f (x) for x = a may not be de ned here, as at point G or it can coincide with f (a ; 0) or with f (a + 0) (point F ) or it can be di erent from f (a ; 0) and f (a + 0) (point A). A: f (x) = 1 1 f (1 ; 0) = 1 f (1 + 0) = 0 (Fig. 2.8, p. 53). 1 + e x;1 B: f (x) = E (x) (Fig. 2.1c, p. 49) f (n ; 0) = n ; 1 f (n + 0) = n (n integer). 1 C: f (x) = nlim f (1 ; 0) = 1 f (1 + 0) = 0 f (1) = 12 . !1 1 + x2n
3. Removable Discontinuity
If it happens that xlim f (x) exists, i.e., f (a ; 0) = f (a + 0), but either the function is not de ned for !a x = a or f (a) 6= xlim f (x) (point D in Fig. 2.9, p. 57), this type of discontinuity is called removable, !a because de ning f (a) = xlim f (x) the function becomes continuous here. We add only one point to !a the curve, or we change the place only of one point at D. The di erent indeterminate expressions for x = a, which have a nite limit examined by l'Hospital's rule or with other methods, are examples of removable discontinuities. p f (x) = 1 +xx ; 1 is an undetermined 00 expression for x = 0, but xlim f (x) = 12 the function !0 8 > p > > < 1 + x ; 1 for x 6= 0 x f (x) = > > 1 > : for x = 0 2 is continuous.
2.1.5.4 Continuity and Discontinuity of Elementary Functions
The elementary functions are continuous on their domains the points of discontinuity do not belong to their domain. We have the following theorems: 1. Polynomials are continuous everywhere. 2. Rational Functions PQ((xx)) with polynomials P (x) and Q(x) are continuous everywhere except the points x, where Q(x) = 0. If at x = a, Q(a) = 0 and P (a) 6= 0, the function tends to 1 on both sides of a we call this point a pole. The function also has a pole if P (a) = 0, but a is a root of the denominator with higher multiplicity than for the numerator (see 1.6.3.1, 2., p. 43). Otherwise the discontinuity is removable. 3. Irrational Functions Roots of polynomials are continuous for every x in their domain. At the end of the domain they can terminate by a nite value if the radicand changes its sign. Roots of rational functions are discontinuous for such values of x where the radicand is discontinuous. 4. Trigonometric Functions The functions sin x and cos x are continuous everywhere tan x and sec x have in nite jumps at the points x = (2n +2 1) the functions cot x and cosec x have in nite jumps at the points x = n (n integer). 5. Inverse Trigonometric Functions The functions arctan x and arccot x are continuous everywhere, arcsin x and arccos x terminate at the end of their domain because of ;1 x +1, and they are continuous here from one side. 6. Exponential Functions ex or ax with a > 0 They are continuous everywhere.
2.1 Notion of Functions 59
7. Logarithmic Function log x with Arbitrary Positive Base The function is continuous for all positive x and terminates at x = 0 because of xlim log x = ;1 by a right-sided limit. !+0 8. Composite Elementary Functions The continuity is to be checked for every point x of every elementary function containing in the composition (see also continuity of composite functions in 2.1.5.5, 2., p. 59). 1 x;2 ep Find the points of discontinuity of the function y = . The exponent x ;1 2 has an 3 x sin 1 ; x 1 1 1 in nite jump at x = 2 for x = 2 also e x;2 has an in nite jump: e x;2 = 0 e x;2 = 1. x=2;0
x=2+0
The function y has a nite denominator at x = 2. Consequently, at x = 2 there is an in nite jump of the same type as at point C in Fig. 2.9, p. 57. p For x = 0 the denominator is also zero, just like for the valuesp of x, for which sin 3 1 ; x is equal to 3 zero. These last ones correspond to the roots of the equation 1 ; x = n or x = 1 ; n33 , where n is an arbitrary integer. The numerator is not equal to zero for these numbers, so at the points x = 0, x = 1, x = 1 3, x = 1 83, x = 1 273, . . . the function has the same type of discontinuity as the point E in Fig. 2.9, p. 57.
2.1.5.5 Properties of Continuous Functions
1. Continuity of Sum, Di erence, Product and Quotient of Continuous Functions If f (x) and g(x) are continuous on the interval a b], then f (x) g(x) , f (x) g(x) are also continuous, and if g(x) 6= 0 on this interval, then f (x) is also continuous. g(x)
2. Continuity of Composite Functions y = f (u(x))
If u(x) is continuous at x = a and f (u) is continuous at u = u(a) then the composite function y = f (u(x)) is continuous at x = a, and
f (u(x)) = f xlim u(x) xlim !a !a
= f (u(a)) (2.31) is valid. This means that a continuous function of a continuous function is also continuous. Remark: The converse sentence is not valid. It is possible that the composite function of discontinuous functions is continuous.
3. Bolzano Theorem
If a function f (x) is continuous on a nite closed interval a b], and f (a) and f (b) have di erent signs, then f (x) has at least one root in this interval, i.e., there exists at least one interior point of this interval c such that: f (c) = 0 with a < c < b: (2.32) The geometric interpretation of this statement is that the graph of a continuous function can go from one side of the x-axis to the other side only if the curve has an intersection point with the x-axis.
4. Intermediate Value Theorem
If a function f (x) is continuous on a connected domain, and at two points a and b of this domain, where a < b, it has di erent values A and B , i.e., f (a) = A f (b) = B A 6= B (2.33a) then for any value C between A and B there is at least one point c between a and b such that f (c) = C (a < c < b A < C < B or A > C > B ): (2.33b) In other words: The function f (x) takes every value between A and B on the interval (a b) at least once. Or: The continuous image of an interval is an interval.
60 2. Functions
5. Existence of an Inverse Function
y
y
j(x )
) f(x
If a one-to-one function is continuous on an interval, it is strictly monotone on this interval. f(x) If a function f (x) is continuous on a connected domain I, and it is strictly monotone increasing or decreasing, then for this f (x) there also exists a conI I tinuous, strictly monotone increasing or decreas0 II x x 0 II ing inverse function '(x) (see also 2.1.3.7, p. 51), a) b) which is de ned on domain II given by the substitution values of f (x) (Fig. 2.10). Figure 2.10 Remark: In order to make sure that the inverse function of f (x) is continuous, f (x) must be continuous on an interval. If we suppose only that the function is strictly monotonic on an interval, and continuous at an interior point c, and f (c) = C , then the inverse function exists, but may be not continuous at C . x) j(
6. Theorem About the Boundedness of a Function
If a function f (x) is continuous on a nite, closed interval a b] then it is bounded on this interval, i.e., there exist two numbers m and M such that m f (x) M for a x b : (2.34)
7. Weierstrass Theorem
If the function f (x) is continuous on the nite, closed interval a b] then f (x) has an absolute maximum M and an absolute minimum m, i.e., there exists in this interval at least one point c and at least one point d such that for all x with a x b: m = f (d) f (x) f (c) = M: (2.35) The di erence between the greatest and smallest value of a continuous function is called its variation in the given interval. The notion of variation can be extended to the case when the function does not have any greatest or smallest value.
2.2 Elementary Functions
Elementary functions are de ned by formulas containing a nite number of operations on the independent variable and constants. The operations are the four basic arithmetical operations, taking powers and roots, the use of an exponential or a logarithm function, or the use of trigonometric functions or inverse trigonometric functions. We distinguish algebraic and transcendental elementary functions. As another type of function, we can de ne the non-elementary functions (see for instance 8.2.5, p. 460).
2.2.1 Algebraic Functions
In an algebraic function the argument x and the function y are connected by an algebraic equation. It has the form p0(x) + p1(x)y + p2(x)y2 + : : : + pn(x)yn = 0 (2.36) where p0, p1 ,. . . , pn are polynomials in x. 3xy3 ; 4xy + x3 ; 1 = 0 i.e., p0(x) = x3 ; 1 , p1(x) = ;4x , p2(x) = 0 , p3(x) = 3x . If it is possible to solve an algebraic equation (2.36) for y, then we have one of the following types of the simplest algebraic functions.
2.2.1.1 Polynomials
We perform only addition, subtraction and multiplication on the argument x: y = anxn + an;1xn;1 + : : : + a0: (2.37) In particular we distinguish y = a as a constant, y = ax + b as a linear function, and y = ax2 + bx + c as a quadratic function.
2.2 Elementary Functions 61
2.2.1.2 Rational Functions
A rational function can always be written in the form of the ratio of two polynomials: n a xn;1 + : : : + a n;1 0 y = banxxm + : m + bm;1 xm;1 + : : : + b0 The special case +b y = ax cx + d is called a homographic or linear fractional function.
(2.38a) (2.38b)
2.2.1.3 Irrational Functions
Besides the operations enumerated for rational functions, the argument x also occurs under the radical sign. q p A: y = 2x + 3 , B: y = 3 (x2 ; 1)px .
2.2.2 Transcendental Functions
Transcendental functions cannot be given by an algebraic equation like (2.36). We introduce the simplest elementary transcendental functions in the following.
2.2.2.1 Exponential Functions
The variable x or an algebraic function of x is in the exponent of a constant base (see 2.6.1, p. 71). A: y = ex , B: y = ax, C: y = 23x2;5x.
2.2.2.2 Logarithmic Functions
The function is the logarithm with a constant base of the variable x or an algebraic function of x (see 2.6.2, p. 71). A: y = ln x , B: y = lg x , C: y = log2 (5x2 ; 3x) .
2.2.2.3 Trigonometric Functions
The variable x or an algebraic function of x occurs under the symbols sin, cos, tan, cot, sec, cosec (see 2.7, p. 74). A: y = sin x, B: y = cos(2x + 3), C: y = tan px. In general, the argument of a trigonometric function is not only an angle or a circular arc as in the geometric de nition, but an arbitrary quantity. The trigonometric functions can be de ned in a purely analytic way without any geometry. For instance we can represent them by an expansion in a series, 2 or, e.g., the sin function as the solution of the di erential equation d y2 + y = 0 with the initial values dx dy = 1 at x = 0. The numerical value of the argument of the trigonometric function is equal y = 0 and dx to the arc in units of radians. When we deal with trigonometric functions, the argument is considered to be given in radian measure.
2.2.2.4 Inverse Trigonometric Functions
The variable x or an algebraic function of x is in the argument of the inverse trigonometric functions (see 2.8, p. 84) arcsin, arccos, etc.
62 2. Functions B: y = arccos p1 ; x .
A: y = arcsin x,
2.2.2.5 Hyperbolic Functions (see 2.9, p. 87).
2.2.2.6 Inverse Hyperbolic Functions (see 2.10, p. 91).
2.2.3 Composite Functions
Composite functions are all possible compositions of the above algebraic and transcendental functions, i.e., if a function has another function as an argument. p x. A: y = ln sin x, B: y = ln x x+2 +arcsin 5ex Such composition of a nite number of elementary functions again yields an elementary function. The examples C in the previous types of functions are also composite functions.
2.3 Polynomials
2.3.1 Linear Function
The graph of the linear function y = ax + b (2.39) (polynomial of degree 1) is a line (Fig. 2.11a). For a > 0 the function is monotone increasing, for a < 0 it is monotone decreasing for!a = 0 it is a polynomial of degree zero, i.e., it is a constant function. The intercepts are at A ; ab 0 and B (0 b) (for details see 3.5.2.4, 1., p. 194). With b = 0 we have direct proportionality y = ax (2.40) graphically it is a line running through the origin (Fig. 2.11b). y
y
y
B
C
a) 0
A
B
x
y
B
A 2 A1 0
b)
0
x
a)
Figure 2.11
A1
C
A2 0
x
x
b)
Figure 2.12
2.3.2 Quadratic Polynomial The polynomial of second degree y = ax2 + bx + c
(2.41) b (quadratic polynomial) de nes a parabola with a vertical axis of symmetry at x = ; 2a (Fig. 2.12). For a > 0 the function is rst decreasing, it has a minimum, then it is increasing again. For a < 0 rst
2.3 Polynomials 63
it is increasing, it has a maximum, p 2 then it!is decreasing again. The intersection points A1 A2 with the ; b b ; 4ac 0 , the intersection point B with the y-axis is at (0 c). The x-axis, if any, are at 2a
!
b2 (for more details about the parabola see 3.5.2.8, extremum point of the curve is at C ; 2ba 4ac4; a p. 203).
B 0
a)
y
y
y
E E A1
C B Eϕ
A1
ϕ
0
x
b)
2.3.3 Cubic Polynomials
A1
x
c)
0 A A3 x 2 D
Figure 2.13
The polynomial of third degree y = ax3 + bx2 + cx + d (2.42) de nes a cubic parabola (Fig. 2.13a,b,c). Both the shape of the curve and the behavior of the function depend on a and the discriminant % = 3ac ; b2 . If % 0 holds (Fig. 2.13a,b), then for a > 0 the function is monotonically increasing, and for a < 0 it is decreasing. If % < 0 the function has exactly one local minimum and one local maximum (Fig. 2.13c). For a > 0 the value of the function rises from ;1 until the maximum, then falls until the minimum, then it rises again to +1 for a < 0 the value of the function falls from +1 until the minimum, then rises until the maximum, then it falls again to ;1. The intersection points with the x-axis are at the values of the real roots of (2.42) for y = 0. The function can have one, two (then there is a point where the x-axis is the tangent line of the curve) or three real roots: A1 A2 and A3. The intersection with the y-axis is at B (0 dp), the!extreme points p point b d + 2b3 ; 9abc (6ac ; 2b2) ;% . ; % of the curve C and D, if any, are at ; 3a 27a2 ! 3 The inection point which is also the center of symmetry of the curve is at E ; b 2b ; 92abc + d . 3a 27a ! dy % At this point the tangent line has the slope tan ' = dx E = 3a .
2.3.4 Polynomials of -th Degree n
The integral rational function of n-th degree y = anxn + an;1xn;1 + : : : + a1x + a0 (2.43) de nes a curve of n-th degree or n-th order (see 3.5.2.3, 5., p. 194) of parabolic type (Fig. 2.14). Case 1, n odd: For an > 0 the value of y changes continuously from ;1 to +1, and for an < 0 from +1 to ;1. The curve can intersect or contact the x-axis up to n times, and there is at least one intersection point (for the solution of an equation of n-th degree see 1.6.3.1, p. 43 and 19.1.2, p. 887). The function (2.43) has none or an even number up to n ; 1 of extreme values, where minima and maxima occur alternately the number of inection points is odd and is between 1 and n ; 2. There are no asymptotes or singularities. Case 2, n even: For an > 0 the value of y changes continuously from +1 through its minimum
64 2. Functions
0
4 3 2
n=2
y 6 5
n=4
n odd n even
−1
4 3
x
1 −1
1
x
1 1 −1 −2 −3 −4
2
a)
n=5 n =3
y
y
x
b)
Figure 2.14 Figure 2.15 until +1 and for an < 0 from ;1 through its maximum until ;1. The curve can intersect or contact the x-axis up to n times, but it is also possible that it never does that. The number of extrema is odd, and maxima and minima alternate the number of inection points is even, and it can also be zero. There are no asymptotes or singularities. If we want to sketch the graph of a function, it is recommended rst to determine the extreme points, the inection points, the values of the rst derivative at these points, then to sketch the tangent lines at these points, and nally to connect these points continuously.
2.3.5 Parabola of -th Degree n
The graph of the function y = axn (2.44) where n > 0, integer, is a parabola of n-th degree, or of n-th order (Fig. 2.15). 1. Special Case a = 1: The curve y = xn goes through the point (0 0) and (1 1) and contacts or intersects the x-axis at the origin. For even n we have a curve symmetric with respect to the y-axis, and with a minimum at the origin. For odd n the curve is symmetric with respect to the origin, and it has an inection point there. There is no asymptote. 2. General Case a 6= 0: We get the curve of y = axnn from the curve of y = xn by stretching the ordinates by the factor jaj. For a < 0 we reect y = jajx with respect to the x-axis.
2.4 Rational Functions
2.4.1 SpecialFractionalLinearFunction(InverseProportionality) The graph of the function (2.45) y = xa is an equilateral hyperbola, whose asymptotes are the coordinate axes (Fig. 2.16). The point of discontinuity is at x = 0 with y = 1. If a > 0 holds, then the function is strictly monotone decreasing in the interval (;1 0) with values from 0 to ;1 and also strictly monotone decreasing in the interval (0 +1) with values from +1 to 0 (curve in the rst and third quadrants). If a < 0, then the function is increasing in the interval (;1 0) with values from 0 to +1 and also increasing in the interval (0 +1) with values from ;1 to 0 (dotted curve in the second and fourth quadrants). The vertices A
2.4 Rational Functions 65
q
q
q
q
and B are at jaj + jaj and jaj ; jaj with the same sign for a > 0 and with di erent sign for a < 0. There are no extrema (for more details about hyperbolas see 3.5.2.7, p. 200). y
y
A’
A 0
B
B’
x
Figure 2.16
2.4.2 Linear Fractional Function The graph of the function b1 y = aa1 xx + +b 2
2
0
C
A
B
x
Figure 2.17
(2.46)
is an equilateral hyperbola, whose ! asymptotes are parallel to the coordinate axes (Fig. 2.17). b 2 a1 The center is at C ; a a . The parameter a in the equality (2.45) corresponds here to ; a%2
2 2 q q 1 2 0 b j % j a + j%j A a b 2 1 with % = a1 b1 . The vertices of the hyperbola A and B are at @; a and a 2 2 q 2 2 q 1 0 @; b2 j%j a1 ; j%j A, where for % < 0 we take the same signs, for % > 0 di erent ones. The a2 a2 point of discontinuity is at x = ; b2 . For % < 0 the values of the function are decreasing from a1 to a2 a2 ;1 and from +1 to aa1 . For % > 0 the values of the function are increasing from aa1 to +1 and from 2 2 ;1 to aa1 . There is no extremum. 2
2.4.3 Curves of Third Degree, Type I
The graph of the function ! 2 + c (b 6= 0 c 6= 0) y = a + xb + xc2 = ax +xbx (2.47) 2 (Fig. 2.18) is a curve of third degree (type I). It has two asymptotes x = 0 and y = a and it has two branches. One of them corresponds to the monotone changing of y while it takes its values between a and +1 or ;1 the other branch goes through three characteristic points:!the intersection point with 2 the asymptote y = a at A ; c a , an extreme point at B ; 2c a ; b and an inection point at b b 4 c 2! 2 b 3 c C ; b a ; 9c . The positions of the branches depend on the signs of b and c, and there are four p2 ! cases (Fig. 2.18). The intersection points D, E with the x-axis, if any, are at ;b 2ba ; 4ac 0
66 2. Functions their number can be two, one (the x-axis is a tangent line) or none, depending on whether b2 ; 4ac > 0 = 0 or < 0 holds. For b = 0 the function (2.47) becomes the function y = a + xc2 (see (Fig. 2.21) the reciprocal power), and for c = 0 it becomes the homographic function y = axx+ b , as a special case of (2.46). y
a
a)
y
A B
C
x
c>0,b0,b>0
a A E
CB A a D E
x
c0
d)
x
c 0: The function is positive and continuous for arbitrary values of x and it is increasing on the interval (;1 ; b ). Here it takes its maximum, 4a , then it is decreasing again in the interval 2a % ! (; 2ba 1). The extreme point A of the curve is at ; 2ba 4%a , the inection points B and C are at p ! ; 2ba 2ap%3 3%a and for the corresponding slopes of the tangent lines (angular coecients ) we
2.4 Rational Functions 67
3=2 get tan ' = a2 %3 (Fig. 2.19a). Case b) = 0: The function is positive for arbitrary values of x, its value rises from 0 to +1, at y = +1. Then its value falls from x = ; 2ba = x0 it has a point of discontinuity (a pole), where xlim !x0 here back to 0 (Fig. 2.19b). Case c) < 0: The value of y rises from 0 to +1, at the point of discontinuity it jumps to ;1, and rises to the maximum, then falls back to ;1 at the other point!of discontinuity it jumps to +1, then it falls to 0. The extreme point A of the curve is at ; 2ba 4%a . The points of discontinuity are
p
at x = ;b 2a ;% (Fig. 2.19c). y
A
B
0
a)
y
y
ϕ1
C ϕ2
∆ 0
x
0
b)
∆ =0
0
x
A
x ∆ 0
c)
Figure 2.19
2.4.5 Curves of Third Degree, Type III
The graph of the function y = ax2 +xbx + c (2.49) is a curve of third degree (type III) which goes through the origin, and has the x-axis (Fig. 2.20) as an asymptote. The behavior of the function depends on the signs of a and of % = 4ac ; b2 , and for % < 0 also on the signs of the roots and of the equation ax2 + bx + c = 0, and for % = 0 also on the sign of b. From the two cases, a > 0 and a < 0, we consider only the rst one because reecting the curve x of y = with respect to the x-axis we get the second one. (;a)x2 ; bx ; c Case a) > 0: The function is continuous everywhere, its value falls from 0 to the minimum, then p ! rises to the maximum, then falls again to 0. r The extreme points of the curve, A and B , are at ac ;b %2 ac there are three inection points (Fig. 2.20a). Case b) = 0: The behavior of the function depends on the sign of b, so we have two cases. In both cases there is a point of discontinuity at x = ; b both curves have one inection point. 2a b > 0: The value of the function falls from 0 to ;1, the function has a point of discontinuity, then the value of the function rises from ;1 to the maximum, then decreases to 0 (Fig. 2.20b1 ). The extreme ! rc 1 point A of the curve is at A + a 2pac + b .
68 2. Functions y
y
yA 0
B 0
x
x A
A ∆ 0
a)
b1)
∆ =0,b 0
y
0
c1 )
y
x
∆ 0
A 0
A
x
∆ 0, α and β negative
c2 )
x
∆ =0,b 0
b2 )
y
B
0
c3 )
0 B
x
∆ 0, α and β positive
Figure 2.20 b < 0: The value of the function falls from 0 to the minimum, then rises to +1, running through the origin, then the function has a point of discontinuity, then thervalue of the function ! falls from +1 to 0 c 1 (Fig. 2.20b2 ). The extreme point A of the curve is at A ; a ; 2pac ; b .
Case c) < 0: The function has two points of discontinuity, at x = and x = its behavior
depends on the signs of and . The signs of and are di erent: The value of the function falls from 0 to ;1, jumps up to +1, then falls again from +1 to ;1, running through the origin, then jumps again up to +1 , then it falls tending to 0 (Fig. 2.20c1 ). The function has no extremum. The signs of and are both negative: The value of the function falls from 0 to ;1, jumps up to +1 , from here it goes through a minimum up to +1 again, jumps down to ;1 , then rises to a maximum, then falls tending to 0 (Fig. 2.20c2 ). The extremum points A and B can be calculated with the same formula as in case a) of 2.4.5. The signs of and are both positive: The value of the function falls from 0 until the minimum, then rises to +1, jumps down to ;1, then it rises to the maximum, then it falls again to ;1, then jumps up to +1 and then it tends to 0 (Fig. 2.20c3). The extremum points A and B can be calculated by the same formula as in case a) of 2.4.5. In all three cases the curve has one inection point.
2.4.6 Reciprocal Powers
The graph of the function y = xan = ax;n (n > 0 integer)
(2.50)
2.5 Irrational Functions 69 y
0
A(1,1)
y
x y= x12 y= x13
0
y= ax+b A
A' a>0 a 0 and for even n the value of the function rises from 0 to +1, then it falls tending to 0, and it is always positive. For odd n it falls from 0 to ;1, it jumps up to +1, then it falls tending to 0. Case b) For a < 0 and for even n the value of the function falls from 0 to ;1, then it tends to 0, and it is always negative. For odd n it rises from 0 up to +1, jumps down to ;1, then it tends to 0. The function does not have any extremum. The larger n is, the quicker the curve approaches the x-axis, and the slower it approaches the y-axis. For even n the curve is symmetric with respect to the y-axis, for odd n it is centrosymmetric and its center of symmetry is the origin. The Fig. 2.21 shows the cases n = 2 and n = 3 for a = 1.
2.5 Irrational Functions
2.5.1 Square Root of a Linear Binomial The union of the curve of the two functions p y = ax + b
!
(2.51)
is a parabola with the x-axis as the symmetry axis. The vertex A is at ; ab 0 , the semifocal chord (see 3.5.2.8, p. 203) is p = a . The domain of the function and the shape of the curve depend on the 2 sign of a (Fig. 2.22) (for more details about the parabola see 3.5.2.8, p. 203).
2.5.2 Square Root of a Quadratic Polynomial
The union of the graphs of the two functions p y = ax2 + bx + c (2.52) is for a < 0 an ellipse, for a > 0 a hyperbola (Fig. 2.23). One of the two symmetry axes is the x-axis, the other one is the line x = ; 2ba . s 1 p ! 0 b ; % b @ The vertices A C and B D are at ; 0 and ; % A, where % = 4ac ; b2 . 2a 2a 4a The domain of the function and the shape of the curve depend on the signs of a and % (Fig. 2.23). For a < 0 and % > 0 the function has only imaginary values, so no curve exists (for more details about
70 2. Functions y y
B
B A
C
C
x
0
x
0
D a)
y A x
0
D b)
a0
Figure 2.23 the ellipse and hyperbola see 3.5.2.6, p. 198 and 3.5.2.7, p. 200).
2.5.3 Power Function We discuss the power function
y = axk = axm=n (m n integer, positive, coprime) (2.53) for k > 0 and for k < 0 (Fig. 2.24). We restrict our investigation for the case a = 1, because for a 6= 1 the curve di ers from the curve of y = xk only by a stretching in the direction of y-axis by a factor jaj, and for a negative a also by a reection to the x-axis. y
y
1/2
y=x
1 0 a)
1 1
y=x 0 1
x 1/2
y=−x
b)
y
1/3
y=x
2/3
1 x
0 c)
y 3/2
y=x
1 1
x
0
1
x 3/2 y=−x
d)
Figure 2.24
Case a) k > 0 y = xm=n: The shape of the curve is represented in four characteristic cases depending on the numbers m and n in Fig. 2.24. The curve goes through the points (0 0) and (1 1). For k > 1 the x-axis is a tangent line of the curve at the origin (Fig. 2.24d), for k < 1 the y-axis is a tangent line also at the origin(Fig. 2.24a,b,c). For even n we may consider the union of the graph of functions y = xk : it has two branches symmetric to the x-axis (Fig. 2.24a,d), for even m the curve is symmetric to the y-axis (Fig. 2.24c). If m and n are both odd, the curve is symmetric with respect to the origin (Fig. 2.24b). So the curves can have a vertex, a cusp or an inection point at the origin (Fig. 2.24). None of them has any asymptote. Case b) k < 0, y = x;m=n: The shape of the curve is represented in three characteristic cases depending on m and n in Fig. 2.25. The curve is a hyperbolic type curve, where the asymptotes coincide with the coordinate axes (Fig. 2.25). The point of discontinuity is at x = 0. The greater jkj is the quicker the curve approaches the x-axis, and the slower it approaches the y-axis. The symmetry properties of the curves are the same as above for k > 0 they depend on whether m and n are even or
2.6 Exponential Functions and Logarithmic Functions 71 y
y y=x
−1/3
1 0
y
−3/2
y=x
y=x 1
1 0
x
1
−2/3
1
x
0
1
x
−3/2
a)
b)
y=−x
c)
Figure 2.25 odd. There is no extreme value.
2.6 Exponential Functions and Logarithmic Functions 2.6.1 Exponential Functions
x
y
y=10 x y=e x y=2
x
y=( 12 ) x y=( 1e ) 1 x y=(10)
The graphical representation of the function y = ax = ebx (a > 0 b = ln a) (2.54) is the exponential curve (Fig. 2.26). For a = e we have the natural exponential curve y = ex : (2.55) The function has only positive values. Its domain is the interval (;1 +1). For a > 1, i.e., for b > 0, the function is strictly monotone increasing and takes its values from 0 until 1. For a < 1, i.e., for b < 0, it is strictly monotone decreasing, its value falls from 1 until 0. The larger jbj is, the greater is the speed of growth and decay. The curve goes through the point (0 1) and approaches asymptotically the x-axis, for b > 0 on the and for b < 0 on the left, and more quickly for greater values of jbj. 1 right x ;x The function y = a = a increases for a < 1 and decreases for a > 1. y y=log2x=lb x y=logex=ln x y=log10x=lg x
1 0
1 0
x
Figure 2.26
2.6.2 Logarithmic Functions
x
y=log1/10 x y=log1/e x y=log1/2 x
Figure 2.27
The function y = log ax (a > 0 a 6= 1) (2.56) de nes the logarithmic curve (Fig. 2.27) the curve is the reection of the exponential curve with respect to the line y = x. For a = e we have the curve of the natural logarithm y = ln x: (2.57)
72 2. Functions The real logarithmic function is de ned only for x > 0. For a > 1 it is strictly monotone increasing and takes its values from ;1 to +1, for a < 1 it is strictly monotone decreasing, and takes its values from +1 to ;1, and the greater j ln aj is, the quicker the growth and decay. The curve goes through the point (1 0) and approaches asymptotically the y-axis, for a > 1 down, for a < 1 up, and again more quickly for larger values of j ln aj.
2.6.3 Error Curve
The function 2 y = e;(ax) (2.58) gives the error curve (Gauss error distribution curve) (Fig. 2.28). Since the function is even, the yaxis is the symmetry axis of the curve and the larger jaj is, the quicker it approaches asymptotically the x-axis. It takes its maximum at zero, and it is equal to one, so the ! extreme point A of the curve is 1 1 at (0 1), the inection points of the curve B C are at p pe . a q 2 The angles of slopes of the tangent lines are here tan ' = a 2=e . y A very important application of the error curve (2.58) is the description of the normal distribution properties of the observational error (see 16.2.4.1, p. 758.): ;x2 y = '(x) = p1 e 22 : (2.59) 2
C
A
B ϕ
ϕ
0
x
Figure 2.28
2.6.4 Exponential Sum The function y = aebx + cedx
(2.60) y
y y b,d>0 b,d 0
0
x
x
b) c < 0
Figure 2.30
2.6.5 Generalized Error Function
The curve of the function b2 b 2 (2.61) y = aebx+cx2 = (ae; 4c2 )ec(x + 2c ) can be considered as the generalization of the error function (2.58) it results in a symmetric curve with respect to the vertical line x = ; b , it has no intersection point with the x-axis, and the intersection 2c point D with the y-axis is at (0 a) (Fig. 2.30a,b). The shape of the curve depends on the signs of a and c. Here we discuss only the case a > 0, because we get the curve for a < 0 by reecting it in the x-axis. Case a) c > 0: The value of the function falls from +1 until the minimum, and! then rises again to b2 +1. It is always positive. The extreme point A of the curve is at ; 2bc ae; 4c and it corresponds to the minimum of the function there is no inection point or asymptote (Fig. 2.30a). ! b2 Case b) c < 0: The x-axis is the asymptote. The extreme point A of the curve is at ; 2bc ae; 4c and it corresponds to the maximum of the function. The inection points B and C are at
74 2. Functions ! c) ;b p;2c ae ;(b24+2 c (Fig 2.30b). 2c
2.6.6 Product of Power and Exponential Functions
We discuss the function y = axb ecx (2.62) only in the case a > 0, because in the case a < 0 we get the curve by reecting it in the x-axis. For a non-integer b the function is de ned only for x > 0, and for an integer b the shape of the curve for negative x can be deduced also from the following cases (Fig. 2.31). Fig. 2.31 shows how the curve behaves for arbitrary parameters. For b > 0 the curve passes through the origin. The tangent line at this point for b > 1 is the x-axis, for b = 1 the line y = x, for 0 < b < 1 the y-axis. For b < 0 the y-axis is an asymptote. For c > 0 the function is increasing and exceeds any value, for c < 0 it tends asymptotically to 0. For di erent signs of b and c the function has an extremum at x = ; bc (point A on the curve). The curve has either no or p one or two inection points at x = ; b c b (points C and D see Fig. 2.31c,e,f,g). y
y
y
y
A
C 0
a)
c>0, b>1
x
0 c>0, b=1 x
0 c>0, 0 360 or x > 180: If the angle is greater than 360 or greater than 180, then we reduce it for a value , for which 0 360 or 0 180 holds, in the following way (n integer): sin(360 n + ) = sin (2.71) cos(360 n + ) = cos (2.72) tan(180 n + ) = tan
(2.73)
cot(180 n + ) = cot :
(2.74)
78 2. Functions Quadrant I II III IV
Table 2.2 Signs of trigonometric functions Angle sin cos tan cot from 0 to 90 + + + + from 90 to 180 + ; ; ; from 180 to 270 ; ; + + from 270 to 360 ; + ; ;
Table 2.3 Values of trigonometric functions for 0 Angle Radian sin cos tan cot 0 0 0 1 0 1 p p p 1 1 3 3 30 3 6 p2 p2 3 1 2 2 1 45 1 4 p2 2 p p 1 3 1 60 3 33 3 2 2 1 90 1 0 1 0 2
30 45 sec 1 p 2 3 3 p 2
sec +
; ;
csc + +
+
; ;
60 and 90: csc
1 2
p 2 p
2 3 3 1 1 2
Table 2.4 Reduction formulas and quadrant relations of trigonometric functions Function x = 90 x = 180 x = 270 x = 360 ; sin x + cos sin ; cos ; sin cos x sin ; cos sin + cos tan x cot tan cot ; tan cot x tan cot tan ; cot
Argument x < 0: If the argument is negative, then we reduce the calculation of functions for positive argument: sin(;) = ; sin (2.75) cos(;) = cos (2.76) tan(;) = ; tan (2.77) cot(;) = ; cot : (2.78) Argument x for 90 < x < 360: If 90 < x < 360 holds, then we reduce the arguments for an acute angle by the reduction formulas given in Table 2.4. The relations between the values of
the functions belonging to the arguments which di er from each other by 90 180 or 270 or which complete each other to 90 180 or 270 are called quadrant relations. The rst and second columns of Table 2.4 give the complementary angle formulas, and the rst and third ones give the supplementary angle formulas. Because x = 90 ; is the complementary angle (see 3.1.1.2, p. 129) of , we call the relations cos = sin x = sin(90 ; ) (2.79a) sin = cos x = cos(90 ; ) (2.79b) complementary angle formulas.
2.7 Trigonometric Functions (Functions of Angles) 79
For + x = 180 the relations between the trigonometric functions for supplementary angles (see 3.1.1.2, p. 129) sin = sin x = sin(180 ; ) (2.80a) ; cos = cos x = cos(180 ; ) (2.80b) are called supplementary angle formulas. Argument x for 0 < x < 90 : We used to get the values of trigonometric functions for acute angles (0 < x < 90) from tables, but today we use calculators. sin(;1000) = ; sin 1000 = ; sin(360 2 + 280) = ; sin 280 = + cos 10 = +0:9848:
4. Angles in Radian Measure
The arguments given in radian measure, i.e., in units of radians, can be easily converted by formula (3.2) (see 3.1.1.5, p. 130).
2.7.2 Important Formulas for Trigonometric Functions
Remark: Trigonometric functions with complex argument z are discussed in 14.5.2, p. 699.
2.7.2.1 Relations Between the Trigonometric Functions of the Same Angle (Addition Theorems) sin2 + cos2 = 1 sec2 ; tan2 = 1 cosec2 ; cot2 = 1
(2.81) (2.82) (2.83)
sin cosec = 1 cos sec = 1 tan cot = 1
(2.84) (2.85) (2.86)
cos = cot : sin (2.88) (2.87) sin cos = tan Some important relations are summarized in Table 2.5 in order to create an easy survey. The square roots are always considered with the sign which corresponds to the quadrant where the argument is.
2.7.2.2 Trigonometric Functions of the Sum and Di erence of Two Angles sin( ) = sin cos cos sin (2.89)
cos( ) = cos cos sin sin (2.90)
tan tan( ) = 1tan tan tan
cot 1 cot( ) = cot cot cot
(2.91)
sin( + + ) = sin cos cos + cos sin cos + cos cos sin ; sin sin sin cos( + + ) = cos cos cos ; sin sin cos ; sin cos sin ; cos sin sin :
(2.92) (2.93) (2.94)
2.7.2.3 Trigonometric Functions of an Integer Multiple of an Angle sin 2 = 2 sin cos sin 3 = 3 sin ; 4 sin3
(2.95) (2.96)
sin 4 = 8 cos3 sin ; 4 cos sin (2.99)
cos 2 = cos2 ; sin2 cos 3 = 4 cos3 ; 3 cos
cos 4 = 8 cos4 ; 8 cos2 + 1
(2.97) (2.98) (2.100)
80 2. Functions Table 2.5 Relations between the trigonometric functions of the same argument in the interval 0 < < 2
sin cos tan cot
sin
cos
tan cot p ; 1 ; cos2 p tan 2 p 1 2 1 + tan 1 + cot p p 1 2 p cot 2 1 ; sin2 ; 1 + tan 1 + cot p 2 1 ; cos 1 p sin 2 ; cos cot 1 ; sin p 1 ; sin2 p cos 1 ; sin tan 1 ; cos2
2 tan tan 2 = 1 ; tan2 3 tan ; tan3 tan 3 = 1 ; 3 tan2 3 tan 4 = 4 tan 2; 4 tan 4 1 ; 6 tan + tan
(2.101) (2.102) (2.103)
2;1 cot 2 = cot2 cot 3 ; 3 cot cot cot 3 = 3 cot2 ; 1 4 ; 6 cot2 + 1 cot 4 = cot4 cot 3 ; 4 cot :
(2.104) (2.105) (2.106)
For larger values of n in order to gain ! a formula for sin n and cos n we use the de Moivre formula. n Using the binomial coecients m (see 1.1.6.4, p. 12) we get: n n! X ik cosn;k sink = (cos + i sin )n k=0 k = cosn +!in cosn;1 sin ! ! ; n2 cosn;2 sin2 ; i n3 cosn;3 sin3 + n4 cosn;4 sin4 + : : :
cos n + i sin n =
therefore, we get
!
!
!
cos n = cosn ; n2 cosn;2 sin2 + n4 cosn;4 sin4 ; n6 cosn;6 sin6 + : : :
!
! sin n = n cosn;1 sin ; n3 cosn;3 sin3 + n5 cosn;5 sin5 ; : : : :
2.7.2.4 Trigonometric Functions of Half-Angles
(2.107)
(2.108) (2.109)
In the following formulas the sign of the square root must be chosen positive or negative, according to the quadrant where the half-angle is.
2.7 Trigonometric Functions (Functions of Angles) 81
s
sin 2 = 12 (1 ; cos )
(2.110)
s
s
cos 2 = 12 (1 + cos )
cos 1 ; cos sin tan 2 = 11 ; + cos = sin = 1 + cos s cot = 1 + cos = 1 + cos = sin : 2 1 ; cos sin 1 ; cos
(2.111) (2.112) (2.113)
2.7.2.5 Sum and Di erence of Two Trigonometric Functions
sin + sin = 2 sin + cos ; (2.114) sin ; sin = 2 cos + sin ; (2.115) 2 2 2 2 cos + cos = 2 cos +2 cos ;2 (2.116) cos ; cos = ;2 sin +2 sin ;2 (2.117) tan tan = sin( ) (2.118) cot cot = sin( ) (2.119) cos cos sin sin
; ) tan + cot = cos( cos sin
+ ) (2.120) cot ; tan = cos( sin cos :
(2.121)
2.7.2.6 Products of Trigonometric Functions sin sin = 12 cos( ; ) ; cos( + )]
cos cos = 12 cos( ; ) + cos( + )] sin cos = 12 sin( ; ) + sin( + )]
sin sin sin = 14 sin( + ; ) + sin( + ; ) + sin( + ; ) ; sin( + + )] sin cos cos = 41 sin( + ; ) ; sin( + ; ) + sin( + ; ) + sin( + + )] sin sin cos = 1 ; cos( + ; ) + cos( + ; ) 4 + cos( + ; ) ; cos( + + )] cos cos cos = 14 cos( + ; ) + cos( + ; ) + cos( + ; ) + cos( + + )]:
(2.122) (2.123) (2.124) (2.125) (2.126) (2.127) (2.128)
82 2. Functions
2.7.2.7 Powers of Trigonometric Functions sin2 = 12 (1 ; cos 2)
(2.129)
cos2 = 12 (1 + cos 2)
(2.130)
sin3 = 41 (3 sin ; sin 3)
(2.131)
cos3 = 14 (cos 3 + 3 cos )
(2.132)
sin4 = 81 (cos 4 ; 4 cos 2 + 3) (2.133)
cos4 = 81 (cos 4 + 4 cos 2 + 3): (2.134)
For large values of n we can express sinn and cosn , from the formulas for cos n and sin n on page 80.
2.7.3 Description of Oscillations 2.7.3.1 Formulation of the Problem
In engineering and physics we often meet quantities depending on time and given in the form u = A sin(!t + '): (2.135) We call also them sinusoidal quantities. Their dependence on time results in a harmonic oscillation. The graphical representation of (2.135) results in a general sine curve, as shown in Fig. 2.39.
A
u
ϕo 0 ω
π ω
A b
t ϕ
a
Figure 2.39 Figure 2.40 The general sine curve di ers from the simple sine curve y = sin x: a) by the amplitude A, i.e., the greatest distance between its points and the time axis t, b) by the period T = 2! , which corresponds to the wavelength (with ! as the frequency of the oscillation, which is called the angular or radial frequency in wave theory), c) by the initial phase or phase shift by the initial angle ' 6= 0. The quantity u(t) can also be written in the form u = a sin !t + b cos !t: (2.136) p Here for a and b we have A = a2 + b2 and tan ' = ab . We can represent the amounts a, b, A and ' as sides and angle of a right triangle (Fig. 2.40).
2.7.3.2 Superposition of Oscillations
In the simplest case we call a superposition of oscillations the addition of two oscillations with the same frequency. It results again in a harmonic oscillation with the same frequency: A1 sin(!t + '1 ) + A2 sin(!t + '2 ) = A sin(!t + ') (2.137a) with q sin '1 + A2 sin '2 A = A1 2 + A2 2 + 2A1 A2 cos('2 ; '1) tan ' = AA1 cos (2.137b) ' + A cos ' 1
1
2
2
2.7 Trigonometric Functions (Functions of Angles) 83
u0=A sinϕ
y
N A2 A u ϕ2 ϕ
i
2.7.3.3 Vector Diagram for Oscillations
y
a
0 ϕ1 a)
ωt
b A1 x
0 b)
P
time axis
where the quantities A and ' can be determined by a vector diagram (Fig. 2.41a). Also a linear combination of several sine functions with the same frequency again yields a general sine function (harmonic oscillation) with the same frequency: X ci Ai sin(!t + 'i) = A sin(!t + '): (2.137c)
u x
The general sine function (2.135, 2.136) can be Figure 2.41 represented easily by the polar coordinates = A ' and by the Cartesian coordinates x = a, y = b (see 3.5.2.1, p. 189) in a plane. The sum of two such quantities then behaves as the sum of two summand vectors (Fig. 2.41a). Similarly the sum of several vectors results in a linear combination of several general sine functions. We call this representation a vector diagram. The quantity u for a given time t can be determined from the vector diagram with the help of Fig. 2.41a: First we put the time axis OP (t) through the origin O, which rotates clockwise around O by a constant angular velocity !. At start t = 0 the axes y and t coincide. Then at any time t the projection ON of the vector u~ onto the time axis is equal to the absolute value of the general sine function u = A sin(!t + '). For time t = 0 the value u0 = A sin ' is the projection onto the y-axis (Fig. 2.41b).
2.7.3.4 Dampingaxof Oscillations The function y = Ae; sin(!x + '0) y
(a x > 0)
D1 A1 B
0
D3 A
E1 C1 C2 E2 A D2 2
3
E3 C3
D5 A5 C6 E5 C5 D A E E4 6 6 6
C4 D4A4
x
(2.138) results in a curve of a damped oscillation (Fig. 2.42). The oscillation proceeds along the x-axis, while the curve asymptotically approaches the x-axis. The sine curve is ;enclosed by the exponential curves y = Ae ax , and it contacts them in the 0points 1 B k + 2 ; '0 A1 A2 : : : Ak = B @ ! 0 1 11 k + 2 ; '0 CC B k C (;1) A exp B @;a AC A. !
Figure 2.42 ! '0 0 . The intersection points with the coordinate axes are B = (0 A sin '0) C1 C2 : : : Ck = k ; ! k ; ' 0+ The extrema D1 , D2 : : : are at x = and the inection points E1, E2 : : : are at x = ! k ; '0 + 2 with tan = ! . ! a The logarithmic decrement of the damping is = ln yi = a , where yi and yi+1 are the ordinates yi+1 ! of two consecutive extrema.
84 2. Functions
2.8 Inverse Trigonometric Functions
We call the inverse of trigonometric functions also cyclometric functions. Because only a one-to-one function can have an inverse, we have to restrict the domain of the trigonometric functions to an interval such that the considered function should be a one-to-one function on it. There are in nitely many such intervals, and for each we can de ne an inverse. We will distinguish them by the index k according to the corresponding interval. Obviously the trigonometric functions are monotonic in these intervals. y p 2 −1 A
y
y B j
0 1 x −p 2
Figure 2.43
A
p p 2
y
p
p
j
−1 0 1 B −p 2
Figure 2.44
j 0
x
x
0j
−p
Figure 2.45
x
Figure 2.46
2.8.1 De nition of the Inverse Trigonometric Functions
First we discuss the inverse of the sine function (Fig. 2.43). The usual notation for it is arc sine x, or arcsin x. The domain of y = sin x will split into monotonic intervals k ; 2 x k + 2 with k = 0 1 2 : : : . Reecting the curve of y = sin x in the line y = x we get the curves of the inverse functions (placed above each other) y = arck sin x with the domain and ranges (2.139a) ;1 x +1 and k ; 2 y k + 2 where k = 0 1 2 : : : : (2.139b) The form y = arck sin x has the same meaning as x = sin y. Similarly, we can get the other inverse trigonometric functions which are represented in Fig. 2.44{2.46. The domains and ranges of the inverse functions can be found in Table 2.6.
2.8.2 Reduction to the Principal Value
p
y
s co
arccot
arc p 2
n
−1 arctan
arctan arccot
ar csi
If we restrict the domain of the trigonometric functions to the intervals which belong to the index 0 in the de nitions above, we call it the principal value of the function and in several textbooks a capital letter is used for the notation, e.g., Sin x. Similarly, the inverse trigonometric functions for k = 0 have the so-called principal value, which is written without the index k, for instance arcsin x arc0 sin x. In Fig. 2.47 the principal values of the inverse functions are presented. Remark: Calculators give the principal value of these inverses. We can get the values of di erent inverses from the principal value by the following formulas: arck sin x = k + (;1)k arcsin x: (2.140)
0 −
1 p 2
Figure 2.47
x
2.8 Inverse Trigonometric Functions 85
k + 1) ; arccos x (k odd), arck cos x = (k + arccos x (k even). arck tan x = k + arctan x: arck cot x = k + arccot x: A: arcsin 0 = 0 arck sin 0 = k. B: arccot 1 = 4 arck cot 1 = 4 + k.
(2.141) (2.142) (2.143)
8 ; + (k + 1) for odd k, < 1 1 C : arccos 2 = 3 arck cos 2 = : 3+ k for even k. 3 Table 2.6 Domains and ranges of the inverses of trigonometric functions Inverse function arc sine y = arck sin x arc cosine y = arck cos x arc tangent y = arck tan x arc cotangent y = arck cot x
Domain
Range
Trigonometric function with the same meaning
;1 x 1
k ; 2 y k + 2
x = sin y
;1 x 1
k y (k + 1)
x = cos y
;1 < x < 1 k ; 2 < y < k + 2
x = tan y
;1 < x < 1
x = cot y
k < y < (k + 1)
k = 0 1 2 : : : . For k = 0 we get the principal value of the inverse functions, which is usually written without an index, e.g., arcsin x arc0 sin x.
2.8.3 Relations Between the Principal Values (
p
2 p 1 ; x (;1 x 0), arcsin x = 2 ; arccos x = arctan p x 2 = ; arccos 1;x arccos 1 ; x2 (0 x 1).
(
p
2 p 1 ; x ( ; 1 x 0), arccos x = 2 ; arcsin x = arccot p x 2 = ; arcsin 1;x arcsin 1 ; x2 (0 x 1). x arctan x = ; arccot x = arcsin p : 2 1 + x2 8 8 > 1 > > ; arccos p1 1+ x2 (x 0) , < arccot x ; (x < 0) < arctan x = > => : arccot 1 > ( x > 0) : arccos p1 1+ x2 (x 0). x arccot x = 2 ; arctan x = arccos p x 2 : 1+x
(2.144) (2.145) (2.146) (2.147) (2.148)
86 2. Functions 8 8 > ; arcsin p 1 1 > (x 0), < arctan x + (x < 0) > < 1 + x2 arccot x = > = > arcsin p 1 : arctan 1 (x 0). (x > 0) > : x 1 + x2
(2.149)
arcsin(;x) = ; arcsin x: arctan(;x) = ; arctan x:
(2.152) (2.153)
2.8.4 Formulas for Negative Arguments
arccos(;x) = ; arccos x: arccot(;x) = ; arccot x:
(2.150) (2.151)
2.8.5 Sum and Dierence of arcsin and arcsin q x
p
arcsin x + arcsin y = arcsin x 1 ; y2 + y 1 ; x2
q
p
= ; arcsin x 1 ; y2 + y 1 ; x2
y
(xy 0 or x2 + y2 1)
(x > 0 y > 0 x2 + y2 > 1)
(2.154a) (2.154b)
q p = ; ; arcsin x 1 ; y2 + y 1 ; x2 (x < 0 y < 0 x2 + y2 > 1): (2.154c) q p arcsin x ; arcsin y = arcsin x 1 ; y2 ; y 1 ; x2 (xy 0 oder x2 + y2 1) (2.155a) q p = ; arcsin x 1 ; y2 ; y 1 ; x2 (x > 0 y < 0 x2 + y2 > 1) (2.155b) q p = ; ; arcsin x 1 ; y2 ; y 1 ; x2 (x < 0 y > 0 x2 + y2 > 1): (2.155c)
2.8.6 Sum and Dierence of arccos and arccos q p
x
y
arccos x + arccos y = arccos xy ; 1 ; x2 1 ; y2
(x + y 0) q p = 2 ; arccos xy ; 1 ; x2 1 ; y2 (x + y < 0): q p arccos x ; arccos y = ; arccos xy + 1 ; x2 1 ; y2 (x y) q p = arccos xy + 1 ; x2 1 ; y2 (x < y):
2.8.7 Sum and Dierence of arctan and arctan x
y (xy < 1) arctan x + arctan y = arctan 1x;+xy y (x > 0 xy > 1) = + arctan 1x;+xy = ; + arctan x + y (x < 0 xy > 1): 1 ; xy x ; y arctan x ; arctan y = arctan 1 + xy (xy > ;1)
(2.156a) (2.156b) (2.157a) (2.157b)
y
(2.158a) (2.158b) (2.158c) (2.159a)
2.9 Hyperbolic Functions 87
= + arctan x ; y (x > 0 xy < ;1) 1 + xy y (x < 0 xy < ;1): = ; + arctan 1x+;xy
2.8.8 Special Relations for arcsin ! arccos arctan p 2 arcsin x = arcsin 2x 1 ; x2 p
= ; arcsin 2x 1 ; x2
p
x
jxj p12
= ; ; arcsin 2x 1 ; x2
x
!
p1 < x 1 2
!
;1 x < ; p12 :
(2.159b) (2.159c) x
(2.160a) (2.160b) (2.160c)
2 arccos x = arccos(2x2 ; 1) (0 x 1) (2.161a) 2 = 2 ; arccos(2x ; 1) (;1 x < 0): (2.161b) 2 x 2 arctan x = arctan (jxj < 1) (2.162a) 1 ; x2 = + arctan 1 ;2xx2 (x > 1) (2.162b) = ; + arctan 2x 2 (x < ;1): (2.162c) 1;x cos(n arccos x) = Tn(x) (n 1) (2.163) where n 1 can also be a fractional number and Tn(x) is given by the equation p 2 n p 2 n x+ x ;1 + x; x ;1 Tn(x) = : (2.164) 2 For any integer n, Tn(x) is a polynomial of x (a Chebyshev polynomial). To study the properties of the Chebyshev polynomials see 19.6.3, p. 923.
2.9 Hyperbolic Functions
2.9.1 De nition of Hyperbolic Functions
Hyperbolic sine, hyperbolic cosine and hyperbolic tangent are de ned by the following formulas:
x ;x sinh x = e ;2 e (2.165) x ;x (2.166) cosh x = e +2 e x ;x e tanh x = eex ; (2.167) + e;x : The geometric de nition in Chapter 3 Geometry (see 3.1.2.2, p. 131), is an analogy to the trigonometric functions.
88 2. Functions Hyperbolic cotangent, hyperbolic secant and hyperbolic cosecant are de ned as reciprocal values of the above hyperbolic functions: x ;x coth x = 1 = ex + e;x (2.168) tanh x e ; e 1 = 2 sech x = cosh (2.169) x ex + e;x cosech x = 1 = x 2 ;x : (2.170) sinh x e ; e The shapes of curves of hyperbolic functions are shown in Fig. 2.48{2.52. y cosh
3
cosh
cosech 2
coth sinh
1 tanh sech −2 tanh
sech
−1 cosech
0
1
2 x
y 4 3 2 1
j −2 −1 0 1 2 x −1 −2 −3 −4
y 6 5 4 3 2A 1 −2 −1 1 2 x
−1
coth sinh
−2
Figure 2.48
Figure 2.49
Figure 2.50
2.9.2 Graphical Representation of the Hyperbolic Functions 2.9.2.1 Hyperbolic Sine
y = sinh x (2.165) is an odd strictly monotone increasing function between ;1 and +1 (Fig. 2.49). The origin is its symmetry center, the inection point, and here the angle of slope of the tangent line is ' = 4 . There is no asymptote.
2.9.2.2 Hyperbolic Cosine
y = cosh x (2.166) is an even function, it is strictly monotone decreasing for x < 0 from +1 to 1, and for x > 0 it is strictly monotone increasing from 1 until +1 (Fig. 2.50). The minimum is at x = 0 and it is equal to 1 (point A(0 1)) it has no asymptote. The curve is symmetric with respect to the 2 y-axis and it always stays above the curve of the quadratic parabola y = 1+ x2 (the broken-line curve). Because the function demonstrate a catenary curve, we call the curve a catenoid (see 2.15.1, p. 105).
2.9.2.3 Hyperbolic Tangent
y = tanh x (2.167) is an odd function, for ;1 < x < +1 strictly monotone increasing from ;1 to +1 (Fig. 2.51). The origin is the center of symmetry, and the inection point, and here the angle of slope
2.9 Hyperbolic Functions 89
of the tangent line is ' = 4 . The asymptotes are the lines y = 1. y 4 3 2 1
y 1 j −3−2 −1 0 1 2 3 x −1
−4 −3 −2 −1 0 1 2 3 4 x −1 −2 −3 −4
Figure 2.51
Figure 2.52
2.9.2.4 Hyperbolic Cotangent
y = coth x (2.168) is an odd function which is not continuous at x = 0 (Fig. 2.52). It is strictly monotone decreasing in the interval ;1 < x < 0 and it takes its values from ;1 until ;1 in the interval 0 < x < +1 it is also strictly monotone decreasing with values from +1 to +1 . It has no inection point, no extreme value. The asymptotes are the lines x = 0 and y = 1 .
2.9.3 Important Formulas for the Hyperbolic Functions
We have similar relations between the hyperbolic functions as between trigonometric functions. We can show the validity of the following formulas directly from the de nitions of hyperbolic functions, or if we consider the de nitions and relations of these functions also for complex arguments, from (2.198){ (2.205), we can calculate them from the formulas known for trigonometric functions.
2.9.3.1 Hyperbolic Functions of One Variable cosh2 x ; sinh2 x = 1
(2.171)
coth2 x ; cosech2 x = 1
(2.172)
sech2 x + tanh2 x = 1
(2.173)
tanh x coth x = 1
(2.174)
sinh x = tanh x cosh x
(2.175)
cosh x = coth x: sinh x
(2.176)
2.9.3.2 Expressing a Hyperbolic Function by Another One with the Same Argument The corresponding formulas are collected in Table 2.7, so we can have a better survey.
2.9.3.3 Formulas for Negative Arguments sinh(;x) = ; sinh x tanh(;x) = ; tanh x
(2.177) (2.178)
cosh(;x) = cosh x coth(;x) = ; coth x:
(2.179) (2.180)
2.9.3.4 Hyperbolic Functions of the Sum and Di erence of Two Arguments (Addition Theorems) sinh(x y) = sinh x cosh y cosh x sinh y cosh(x y) = cosh x cosh y sinh x sinh y
(2.181) (2.182)
90 2. Functions Table 2.7 Relations between two hyperbolic functions with the same arguments for x > 0 sinh x
sinh x cosh x tanh x coth x
tanh x coth x ; cosh2 x ; 1 q tanh x 2 q 12 1 ; tanh x coth x ; 1 q 2 q 1 2 q coth2 x sinh x + 1 ; 1 ; tanh x coth x ; 1 q 2 cosh x ; 1 1 q sinh2 x ; cosh x coth x sinh x + 1 q 2 sinh x + 1 q cosh x 1 ; sinh x tanh x cosh2 x ; 1
x tanh y tanh(x y) = 1tanh tanh x tanh y
q
cosh x
coth x coth y : (2.183) coth(x y) = 1coth x coth y
(2.184)
2.9.3.5 Hyperbolic Functions of Double Arguments sinh 2x = 2 sinh x cosh x cosh 2x = sinh2 x + cosh2 x
(2.185) (2.186)
tanh 2x = 2 tanh x2 1 + tanh x coth2 x : coth 2x = 1 + 2 coth x
2.9.3.6 De Moivre Formula for Hyperbolic Functions (cosh x sinh x)n = cosh nx sinh nx:
2.9.3.7 Hyperbolic Functions of Half-Argument s
s
sinh x2 = 12 (cosh x ; 1) (2.190) cosh x2 = 12 (cosh x + 1) The sign of the square root in (2.190) is positive for x > 0 and negative for x < 0. sinh x = cosh x + 1 : tanh x = cosh x ; 1 = sinh x (2.192) coth x2 = cosh x;1 sinh x 2 sinh x cosh x + 1
(2.187) (2.188) (2.189)
(2.191) (2.193)
2.9.3.8 Sum and Di erence of Hyperbolic Functions sinh x sinh y = 2 sinh x y cosh x y 2 2 x + y x cosh x + cosh y = 2 cosh 2 cosh ;2 y cosh x ; cosh y = 2 sinh x +2 y sinh x ;2 y
(2.194) (2.195) (2.196)
2.10 Area Functions 91
sinh(x y) : tanh x tanh y = cosh x cosh y
(2.197)
2.9.3.9 Relation Between Hyperbolic and Trigonometric Functions with Complex Arguments z sin z = ;i sinh iz (2.198) sinh z = ;i sin iz (2.202) cos z = cosh iz (2.199) cosh z = cos iz (2.203) tan z = ;i tanh iz (2.200) tanh z = ;i tan iz (2.204) cot z = i coth iz (2.201) coth z = i cot iz: (2.205) Every relation between hyperbolic functions, which contains x or ax but not ax + b, can be derived from the corresponding trigonometric relation with the substitution i sinh x for sin and cosh x for cos . A: cos2 + sin2 = 1 cosh2 x + i2 sinh2 x = 1 or cosh2 x ; sinh2 x = 1. B: sin 2 = 2 sin cos i sinh 2x = 2i sinh x cosh x or sinh 2x = 2 sinh x cosh x.
2.10 Area Functions 2.10.1 De nitions
The area functions are the inverse functions of the hyperbolic functions, i.e., the inverse hyperbolic functions. The functions sinh x , tanh x, and coth x are strictly monotone, so they have unique inverses without any restriction the function cosh x has two monotonic intervals so we can consider two inverse functions. The name area refers to the fact that the geometric de nition of the functions is the area of certain hyperbolic sectors (see 3.1.2.2, p. 131).
2.10.1.1 Area Sine
The function y = Arsinh x (2.206) (Fig. 2.53) is an odd, strictly monotone increasing function, with domain and range given in Table 2.8. It is equivalent to the expression x = sinh y . The origin is the center of symmetry and the inection point of the curve, where the angle of slope of the tangent line is ' = . 4
2.10.1.2 Area Cosine
The functions y = Arcosh x and y = ; Arcosh x (2.207) (Fig. 2.54) or x = cosh y have the domain and range given in Table 2.8 they are de ned only for x 1. The function curve starts at the point A(1 0) with a vertical tangent line and the function increases or decreases strictly monotonically respectively. y 2 1 −4 −3 −2 −1
y j 0 1 −1
2 1
2
−2
Figure 2.53
3
4
x
y=Arcosh x
1A −2 −1 0 2 3 4 5 6 x −1 −2 y=-Arcosh x
Figure 2.54
92 2. Functions Table 2.8 Domains and ranges of the area functions Hyperbolic function Area function Domain Range with same meaning area sine y = Arsinh x ;1 < x < 1 ;1 < y < 1 x = sinh y area cosine y = Arcosh x y 0): (2.225b) The Fourier transform of the damped oscillation is the Lorentz or Breit{Wigner curve (see 15.3.1.4, p. 731).
2.11.3 Cartesian Folium (Folium of Descartes)
The equation x3 + y3 = 3axy (a > 0) or (2.226a) 2 3 at 3 at in parametric form x = 1 + t3 y = 1 + t3 with t = tan 0 ;1 < t < ;1 and ; 1 < t < 1) (2.226b) gives the Cartesian folium curve represented in Fig. 2.59. The origin is a double point because the curve passes through it twice, and here both coordinate axes are tangent lines. At the origin the radius of curvature for both branches of the curve is r = 32a . The equation of the asymptote is x + y + a = 0. 2 The vertex A has the coordinates A 32 a 32 a . The area of the loop is S1 = 32a . The area S2 between
2.11 Curves of Order Three (Cubic Curves) 95
the curve and the asymptote has the same value.
2.11.4 Cissoid
3 The equation y2 = a x; x (a > 0)
(2.227a)
2 at3 or in parametric form x = 1 at + t2 y = 1 + t2 with t = tan 0 ;1 < t < 1)
(2.227b)
2 or with polar coordinates = a sin ' (a > 0) (2.227c) cos ' (Fig. 2.60) describes the locus of the points P for which 0P = MQ (2.228) is valid. Here M is the second intersection point of the line 0P with the drawn circle of radius a2 , and Q is the intersection point of the line 0P with the asymptote x = a. The area between the curve and the asymptote is equal to S = 34 a2 . y
y a
P2
Q P
P1 A
M
0
x
M x
0 a
Figure 2.60
Figure 2.61
2.11.5 Strophoide
Strophoide is the locus of the points P1 and P2 , which are on an arbitrary half-line starting at A (A is on the negative x-axis) and for which the equalities MP 1 = MP 2 = 0M (2.229) are valid. Here M is the intersection point with the y-axis (Fig. 2.61). The equation of the strophoide in Cartesian, and in polar coordinates, and in parametric form is: x 2' = ;a cos (2.230b) (2.230a) y2 = x2 aa + cos ' (a > 0) ; x (a > 0) 2 1 2 x = a tt2 ; y = at t2 ; 1 with t = tan 0 ;1 < t < 1): (2.230c) +1 t +1 The origin is a double point with tangent lines y = x. The asymptote has the equation x = a. The vertex is A(;a 0). The area of the loop is S1 = 2a2 ; 12 a2 , and the area between the curve and the
96 2. Functions asymptote is S2 = 2a2 + 1 a2 . 2
2.12 Curves of Order Four (Quartics) 2.12.1 Conchoid of Nicomedes
The Conchoid of Nicomedes (Fig. 2.62) is the locus of the points P , for which 0P = 0M l (2.231) holds, where M is the intersection point of the line between 0P1 and 0P2 with the asymptote x = a. The \+" sign belongs to the outer branch of the curve, the \;" sign belongs to the inner one. The equations for the conchoid of Nicomedes are the following in Cartesian coordinates, in parametric form and in polar coordinates: (x ; a)2 (x2 + y2) ; l2x2 = 0 (a > 0 l > 0) (2.232a) x = a + l cos ' y = a tan ' + l sin ' (right branch: ; 2 < ' < 2 left branch: 2 < ' < 32 ) (2.232b) a = cos ' l (\+" sign: right branch, \;" sign: left branch,) (2.232c) 1. Right Branch: The asymptote is x = a. The vertex A is at (a + l 0), the inection points B , C have as x-coordinate the greatest root of the equation x3 ; 3a2 x + 2a(a2 ; l2 ) = 0. The area between the right branch and the asymptote is S = 1 . 2. Left Branch: The asymptote is x = a . The vertex D is at (a ; l 0) . The origin is a singular point, whose type depends on a and l: Case a) For l < a it is an isolated point (Fig. 2.62a). The curve 3has two2 further 2inection points E and F , whose abscissa is the second greatest root of the equation x ; 3a x + 2a(a ; l2) = 0. Case b) For l > a the origin is a double point (Fig. 2.62b). The curve has a maximum p 2 and2a minimum p3 2 value at x = a ; al . At the origin the slopes of the tangent lines are tan = l ; a . Here the a p2 2 l ; a l . radius of curvature is r0 = 2a Case c) For l = a the origin is a cuspidal point (Fig. 2.62c).
2.12.2 General Conchoid
The conchoid of Nicomedes is a special case of the general conchoid. We get the conchoid of a given curve if we elongate the length of the position vector of every point by a given constant segment l. If we consider a curve in a polar coordinate system with an equation = f ('), the equation of its conchoid is = f (') l: (2.233) So, the conchoid of Nicomedes is the conchoid of the line.
2.12.3 Pascal's Limacon
The conchoid of a circle is called the Pascal limacon (Fig. 2.63) if in (2.231) the origin is on the perimeter of the circle, which is a further special case of the general conchoid (see 2.12.2, p. 96). The equations in the Cartesian and in the polar coordinate systems and in parametric form are the following (see also (2.245c), p. 103): (x2 + y2 ; ax)2 = l2 (x2 + y2) (a > 0 l > 0) (2.234a)
2.12 Curves of Order Four (Quartics) 97 y
y
B
B
P1 M B E A 0D F x C a l
P1
P1
P2
a)
y
D
P2 0
M A
x
a
C
la
c)
l=a
Figure 2.62 y
C l
B P
l
M
0
a
l D
a)
_ 2a l>
P
y
A
I G B 0 H K
C y
x
A a
a 0 l > 0 0 ' < 2) (2.234c) with a as the diameter of the circle. The vertices A B are at (a l 0). The shape of the curve depends on the quantities a and l, as we can see in Fig. 2.63 and 2.64. a) Extreme Points and Inection Points: For curve has four extreme points C , D, E , F p a2 > l the 2! ; l l + 8 a for a l it has two they are at cos ' = . For a < l < 2a there exist two inection 4a ! 2 2 points G and H at cos ' = ; 2a3al+ l . p 2 2! 2 b) Double Tangent: For l < 2a, at the points I and K at ; 4l a l 4a4a; l there is a double tangent. c) Singular Points: The origin is a singular point: For a < l it is an isolated point, for a > l it is a
98 2. Functions
p
2 2 double point and the slopes of the tangent lines are tan = a l; l , here the radius of curvature p is r0 = 12 a2 ; l2 . For a = l the origin is a cuspidal point then we call the curve a cardioid (see also 2.13.3, p. 101). 2 The area of the limacon is S = a2 + l2, where in the case a > l (Fig. 2.63c) the area of the inside loop is counted twice.
2.12.4 Cardioid y I 0 K
C M A a
l D
a=1
Figure 2.64
x
The cardioid (Fig. 2.64) can be de ned in two di erent ways, as: 1. Special case of the Pascal limacon with 0P = 0M a (2.235) where a is the diameter of the circle. 2. Special case of the epicycloid with the same diameter a for the xed and for the moving circle (see 2.13.3, p. 101). The equation is (x2 + y2)2 ; 2ax(x2 + y2) = a2y2 (a > 0) (2.236a) and the parametric form, and the equation in polar coordinates are: x = a cos '(1 + cos ') y = a sin '(1 + cos ') (a > 0 0 ' < 2 ) (2.236b) = a(1 + cos ') (a > 0): (2.236c)
The origin is a cuspidal point. The vertex A is at (2a 0) extreme points C and D are at cos ' = 1 with 2 p ! 3 3 3 3 2 coordinates 4 a 4 a . The area is S = 2 a , i.e., six times the area of a circle with diameter a. The length of the curve is L = 8a.
2.12.5 Cassinian Curve
The locus of the points P , for which the product of the distances from two xed points F1 and F2 with coordinates (c 0) and (;c 0) resp., is equal to a constant a2 6= 0, is called a Cassinian curve (Fig. 2.65): F1P F2P = a2 : (2.237) The equations in Cartesian and polar coordinates are: (x2 + y2)2 ; 2c2 (x2 ; y2) = a4 ; c4 (a > 0 c > 0) (2.238a)
q 2 = c2 cos 2' c4 cos2 2' + (a4 ; c4 ) (a > 0 c > 0): (2.238b) The shape of the curve depends on the quantities a and c : p Case a > c 2: For a > c 2 the curve is an ovalp whose shape resembles an ellipse (Fig. 2.65a). The intersectionppoints A C with the x-axis are ( a2 + c2 0), the intersection points B D with the y-axis are (0 a2 ; c2). p p Case a = c 2: For a = c 2 the curve is of the same type with A C ( c 3 0) and B D (0 c), where the curvature at the points B and D is equal to 0, i.e., there is a narrow contact with the lines y = c.
p
p
2.12 Curves of Order Four (Quartics) 99 y
a
B
a)
G
c
C F2
y
P 0
A F1
D a>c 2
x
C
b)
L
BP
a
y
a E P F1 A c x I
G
E
c A x 0 F1 F2 M I KN D
Q
C F2 K
c 0 ;1 < t < 1) (2.241a) where a is the radius of the circle and t is the angle 0): (2.241b) The curve is periodic with period 0O1 = 2a. At 0, O1, O2 : : :, Ok = (2ka 0) we have cusps, the very tices are at Ak = ((2k + 1)a 2a). A1 A2 The arclength of 0P is 2a L = 8a sin2(t=4), the length of one P C C1 arch is L0A1 O1 = 8a. The area of one t arch is S = 3a2. The radius of cur2π a 4π a vature is r = 4a sin 21 t , at the vertices B πa 3π a O2 0 O1 x rA = 4a. The evolute of a cycloid (see 3.6.1.6, p. 236) is a congruent cycloid, which is denoted in Fig. 2.67 by the broken line. Figure 2.67
2.13.2 Prolate and Curtate Cycloids or Trochoids y
C D0 B0
a)
0
A2
A1
P
t C1 M
λ>1
D2
D1 B1
O1
B2
O2 x
y C E B0 1
b)
0 λ1, a>0
x
0
b)
l 0
Figure 2.71 y
y
P 0
a)
C
P
j
C
j
x
l >1, a 0 ;1 < ' < 1) :
104 2. Functions
0
P2
P2
P1 j1
P1
a
j2
x
A1
j1 j2
A2 K
x
Figure 2.73 Figure 2.74 The curve has two branches in a symmetric position with respect to the y-axis. Every ray 0K intersects _ the curve at the points 0, A1 , A2 ,. . . ,An ,. . . their distance is Ai Ai+1 = 2a. The arclength 0P is q 2L tends to 1. The area of the L = a2 ' '2 + 1 + Arsinh ' , where for large ' the expression a' 2 2 2 3=2 a 3 3 sector P10P2 is S = 6 ('2 ; '1 ). The radius of curvature is r = a (''2++1)2 and at the origin r0 = a2 .
2.14.2 Hyperbolic Spiral
The equation of the hyperbolic spiral in polar coordinates is = 'a (a > 0 ;1 < ' < 0 0 < ' < 1) :
(2.247)
The curve of the hyperbolic spiral (Fig. 2.74) has two branches in a symmetric position with respect to the y-axis. The line y = a is the asymptote for both! branches, and the origin is an asymptotic 2 2 point. The area of the sector P10P2 is S = a2 '1 ; '1 , and 'lim S = 2a' is valid. The radius of 2 !1 curvature is r = a '
p1 + '2 !3 '
1
2
1
.
2.14.3 Logarithmic Spiral
The logarithmic spiral is a curve (Fig. 2.75) which intersects all the rays starting at the origin 0 at the same angle . The equation of the logarithmic spiral in polar coordinates is = aek' (a > 0 ;1 < ' < 1) (2.248) _ where point of the curve. The arclengthp P1P2 is L = p k2 = cot . The origin is the asymptotic 1 + k ( ; ) , the limit of the arclength 0_P calculated from the origin is L = 1 + k2 . The 2 1 0 k k p 2 radius of curvature is r = 1 + k = L0 k. Special case of a circle: For = 2 we have k = 0, and the curve is a circle.
2.14.4 Evolvent of the Circle
The evolvent of the circle is a curve (Fig. 2.76) which is described by the endpoint of a string while _ we roll it o a circle, and we always keep it tight, so that AB = BP . The equation of the evolvent of the
2.15 Various Other Curves 105 circle is in parametric form x = a cos ' + a' sin ' y = a sin ' ; a' cos ' (2.249) where a is the radius of the circle, and ' = R ' > . 6. Hollow Cylinder With the notation R for the outside radius and r for the inside one, = R ; r for the di erence of the radii, and % = R 2+ r for the mean radius (Fig. 3.69) we have: V = h(R 2 ; r2) = h (2 R ; ) = h (2 r + ) = 2 h %:
(3.147)
R
2a
h
R ϕ R
r
h
b
Figure 3.68
Figure 3.69
156 3. Geometry 7. Conical Surface arises by moving a line, the generating line, along a curve, the direction curve so that it always goes through a xed point, the vertex (Fig. 3.70). 8. Cone (Fig. 3.71) is bounded by a conical surface with a closed direction curve and a base cut out from a plane by the surface. For an arbitrary cone V = h A3 G :
(3.148)
h
gen erat line ing
directing curve
l
h R
vertex
Figure 3.70
Figure 3.71
r h
R
H
l
Figure 3.72
R
Figure 3.73
Figure 3.74
9. Right Circular Cone has a circle as base and its vertex is right above the centre of the circle (Fig. 3.72). With l as the length of the apothem and R as the radius of the base we have: p 2 2 1 2 S = R(R + l) : (3.151) V = 3 R h (3.149)
M = R l = R R + h (3.150)
10. Frustum of Right Cone or Truncated Cone (Fig. 3.73) q M = l (R + r ) l = h2 + (R ; r)2 (3.152) V = 3h R 2 + r2 + R r
(3.154)
H = h + Rh;r r :
(3.153) (3.155)
11. Conic Sections see 3.5.2.9, p. 205. 12. Sphere (Fig. 3.74) with radius R and diameter D = 2R. Every plane section of it is a circle. A plane section through the center results in a great circle (see 3.4.1.1, p. 158) with radius R . Only one great circle can be tted through two surface points of the sphere if they are not the endpoints of the same diameter. The shortest connecting surface curve between two surface points is the arc of the great circle between them (see 3.4.1.1, p. 158). Formulas for the surface and for the volume of the sphere:
3.3 Stereometry 157
S = 4 R 2 12:57 R 2
p3
p3
S = 36 V 2 4:836 V 2 3 3 V = D 6 0:5236 D s p R = 12 S 0:2821 S
S = D2 3:142 D2 V = 43 R 3 4:189 R 3 s 3 p 1 V = 6 S 0:09403 S 3 s p R = 3 34 V 0:6204 3 V :
(3.156a) (3.156c) (3.157b) (3.158a)
13. Spherical Sector (Fig. 3.75) S = R(2 h + a)
14. Spherical Cap (Fig. 3.76) a2 = h(2 R ; h) M = 2 R h = a2 + h2
2 V = 2 3R h :
(3.159) (3.161) (3.163)
(3.156b) (3.157a) (3.157c) (3.158b) (3.160)
V = 16 h 3 a2 + h2 = 13 h2(3 R ; h) (3.162) S = 2 R h + a2 = h2 + 2 a2 : (3.164)
15. Spherical Layer (Fig. 3.77) 2 2 2 !2 R 2 = a2 + a ; 2b h ; h (3.165)
2b
h
h
h
V = 61 h 3 a2 + 3 b 2 + h2 (3.166) M = 2Rh (3.167) S = 2 R h + a2 + b 2 : (3.168) If V1 is the volume of a truncated cone written in a spherical layer (Fig. 3.78) and l is the length of its apothem then we have V ; V1 = 16 h l 2: (3.169)
2a
Figure 3.75
R
R
R
2a
Figure 3.76
2a
Figure 3.77
16. Torus (Fig. 3.79) is the solid we get by rotating a circle around an axis which is in the plane of the circle but does not intersect it. S = 42 R r 39:48 R r (3.170a)
S = 2 D d 9:870 D d (3.170b) 1 V = 22 R r2 19:74 R r2 (3.171a) V = 4 2 D d 2 2:467 D d 2: (3.171b) 17. Barrel (Fig. 3.80) arises by rotation of a generating curve a circular barrel by rotation of a circular segment, a parabolic barrel by rotation of a parabolic segment. For the circular barrel we have the approximation formulas
158 3. Geometry d
R
l
h
D
Figure 3.78
h
r
r
2
Figure 3.79
V 0:262 h 2 D 2 + d (3.172a) or V 0:0873 h(2 D + d)2 for the parabolic barrel V = 15h 2 D 2 + D d + 34 d 2 0:05236h 8 D 2 + 4 D d + 3 d 2 :
Figure 3.80 (3.172b) (3.173)
3.4 Spherical Trigonometry
For geodesic measures which extend over great distances we have to consider the spherical shape of the Earth. For this we need spherical geometry. In particular we need formulas for spherical triangles, i.e., triangles lying on a sphere. This was also realized by the ancient Greeks, so besides the trigonometry of the plane the trigonometry of the sphere was developed, and we consider Hipparchus (around 150 BC) as the founder of spherical geometry.
3.4.1 Basic Concepts of Geometry on the Sphere 3.4.1.1 Curve, Arc, and Angle on the Sphere
1. Spherical Curves, Great Circle and Small Circle
Curves on the surface of a sphere are called spherical curves. Important spherical curves are great circles and small circles. They are intersection circles of a plane passing through the sphere, the socalled intersecting plane (Fig. 3.81): If a sphere of radius R is intersected by a plane K at distance h from the center O of the sphere, then for radius p r of the intersection circle we have r = R2 ; h2 (0 h R): (3.174) For h = 0 the intersecting plane goes through the center of the sphere, and r takes the greatest possible value. In this case the intersection circle g in the plane ; is called a great circle. Every other intersection circle, with 0 < h < R, is called a small circle, for instance the circle k in Fig. 3.81. For h = R the plane K has only one common point with the sphere. Then it is called a tangent plane. On the Earth the equator and the meridians with their countermeridians { which are their reections with respect to the Earth's axis { represent great circles. The parallels of latitude are small circles (see also 3.4.1.2, p. 160).
2. Spherical Distance
Through two points A and B of the surface of the sphere, which are not opposite points, i.e., they are not the endpoints of the same diameter, in nitely many small circles can be drawn, but only one great circle (with the plane of the great circle g). Consider two small circles k1 k2 through A and B and turn them into the plane of the great circle passing through A and B (Fig. 3.82). The great circle has the greatest radius and so the smallest curvature. So the shorter arc of the great circle is the shortest connection between A and B . It is the shortest connection between A and B on the surface of the sphere, and it is called the spherical distance.
3.4 Spherical Trigonometry 159 Κ r
Γ
k
R
A
A
R. arc e 0g
R
k1
k2 0
B
g
e
R 0
g
B
h
Figure 3.81
3. Geodesic Lines
Figure 3.82
Figure 3.83
Geodesic lines are the curves on a surface which are the shortest connections between two points of the surface (see 3.6.3.6, p. 250). In the plane the straight lines, on the sphere the great circles, are the geodesic lines (see also 3.4.1.2, p. 160).
4. Measurement of the Spherical Distance
The spherical distance of two points can be expressed as a measure of length or as a measure of angle (Fig. 3.83). 1. Spherical Distance as a Measure of Angle is the angle between the radii 0A and 0B measured at the center 0. This angle determines the spherical distance uniquely, and in the following we denote it by a lowercase Latin letter. The notation can be given at the center or on the great circle arc. 2. Spherical Distance as a Measure of Length is the length of the great circle arc between A _ and B . It is denoted by AB (arc AB ). 3. Conversions from Measure of Angle into Measure of Length and conversely can be done by the formulas _ _ AB = R arc e = R %e (3.175a) e =AB R% : (3.175b) Here e denotes the angle given in degrees and arc e denotes the angle in radian (see radian measure 3.1.1.5, p. 130). The conversion factor % is equal to 0 00 % = 1 rad = 180 (3.175c) = 57:2958 = 3438 = 206265 : The determinations of the distance as a measure of length or angle are equivalent but in spherical trigonometry the spherical distances are given mostly as a measure of angle. A: For spherical calculations on the Earth's surface we consider a sphere with the same volume as the biaxial reference ellipsoid of Krassowski. This radius of the Earth is R = 6371:110 km, and consequently we have 1 =^ 111:2 km, 10 =^ 1853:3 m = 1 oldseamile. Today 1 seamile = 1852 m. _ B: The spherical distance between Dresden and St. Petersburg is AB = 1433 km or 1433 km 57:3 = 12:89 = 12530. e = 6371 km
5. Intersection Angle, Course Angle, Azimuth
The intersection angle between spherical curves is the angle between their tangent lines at the intersection point P1 . If one of them is a meridian, the intersection angle with the curve segment to the north
a)
P2
di an
N
an
N
P1
a
(+)
b)
N
me ri
rid i
n ia
me
(-) a P1
me rid
160 3. Geometry
c)
d
P1
Figure 3.84 from P1 is called the course angle in navigation. To distinguish the inclination of the curve to the east or to the west, we assign a sign to the course angle according to Fig. 3.84a,b and we restrict it to the interval ;90 < 90. The course angle is an oriented angle, i.e., it has a sign. It is independent of the orientation of the curve { of its sense. The orientation of the curve from P1 to P2, as in Fig. 3.84c, can be described by the azimuth : It is the intersection angle between the northern part of the meridian passing through P1 and the curve arc from P1 to P2. We restrict the azimuth to the interval 0 < 360: Remark: In navigation the position coordinates are usually given in sexagesimal degrees the spherical distances and also the course angles and the azimuth are given in decimal degrees.
3.4.1.2 Special Coordinate Systems 1. Geographical Coordinates
To determine points P on the Earth's surface we use geographical coordinates (Fig. 3.85), i.e., spherical coordinates with the radius of the Earth, the geographical longitude and the geographical latitude '. To determine the degree of longitude we subdivide the surface of the Earth by half great circles from the north pole to the south pole, by so-called meridians. The zero meridian goes through the observatory of Greenwich. From here we count the east longitude with the help of 180 meridians and the west longitude with 180 meridians. At the equator they are at a distance of 111 km of each other. East longitudes are given as positive, west longitudes are given as negative values. So ;180 < 180. N
N l
Equator
+x
P
zero meridian j
-y
Dx
l geographic longitude l
geographic latitude j
parallel to the P1 central Dx' meridian P2 +y central point
a)
-x
central meridian
+x +x2 +x1 −y
Dx
P1 Dx’ P2 +y
−x b)
Figure 3.85 Figure 3.86 To determine the degree of latitude the Earth's surface is divided by small circles parallel to the equator. Starting from the equator we count 90 degrees of latitude to the north, the northings, and 90 southern latitudes. Northings are positive, southern latitudes are negative. So ;90 ' 90.
2. Soldner Coordinates
The right-angled Soldner coordinates and Gauss{Kruger coordinates are important in wide surface surveys. To map part of the curved Earth's surface onto a right-angled coordinate system in a plane, distance preserving in the ordinate direction, according to Soldner we place the x-axis on a meridian (we call it a central meridian), and the origin at a well-measured center point (Fig. 3.86a). The ordinate y of a point P is the segment between P and the foot of the spherical orthogonal (great circle) on the
3.4 Spherical Trigonometry 161
central meridian. The abscissa x of the point P is the segment of a circle between P and the main parallel passing through the center, where the circle is in a parallel plane to the central meridian (Fig. 3.86b). Transferring the spherical abscissae and ordinates into the plane coordinate system the segment %x is stretched and the directions are distorted. The coecient of elongation a in the direction of the abscissa is 2 a = %%xx0 = 1 + 2yR2 R = 6371 km: (3.176) To moderate the stretching, the system may not be extended more than 64 km on both sides of the central meridian. A segment of 1 km length has an elongation of 0:05 m at y = 64 km.
3. Gauss{Kruger Coordinates
To map part of the curved Earth's surface onto the plane with an angle-preserving (conformal) mapping, rst we prepare the subdivision of the Gauss{Kruger system into meridian zones. For Germany these mid-meridians are at 6 9 12, and 15 east longitude (Fig. 3.87a). The origin of every meridian zone is at the intersection point of the mid-meridian and the equator. In the north{south direction we consider the total range, in the east{west direction we consider a 1 400 wide strip on both sides. In Germany it is ca. 100 km. The overlap is 200, which is here nearly 20 km. y - elongation The coecient of elongation y right value high value a in the abscissa direction x - elon(Fig. 3.87b) is the same as P gation in the Soldner system (3.176). Dx To keep the mapping angleDx' 7° 8° 9° 10° 11° preserving, we add to the orEquator dinates the quantity b: a) b) 50 km 3 b = 6yR2 : (3.177) Figure 3.87
3.4.1.3 Spherical Lune or Biangle
Suppose there are two planes ;1 and ;2 passing through the endpoints A and B of a diameter of the sphere and enclosing an angle (Fig. 3.88) and so de ning two great circles g1 and g2. The part of the surface of the sphere bounded by the halves of the great circles is called a spherical lune or biangle or spherical digon. The sides of the spherical biangle are de ned by the spherical distances between A and B on the great circles. Both are 180. We consider the angles between the tangents of the great circles g1 and g2 at the points A and B as the angles of the spherical biangle. They are the same as the so-called dihedral angle between the planes ;1 and ;2. If C and D are the bisecting points of both great circle arcs A and B , the angle can be expressed as the spherical distance of C and D. The area Ab of the spherical biangle is proportional to the surface area of the sphere just as the angle to 360 . Therefore the area is R2 = 2R2 = 2 R2 arc with the conversion factor % as in (3.175c): Ab = 4360 (3.178) %
3.4.1.4 Spherical Triangle
Consider three points A B , and C on the surface of a sphere, not on the same great circle. If we connect every two of them by a great circle (Fig. 3.89), we get a spherical triangle ABC . The sides of the triangle are de ned as the spherical distances of the points, i.e., they represent the angles at the center between the radii 0A, 0B , and 0C . They are denoted by a b, and c, and in the following they are given in angle measure, independently of whether they are denoted at the center as angles or on the surface as great circle arcs. The angles of the spherical triangle are the angles between every two planes of the great circles. They are denoted by , and . The order of the notation of the points, sides, and angles of the spherical triangle follows the same
162 3. Geometry α
C
A
α
b γ a α
g1 g2 0 R α Cα D Γ1
A
Γ2 B
c β B
90° P1 left pole
b ca 0
g polar
P2 right pole
Figure 3.88 Figure 3.89 Figure 3.90 scheme as for triangles of the plane. A spherical triangle is called a right-sided triangle if at least one side is equal to 90. There is an analogy with the right-angled triangles of the plane.
3.4.1.5 Polar Triangle
1. Poles and Polar The endpoints of a diameter P1 and P2 are called poles, and the great circle g being perpendicular to this diameter is called polar (Fig. 3.90). The spherical distance between a pole
and any point of the great circle g is 90 . The orientation of the polar is de ned arbitrarily: Traversing the polar along the chosen direction we have a left pole to the left and a right pole to the right. 2. Polar Triangle A0 B 0C 0 of a given spherical triangle A B C is a spherical triangle such that the vertices of the original triangle are poles for its sides (Fig. 3.91). For every spherical triangle ABC there exists one polar triangle A0B 0 C 0. If the triangle A0B 0 C 0 is the polar triangle of the spherical triangle ABC , then the triangle ABC is the polar triangle of the triangle A0 B 0C 0 . The angles of a spherical triangle and the corresponding sides of its polar triangle are supplementary angles, and the sides of the spherical triangle and the corresponding angles of its polar triangle are supplementary angles: a0 = 180 ; b0 = 180 ; c0 = 180 ; (3.179a) 0 0 0 = 180 ; a = 180 ; b = 180 ; c: (3.179b) C
C’ a’ b’
γ
90
°
b’
γ’
A’
a
β B
c
a’
b
γ’
C a
°
α α’ A
90
b
c’
Figure 3.91
β’
A
c
B c’
B’
α’ α A
γ
C
β B
a)
β’
b) Figure 3.92
3.4.1.6 Euler Triangles and Non-Euler Triangles
The vertices A B C of a spherical triangle divide every great circle into two, usually di erent parts. Consequently there are several di erent triangles with the same vertices, e.g., also the triangle with sides a0 b c and the shadowed surface in Fig. 3.92a. According to the de nition of Euler we should always choose the arc which is smaller than 180 as a side of the spherical triangle. This corresponds
3.4 Spherical Trigonometry 163
to the de nition of the sides as spherical distances between the vertices. Considering this, we call the spherical triangles all of whose sides and angles are less than 180 Euler triangles, otherwise we call them non-Euler triangles. In Fig. 3.92b there is an Euler triangle and a non-Euler triangle.
3.4.1.7 Trihedral Angle
This is a three-sided solid formed by three edge sa sb sc starting at a vertex O (Fig. 3.93a). As sides of the trihedral angle we de ne the angles a b, and c, every of them enclosed by two edges. The regions between two edges are called the faces of the trihedral angle. The angles of the trihedral angle are , and , the angles between the faces. If the vertex of a trihedral angle is at the center O of a sphere, it cuts out a spherical triangle of the surface (Fig. 3.93b). The sides and the angles of the spherical triangle and the corresponding trihedral angle are coincident, so every theorem derived for a trihedral angle is valid for the corresponding spherical triangle, and conversely. sa A 2 a
sa a b
sb
a)
g
c b a
sc
0
B2 b
sb
b)
A1 a B1 b g
g sc
c b a
0
C1
C2
Figure 3.93
3.4.2 Basic Properties of Spherical Triangles 3.4.2.1 General Statements
For an Euler triangle with sides a b, and c, whose opposite angles are , and , the following statements are valid: 1. Sum of the Sides The sum of the sides is between 0 and 360: 0 < a + b + c < 360: (3.180) 2. Sum of two Sides The sum of two sides is greater than the third one, e.g., a + b > c: (3.181) 3. Di erence of Two Sides The di erence of two sides is smaller than the third one, e.g., ja ; bj < c: (3.182) 4. Sum of the Angles The sum of the angles is between 180 and 540: 180 < + + < 540: (3.183) 5. Spherical Excess The di erence = + + ; 180 (3.184) is called the spherical excess. 6. Sum of Two Angles The sum of two angles is less than the third one increased by 180, e.g., + < + 180: (3.185) 7. Opposite Sides and Angles Opposite to a greater side there is a greater angle, and conversely. 8. Area The area AT of a spherical triangle can be expressed by the spherical excess and by the radius of the sphere R with the formula = R 2 = R 2 arc : (3.186a) AT = R 2 180 %
164 3. Geometry Here % is the conversion factor (3.175c). From the theorem of Girard, with AS as the surface area of the sphere, we have AS : (3.186b) AT = 720 If we know the sides and not the excess, then can be calculated by the formula of L'Huilier (3.201).
3.4.2.2 Fundamental Formulas and Applications
The notation for the quantities of this paragraph corresponds to those of Fig. 3.89.
1. Sine Law
sin a = sin (3.187a) sin b sin sin c sin sin b sin sin c = sin (3.187b) sin a = sin : (3.187c) The equations from (3.187a) to (3.187c) can also be written as proportions, i.e., in a spherical triangle the sines of the sides are related as the sines of the opposite angles: sin a sin b sin c (3.187d) sin = sin = sin : The sine law of spherical trigonometry corresponds to the sine law of plane trigonometry.
2. Cosine Law, or Cosine Law for Sides
cos a = cos b cos c + sin b sin c cos (3.188a) cos b = cos c cos a + sin c sin a cos (3.188b) cos c = cos a cos b + sin a sin b cos : (3.188c) The cosine law for sides in spherical trigonometry corresponds to the cosine law of plane trigonometry. From the notation we can see that the cosine law contains the three sides of the spherical triangle.
3. Sine{Cosine Law
sin a cos = cos b sin c ; sin b cos c cos (3.189a) sin a cos = cos c sin b ; sin c cos b cos : (3.189b) We can get four more equations by cyclic change of the quantities (Fig. 3.34). The sine{cosine law corresponds to the projection rule of plane trigonometry. Because it contains ve quantities of the spherical triangle, we do not use it directly for solving problems of spherical triangles, but we usually use it for the derivation of further equations.
4. Cosine Law for Angles of a Spherical Triangle
cos = ; cos cos + sin sin cos a (3.190a) cos = ; cos cos + sin sin cos b (3.190b) cos = ; cos cos + sin sin cos c: (3.190c) This cosine rule contains the three angles of the spherical triangle and one of the sides. With this law we can easily express an angle by the opposite side with the angles on it, or a side by the angles consequently every side can be expressed by the angles. Contrary to this, for plane triangles we get the third angle from the sum of 180. Remark: It is not possible to determine any side of a plane triangle from the angles, because there are in nitely many similar triangles.
5. Polar Sine{Cosine Law
sin cos b = cos sin + sin cos cos a (3.191a) sin cos c = cos sin + sin cos cos a: (3.191b) Four more equations can be get by cyclic change of the quantities (Fig. 3.34). Just as for the cosine law for angles, also the polar sine{cosine law is not usually used for direct calculations for spherical triangles, but to derive further formulas.
3.4 Spherical Trigonometry 165
6. Half-Angle Formulas
To determine an angle of a spherical triangle from the sides we can use the cosine law for sides. The half-angle formulas allow us to calculate the angles by their tangents, similarly to the half-angle formulas of plane trigonometry:
v u u
tan = t sin(s ; b) sin(s ; c) (3.192a) 2 sin s sin(s ; a)
v u u
tan = t sin(s ; c) sin(s ; a) (3.192b) 2 sin s sin(s ; b)
v u u s = a + 2b + c : (3.192d) tan = t sin(s ; a) sin(s ; b) (3.192c) 2 sin s sin(s ; c) If from the three sides of a spherical triangle we want to determine all the three angles, the following calculations are useful: tan 2 = sin(sk; a) (3.193a) tan 2 = sin(sk; b) (3.193b) k tan = with 2 sin(s ; c)
(3.193c)
s
k = sin(s ; a) sin(sins ;s b) sin(s ; c) (3.193d)
s = a + 2b + c :
(3.193e)
7. Half-Side Formulas
With the half-side formulas we can tell one side or all of the sides of a spherical triangle from its three angles:
v u u
; ) cos( ; ) cot a2 = t cos( ; cos cos( ; ) (3.194a)
or
v u u ; ) cos( ; ) c cot 2 = t cos( ; cos cos( ; ) (3.194c) 0 cot a2 = cos(k ; )
(3.195a)
v u u
; ) cos( ; ) cot 2b = t cos( ; cos cos( ; ) (3.194b) = + 2 +
(3.194d)
0 cot 2b = cos(k ; )
(3.195b)
k0 cot c = with 2 cos( ; )
s ; ) cos( ; ) (3.195d) k0 = cos( ; ) cos( ; cos
(3.195c)
= + 2 + :
(3.195e)
Because for the sum of the angles of a spherical triangle according to (3.183): 180 < 2 < 540 or 90 < < 270 (3.196) holds, cos < 0 must always be valid. Because of the requirements for Euler triangles all the roots are real.
166 3. Geometry N
N g
a
λ2 γ =λ 2-λ 1
°-2j b = 90
b= 90°−ϕ 1
°- j 1
P2
90
c=e
ϕ1
=
λ1
P1
b d1 P1
ϕ2 P2 a c = e d2
Figure 3.94
Figure 3.95
8. Applications of the Fundamental Formulas of Spherical Geometry
With the help of the given fundamental formulas we can determine, for instance distances, the azimuth, and course angles on the Earth. A: Determine the shortest distance between Dresden (1 = 13460 = 13:77 '1 = 51160 = 51:27) and Peking (2 = 116290 = 116:48 '2 = 39450 = 39:75). Solution: The geographical coordinates (1 '1) (2 '2) and the north pole N (Fig. 3.94) result in two sides of the triangle P1P2N a = 90 ; '2 and b = 90 ; '1 lying on meridians, and also the angle between them = 2 ; 1. For c = e it follows from the cosine law (3.188a) cos c = (cos a cos b + sin a sin b cos c) cos e = cos(90 ; '1) cos(90 ; '2) + sin(90 ; '1) sin(90 ; '2 ) cos(2 ; 1) = sin '1 sin '2 + cos '1 cos '2 cos(2 ; 1) (3.197) _ i.e., cos e = 0:49883 ; 0:10583 = 0:39300 e = 66:86 . The great circle segment P1P2 has length 7434:618 km 7435 km using (3.175a). For comparison: Berlin - Peking 7371 km B: Calculate the course angles 1 and 2 at departure and at arrival, and also the distance in sea miles of a voyage from Bombay (1 = 72480 '1 = 19000) to Dar es Saalam (2 = 39280 '2 = ;6 490) along a great circle. Solution: The calculation of the two sides a = 90 ; '1 = 71000 b = 90 ; '2 = 96490 and the enclosed angle = 1 ; 2 = 33 200 in the spherical triangle P1P2N with the help of the geographical coordinates (1 '1) (2 '2) (Fig. 3.95) and the cosine law (3.188c) cos c = cos e = cos a cos b + _ _ sin a sin b cos yields P1P2 = e = 41:777, and because 10 1 sm we have P1P2 2507 sm. With the cosine law for sides we get b cos c = 51:248 and = arccos cos b ; cos a cos c = 125:018: = arccos cos asin; bcos sin c sin a sin c Therefore, we get 1 = 360 ; = 234:982 and 2 = 180 + = 231:248: Remark: It makes sense to use the sine law to determine sides and angles only if it is already obvious from the problem that the angles are acute or obtuse.
3.4.2.3 Further Formulas 1. Delambre Equations
Analogously to the Mollweide formulas of plane trigonometry the corresponding formulas of Delambre are valid for spherical triangles:
3.4 Spherical Trigonometry 167
cos ; sin a + b sin ; sin a ; b 2 = 2c 2 = 2 (3.198a) (3.198b) sin sin cos sin c 2 2 2 2 cos +2 cos a +2 b sin +2 cos a ;2 b = (3.198c) = : (3.198d) sin 2 cos 2c cos 2 cos 2c Because for every equation two more equations exist by cyclic changes, altogether there are 12 Delambre equations.
2. Neper Equations and Tangent Law
sin a ;2 b cos a ;2 b ; + tan 2 = a + b cot 2 (3.199a) tan 2 = a + b cot 2 (3.199b) sin 2 cos 2 sin ;2 cos ;2 a ; b c a + b tan 2 = + tan 2 (3.199c) tan 2 = + tan 2c : (3.199d) sin 2 cos 2 These equations are also called Neper analogies. From these we can derive formulas analogous to the tangent law of plane trigonometry: tan a ;2 b tan ;2 tan b ;2 c tan ;2 = (3.200a) = (3.200b) tan a +2 b tan +2 tan b +2 c tan +2 tan c ; a tan ; 2 = 2 : (3.200c) tan c +2 a tan +2
3. L'Huilier Equations
We can calculate the area of a spherical triangle with the help of the excess . It can be calculated from the known angles according to (3.184), or if the three sides a b c are known, with the formulas about the angles from (3.193a) to (3.193e). The L'Huilier equation makes possible the direct calculation of from the sides: s tan 4 = tan 2s tan s ;2 a tan s ;2 b tan s ;2 c : (3.201) This equation corresponds to the Heron formula of plane trigonometry.
3.4.3 Calculation of Spherical Triangles
3.4.3.1 Basic Problems, Accuracy Observations
The di erent cases occurring most often in calculations of spherical triangles are arranged into so-called basic problems . For every basic problem for acute-angled spherical triangles there are several ways to solve it, and it depends on whether we base our calculations only on the formulas from (3.187a) to (3.191b) or also on the formulas from (3.192a) to (3.201), and also whether we are looking for only one
168 3. Geometry or more quantities of the triangle. Formulas containing the tangent function yield numerically more accurate results, especially in comparison to the determination of de ning quantities with the sine function if they are close to 90, and with the cosine function if their value is close to 0 or 180. For Euler triangles the quantities calculated with the sine law are bivalued, since the sine function is positive in both of the rst quadrants, while the results obtained from other functions are unique.
3.4.3.2 Right-Angled Spherical Triangles 1. Special Formulas
In a right-angled spherical triangle one of the angles is 90. The sides and angles are denoted analogously to the plane right-angled triangle. If as in Fig. 3.96 is a right angle, the side c is called the hypotenuse, a and b the legs and and are the leg angles. From the equations from (3.187a) to (3.201) it follows for = 90: sin a = sin sin c (3.202a) sin b = sin sin c (3.202b) cos c = cos a cos b
(3.202c)
cos c = cot cot
(3.202d)
tan a = cos tan c
(3.202e)
tan b = cos tan c
(3.202f)
tan b = sin a tan
(3.202g)
tan a = sin b tan
(3.202h)
cos = sin cos a
(3.202i)
cos = sin cos b:
(3.202j)
c α
β a 90° b
b α
Figure 3.96
90° a c
90°- a
90°- b β
α
c
β
Figure 3.97
If in certain problems other sides or angles are given, for instance instead of the quantities b , we get the necessary equations by cyclic change of these quantities. For calculations in a right-angled spherical triangle we usually start with three given quantities, the angle = 90 and two other quantities. There are six basic problems, which are represented in Table 3.8. Table 3.8 De ning quantities of a spherical right-angled triangle
Basic Given de ning problem quantities 1. 2. 3. 4. 5. 6.
Number of the formula to determine other quantities
Hypotenuse and a leg c a (3.202a), (3.202e), b (3.202c) Two legs a b (3.202h), (3.202g), c (3.202c) Hypotenuse and an angle c a (3.202a), b (3.202f), (3.202d) Leg and the angles on it a c (3.202e), b (3.202j), (3.202i) Leg and the angle opposite to it a b (3.202h), c (3.202a), (3.202i) Two angles a (3.202i), b (3.202j), c (3.202d)
3.4 Spherical Trigonometry 169
2. Neper Rule
The Neper rule summarizes the equations (3.202a) to (3.202j). If the ve determining quantities of a right-angled spherical triangle, not considering the right angle, are arranged along a circle in the same order as they are in the triangle, and if the legs are replaced by their complementary angles 90 ; a, 90 ; b, (Fig. 3.97), then the following is valid: 1. The cosine of every de ning quantity is equal to the product of the cotangent values of its neighboring quantities. 2. The cosine of every de ning quantity is equal to the product of the sine values of the non-neighboring quantities. tan b (see (3.202f)) . A: cos = cot(90 ; b) cot c = tan c B: cos(90 ; a) = sin c sin = sin a (see (3.202a)). C: Map the sphere onto a cylinder which contacts the sphere along a meridian, the so-called central meridian. This meridian and the equator form the axis of the Gauss{Kruger system (Fig. 3.98a,b). c=
x geogr. gitter N' north north k' g y P' Q' g'
k
90
N
a g b=h P
°-j
central meridian
Q
g
m
Equato r
a)
α A
m' b)
γ C a
b c
β B
y
Figure 3.98
Figure 3.99
Solution: A point P of the surface of the sphere will have the corresponding point P 0 on the plane.
The great circle g passing through the point P perpendicular to the central meridian is mapped into a line g0 perpendicular to the x-axis, and the small circle k passing through P parallel to the given meridian becomes a line k0 parallel to the x-axis (grid meridian). The image of the meridian m through P is not line but a curve m0 (true meridian). The upward direction of the tangent of m0 at P 0 gives the geographical north, the upward direction of k0 gives the grid north direction. The angle between the two north directions is called the meridian convergence. In a right-angled spherical triangle QPN with c = 90 ; ' , and b = we get from = 90 ; . The tan Neper rule yields cos = tan b or cos(90 ; ) = sin = tan tan '. Because and tan c tan(90 ; ') are mostly small, we consider sin tan consequently = tan ' is valid. The length deviation of this cylinder sketch is pretty small for small distances , and we can substitute = y=R, where y is the ordinate of P . We get = (y=R) tan '. The conversion of from radian into degree measure yields a meridian convergence of = 0:018706 or = 10401900 at ' = 50 y = 100 km.
3.4.3.3 Spherical Triangles with Oblique Angles
For three given quantities, we distinguish between six basic problems, just as we did for right-angled spherical triangles. The notation for the angles is and a b c for the opposite sides (Fig. 3.99). Tables 3.9, 3.10, 3.11, and 3.12 summarize, which formulas should be used for which de ning quantities in the case of the six basic problems. Problems 3 4 5, and 6 can also be solved by decomposing the general triangle into two right-angled triangles. To do this for problems 3 and 4 (Fig. 3.100, Fig. 3.101) we use the spherical perpendicular from B to AC to the point D, and for problems 5 and 6 (Fig. 3.102) from C to AB to the point D.
170 3. Geometry u b-u α A
D
γ
C
γ
a
b D
v c
B
α A
v c
C a β-µ β µ B
α
b
ϕ
u
D
A
C γ ψ a c
v β B
Figure 3.100 Figure 3.101 Figure 3.102 In the headings of Tables 3.9, 3.10, 3.11, and 3.12 the given sides and angles are denoted by S and W respectively. This means for instance SWS: Two sides and the enclosed angle are given. Table 3.9 First and second basic problems for spherical oblique triangles
First basic problem Given: 3 sides a, b, c
Conditions: 0 < a + b + c < 360 a + b > c a + c > b b + c > a:
SSS
Second basic problem Given: 3 angles , ,
WWW
Conditions: 180 < + + < 540 + < 180 + + < 180 + + < 180 + : Solution 1: Required : Solution 1: Required a : cos a ; cos b cos c cos or cos = or cos a = cos sin+ cos sin b sin c sin s s sin( s ; b ) sin( s ; c ) a cos( ; ) cos( ; ) tan 2 = cot 2 = ; cos cos( sin s sin(s ; a) ; ) s = a + 2b + c : = + 2 + : Solution 2:sRequired , , : Solution s 2: Required a, b, c : ; ) cos( ; ) : sin( s ; a ) sin( s ; b ) sin( s ; c ) 0 k = cos( ; ) cos( k = ; cos sin s 0 0 tan 2 = sin(sk; a) tan 2 = sin(sk; b) cot a2 = cos(k ; ) cot 2b = cos(k ; ) k0 tan 2 = sin(sk; c) : cot c = : 2 cos( ; ) Checking: (s ; a) + (s ; b) + (s ; c) = s Checking: ( ; ) + ( ; ) + ( ; ) = tan 2 tan 2 tan 2 sin s = k: cot a cot b cot c (; cos ) = k0: 2 2 2
A Tetrahedron: A tetrahedron has base ABC and vertex S (Fig. 3.103). The faces ABS and BCS intersect each other at an angle 74180 BCS and CAS at an angle 63400, and CAS and ABS at 80000. How big are the angles between every two of the edges AS , BS , and CS ? Solution: From a spherical surface around the vertex S of the pyramid, the trihedral angle (Fig. 3.103) cuts out a spherical triangle with sides a b c. The angles between the faces are the angles of the spherical triangle, the angles between the edges, we are looking for, are the sides. The determination of the angles a b c corresponds to the second problem. Solution 2 in Table 3.9 yields: = 108590 ; = 28590 ; = 34410 ; = 45190 k0 = 1:246983 cot(a=2) = 1:425514 cot(b=2) = 1:516440 cot(c=2) = 1:773328.
3.4 Spherical Trigonometry 171
90° −ϕ 1
S
α c
b
P1
a β
δ1
ϕ 90°− 0
ε1
γ
N 90 °−ϕ ∆λ ∆λ 1 ∆λ 2 2 ε2 δ 2 P2 e ξ2
ξ1
C
e1
e2
P0
A B
Figure 3.103 Figure 3.104 Table 3.10 Third basic problem for spherical oblique triangles
Third basic problem Given: 2 sides and the enclosed angle, e.g., a, b,
SWS
Condition: none
sin a ; b 2 Solution 1: Required c, or c and : tan ; = a + b cot 2 2 cos c = cos a cos b + sin a sin b cos sin a sin : 2 sin = sinsin c (;90 < ;2 < 90) can be in quadrant I. or II. We apply the theorem: = +2 + ;2 = +2 ; ;2 : Larger angle is opposite Solution 4: Required , , c : to the larger side or a;b checking calculation: + = cos 2 cos 2 = Z tan in q. I. cos a ; cos b cos c >< 0 ! in q. II. 2 cos a +2 b sin 2 N Solution 2: Required , or and c : tan u = tan a cos sin a ;2 b cos 2 Z 0 ; tan 2 = a + b = N 0 sin u tan = tan sin 2 sin 2 sin(b ; u) tan( b ; u ) (;90 < +2 < 90) tan c = : cos Solution 3: Required and (or) : = + + ; = + ; ; 2 2 2 2 cos a ; b 0 c Z Z c tan + = a +2 b cot : cos 2 = + sin 2 = ; : 2 2 cos 2 sin 2 sin 2 Checking: Double calculation of c :
B Radio Bearing: In the case of a radio bearing two xed stations P1(1 '1) and P2(2 '2) receive the azimuths 1 and 2 via the radio waves emitted by a ship (Fig. 3.104). We have to determine the geographical coordinates of the position P0 of the ship. The problem, known by marines as the shore-
172 3. Geometry to-ship bearing, is an intersection problem on the sphere, and it will be solved similar to the intersection problem in the plane (see 3.2.2.3, p. 147). 1. Calculation in triangle P1P2N : In triangle P1P2N the sides P1 N = 90 ; '1 P2N = 90 ; '2 and the angle 1 0 solution. a sin > 1 0 solution. 1: sinsin 1: sinsin a sin b sin sin a sin 2: sin a = 1 1 solution = 90 : 2: sin = 1 1 solution b = 90: b sin < 1 : a sin < 1 : 3: sinsin 3: sinsin a 3:1: sin a > sin b : 3:1: sin > sin : 3:1:1: b < 90 1 solution 1: 3:1:1: < 90 1 solution b1 : 3:1:2: b > 90 1 solution 2: 3:1:2: > 90 1 solution b2 : 3:2: sin a < sin b : 3:2: sin < sin : a < 90 < 90 2 solutions 90 < 90 2 solutions 3:2:1: a > 90 > 90 3:2:1: aa < 1 2: > 90 > 90 b1 b2 : 90 > 90 90 > 90 3:2:2: aa < 3:2:2: aa < > 90 < 90 0 solution. > 90 < 90 0 solution. Further calculations with one angle Further calculations with one side or with two angles : or with two sides b : Method 1: Method 2: tan u = tan b cos cos +2 cos a ;2 b c a + b + tan 2 = tan 2 tan 2 = cot 2 tan v = tan a cos cos ;2 sin a +2 b c = u+v cot ' = cos b tan sin + sin a ; b cot = cos a tan = tan a ;2 b ;2 : = cot ;2 a +2 b : = '+: sin 2 sin 2 c Checking: Double calculation of and 2 2
1. Orthodrome
1. Notion The geodesic lines of the surface of the sphere { which are curves, connecting two points A and B by the shortest path { are called orthodromes or great circles (see 3.4.1.1, 3., p. 159). 2. Equation of the Orthodrome Moving on an orthodrome { except for meridians and the equator
{ needs a continuous change of course angle. These orthodromes with position-dependent course angles can be given uniquely by their point closest to the north pole PN(N 'N), where 'N > 0 holds. The orthodrome has the course angle N = 90 at the point closest to the north pole. The equation of the orthodrome through PN and the running point Q( '), whose relative position to PN is arbitrary, can be given by the Neper rule according to Fig. 3.105 as: tan 'N cos( ; N) = tan ': (3.203)
174 3. Geometry Point Closest to the North Pole: The coordinates of the point closest to the north pole PN(N 'N) of an orthodrome with a course angle A (A = 6 0) at the point A(A 'A) ('A =6 90) can be calculated by the Neper rule considering the relative position of PN and the sign of A according to Fig. 3.106
as:
'A : (3.204b) 'N = arccos(sin jAj cos 'A) (3.204a) and N = A + sign(A) arccos tan tan ' N
Remark: If a calculated geographical distance is not in the domain ;180 < 180, then for = 6 k 180 (k 2 IN) the reduced geographical distance red is:
! red = 2 arctan tan 2 : We call this the reduction of the angle in the domain.
N
lN - l
N
(3.205)
o
j 90 -
o ϕ A 90 -
R
. PN d N
A 0
αΑ
-ϕ
N
Figure 3.105
PN
o
-j
R
R
.
90
o
90
0
a Q
λN - λA
R
Figure 3.106
Intersection Points with the Equator: The intersection points PE1 (E1 0) and PE2 (E2 0) of the orthodrome with the equator can be calculated from (3.203) because tan 'N cos(E ; N) = 0 ( =
1 2) must hold: E = N 90 ( = 1 2): (3.206) Remark: In certain cases an angle reduction is needed according to (3.205). 3. Arclength If the orthodrome goes through the points A(A 'A) and B (B 'B ), the cosine law for sides yields for the spherical distance d or for the arclength between the two points: d = arccos sin 'A sin 'B + cos 'A cos 'B cos(B ; A)]: (3.207a) We can convert this central angle into a length considering the radius of the Earth R: R : d = arccos sin 'A sin 'B + cos 'A cos 'B cos(B ; A)] 180 (3.207b) 4. Course Angle If we use the sine and cosine laws for sides to calculate sin A and cos A, we get the nal result for the course angle A after division: 'A cos 'B sin(B ; A) : A = arctan cossin (3.208) 'B ; sin 'A cos d Remark: With the formulas (3.207a), (3.208), (3.204a) and (3.204b), the coordinates of the point closest to the north pole PN can be calculated for the orthodrome given by the two points A and B . 5. Intersection Pointwith a ParallelCircle For the intersection points X1 (X1 'X ) and X2(X2 'X ) of an orthodrome with the parallel circle ' = 'X we get from (3.203): 'X X = N arccos tan (3.209) tan 'N ( = 1 2):
3.4 Spherical Trigonometry 175
From the Neper rule we have for the intersection angles X1 and X2 , by which an orthodrome with a point closest to the north pole PN(N 'N) intersects the parallel circle ' = 'X : cos 'N jX j = arcsin cos ( = 1 2): (3.210) 'X The argument in the arc sine function must be extremal with respect to the variable 'X for the minimal course angle jminj. We get: sin 'X = 0 ) 'X = 0 , i.e., the absolute value of the course angle is minimal at the intersection points of the equator: jXmin j = 90 ; 'N: (3.211) Remark 1: Solutions of (3.209) exist only for j'X j 'N. Remark 2: In certain cases a reduction of the angle is needed according to (3.205). 6. Intersection Point with a Meridian We get for the intersection point Y (Y 'Y ) of an orthodrome with the meridian = Y according to (3.203): 'Y = arctan tan 'N cos(Y ; N)]: (3.212)
2. Small Circle
1. Notion The de nition of small a circle on the surface of the sphere must be discussed here in more detail than in 3.4.1.1, p. 158: The small circle is the locus of the points of the surface (of the sphere) at a spherical distance r (r < 90) from a point xed on the surface M (M 'M ) (Fig. 3.107). We denote the spherical center by M r is called the spherical radius of the small circle. The plane of the small circle is the base of a spherical cap with an altitude h (see 3.3.4, p. 157). The spherical center M is above the center of the small circle in the plane of the small circle. In the plane the circle has the planar radius of the small circle r0 (Fig. 3.108). Hence, parallel circles are special small circles with 'M = 90 . For r ! 90 the small circle tends to an orthodrome. N
spherical triangle
r Q
r
σ
r
PN M
0
r
d
A k
s
M
}
h
.
r0
B 0
k
R
small circle plane
Figure 3.107
Figure 3.108 2. Equations of Small Circles As de ning parameters we can use either M and r or the small circle point PN(N 'N) nearest the north pole and r. If the running point on the small circle is Q( '), we get the equation of the small circle with the cosine law for sides corresponding to Fig. 3.107: cos r = sin ' sin 'M + cos ' cos 'M cos( ; M ): (3.213a) From here, because of 'M = 'N ; r and M = N, we get: cos r = sin ' sin('N ; r) + cos ' cos('N ; r) cos( ; N): (3.213b) A: For 'M = 90 we get the parallel circles from (3.213a), since cos r = sin ' ) sin(90 ; r) = sin ' ) ' = const. B: For r ! 90 we get orthodromes from (3.213b). 3. Arclength The arclength s between two points A(A 'A) and B (B 'B ) on a small circle k can r0 cos d = cos2 r + sin2 r cos be calculated corresponding to Fig. 3.108 from the equalities s = 2360
176 3. Geometry and r0 = R sin r:
2 r R s = sin r arccos cos dsin;2cos (3.214) r 180 : For r ! 90 the small circle becomes an orthodrome, and from (3.214) and (3.207b) it follows that s = d. 4. Course Angle According to Fig. 3.109 the orthodrome passing through the points A(A 'A) and M (M 'M ) intersects perpendicularly the small circle with radius r. For the course angle Orth of the orthodrome after (3.208) 'A cos 'M sin(M ; A) Orth = arctan cos sin (3.215a) 'M ; sin 'A cos r is valid. So, we get for the required course angle A of the small circle at the point A: A = (jOrthj ; 90) sign(Orth ): (3.215b)
αOrth
N
N αΑ
A
r M
k
R Orth. R
Figure 3.109
T1
PN r M
T2
k
R
Figure 3.110 5. Intersection Points with a Parallel Circle For the geographical longitude of the intersection points X1 (X1 'X ) and X2(X2 'X ) of the small circle with the parallel circle ' = 'X we get from (3.213a): 'X sin 'M ( = 1 2): (3.216) X = M arccos cos rcos;'sin cos 'M X Remark: In some cases an angle reduction is needed according to (3.205). 6. Tangential Point The small circle is touched by two meridians, the tangential meridians, at the tangential points T1 (T1 'T ) and T2 (T2 'T ) (Fig. 3.110). Because for them the argument of the arc cosine in (3.216) must be extremal with respect to the variable 'X , we get: 'X sin 'M ( = 1 2): (3.217b) 'M = arccos cos rcos;'sin cos 'T = arcsin sin 'M cos r (3.217a) T M X Remark: In certain cases an angle reduction is needed according to (3.205). 7. Intersection Points with a Meridian The calculation of the geographical latitudes of the intersection points Y1(Y 'Y1 ) and Y2(Y 'Y2 ) of the small circle with meridian = Y can be done according to (3.213a) with the equations p 2 2 2 A +B ;C ( = 1 2) (3.218a) 'Y = arcsin ;AC BA2 + B2 where we use the notation: A = sin 'M B = cos 'M cos(Y ; M ) C = ; cos r: (3.218b) For A2 + B 2 > C 2, in general, there are two di erent solutions, from which one is missing if a pole is on the small circle.
3.4 Spherical Trigonometry 177
If A2 + B 2 = C 2 holds and there is no pole on the small circle, then the meridian touches the small circle at a tangential point with geographical latitude 'Y1 = 'Y2 = 'T .
3. Loxodrome
1. Notion A spherical curve, intersecting all meridians with the same course angle, is called a loxodrome or spherical helix. So, parallels ( = 90 ) and meridians ( = 0 ) are special loxodromes. 2. Equation of the Loxodrome Fig. 3.111 shows a loxodrome with course angle through the running point Q( ') and the in nitesimally close point P ( + d ' + d'). The right-angled spherical triangle QCP can be considered as a plane triangle because of its small size. Then: ' d tan d' tan = R cos (3.219a) R d' ) d = cos ' : Considering that the loxodrome must go through the point A(A 'A), therefore, we get the equation of the loxodrome by integration:
tan 45 + ' 180 2 ; A = tan ln ( 6= 90): (3.219b) tan 45 + '2A In particular if A is the intersection point PE(E 0) of the loxodrome with the equator, then: ; E = tan ln tan 45 + '2 180 ( 6= 90) : (3.219c) Remark: The calculation of E can be done with (3.224). N
N
α
ds Q(λ,ϕ) α
Rcos
P(λ+dλ,ϕ+dϕ) dϕ .R C ϕ dλ
ϕ
λ
P(ϕ=0,λ=0)
Figure 3.111
3. Arclength From Fig. 3.111 we nd the di erential relation cos = R d' ) ds = R d' : ds cos
Figure 3.112
(3.220a) Integration with respect to ' results in the arclength s of the arc segment with the endpoints A(A 'A) and B (B 'B ): R s = j'Bcos;'Aj 180 ( 6= 90) : (3.220b) If A is the starting point and B is the endpoint, then from the given values A and s we can determine step-by-step rst 'B from (3.220b), then B from (3.219b). Approximation Formulas: According to Fig. 3.111, with Q = A and P = B we get an approximation for the arclength l with the arithmetical mean of the geographical latitudes with (3.221a) and (3.221b):
178 3. Geometry cos 'A + 'B (B ; A) R 2 180 : l cos 'A + 'B R : l = sin 2 (B ; A) 180 sin =
(3.221a) (3.221b)
4. Course Angle The course angle of the loxodrome through the points A(A 'A) and B (B 'B ),
or through A(A 'A) and its equator intersection point PE(E 0 ) are given according to (3.219b) and (3.219c) by: = arctan (B ; A)'B 180 (3.222a) tan 45 + 2 ln tan 45 + '2A : = arctan (A ; E)'A 180 (3.222b) ln tan 45 + 2
5. Intersection Point with a Parallel Circle Suppose a loxodrome passes through the point A(A 'A) with a course angle . The intersection point X (X 'X ) of the loxodrome with a parallel circle ' = 'X is calculated from (3.219b): tan 45 + 'X 180 2 X = A + tan ln ( 6= 90): (3.223) tan 45 + '2A With (3.223) we calculate the intersection point with the equator PE(E 0): E = A ; tan ln tan 45 + '2A 180 ( 6= 90) : (3.224) Remark: In certain cases an angle reduction is needed according to (3.205). 6. Intersection Point with a Meridian Loxodromes { except parallel circles and meridians { wind in a spiral form around the pole (Fig. 3.112). The in nitely many intersection points Y (Y 'Y ) ( 2 Z) of the loxodrome passing through A(A 'A) with course angle with the meridian = Y can be calculated from (3.219b): ( "
) tan 45 + 'A ; 90 ( 2 Z): (3.225a) A + 360 'Y = 2 arctan exp Y ; tan 180 2 If A is the equator intersection point PE(E 0) of the loxodrome, then we have simply: "
; 90 ( 2 Z): E + 360 'Y = 2 arctan exp Y ; tan 180
(3.225b)
4. Intersection Points of Spherical Curves
1. Intersection Points of Two Orthodromes Suppose the considered orthodromes have points PN1 (N1 'N1 ) and PN2 (N2 'N2 ) closest to the north pole, where PN1 6= PN2 holds. Substituting the intersection point S (S 'S ) in both orthodrome equations we get an equation system tan 'N1 cos(S ; N1 ) = tan 'S (3.226a)
tan 'N2 cos(S ; N2 ) = tan 'S : (3.226b)
3.4 Spherical Trigonometry 179
By elimination of 'S and using the addition law for the cosine function we get: 'N1 cos N1 ; tan 'N2 cos N2 tan S = ; tan (3.227) tan 'N1 sin N1 ; tan 'N2 sin N2 : The equation (3.227) has two solutions S1 and S2 in the domain ;180 < 180 of the geographical longitude. The corresponding geographical latitudes can be got from (3.226a): 'S = arctan tan 'N1 cos(S ; N1 )] ( = 1 2): (3.228) The intersection points S1 and S2 are antipodal points, i.e., they are the mirror images of each other with respect to the centre of the sphere. 2. Intersection Points of Two Loxodromes Suppose the considered loxodromes have equator intersection points PE1 (E1 0) and PE2 (E2 0) and the course angles 1 und 2 (1 6= 2). Substituting the intersection point S (S 'S ) in both loxodrome equations we get the equation system: S ; E1 = tan 1 ln tan 45 + '2S 180 (1 6= 90) (3.229a) S ; E2 = tan 2 ln tan 45 + '2S 180 (2 6= 90): (3.229b) By elimination of S and expressing 'S we get an equation with in nitely many solutions: "
; 90 ( 2 Z): 1 ; E2 + 360 'S = 2 arctan exp Etan (3.230) 2 ; tan 1 180 The corresponding geographical longitudes S can be found by substituting 'S in (3.229a): S = E1 + tan 1 ln tan 45 + '2S 180 (1 6= 90) ( 2 Z): (3.231) Remark: In certain cases an angle reduction is needed according to (3.205).
180 3. Geometry
3.5 Vector Algebra and Analytical Geometry 3.5.1 Vector Algebra
3.5.1.1 Denition of Vectors 1. Scalars and Vectors
Quantities whose values are real numbers are called scalars. Examples are mass, temperature, energy, and work (for scalar invariant see 3.5.1.5, p. 184, 3.5.3.1, p. 212 and 4.3.5.2, p. 270). Quantities which can be completely described by a magnitude and by a direction in space are called vectors. Examples are power, velocity, acceleration, angular velocity, angular acceleration, and electrical and magnetic force. We represent vectors by directed line segments in space. In this book we denote the vectors of three-dimensional Euclidean space by ~a, and in matrix theory by a (see also 4.1.3, p. 253).
2. Polar and Axial Vectors
Polar vectors represent quantities with magnitude and direction in space, such as speed and acceleration axial vectors represent quantities with magnitude, direction in space, and direction of rotation, such as angular velocity and angular acceleration. In notation we distinguish them by a polar or by an axial arrow (Fig. 3.113). In mathematical discussion we treat them in the same way.
3. Magnitude or Absolute Value and Direction in Space
For the quantitative description of vectors ~a or a, as line segments between the initial and endpoint A and B resp., we have the magnitude, i.e., the absolute value j~aj, the length of the line segment, and the direction in space, which is given by a set of angles.
4. Equality of Vectors
Two vectors ~a and ~b are equal if their magnitudes are the same, and they have the same direction, i.e., if they are parallel and oriented identically. Opposite and equal vectors are of the same magnitude, but oppositely directed: ;! ;! ;! ;! AB = ~a BA = ;~a but jAB j = jBAj: (3.232) Axial vectors have opposite and equal directions of rotation in this case. B
B
P(x,y,z)
k
a a) A
z
b b) A
Figure 3.113
B
r
i 0 x
j y
A a)
a
C b c f
Figure 3.114
C
D d e E F
D b b) A a
c d B
Figure 3.115
5. Free Vectors, Bound or Fixed Vectors, Sliding Vectors
A free vector is considered to be the same, i.e. its properties do not change, if it is translated parallel to itself, so its initial point can be an arbitrary point of space. If the properties of a vector belong to a certain initial point, we call it a bound or xed vector. A sliding vector can be translated only along the line it is already in. In mathematics we deal with free vectors.
6. Special Vectors 0
a) Unit Vector ~a = ~e is a vector with length or absolute value equal to 1. With it we can express the vector ~a as a product of the magnitude and of a unit vector having the same direction as ~a: ~a = ~e j~aj: (3.233) We often use the unit vectors ~i, ~j, ~k or ~ei, ~ej , ~ek (Fig. 3.114) to denote the three coordinate axes in
the direction of increasing coordinate values.
3.5 Vector Algebra and Analytical Geometry 181
In Fig. 3.114 the directions given by the three unit vectors form an orthogonal triple. These unit vectors de ne an orthogonal coordinate system because for their scalar products e~ie~j = e~ie~k = e~j e~k = 0 (3.234) is valid. Because also e~ie~i = e~je~j = e~k e~k = 1 (3.235) holds, we call it an orthonormal coordinate system. (For more about scalar product see (3.247).) b) Null Vector or zero vector is the vector whose magnitude is equal to 0, i.e., its initial and endpoint coincide, and it has no direction. ;! c) Radius Vector ~r or position vector of a point P is the vector 0P with the initial point at the origin and endpoint at P (Fig. 3.114). In this case we call the origin also a pole or polar point. The point P is de ned uniquely by its radius vector. d) Collinear Vectors are parallel to the same line. e) Coplanar Vectors are parallel to the same plane. They satisfy the equality (3.259).
3.5.1.2 Calculation Rules for Vectors
1. Sum of Vectors
a) The Sum of Two Vectors AB = ~a and AD = ~b can be represented also as the diagonal of the ;! parallelogram ABCD, as the vector AC = ~c in Fig. 3.115b. The most important properties of the ;!
;!
sum of two vectors are the commutative law and the triangle inequality: ~a + b~ = ~b + ~a j ~a + ~b j j ~a j + j ~b j: (3.236a) ;! b) The Sum of Several Vectors ~a ~b ~c : : : ~e is the vector ~f = AF , which closes the broken line composed of the vectors from ~a to ~e as in Fig. 3.115a. Important properties of the sum of several vectors are the commutative law and the associative law of addition. For three vectors we have: ~a + ~b + ~c = ~c + ~b + ~a (~a + ~b) + ~c = ~a + (~b + ~c): (3.236b) c) The Di erence of Two Vectors ~a ; ~b can be considered as the sum of the vectors ~a und ;~b, i.e., ~a ; ~b = ~a + (;~b) = ~d (3.236c) which is the other diagonal of the parallelogram (Fig. 3.115b). The most important properties of the di erence of two vectors are: ~a ; ~a = ~0 (null vector) j ~a ; ~b j j j ~a j ; j ~b j j: (3.236d) a
w
a)
u
v
au
gw bv
bv
a u
c)
b)
B
v A d)
au
Figure 3.116
2. Multiplication of a Vector by a Scalar, Linear Combination
The products ~a and ~a are equal to each other and they are parallel (collinear) to ~a. The length (absolute value) of the product vector is equal to jjj~aj. For > 0 the product vector has the same direction as ~a for < 0 it has the opposite one. The most important properties of the product of vectors by scalars are: (3.237a) ~a = ~a ~a = ~a ( + ) ~a = ~a + ~a (~a + ~b) = ~a + ~b:
182 3. Geometry The linear combination of the vectors ~a ~b ~c : : : ~d with the scalars : : : is the vector ~k = ~a + ~b + + ~d: (3.237b)
3. Decomposition of Vectors
In three-dimensional space every vector ~a can be decomposed uniquely into a sum of three vectors, ~ (Fig. 3.116a,b): which are parallel to the three given non-coplanar vectors ~u ~v w ~a = ~u + ~v + w ~: (3.238a) ~ are called the components of the decomposition, the scalar factors , The summands ~u, ~v and w and are the coecients. When all the vectors are parallel to a plane we can write ~a = ~u + ~v (3.238b) with two non-collinear vectors ~u and ~v being parallel to the same plane (Fig. 3.116c,d).
3.5.1.3 Coordinates of a Vector z a axi
azk
;!
1. Cartesian Coordinates According to (3.238a) every vector AB = ~a can be decomposed uniquely into a sum of vectors parallel to the basis vectors of the coordinate system ~i ~j ~k or ~ei ~ej ~ek : ~a = ax~i + ay~j + az~k = ax~ei + ay~ej + az~ek (3.239a)
where the scalars ax, ay and az are the Cartesian coordinates of the vector ~a in the system with the unit vectors ~ei, ~ej and ~ek . We also write i 0 ~a = fax ay az g or ~a(ax ay az ): (3.239b) j y The three directions de ned by the unit vectors form an orthogonal direcx tion triple. The components of a vector are the projections of this vector on the coordinate axes (Fig. 3.117). Figure 3.117 The coordinates of a linear combination of several vectors are the same linear combination of the coordinates of these vectors, so the vector equation (3.237b) corresponds to the following coordinate equations: kx = ax + bx + + dx ky = ay + by + + dy (3.240) kz = az + bz + + dz : For the coordinates of the sum and of the di erence of two vectors ~c = ~a ~b (3.241a) the equalities cx = ax bx cy = ay by cz = az az (3.241b) are valid. The radius vector ~r of the point P (x y z) has the Cartesian coordinates of this point: rx = x ry = y rz = z ~r = x~i + y ~j + z ~k: (3.242) 2. Ane Coordinates are a generalization of Cartesian coordinates with respect to a system of linearly independent but not necessarily orthogonal vectors, i.e., to three non-coplanar basis vectors ~e1 ~e2 ~e3. The coecients are a1 a2 a3, where the upper indices are not exponents. Similarly to (3.239a,b) we have for ~a n o or ~a = a1 a2 a3 or ~a a1 a2 a3 : (3.243b) ~a = a1 ~e1 + a2 ~e2 + a3 ~e3 (3.243a) This notation is especially suitable as the scalars a1 a2 a3 are the contravariant coordinates of a vector (see 3.5.1.8, p. 187). For ~e1 = ~i, ~e2 = ~j, ~e3 = ~k the formulas (3.243a,b) become (3.239a,c). For the linear combination of vectors (3.237b) just as for the sum and di erence of two vectors (3.241a,b) in k
ay j
3.5 Vector Algebra and Analytical Geometry 183
analogy to (3.240) the same coordinate equations are valid: k1 = a1 + b1 + + d1 k2 = a2 + b2 + + d2 k3 = a3 + b3 + + d3 c1 = a1 b1 c2 = a2 b2 c3 = a3 b3 :
(3.244) (3.245)
3.5.1.4 Directional Coecient
The directional coecient of a vector ~a along a vector ~b is the scalar product ab = ~a ~b0 = j~aj cos ' (3.246) b ~ 0 ~ ~ where b = ~ is the unit vector in the direction of b and ' is the angle between ~a and b. jbj The directional coecient represents the projection of ~a on ~b. In the Cartesian coordinate system the directional coecients of the vector ~a along the x y z axes are the coordinates ax ay az . This statement is usually not true in a non-orthonormal coordinate system.
3.5.1.5 Scalar Product and Vector Product 1. Scalar product
The scalar product or dot product of two vectors ~a and b ~b is de ned by the equation j ~ ~ ~ ~ ~a b = ~a b = (~a b) = j~a j jb j cos ' (3.247) a where ' is the angle between ~a and ~b considering them with a common initial point (Fig. 3.118). The value of a scalar product is a scalar. Figure 3.118
c
b j a
Figure 3.119
2. Vector Product
or cross product of the two vectors ~a and ~b is a vector ~c such that it is perpendicular to the vectors ~a and ~b, and in the order ~a, ~b, and ~c the vectors form a right-hand system (Fig. 3.119). If the vectors have the same initial point, and we look at the plane of ~a and ~b from the endpoint of ~c, then the shortest rotation of ~a in the direction of ~b is counterclockwise. The vectors ~a, ~b, and ~c have the same arrangement as the thumb, the fore nger, and the middle nger of the right hand. Therefore this is called the right-hand rule. The vector product ~a ~b = ~a ~b ] = ~c (3.248a) has magnitude j~c j = j~a j j~b j sin ' (3.248b) where ' is the angle between ~a and ~b. Numerically the length of ~c is equal to the area of the parallelogram de ned by the vectors ~a and ~b.
3. Properties of the Products of Vectors
a) The Scalar Product is commutative: ~a ~b = ~b ~a: (3.249) b) The Vector Product is anticommutative (changes its sign if we interchange the factors): ~a ~b = ;(~b ~a): (3.250)
184 3. Geometry c) Multiplication by a Scalar Scalars can be factored out: (~a ~b ) = (~a) ~b (3.251a) (~a ~b ) = (~a) ~b: (3.251b) d) Associativity The scalar and vector products are not associative: ~a (~b ~c ) = 6 (~a ~b ) ~c (3.252a) ~a (~b ~c) = 6 (~a ~b ) ~c: (3.252b) e) Distributivity The scalar and vector products are distributive over addition: ~a (~b + ~c ) = ~a ~b + ~a ~c (3.253a) ~a (~b + ~c ) = ~a ~b + ~a ~c and (~b + ~c ) ~a = ~b ~a + ~c ~a: (3.253b) f) Orthogonality of Two Vectors Two vectors are perpendicular to each other (~a ? ~b) if the equality ~a ~b = 0 holds, and neither ~a nor ~b are null vectors. (3.254) ~ g) Collinearity of Two Vectors Two vectors are collinear (~a k b) if the equality ~a ~b = ~0 holds, and neither ~a nor ~b are null vectors. (3.255) h) Multiplication of the same vectors: ~a ~a = ~a2 = a2 ~a ~a = ~0: (3.256) i) Linear Combinations of Vectors can be multiplied in the same way as scalar polynomials (be-
cause of the distributive property), only we have to be careful with the vector product. If we interchange the factors, we have to change the sign. A: ( 3~a +5~b ; 2~c) (~a ; 2~b ; 4~c) = 3~a2 + 5~b~a ; 2~c~a ; 6~a~b ; 10~b 2 + 4~c~b ; 12~a~c ; 20~b~c + 8~c 2 = 3~a2 ; 10~b 2 + 8~c 2 ; ~a~b ; 14~a~c ; 16~b~c: B: ( 3~a +5~b ; 2~c) (~a ; 2~b ; 4~c) = 3~a ~a + 5~b ~a ; 2~c ~a ; 6~a ~b ; 10~b b~ + 4~c ~b ; 12~a ~c ; 20~b ~c + 8~c ~c = 0 ; 5~a ~b + 2~a ~c ; 6~a ~b + 0 ; 4~b ~c ; 12~a ~c ; 20~b ~c + 0 = ;11~a ~b ; 10~a ~c ; 24~b ~c = 11~b ~a + 10~c ~a + 24~c ~b: j) Scalar Invariant is a scalar quantity if it does not change its value under a translation or a rotation of the coordinate system. The scalar product of two vectors is a scalar invariant. A: The coordinates of a vector ~a = fa1 a2 a3g are not scalar invariants, because in di erent coordinate systems they can have di erent values. B: The length of a vector ~a is a scalar invariant, because it has the same value in di erent coordinate systems. C: Since the scalar product of two vectors is a scalar invariant, the scalar product of a vector by itself is also a scalar invariant, i.e., ~a ~a = j~aj2 cos ' = j~aj2 because ' = 0.
3.5.1.6 Combination of Vector Products 1. Double Vector Product
The double vector product ~a (~b ~c) results in a vector coplanar to ~b and ~c: ~a (~b ~c) = ~b (~a ~c) ; ~c (~a ~b ):
2. Mixed Product
(3.257)
The mixed product (~a ~b) ~c, which is also called the triple product, results in a scalar whose absolute value is numerically equal to the volume of the parallelepipedon de ned by the three vectors the result is positive if ~a, ~b, and ~c form a right-hand system, negative otherwise. Parentheses and crosses can be
3.5 Vector Algebra and Analytical Geometry 185
omitted: (~a ~b)~c = ~a ~b ~c = ~b ~c ~a = ~c ~a~b = ;~a~c ~b = ;~b ~a~c = ;~c ~b ~a: (3.258) The interchange of any two terms results in a change of sign the cyclic permutation of all three terms does not a ect the result. For coplanar vectors, i.e., if ~a is parallel to the plane de ned by ~b and ~c, we have: ~a (~b ~c) = 0: (3.259)
3. Formulas for Multiple Products
a) Lagrange Identity: (~a ~b)(~c ~d) = (~a ~c) (~b ~d) ; (~b ~c) (~a ~d) ~a~e ~a~f ~a~g b) ~a ~b ~c ~e ~f ~g = ~b~e ~b~f ~b~g : ~c ~e ~c~f ~c ~g
(3.260) (3.261)
4. Formulas for Products in Cartesian Coordinates If the vectors ~a, ~b, ~c are given by Cartesian coordinates as ~a = fax ay az g ~b = fbx by bz g ~c = fcx cy cz g then we can calculate the products by the following formulas: 1. Scalar Product: ~a b~ = axbx + ay by + az bz :
2. Vector Product: ~a ~b = (ay bz ; azby ) ~i + (az bx ; axbz ) ~j + (axby ; ay bx) ~k ~i ~j ~k
= ax ay az : bx by bz a a a x y z 3. Mixed Product: ~a ~b ~c = bx by bz : cx cy cz
(3.262) (3.263) (3.264) (3.265)
5. Formulas for Products in A ne Coordinates
1. Metric Coecients and Reciprocal System of Vectors If we have the ane coordinates of two vectors ~a and ~b in the system of ~e1 , ~e2, ~e3 , i.e., ~a = a1 ~e1 + a2 ~e2 + a3 ~e3 ~b = b1 ~e1 + b2 ~e2 + b3 ~e3 (3.266) are given, and we want to calculate the scalar product ~a ~b = a1 b1 ~e1 ~e1 + a2 b2 ~e2 ~e2 + a 3 b3 ~e3 ~e3 + a1 b2 + a2 b1 ~e1 ~e2 + a2 b3 + a3 b2 ~e2 ~e3 + a3 b1 + a1 b3 ~e3 ~e1 (3.267) or the vector product ~a ~b = a2 b3 ; a3 b2 ~e2 ~e3 + a3 b1 ; a1 b3 ~e3 ~e1 + a1 b2 ; a2 b1 ~e1 ~e2 (3.268a) with the equalities ~e1 ~e1 = ~e2 ~e2 = ~e3 ~e3 = ~0 (3.268b) then we have to know the products in pairs of coordinate vectors. For the scalar product these are the six metric coecients (numbers) g11 = ~e1 ~e1 g22 = ~e2 ~e2 g33 = ~e3 ~e3 g12 = ~e1 ~e2 = ~e2 ~e1 g23 = ~e2 ~e3 = ~e3 ~e2 g31 = ~e3 ~e1 = ~e1 ~e3 (3.269)
186 3. Geometry and for the vector product the three vectors ~e 1 = (~e2 ~e3 ) ~e 2 = (~e3 ~e1 ) ~e 3 = (~e1 ~e2) (3.270a) which are the three reciprocal vectors with respect to ~e1, ~e2 , ~e3, where the coecient 1 = (3.270b) ~e1 ~e2 ~e3 is the reciprocal value of the mixed product of the coordinate vectors. This notation serves only as a shorter way of writing in the following. With the help of the multiplication Tables 3.13 and 3.14 for the basis vectors calculations with the coecients will be easy to perform.
~e1 ~e2 ~e1 g11 g12 ~e2 g21 g22 ~e3 g31 g32 (gki = gik )
~e3 g13 g23 g33
Table 3.14 Vector product of basis vectors Multipliers ~e1 ~e2 ~e3 ~ e 3 ;~e 2 ~e1 0 3 1 ~ ~ e e ~e2 ; 0 2 1 ~e3 ~e ;~e 0
Multiplicands
Table 3.13 Scalar product of basis vectors
2. Application to Cartesian Coordinates The Cartesian coordinates are a special case of ane coordinates. From Tables 3.15 and 3.16 we have for the basis vectors ~e1 = ~i ~e2 = ~j ~e3 = ~k (3.271a) with the metric coecients g11 = g22 = g33 = 1 g12 = g23 = g31 = 0 ) = ~~1~ = 1 (3.271b) ijk
and the reciprocal basis vectors ~e 1 = ~i ~e 2 = ~j ~e 3 = ~k: (3.271c) So the basis vectors coincide with the reciprocal basis vectors of the coordinate system, or, in other words, in the Cartesian coordinate system the basis vector system is its own reciprocal system.
3. Scalar Product of Vectors Given by Coordinates 3 X 3 X
m=1 n=1
gmnam bn = g a b :
Table 3.15 Scalar product of reciprocal basis vectors
~i ~j ~k
~i
~j
~k
1 0 0
0 1 0
0 0 1
(3.272) Table 3.16 Vector product of reciprocal basis vectors Multipliers ~i ~j ~k ~i 0 ~k ;~j ~j ; ~k 0 ~i ~k ~j ;~i 0
Multiplicands
~a~b =
3.5 Vector Algebra and Analytical Geometry 187
For Cartesian coordinates (3.272) coincides with (3.263). After the second equality in (3.272) we applied a shorter notation for the sum which is often used in tensor calculations (see 4.3.1, 2., p. 262): instead of the complete sum we write only a characteristic term so that the sum should be calculated for repeated indices, i.e., for the indices appearing once down and once up. Sometimes the summation indices are denoted by Greek letters here they have the values from 1 until 3. Consequently we have g a b = g11a1 b1 + g12a1 b2 + g13a1 b3 + g21 a2b1 + g22 a2 b2 + g23 a2 b3 + g31a3 b1 + g32a3 b2 + g33a3 b3 : (3.273) 4. Vector Product of Vectors Given by Coordinates In accordance with (3.268a) 1 2 3 ~e ~e ~e ~a ~b = ~e1 ~e2 ~e3 a1 a2 a3 b1 b2 b3 h i = ~e1 ~e2 ~e3 (a2 b3 ; a3 b2 )~e 1 + (a3 b1 ; a1 b3) ~e 2 + (a1 b2 ; a2 b1)~e 3 (3.274) is valid. For Cartesian coordinates (3.274) coincides with (3.264).
5. Mixed Product of Vectors Given by Coordinates In accordance with (3.268a) we have 1 2 3 a a a ~a ~b ~c = ~e1 ~e2 ~e3 b1 b2 b3 : c1 c2 c3
(3.275)
For Cartesian coordinates (3.275) coincides with (3.265).
3.5.1.7 Vector Equations
Table 3.17 contains a summary of the simplest vector equations. In this table ~a, ~b, ~c are given vectors, ~x is the unknown vector, , , are given scalars, and x, y, z are the unknown scalars we are looking for.
3.5.1.8 Covariant and Contravariant Coordinates of a Vector
1. De nitions The ane coordinates a1 , a2, a3 of a vector ~a in a system with basis vectors ~e1, ~e2 , ~e3 , de ned by the formula ~a = a1 ~e1 + a2 ~e2 + a3 ~e3 = a ~e (3.276) are also called contravariant coordinates of this vector. The covariant coordinates are the coecients in the decomposition with the basis vectors ~e 1 , ~e 2, ~e 3, i.e., with the reciprocal basis vectors of ~e1, ~e2 , ~e3 . With the covariant coordinates a1 , a2, a3 of the vector ~a we have ~a = a1 ~e 1 + a2 ~e 2 + a3 ~e 3 = a ~e : (3.277) In the Cartesian coordinate system the covariant and contravariant coordinates of a vector coincide. 2. Representation of Coordinates with Scalar Product The covariant coordinates of a vector ~a are equal to the scalar product of this vector with the corre-
sponding basis vectors of the coordinate system: a1 = ~a ~e1 a2 = ~a ~e2 a3 = ~a ~e3 : (3.278) The contravariant coordinates of a vector ~a are equal to the scalar product of this vector with the corresponding basis vectors: a1 = ~a ~e 1 a2 = ~a ~e 2 a3 = ~a ~e 3: (3.279) In Cartesian coordinates (3.278) and (3.279) are coincident: ax = ~a~i ay = ~a~j az = ~a ~k: (3.280)
188 3. Geometry Table 3.17 Vector equations ~x unknown vector ~a , ~b , ~c , ~d given vectors x , y , z unknown scalars , , given scalars
Equation 1. ~x + ~a = ~b 2. ~x = ~a 3. ~x ~a = 4. 5. 6. 7. 8.
Solution ~x = ~b ; ~a ~x = ~a
Indeterminate equation if we consider all vectors ~x satisfying the equation, with the same initial point, then the endpoints form a plane perpendicular to the vector ~a. Equation 3. is called the vector equation of this plane. ~x ~a = ~b (~b ? ~a) Indeterminate equation if we consider all vectors ~x satisfying the equation, with the same initial point, then the endpoints form a line parallel to ~a. Equation 4. is called the vector equation of this line. ( ~ ~ ~x ~a = ~ x = ~a +a2a b (a = j~aj) ~x ~a = ~b (~b ? ~a) 8 > < ~x ~a = ~ ~ ~ ~x = (b ~c) + (~c~ ~a) + (~a b) = ~a~ + ~b~ + ~c~ > : ~~xx ~bc == ~ab ~c where ~a~, ~b~ , ~c~ are the reciprocal vectors of ~a, ~b, ~c (see 3.5.1.6, 1., p. 185). ~~ ~ ~~ ~d = x~a + y ~b + z~c x = d~b ~c y = ~a ~d ~c z = ~a b~ d ~ab ~c ~a b ~c ~a b ~c ~d = x(~b ~c) ~d ~a ~d ~b ~d ~c x = y = z = ~ +y (~c ~a) + z (~a b) ~a ~b ~c ~a ~b ~c ~a b~ ~c
3. Representation of the Scalar Product in Coordinates
The determination of the scalar product of two vectors by their contravariant coordinates yields the formula (3.272). The corresponding formula for covariant coordinates is: ~a ~b = g a b (3.281) where gmn = ~e m ~e n are the metric coecients in the system with the reciprocal vectors. Their relation with the coecients gmn is m+n mn gmn = (;g1) g gA (3.282) g11 g12 g13 21 22 23 g31 g32 g33 where Amn is the subdeterminant of the determinant in the denominator we get it by deleting the row and column of the element gmn. If the vector ~a is given by covariant coordinates, and the vector ~b by contravariant coordinates, then their scalar product is ~a ~b = a1 b1 + a2 b2 + a3 b3 = a b (3.283) and analogously we have ~a ~b = a b : (3.284)
3.5 Vector Algebra and Analytical Geometry 189
3.5.1.9 Geometric Applications of Vector Algebra
In Table 3.18 we demonstrate some geometric applications of vector algebra. Other applications from analytic geometry, such as vector equations of the plane and of the line, are demonstrated in 3.5.1.7, p. 188 and 3.5.3.4, p. 214 . and on the subsequent pages. Table 3.18 Geometric application of vector algebra
Determination
Vector formula p a = ~a 2
Formula with coordinates (in Cartesian coordinates)
Length of the vector ~a Area of the parallelogram determined by S = ~a ~b the vectors ~a and ~b Volume of the parallelepiped determined by the V = ~a~b ~c vectors ~a, ~b, ~c Angle between the ~ cos ' = q~ab vectors ~a and ~b ~a2~b2
q a = a2x + a2y + a2z
v u 2 2 2 u S = t abyy abzz + abzz abxx + abxx abyy a a a x y z V = bx by bz cx cy cz cos ' = q 2 axb2x + a2y bqy +2 az bz2 2 ax + ay + az bx + by + bz
3.5.2 Analytical Geometry of the Plane
3.5.2.1 Basic Concepts, Coordinate Systems in the Plane
The position of every point P of a plane can be given by an arbitrary coordinate system. The numbers determining the position of the point are called coordinates. Mostly we use Cartesian coordinates and polar coordinates.
1. Cartesian or Descartes Coordinates
The Cartesian coordinates of a point P are the signed distances of this point, given in a certain measure, from two coordinate axes perpendicular to each other (Fig. 3.120). The intersection point 0 of the coordinate axes is called the origin. The horizontal coordinate axis, usually the x-axis , is usually called the axis of abscissae, the vertical coordinate axis, usually the y-axis , is the axis of ordinates. y b
0
P(a,b)
II − −
a x
a) III
y
I
+ + 0 − −
+ + IV
x
I II III IV x + − − + y + + − − b)
Figure 3.120 Figure 3.121 The positive direction is given on these axes: on the x-axis usually to the right, on the y-axis upwards. The coordinates of a point P are positive or negative according to which half-axis the projections of the point fall (Fig. 3.121). The coordinates x and y are called the abscissa and the ordinate of the point P , respectively. We de ne the point with abscissa a and ordinate b with the notation P (a b). The x y plane is divided into four quadrants I, II, III, and IV by the coordinate axes (Fig. 3.121,a).
190 3. Geometry
2. Polar Coordinates
The polar coordinates of a point P (Fig. 3.122) are the radius , i.e., the distance of the point from a given point, the pole 0, and the polar angle ', i.e., the angle between the line 0P and a given oriented half-line passing through the pole, the polar axis. The pole is also called the origin. The polar angle is positive if it is measured counterclockwise from the polar axis, otherwise it is negative. v=b2
r 0
u=a1
v=b3
P(r,j)
u=a2
P
u=a3
v=b1
j
Figure 3.122
Figure 3.123
3. Curvilinear Coordinate System
This system consists of two one-parameter families of curves in the plane, the family of coordinate curves (Fig. 3.123). Exactly one curve of both families passes through every point of the plane. They intersect each other at this point. The parameters corresponding to this point are its curvilinear coordinates. In Fig. 3.123 the point P has curvilinear coordinates u = a1 and v = b3 . In the Cartesian coordinate system the coordinate curves are straight lines parallel to the coordinate axes in the polar coordinate system the coordinate curves are concentric circles with the center at the pole, and half-lines starting at the pole. y
y' P
y
x'
b 0
a
x
x' x
Figure 3.124
y
0'
y’
y
y'
y’
0
x
x’ ϕ
x’ x
Figure 3.125
3.5.2.2 Coordinate Transformations
Under transformation of a Cartesian coordinate system into another one, the coordinates change according to certain rules.
1. Parallel Translation of Coordinate Axes
We shift the axis of the abscissae by a, and the axis of the ordinates by b (Fig. 3.124). Suppose a point P has coordinates x, y before the translation, and it has the coordinates x0 , y0 after it. The old coordinates of the new origin 00 are a, b. The relations between the old and the new coordinates are the following: x = x0 + a y = y0 + b (3.285a) x0 = x ; a y0 = y ; b: (3.285b)
2. Rotation of Coordinate Axes
Rotation by an angle ' (Fig. 3.125) yields the following changes in the coordinates: x = x0 cos ' ; y0 sin ' y = x0 sin ' + y0 cos ' x0 = x cos ' + y sin ' y0 = ;x sin ' + y cos ':
(3.286a) (3.286b)
3.5 Vector Algebra and Analytical Geometry 191
The coecient matrix belonging to (3.286a) ! ! ! ! ' ; sin ' x = D x0 and x0 = D;1 x D = cos with (3.286c) 0 0 sin ' cos ' y y y y is called the rotation matrix. In general, transformation of a coordinate system into another can be performed in two steps, a translation and a rotation of the coordinate axes. y
ϕ
P2(x2 ,y2)
P1
d
ρ
y
0
y
P
x
x
r1
P1(x1 ,y1) x
0
Figure 3.126
Figure 3.127
0
j1
d P2
j2
r2
Figure 3.128
3. Transforming Cartesian Coordinates into Polar Coordinates and Conversely
We suppose in the following that the origin coincides with the pole, and the axis of abscissae coincides with the polar axis (Fig. 3.126): x = (') cos ' y = (') sin ' (; < ' 0) (3.287a) 8 y > arctan x + for x < 0 > > y > > for x > 0 > < arctan q x '=> (3.287c) = x2 + y2 (3.287b) for x = 0 and y > 0 > > ;2 for x = 0 and y < 0 > > 2 : inde ned for x = y = 0:
3.5.2.3 Special Notation in the Plane 1. Distance Between Two Points
If the two points given in Cartesian coordinates as P1 (x1 y1) and P2 (x2 y2) (Fig. 3.127), then their distance is q (3.288) d = (x2 ; x1 )2 + (y2 ; y1)2 : If they are given in polar coordinates as P1 ( 1 '1) and P2 ( 2 '2) (Fig. 3.128), their distance is
q d = 21 + 22 ; 2 1 2 cos ('2 ; '1 ):
2. Coordinates of Center of Mass
(3.289)
The coordinates (x y) of the center of mass of a system of material points Mi (xi yi) with masses mi (i = 1 2 : : : n) are calculated by the following formula: P P (3.290) x = Pmmixi y = Pmmi yi : i i
192 3. Geometry
3. Division of a Line Segment
P1P = m = 1. Division in a Given Ratio The coordinates of the point P with division ratio PP n 2 (Fig. 3.129a) of the line segment P1P2 are calculated by the formulas
mx2 x1 + x2 + my2 = y1 + y2 : x = nxn1 + (3.291a) y = nyn1 + (3.291b) +m = 1+ m 1+ For the midpoint M of the segment P1 P2, because of = 1, we have x = x1 +2 x2 (3.291c) y = y1 +2 y2 : (3.291d) The sign of the segments P1P and PP 2 can be de ned. Their signs are positive or negative depending on whether their directions are coincident with P1P2 or not. Then formulas (3.291a,b,c,d) result in a point outside of the segment P1P2 in the case < 0. We call this an external division. If P is inside the segment P1 P2, we call it an internal division. We de ne a) = 0 if P = P1, b) = 1 if P = P2 and c) = ;1 if P is an in nite or improper point of the line g, i.e., if P is in nitely far from P1P2 on g. The shape of is shown in Fig. 3.129b. For a point P , for which P2 is the midpoint of the segment P1P , = P1 P = ;2 holds. PP 2 2. Harmonic Division If the internal and external division of a line segment have the same absolute value jj, we call it harmonic division. Denote by Pi and Pa the points of the internal and external division respectively, and by i and a the internal and external devisions. Then we have P1Pi = = P1Pa = ; or i + a = 0: (3.292b) (3.292a) a PiP2 i PaP2 y
l
P2(x2 ,y2) g n P(x,y) m
1 P 3
x=g
−1
P1(x1 ,y1) a) 0
P1 P2 1M 2
>
0
x
b)
−2
Figure 3.129 If M denotes the midpoint of the segment P1 P2 at a distance b from P1 (Fig. 3.130), and the distances of Pi and Pa from M are denoted by xi and xa , then we have b + xi xa + b xi b 2 (3.293) b ; xi = xi ; b or b = xa i.e., xixa = b : The name harmonic division is in connection with the harmonic mean (see 1.2.5.3, p. 20). In Fig. 3.131 the harmonic division is represented for = 5 : 1, analogously to Fig. 3.14. The harmonic mean r of the segments P1Pi = p and P1 Pa = q according to (3.292a) equals in accordance with (1.67b), p. 20, to r = p2+pqq see Fig. 3.132: (3.294)
3.5 Vector Algebra and Analytical Geometry 193 P1
M
Pi xi
b
Pa
xa
P1
Figure 3.130
p1
q P1
Pi p
P2
r
P2
Pa
Pi
p2
P2
Pa
Figure 3.131
q
Figure 3.132 3. Golden Section of a segment a is its division into two parts x and a ; x such that the part x and the whole segment a have the same ratio as the parts a ; x and x: x = a;x: (3.295a) a x In this case x is the geometric mean of a and a ; x, and we have (see also golden section p. 2): p q x = a(a ; x) (3.295b) x = a( 52; 1) 0:618 a: (3.295c) The part x of the segment can be geometrically constructed as shown in Fig. 3.133. The segment x is also the length of the side a of a regular decagon whose circumcircle has radius a. x 2 The problem, to separate a square from a rectangle with the ratio C of sides given as in (3.295a) so that for the remaining rectangle a A B (3.295c) should be valid, also produces the equation of the golden Figure 3.133 section.
4. Areas
y
P1(x1 ,y1) 0
1. Area of a Triangle If the vertices are given by P1 (x1 y1), P2 (x2 y2), and P3 (x3 y3) (Fig. 3.134), then we can calculate the
P3(x3 ,y3)
P2(x2 ,y2) x
area by the formula x y 1 1 1 S = 21 x2 y2 1 = 12 x1 (y2 ; y3) + x2 (y3 ; y1) + x3 (y1 ; y2)] x3 y3 1 = 21 (x1 ; x2 ) (y1 + y2) + (x2 ; x3 ) (y2 + y3) + (x3 ; x1 ) (y3 + y1)] : (3.296)
Figure 3.134 Three points are on the same line if x y 1 x1 y1 1 (3.297) 2 2 = 0: x3 y3 1 2. Area of a Polygon If the vertices are given by P1 (x1 y1), P2 (x2 y2), : : :, Pn (xn yn), then the area is S = 12 (x1 ; x2 ) (y1 + y2) + (x2 ; x3 ) (y2 + y3) + + (xn ; x1 ) (yn + y1)] : (3.298) The formulas (3.296) and (3.298) result in a positive area if the vertices are enumerated counterclockwise, otherwise the area is negative.
194 3. Geometry
5. Equation of a Curve
Every equation F (x y) = 0 for the coordinates x and y corresponds to a curve, which has the property that every point P satis es the equation, and conversely, every point whose coordinates satisfy the equation is on the curve. The set of these points is also called the geometric locus or simply locus. If there is no real point in the plane satisfying the equation F (x y) = 0, then there is no real curve, and we talk about an imaginary curve: A: x2 + y2+ 1 = 0 B: y = ln 1 ; x2 ; cosh x : The curve corresponding to the equality F (x y) = 0 is called an algebraic curve if F (x y) is a polynomial, and the degree of the polynomial is the order or degree of the curve (see 2.3.4, p. 63). If the equation of the curve cannot be transformed into the form F (x y) = 0 with a polynomial expression F (x y), then the curve is called a transcendental curve. The equation of a curve can be de ned in the same way in any coordinate system. But from now on, we talk about the Cartesian coordinate system only, except when stated otherwise.
3.5.2.4 Line
1. Equation of the Line
Every equation that is linear in the coordinates is the equation of a line, and conversely, the equation of every line is a linear equation of the coordinates.
1. General Equation of Line
y
Ax+C=0
Ax + By + C = 0 (A B C const): (3.299) For A = 0 (Fig. 3.135) the line is parallel to the x-axis, for B = 0 it is parallel to the y-axis, for C = 0 it passes through the origin. 2. Equation of the Line with Slope (or Angular Coecient) Every line that is not parallel to the y-axis can be represented by an equation written in the form y = kx + b (k b const): (3.300) The quantity k is called the angular coecient or slope of the line it is equal to the tangent of the angle between the line and the positive direction of the x-axis (Fig. 3.136). The line cuts out the segment b from the y-axis. Both the tangent and the value of b can be negative, depending on the position of the line. 3. Equation of a Line Passing Through a Given Point The equation of a line which goes through a given point P1 (x1 y1) in a given direction (Fig. 3.137) is y ; y1 = k (x ; x1 ) with k = tan : (3.301) Ax By+C=0
0
x
Figure 3.135
y
y
0
y= +B
b d 0
k=tan d
Figure 3.136
x
P1(x1 ,y1) d 0
x
Figure 3.137
4. Equation of a Line Passing Through Two Given Points If two points of the line P1 (x1 y1), P2 (x2 y2) are given (Fig. 3.138), then the equation of the line is y ; y1 x ; x1 (3.302) y2 ; y1 = x2 ; x1 :
3.5 Vector Algebra and Analytical Geometry 195
5. Intercept Equation of a Line If a line cuts out the segments a and b from the coordinate axes, considering them with sign, the equation of the line is (Fig. 3.139) x + y = 1: a b
(3.303)
y
y
y P2(x2 ,y2)
P1(x1 ,y1)
b
d p
P1(x1 ,y1) 0
x
0
Figure 3.138
a
x
0
Figure 3.139
a
x
Figure 3.140
6. Normal Form of the Equation of the Line (Hessian Normal Form) With p as the distance of the line from the origin, and with as the angle between the x-axis and the normal of the line passing through the origin (Fig. 3.140), with p > 0, and 0 < 2, the Hessian normal form is x cos + y sin ; p = 0: (3.304)
We can get the Hessian normal form from the general equation if we multiply (3.299) by the normalizing factor = p 21 2 : (3.305) A +B The sign of must be the opposite to that of C in (3.299). 7. Equation of a Line in Polar Coordinates (Fig. 3.141) With p as the distance of the line from the pole (normal segment from the pole to the line), and with as the angle between the polar axis and the normal of the line passing through the pole, the equation of the line is = cos ('p ; ) : (3.306)
2. Distance of a Point from a Line
We get the distance d of a point P1 (x1 y1) from a line (Fig. 3.140) by substituting the coordinates of the point into the left-hand side of the Hessian normal form (3.304): d = x1 cos + y1 sin ; p: (3.307) If P1 and the origin are on di erent sides of the line, we get d > 0, otherwise d < 0. y
y
ρ
p 0
α
P
P(x0 ,y0)
ϕ 0
Figure 3.141
3. Intersection Point of Lines
x
Figure 3.142
0
x
Figure 3.143
1. Intersection Point of Two Lines In order to get the coordinates (x0 y0) of the intersection point of two lines we have to solve the system of equations given by the equation. If the lines are given by the equations A1x + B1 y + C1 = 0 A2 x + B2 y + C2 = 0 (3.308a)
196 3. Geometry then the solution is C A B C 1 1 1 1 B2 C2 y = C2 A2 : (3.308b) x0 = A 0 A1 B1 1 B1 A2 B2 A2 B2 A B B1 C 1 1 1 1 If A2 B2 = 0 holds, the lines are parallel. If A A2 = B2 = C2 holds, the lines are coincident. 2. Pencil of Lines If a third line with equation A3x + B3 y + C3 = 0 (3.309a) passes through the intersection point of the rst two lines (Fig. 3.142), then the relation A B C A1 B1 C1 (3.309b) 2 2 2 = 0 A3 B3 C3 must be satis ed. The equation (A1x + B1y + C1) + (A2x + B2y + C2 ) = 0 (;1 < < +1) (3.309c) describes all the lines passing through the intersection point P0(x0 y0) of the two lines (3.308a). By (3.309c) we de ne a pencil of lines with center P0(x0 y0). If the equations of the rst two lines are given in normal form, then for = 1 we get the equations of the bisectrices of the angles at the intersection point (Fig. 3.143). y
j 0
y
k2
k2=k1
k1
A
x
Figure 3.144
4. Angle Between Two Lines
y
k1 0
a)
x
k2=- 1 k1 90
0
b) Figure 3.145
o
k1 x
In Fig. 3.144 there are two intersecting lines. If their equations are given in the general form A1x + B1 y + C1 = 0 and A2 x + B2 y + C2 = 0 (3.310a) then for the angle ' we have tan ' = A1 B2 ; A2 B1 (3.310b) A1 A2 + B1B2 cos ' = q A2 1 A2 2+qB12B2 2 (3.310c) sin ' = q A2 1 B2 2;qA22B1 2 : (3.310d) A1 + B1 A2 + B2 A1 + B1 A2 + B2 With the slopes k1 and k2 we have 2 ; k1 tan ' = 1k+ (3.310e) k1 k2
3.5 Vector Algebra and Analytical Geometry 197
qk1 cos ' = q 1 +2kq1k2 2 (3.310f) sin ' = q k2 ; : (3.310g) 1 + k1 1 + k2 1 + k12 1 + k22 Here we consider the angle ' in the counterclockwise direction from the rst line to the second one. B1 1 For parallel lines (Fig. 3.145a) the equalities A A2 = B2 or k1 = k2 are valid. For perpendicular (orthogonal) lines (Fig. 3.145b) we have A1 A2 + B1 B2 = 0 or k2 = ;1=k1 .
3.5.2.5 Circle
1. De nition of the Circle The locus of points at the same given distance from a given point is called a circle. The given distance is called the radius and the given point is called the center of the circle. 2. Equation of the Circle in Cartesian Coordinates The equation of the circle in Cartesian coordinates when its center is at the origin (Fig. 3.146a) is x2 + y 2 = R 2 : (3.311a) If the center is at the point C (x0 y0) (Fig. 3.146b), then the equation is (x ; x0 )2 + (y ; y0)2 = R2: (3.311b) The general equation of second degree ax2 + 2bxy + cy2 + 2dx + 2ey + f = 0 (3.312a) is the equation of a circle only if b = 0 and a = c. In this case the equation can always be transformed into the form x2 + y2 + 2mx + 2ny + q = 0: (3.312b) For the radius and the coordinates of the center of the circle we have the equalities q x0 = ;m y0 = ;n: (3.313b) R = m2 + n2 ; q (3.313a) If q > m2 + n2 holds, the equation de nes an imaginary curve, if q = m2 + n2 the curve has one single point P (x0 y0). y
y P
P(x,y)
R
R x
y0
b) 0
Figure 3.146
3. Parametric Representation of the Circle
C(x0 ,y0)
x0
y0
x
0
C(x0 ,y0)
R
P 0
a)
y
t
x0
x
Figure 3.147
x = x0 + R cos t y = y0 + R sin t (3.314) where t is the angle between the moving radius and the positive direction of the x-axis (Fig. 3.147). 4. Equation of the Circlein Polar Coordinates in the general case corresponding to Fig. 3.148: 2 ; 2 0 cos (' ; '0) + 20 = R2 : (3.315a) If the center is on the polar axis and the circle goes through the origin (Fig. 3.149) the equation has the form = 2R cos ': (3.315b)
198 3. Geometry P R r 0
j j0
y
P
r
R
j
0
2R
r0
Figure 3.148
P(x0,y0) . x
0
Figure 3.149
Figure 3.150
5. Tangent of a Circle The tangent of a circle, given by (3.311a) at the point P (x0 y0) (Fig. 3.150) has the form xx0 + yy0 = R2:
(3.316)
3.5.2.6 Ellipse
1. Elements of the Ellipse In Fig. 3.151, AB = 2a is the major paxis, CD = 2b is the minor axis, A, B , C , D are the vertices, F1 , F2 are the foci at a distance c = a2 ; b2 on both sides from 2 the midpoint, e = c=a < 1 is the numerical eccentricity, and p = b =a is the semifocal chord, i.e., the half-length of the chord which is parallel to the minor axis and goes through a focus. P(x,y)
A F2
0 2c
r1 p
C 2a
Figure 3.151
B F1
x
directrice
2b
r2
d2
F2 d
y
d1
P
0
F1
directrice
y D
x
d
Figure 3.152 2. Equation of the Ellipse If the coordinate axes and the axes of the ellipse are coincident, the equation of the ellipse has the normal form. This equation and the equation in parametric form are x2 + y 2 = 1 x = a cos t y = b sin t: (3.317b) (3.317a) a2 b2 For the equation of the ellipse in polar coordinates see 6., p. 207. 3. De nition of the Ellipse, Focal Properties The ellipse is the locus of points for which the sum of the distances from two given points, the foci, is a constant, and equal to 2a. These distances, which are also called the focal radii of the points of the ellipse, can be expressed as a function of the coordinate x from the equalities r1 = F1 P = a ; ex r2 = F2 P = a + ex r1 + r2 = 2a: (3.318) Also here, and in the following formulas in Cartesian coordinates, we suppose that the ellipse is given in normal form. 4. Directrices of an Ellipse are lines parallel to the minor axis at distance d = a=e from it (Fig. 3.152). Every point P (x y) of the ellipse satis es the equalities r1 r2 (3.319) d1 = d2 = e and this property can also be taken as a de nition of the ellipse.
3.5 Vector Algebra and Analytical Geometry 199
5. Diameter of the Ellipse The chords passing through the midpoint of the ellipse are called diameters of the ellipse. The midpoint of the ellipse is also the midpoint of the diameter (Fig. 3.153).
The locus of the midpoints of all chords parallel to the same diameter is also a diameter it is called the conjugate diameter of the rst one. For k and k0 as slopes of two conjugate diameters the equality 2 kk0 = ;ab2 (3.320) holds. If 2a1 and 2b1 are the lengths of two conjugate diameters and and are the acute angles between the diameters and the major axis, where k = ; tan and k0 = tan hold, then we have the Apollonius theorem in the form a1b1 sin ( + ) = ab a21 + b21 = a2 + b2 : (3.321) y
y
2a
y P(x0 ,y0)
1
0
β
x
F2
0
P(x,y)
r1
N
F1
x
B
0
x
N(x,-y)
2b
1
α
r2
Figure 3.153
Figure 3.154
6. Tangent of the Ellipse at the point P (x0 y0) is given by the equation
Figure 3.155
xx0 + yy0 = 1: (3.322) a2 b2 The normal and tangent lines at a point P of the ellipse (Fig. 3.154) are bisectors of the interior and exterior angles of the radii connecting the point P with the foci. The line Ax + By + C = 0 is a tangent line of the ellipse if the equation A2a2 + B 2 b2 ; C 2 = 0 (3.323) is satis ed. 7. Radius of Curvature of the Ellipse (Fig. 3.154) If u denotes the angle between the tangent line and the radius vector connecting the point of contact P (x0 y0) with a focus, then the radius of curvature is 3 3 2 y2 ! 2 x ( r 1 r2 ) 2 0 0 2 2 R = a b a4 + b4 = a b = sinp3u : (3.324) 2 At the vertices A and B (Fig. 3.151) and at C and D the radii are RA = RB = b = p and RC = a 2 RD = ab .
8. Areas of the Ellipse (Fig. 3.155) a) Ellipse: S = a b:
b) Sector of the Ellipse B0P: SB0P = ab2 arccos xa :
(3.325b)
(3.325a)
c) Segment of the Ellipse PBN: SPBN = a b arccos xa ; x y: (3.325c)
200 3. Geometry 9. Arc and Perimeter of the Ellipse The arclength between two points A and B of the ellipse cannot be calculated in an elementary way as for the parabola, but only with an incomplete elliptic integral of the second kind E (k ') (see 8.2.2.2, 2., p. 449). The perimeter of the ellipse (seealso 8.2.5, 7., 462) can be calculated by a complete elliptic integral of p the second kind E (e) = E e with the numerical eccentricity e = a2 ; b2 =a and with ' = (for 2 2 one quadrant of the perimeter), and it is " 2 2 4
2 6 L = 4aE (e) = 2a 1 ; 12 e2 ; 12 34 e3 ; 12 34 56 e5 ; : (3.326a) b) If we substitute = ((aa ; + b) , then we have "
2 4 6 8 L = (a + b) 1 + 4 + 64 + + 25 + (3.326b) 256 16384 and an approximate value is h p i 64 ; 34 : L 1 5(a + b) ; ab L (a + b) 64 (3.326c) ; 162 For a = 1:5 b = 1 the formula (3.326c) results in the value 7:93. while the better approximation with the complete elliptic integral of the second kind (see 8.1.4.3, p. 438) results in the value 7:98.
3.5.2.7 Hyperbola
1. Elements of the Hyperbola In Fig. 3.156
AB = 2a is the real axis A, B are the vertices 0 the midpoint F1 and F2 are the foci at a distance c > a from the midpoint p on the real axis on both sides CD = 2b = 2 c2 ; a2 is the imaginary axis p = b2 =a the semifocal chord of the hyperbola, i.e., the half-length of the chord which is perpendicular to the real axis and goes through a focus e = c=a > 1 is the numerical eccentricity. 2. Equation of the Hyperbola The equation of the hyperbola in normal form, i.e., for coincident x and real axes, and the equation in parametric form are
y D 2b
r2 p
F2
A
0
B
P(x,y) r1 F1
x
C 2a 2c
Figure 3.156
x2 ; y2 = 1 (3.327a) x = a cosh t y = b sinh t (3.327b) or x = a y = b tan t: (3.327c) cos t a2 b2 In polar coordinates see 3.5.2.9, 6., p. 207. 3. De nition of the Hyperbola, Focal Properties The hyperbola is the locus of points for which the di erence of the distances from two given points, the foci, is a constant 2a. The points for which r1 ; r2 = 2a belong to one branch of the hyperbola (in Fig. 3.156 on the left), the others with r2 ; r1 = 2a belong to the other branch (in Fig. 3.156 on the right). These distances, also called the focal radii, can be calculated from the formulas r1 = (ex ; a) r2 = (ex + a) r2 ; r1 = 2a (3.328) where the upper sign is valid for the right branch, the lower one for the left branch. Here and in the following formulas for hyperbolas in Cartesian coordinates we suppose that the hyperbola is given in normal form.
3.5 Vector Algebra and Analytical Geometry 201 y
d2
y d1
0
directrice
directrice
F2
r2
P r1
r2
F1
P(x0,y0) r1
x
F2
A
0
B
F1
T N x
d
d
Figure 3.157
Figure 3.158 4. Directrices of the Hyperbola are the lines perpendicular to the real axis at a distance d = a=c from the midpoint (Fig. 3.157). Every point of the hyperbola P (x y) satis es the equalities r1 r2 (3.329) d1 = d2 = e: 5. Tangent of the Hyperbola at the point P (x0 y0) is given by the equation xx0 ; yy0 = 1: (3.330) a2 b2 The normal and tangent lines of the hyperbola at the point P (Fig. 3.158) are bisectors of the interior and exterior angles between the radii connecting the point P with the foci. The line Ax + By + C = 0 is a tangent line if the equation A2a2 ; B 2b2 ; C 2 = 0 (3.331) is satis ed. y
y
T G P
d
0
F
x
0
x
T1
Figure 3.159
Figure 3.160
6. Asymptotes of the Hyperbola are the lines (Fig. 3.159) approached in nitely closely by the branches of the hyperbola for x ! 1. (For the de nition of asymptotes see 3.6.1.4, p. 234.) The slopes of the asymptotes are k = tan = b=a. The equations of the asymptotes are ! y = ab x: (3.332) A tangent is intersected by the asymptotes, and they form a segment of the tangent of the hyperbola , i.e., the segment TT 1 (Fig. 3.159). The midpoint of the segment of the tangent is the point of contact P , so TP = T1P holds. The area of the triangle T 0T1 between the tangent and the asymptotes for any point of contact P is the same, and is SD = a b: (3.333)
202 3. Geometry The area of the parallelogram 0FPG, determined by the asymptotes and two lines parallel to the asymptotes and passing through the point P , is for any point of contact P 2 2 2 SP = (a +4 b ) = c4 : (3.334) 7. Conjugate Hyperbolas (Fig. 3.160) have the equations x2 ; y2 = 1 and y2 ; x2 = 1 (3.335) a2 b2 b2 a2 where the second is represented in Fig. 3.160 by the dotted line. They have the same asymptotes, hence the real axis of one of them is the imaginary axis of the other one and conversely. y
y
2a
1
d
β
0 α
x
0
G P(x,y) A
N
x
2b 1
Figure 3.161
Figure 3.162 8. Diameters of the Hyperbola (Fig. 3.161) are the chords between the two branches of the hyperbola passing through the midpoint, which is their midpoint too. Two diameters with slopes k and k0 are called conjugate if one of them belongs to a hyperbola and the other one belongs to its conjugate, and kk0 = b2 =a2 holds. The midpoints of the chords parallel to a diameter are on its conjugate diameter (Fig. 3.161). From two conjugate diameters the one with jkj < b=a intersects the hyperbola. If the lengths of two conjugate diameters are 2a1 and 2b1 , and the acute angles between the diameters and the real axis are and < , then the equalities a21 ; b21 = a2 ; b2 ab = a1b1 sin( ; ) (3.336) are valid. 9. Radius of Curvature of the Hyperbola At the point P (x0 y0) the radius of curvature of the hyperbola is 2 2 !3=2 r r 3=2 R = a2 b2 xa40 + yb40 = 1 2 = p3 (3.337a) ab sin u where u is the angle between the tangent and the radius vector connecting the point of contact with a focus. At the vertices A and B (Fig. 3.156) the radius of curvature is 2 RA = RB = p = ba : (3.337b)
10. Areas in the Hyperbola (Fig. 3.162) a) Segment APN: SAPN = xy ; ab ln xa + yb = x y ; a b Arcosh xa : b) Area 0APG:
(3.338a)
S0APG = ab4 + ab2 ln 2cd : (3.338b) The line segment PG is parallel to the lower asymptote, c is the focal distance and d = 0G.
3.5 Vector Algebra and Analytical Geometry 203
11. Arc of the Hyperbola The arclength between two points A and B of the hyperbola cannot be calculated in an elementary way like the parabola, but we can calculate it by an incomplete elliptic integral of the second kind E (k ') (see p. 449), analogously to the arclength of the ellipse (see p. 200). 12. Equilateral Hyperbola has axes with the same length a = b, so its equation is x2 ; y2 = a2 : (3.339a) The asymptotes of the equilateral hyperbola are perpendicular to each other. If the asymptotes coincide with the coordinate axes (Fig. 3.163), then the equation is 2 x y = a2 : (3.339b) y
N
y
K p
a
0
x
0 F N'
Figure 3.163
3.5.2.8 Parabola
1. Elements of the Parabola
y
p P(x,y) x
p 2
0
x (x0 ,y0)
Figure 3.164
Figure 3.165
In Fig. 3.164 the x-axis coincides with the axis of the parabola, 0 is the vertex of the parabola, F is the focus of the parabola which is on the x-axis at a distance p=2 from the origin, where p is called the semifocal chord of the parabola. We denote the directrix by NN 0 , which is the line perpendicular to the axis of the parabola and intersects the axis at a distance p=2 from the origin on the opposite side as the focus. So the semifocal chord is equal to half of the length of the chord which is perpendicular to the axis and passes through the focus. The numerical eccentricity of the parabola is equal to 1 (see 3.5.2.9, 4., p. 207). 2. Equation of the Parabola If the origin is the vertex of the parabola and the x-axis is the axis of the parabola with the vertex on the left-hand side, then the normal form of the equation of the parabola is y 2 = 2p x : (3.340) For the equation of the parabola in polar coordinates see 3.5.2.9, 6., p. 207. For a parabola with vertical axis (Fig. 3.165) the equation is y = ax2 + bx + c: (3.341a) The parameter of a parabola given in this form is p = 2j1aj : (3.341b) If a > 0 holds, the parabola is open up, for a < 0 it is open down. The coordinates of the vertex are 2 x0 = ; 2ba y0 = 4ac4;a b : (3.341c) 3. Properties of the Parabola (De nition of the Parabola) The parabola is the locus of points P (x y) whose distance from a given point, the focus, is equal to its distance from a given line, the directrix (Fig. 3.164). Here and in the following formulas in Cartesian coordinates we suppose the normal form of the equation of the parabola. Then we have the equation (3.342) PF = PK = x + p2
204 3. Geometry where PF is the radius vector whose initial point is at the focus and endpoint is a point of the parabola. 4. Diameter of the Parabola is a line which is parallel to the axis of the parabola (Fig. 3.166). A diameter of the parabola halves the chords which are parallel to the tangent line belonging to the endpoint of the diameter (Fig. 3.166). With slope k of the chords the equation of the diameter is (3.343) y = kp : y
y
P(x 0
0
x
S u T 0
Figure 3.166
y
) ,y 0
M 0
N FM
x
Q
Figure 3.167
N
P(x,y) R
x
Figure 3.168
5. Tangent of the Parabola (Fig.3.167) The equation of the tangent of the parabola at the point
P (x0 y0) is yy0 = p (x + x0 ) : (3.344) Tangent and normal lines are bisectors of the angles between the radius starting at the focus and the diameter starting at the point of contact. The tangent at the vertex, i.e., the y-axis, halves the segment of the tangent line between the point of contact and its intersection point with the axis of the parabola, the x-axis: TS = SP T 0 = 0M = x0 TF = FP: (3.345) A line with equation y = kx + b is a tangent line of the parabola if p = 2 b k: (3.346) 6. Radius of Curvature of the Parabola at the point P (x1 y1) with ln as the length of the normal PN (Fig. 3.167) is 3=2 3 R = (p +p2xp1 ) = sinp3u = pln2 (3.347a) and at the vertex 0 it is R = p: (3.347b)
7. Areas in the Parabola (Fig. 3.168) a) Parabolic Segment P0N:
S0PN = 23 SMQNP (MQNP is a parallelogram):
(3.348a)
S0PR = 2xy 3 :
(3.348b)
2v s s ! !3 u u p 2 x 2 x 2 x 2 x t 4 l0P = 2 p 1 + p + ln p + 1 + p 5
(3.349a)
b) Area 0PR (Area under the Parabola Curve): 8. Length of Parabolic Arc from the vertex 0 to the point P (x y)
3.5 Vector Algebra and Analytical Geometry 205
s
s
= ; x x + p + p Arsinh 2x : 2 2 p x For small values of we have the approximation 2 y !2 !43 l0P y 41 + 23 xy ; 52 xy 5 :
(3.349b)
(3.349c)
Table 3.19 Equation of curves of second order. Central curves ( 6= 0) 1
Quantities and
% 6= 0 Central curves 6= 0
>0 0: imaginary 2
%=0
A pair of imaginary 2 lines with real common point
% 6= 0 %=0
Hyperbola A pair of intersecting lines
Required coordinate transformations
Normal form of the equation after the transformation
1. Translation of the origin to the center of the curve, whose coordinates are a0 x02 + c0y02 + % = 0 be ; cd bd ; ae x0 = y0 = . q 2. Rotation of the coordinate axes by a + c + (a ; c)2 + 4b2 2 b 0 a= the angle with tan 2 = a ; c . 2 The sign of sin 2 must coincide with q the sign of 2b. Here the slope a + c ; (a ; c)2 + 4b2 c0 = of the new x0 -axis is q 2 (a0 and c0 are the roots of the quadratic c ; a + (c ; a)2 + 4b2 2 k= . equation u ; Su + = 0.) 2b 1 % and S are numbers given in (3.350b). 2 The equation of the curve corresponds to an imaginary curve.
3.5.2.9 Quadratic Curves (Curves of Second Order or Conic Sections)
1. General Equation of Quadratic Curves (Curves of Second Order or Degree)
The ellipse, its special case, the circle, the hyperbola, the parabola or two lines as a singular conic section are de ned by the general equation of a quadratic curve (curve of second order) a x2 + 2 b x y + c y2 + 2 d x + 2 e y + f = 0: (3.350a) We can reduce this equation to normal form with the help of the coordinate transformations given in Tables 3.19 and 3.20.
206 3. Geometry Remark 1: The coecients in (3.350a) are not the parameters of the special conic sections. Remark 2: If two coecients (a and b or b and c) are equal to zero, the required coordinate transformation is reduced to a translation of the coordinate axes. The equation cy2 + 2dx + 2ey + f = 0 can be written in the form (y ; y0)2 = 2p(x ; x0) the equation ax2 + 2dx + 2ey + f = 0 can be written in the form (x ; x0 )2 = 2p (y ; y0). Table 3.20 Equations of curves of second order. Parabolic curves ( = 0)
Quantities and Parabolic curves 1
=0
% 6= 0 %=0
Required coordinate transformation
Shape of the curve
Parabola Two lines: Parallel lines for d2 ; af > 0, Double line for d2 ; af = 0, Imaginary 2 lines for d2 ; af < 0.
Normal form of the equation after the transformation
1. Translation of the origin to the vertex of the parabola whose coordinates x0 and y0 are de ned by the equations ax0 + by0 + ad S+ be = 0 and y02 = 2px0 ! ! be x + e + ae ; bd y + f = 0 . d + dc ; 0 0 p ; bd p = ae S S S a2 + b2 2. Rotation of the coordinate axes by the angle with tan = ; ab the sign of sin must di er from the sign of a. Rotation of the coordinate axes by the angle Sy02 + 2 pad + be y0 + f = 0 can be transa2 + b2 with tan = ; a the sign of sin must di er b formed into the form from the sign of a. (y0 ; y00 ) (y0 ; y10 ) = 0. 1 2
In the case = 0 we suppose that none of the coecients a b c are equal to zero. The equation of the curve corresponds to an imaginary curve.
2. Invariants of Quadratic Curves
are the three quantities a b d % = b c e = ab bc S = a + c: (3.350b) d e f They do not change during a rotation of the coordinate system, i.e., if after a coordinate transformation the equation of the curve has the form (3.350c) a0x0 2 + 2b0x0 y0 + c0 y02 + 2d0x0 + 2e0 y0 + f 0 = 0 then the calculation of these three quantities %, , and S with the new constants will yield the same values.
3.5 Vector Algebra and Analytical Geometry 207
3. Shape of the Quadratic Curves (Conic Sections)
If a right circular cone is intersected by a plane, the result is a conic section. If the plane does not pass through the vertex of the cone, we get a hyperbola, a parabola, or an ellipse depending on whether the plane is parallel to two, one, or none of the generators of the cone. If the plane goes through the vertex, we get a singular conic section with % = 0. As a conic section of a cylinder, i.e., a singular cone whose vertex is at in nity, we get parallel lines. We can determine the shape of a conic section with the help of Tables 3.19 and 3.20. P
directrice
K F
Figure 3.169
4. General Properties of Curves of Second Degree
The locus of every point P (Fig. 3.169) with constant ratio e of the distance to a xed point F , the focus, and the distance from a given line, the directrix, is a curve of second order with numerical eccentricity e. For e < 1 it is an ellipse, for e = 1 it is a parabola, for e > 1 it is a hyperbola.
5. Determination of a Curve Through Five Points
There is one and only one curve of second degree passing through ve given points. If three of these points are on the same line, we have a singular or degenerate conic section.
6. Polar Equation of Curves of Second Degree
All curves of second degree can be described by the polar equation = 1 + epcos ' (3.351) where p is the semifocal chord and e is the eccentricity. Here the pole is at the focus, while the polar axis is directed from the focus to the closer vertex. For the hyperbola this equation de nes only one branch.
3.5.3 Analytical Geometry of Space
3.5.3.1 Basic Concepts, Spatial Coordinate Systems
Every point P in space can be determined by a coordinate system. The directions of the coordinate lines are given by the directions of the unit vectors. In Fig. 3.170a the relations of a Cartesian coordinate system are represented. We distinguish right-angled and oblique coordinate systems where the unit vectors are perpendicular or oblique to each other. Another important di erence is whether it is a right-handed or a left-handed coordinate system. The most common spatial coordinate systems are the Cartesian coordinate system, the spherical polar coordinate system, and the cylindrical polar coordinate system.
1. Right- and Left-Handed Systems
Depending on the successive order of the positive coordinate directions we distinguish right systems and left systems or right-handed and left-handed coordinate systems. A right system has for instance three non-coplanar unit vectors with indices in alphabetical order ~ei ~ej ~ek . They form a right-handed system if the rotation of one of them around the origin into the next one in alphabetical order in the shortest direction is a counterclockwise rotation. This is represented symbolically in Fig. 3.34, p. 142 we substitute the notation a b c for the indices i j k. A left system consequently requires a clockwise rotation. Right- and left-handed systems can be transformed into each other by interchanging two unit vectors. The interchange of two unit vectors changes its orientation: A right system becomes a left system, and conversely, a left system becomes a right system. A very important way to interchange vectors is the cyclic permutation, where the orientation remains unchanged. As in Fig. 3.34 the interchange the vectors of a right system by cyclic permutation yields a rotation in a counterclockwise direction, i.e., according to the scheme (i ! j ! k ! i j ! k ! i ! j k ! i ! j ! k). In a left system the interchange of the vectors by cyclic permutation follows a clockwise rotation, i.e., according to the scheme (i ! k ! j ! i k ! j ! i ! k j ! i ! k ! j ).
208 3. Geometry z
z
P(x,y,z) r
ek ei a)
x
ej
0 y
P(y,x,z) r
ek
z ej x
y
b)
y
ei
0 x
z y
x
Figure 3.170 A right system is not superposable on a left system. The reection of a right system with respect to the origin is a left system (see 4.3.5.1, p. 269). A: The Cartesian coordinate system with coordinate axes x y z is a right system (Fig. 3.170a). B: The Cartesian coordinate system with coordinate axes x z y is a left system (Fig. 3.170b). C: From the right system ~ei ~ej ~ek we get the left system ~ei ~ek ~ej by interchanging the vectors ~ej and ~ek . D: By cyclic permutation we get from the right system ~ei ~ej ~ek the right system ~ej ~ek ~ei and from this one ~ek ~ei ~ej , a right system again. Table 3.21 Coordinate signs in the octants Octant I II III IV V VI VII VIII x + ; ; + + ; ; + y + + ; ; + + ; ; z + + + + ; ; ; ;
2. Cartesian Coordinates
of a point P are its distances from three mutually orthogonal planes in a certain measuring unit, with given signs. They represent the projections of the radius vector ~r of the point P (see 3.5.1.1, 6., p. 181) onto three mutually perpendicular coordinate axes (Fig. 3.170). The interI IV section point of the planes O, which is the intersection point of the axes too, is called the origin. The coordinates 0 x, y, and z are called the abscissa, ordinate, and appliy cate. The written form P (a b c) means that the point P VII VI has coordinates x = a, y = b, z = c. The signs of the cox ordinates are determined by the octant where the point P lies (Fig. 3.171, Table 3.21). VIII In a right-handed Cartesian coordinate system (Fig. V 3.170a) for orthogonal unit vectors given in the order ~ei ~ej ~ek the equalities Figure 3.171 ~ei ~ej = ~ek ~ej ~ek = ~ei ~ek ~ei = ~ej (3.352a) hold, i.e., the right-hand law is valid (see 3.5.1.5, p. 183). The three formulas transform into each other under cyclic permutations of the unit vectors. In a left-handed Cartesian coordinate system (Fig. 3.170b) the equations ~ei ~ej = ;~ek ~ej ~ek = ;~ei ~ek ~ei = ;~ej (3.352b) are valid. The negative sign of the vector product arises from the left-handed order of the unit vectors, see Fig. 3.170b, i.e., from their clockwise arrangement. z
III
II
3.5 Vector Algebra and Analytical Geometry 209
Notice that in both cases the equations ~ei ~ei = ~ej ~ej = ~ek ~ek = ~0 (3.352c) are valid. Usually we work with right-handed coordinate systems the formulas do not depend on this choice. In geodesy we usually use left-handed coordinate systems (see 3.2.2.1, p. 143).
3. Coordinate Surfaces and Coordinate Curves
Coordinate Surfaces have one constant coordinate. In a Cartesian coordinate system they are planes parallel to the other two coordinate axes. By the three coordinate surfaces x = 0, y = 0, and z = 0 three-dimensional space is divided into eight octants (Fig. 3.171). Coordinate lines or coordinate curves are curves with one changing coordinate while the others are constants. In Cartesian systems they are lines parallel to the coordinate axes. The coordinate surfaces intersect each other in a coordinate line.
4. Curvilinear Three-Dimensional Coordinate System
arises if three families of surfaces are given such that for any point of space there is exactly one surface from every system passing through it. The position of a point will be given by the parameter values of the surfaces passing through it. The most often used curvilinear coordinate systems are the cylindrical polar and the spherical polar coordinate systems.
5. Cylindrical Polar Coordinates
(Fig. 3.172) are: The polar coordinates and ' of the projection of the point P to the x y plane and the applicate z of the point P . The coordinate surfaces in a cylindrical polar coordinate system are: The cylinder surfaces with radius % = const the half-planes starting from the z-axis, ' = const and the planes being perpendicular to the z-axis, z = const.
The intersection curves of these coordinate surfaces are the coordinate curves. The transformation formulas between the Cartesian coordinate system and the cylindrical polar coordinate system are (see also Table 3.22): x = % cos ' y = % sin ' z = z (3.353a) q y y % = x2 + y2 ' = arctan x = arcsin % for x > 0: (3.353b) For the required distinction of cases with respect to ' see (3.287c), p. 191. z z r
P 0 x
J 0 j
z
j r
Figure 3.172
y
P
y
x
Figure 3.173
6. Spherical Coordinates or Spherical Polar Coordinates
contain: The length r of the radius vector ~r of the point P , the angle between the z-axis and the radius vector ~r and the angle ' between the x-axis and the projection of ~r on the x y plane. The positive directions (Fig. 3.173) here are for ~r from the origin to the point P , for from the z-axis
210 3. Geometry to ~r, and for ' from the x-axis to the projection of ~r to the x y plane. With the values 0 r < 1 0 , and ; < ' every point of space can be described. Coordinate surfaces are: Spheres with the origin 0 as center and with radius r = const, circular cones with = const, with vertex at the origin, and the z-axis as the axis and closed half-planes starting at the z-axis with ' = const. The intersection curves of these surfaces are the coordinate curves. The transformation formulas between Cartesian coordinates and spherical polar coordinates (see also Table 3.22) are: x = r sin cos ' y = r sin sin ' z = r cos (3.354a) px2 + y2 q y r = x2 + y2 + z2 = arctan z ' = arctan x : (3.354b) For the required distinction of cases with respect to ' see (3.287c), p. 191. Table 3.22 Connections between Cartesian, cylindrical, and spherical polar coordinates
Cartesian coordinates Cylindrical polar coordinates Spherical polar coordinates x= y= zp= x2 + y2 arctan y x =z px2 + y2 + z2 p2 2 arctan x z+ y arctan xy
= % cos ' = % sin ' =z =% =' =z p = %2 + z2 = arctan %z ='
= r sin cos ' = r sin sin ' = r cos = r sin =' = r cos =r = ='
7. Direction in Space
A direction in space can be determined by a unit vector ~t 0 (see 3.5.1.1, 6., p. 180) whose coordinates are the direction cosines, i.e., the cosines of the angles between the vector and the positive coordinate axes (Fig. 3.174) l = cos m = cos n = cos l2 + m2 + n2 = 1: (3.355a) The angle ' between two directions given by their direction cosines l1 , m1 , n1 and l2, m2 , n2 can be calculated by the formula cos ' = l1 l2 + m1 m2 + n1 n2: (3.355b) Two directions are perpendicular to each other if l1l2 + m1 m2 + n1n2 = 0: (3.355c)
3.5.3.2 Transformation of Orthogonal Coordinates 1. Parallel Translation
If the original coordinates are x, y, z, and the new coordinates are x0 , y0, z0 , and a, b, c are the coordinates of the new origin in the original coordinate system (Fig. 3.175), then we have x = x0 + a y = y0 + b z = z0 + c x0 = x ; a y0 = y ; b z0 = z ; c: (3.356)
3.5 Vector Algebra and Analytical Geometry 211 z
z g a
z'
t0 b
0
y' y
b x'
x
x
c
a
y
Figure 3.174
Figure 3.175
2. Rotation of the Cordinate Axes
If the direction cosines of the new axes are given as in Table 3.23 see also (Fig. 3.176), then we have for the old and new coordinates x = l1x0 + l2y0 + l3z0 x0 = l1 x + m1 y + n1 z y = m1x0 + m2 y0 + m3z0 y0 = l2 x + m2 y + n2 z z = n1x0 + n2 y0 + n3 z0 (3.357a) z0 = l3 x + m3 y + n3 z: (3.357b) The coecient matrix of the system (3.357a), which is called the rotation matrix D, and the determinant % of the transformation are l l l 0l l l 1 1 2 3 1 2 3 @ A D = m1 m2 m3 (3.357c) det D = % = m1 m2 m3 : (3.357d) n1 n2 n3 n1 n2 n3 Table 3.23 Notation for the direction cosines under coordinate transformation
With respect to the old axes x y z
x0 l1 m1 n1
Direction cosine of the new axes y0 l2 m2 n2
z0 l3 m3 n3
3. Properties of the Transformation Determinant
a) % = 1, with a positive sign if it remains left- or right-handed, as it was, and with negative sign if it changes its orientation. b) The sum of the squares of the elements of a row or column is always equal to one. c) The sum of the products of the corresponding elements of two di erent rows or columns is equal to zero (see 4.1.4, 9., p. 257). d) Every element can be written as the product of % = 1 and its adjoint (see 4.2.1, p. 259).
4. Euler's Angles
The position of the new coordinate system with respect to the old one can be uniquely determined by three angles which were introduced by Euler (Fig. 3.176). a) The nutation angle is the angle between the positive directions of the z-axis and the z0 -axis it has the limits 0 < . b) The precession angle is the0 angle between the positive direction of the x-axis and the intersection line K of the planes x y and x y0. The positive direction of K is chosen depending on whether the z-axis, the z0 -axis and K form a direction triplet with the same orientation as the coordinate axes (see 3.5.1.3, 2., p. 182). The angle is measured from the x-axis to the direction of the y-axis the limits
212 3. Geometry z
J z
P2(x2 ,y2 ,z2)
y'
d
z'
J x
y
x j '
P(x,y,z)
y
K
Figure 3.176
P1(x1 ,y1 ,z1) y x
Figure 3.177
are 0 < . c) The rotation angle ' is the angle between the positive x0-direction and the intersection line K it has the limits 0 ' < 2. If instead of functions of angles we use the letters cos = c1 cos = c2 cos ' = c3 sin = s1 sin = s2 sin ' = s3 (3.358a) then we get l1 = c2 c3 ; c1s2 s3 m1 = s2 c3 + c1 c2s3 n1 = s1s3 l2 = ;c2 s3 ; c1 s2c3 m2 = ;s2 s3 + c1c2 c3 n2 = s1c3 (3.358b) l3 = s1 s 2 m3 = ;s1 c2 n3 = c1:
5. Scalar Invariant
This is a scalar which keeps its value during translation and rotation. The scalar product of two vectors is a scalar invariant (see 3.5.1.5, 3., p. 184). A: The components of a vector ~a = fa1 a2 a3g are not scalar invariants, because they change their values during translation and rotation. q B: The length of a vector ~a = fa1 a2 a3 g, i.e., the quantity a21 + a22 + a23 , is a scalar invariant. C: The scalar product of a vector with itself is a scalar invariant: ~a~a = ~a2 = j~aj2 cos ' = j~a2 j, because ' = 0.
3.5.3.3 Special Quantities in Space 1. Distance Between Two Points
The distance between the points P1 (x1 y1 z1) and P2 (x2 y2 z2 ) in Fig. 3.177 is
q d = (x2 ; x1 )2 + (y2 ; y1)2 + (z2 ; z1 )2 : (3.359a) The direction cosines of the segment between the points can be calculated by the formulas cos = x2 ;d x1 cos = y2 ;d y1 cos = z2 ;d z1 : (3.359b)
2. Division of a Segment
The coordinates of the point P (x y z) dividing the segment between the points P1 (x1 y1 z1 ) and P2 (x2 y2 z2 ) in a given ratio P1P = m = (3.360) PP 2 n
3.5 Vector Algebra and Analytical Geometry 213
mx2 = x1 + x2 are given by the formulas x = nxn1 + (3.361a) +m 1+ my2 y1 + y2 (3.361b) mz2 z1 + z2 y = nyn1 + z = nzn1 + (3.361c) +m = 1+ +m = 1+ : The midpoint of the segment is given by xm = x1 +2 x2 ym = y1 +2 y2 zm = z1 +2 z2 : (3.362) The coordinates of the center of mass (often called incorrectly the center of gravity) of a system of n material points with mass mi are calculated by the following formulas, where the sum index i changes from 1 to nP: P P x" = Pmmixi y" = Pmmi yi z" = Pmmi zi : (3.363) i
z
0
i
P P2
P3 P1 y
x
Figure 3.178
i
3. System of Four Points
Four points P (x y z) , P1 (x1 y1 z1 ) , P2 (x2 y2 z2) and P3 (x3 y3 z3 ) can form a tetrahedron (Fig. 3.178) or they are in a plane. The volume of a tetrahedron can be calculated by the formula x y z 1 x ; x1 y ; y1 z ; z1 1 V = 6 xx12 yy12 zz12 11 = 16 x ; x2 y ; y2 z ; z2 (3.364) x ; x3 y ; y3 z ; z3 x3 y3 z3 1 ;!
;!
;!
where it has a positive value V > 0 if the orientation of the three vectors PP 1 , PP 2 , PP 3 is the same as the coordinate axes (see 3.5.1.3, 2., p. 182). Otherwise it is negative. The four points are in the same plane if and only if x y z 1 x y z 1 1 1 1 = 0 holds. (3.365) x2 y2 z2 1 x3 y3 z3 1
4. Equation of a Surface
Every equation F (x y z ) = 0 (3.366) corresponds to a surface with the property that the coordinates of every point P satisfy this equation. Conversely, every point whose coordinates satisfy the equation is a point of this surface. The equation (3.366) is called the equation of this surface. If there is no real point in the space satisfying equation (3.366), then there is no real surface. 1. The Equation of a Cylindrical Surface (see 3.3.4, p. 154) whose generating lines are parallel to the x-axis contains no x coordinate: F (y z) = 0. Similarly, the equations of the cylindrical surfaces with generating lines parallel to the y or to the z axes contain no y or z coordinates: F (x z) = 0 or F (x y) = 0 resp. The equation F (x y) = 0 describes the intersection curve between the cylinder and the x y plane. If the direction cosines, or the proportional quantities l, m, n of the generating line of a cylinder are given, then the equation has the form (3.367) F (nx ; lz ny ; mz) = 0: 2. The Equation of a Rotationally Symmetric Surface, i.e., a surface which is created by the rotation of a curve z = f (x) given in the x z plane around the z-axis (Fig. 3.179), will have the form
214 3. Geometry z=f
q
x2 + y 2 :
(3.368)
We can get the equations of rotationally symmetric surfaces also in the case of other variables similarly. The equation of a conical surface, whose vertex is at the origin (see 3.3.4, p. 156), has the form F (x y z) = 0 , where F is a homogeneous function of the coordinates (see 2.18.2.5, 4., p. 120).
5. Equation of a Space Curve
z
x
y
A space curve can be de ned by three parametric equations Figure 3.179 x = '1 (t) y = '2(t) z = '3 (t) : (3.369) To every value of the parameter t, which does not necessarily have a geometrical meaning, there corresponds a point of the curve. Another method to de ne a space curve is the determination by two equations F1(x y z) = 0 F2(x y z) = 0 : (3.370) Both de ne a surface. The space curve contains all points whose coordinates satisfy both equations, i.e., the space curve is the intersection curve of the given surfaces. In general, every equation in the form F1 + F2 = 0 (3.371) for arbitrary de nes a surface which goes through the considered curve, so it can substitute any of the equations (3.370).
3.5.3.4 Line and Plane in Space 1. Equations of the Plane
Every equation linear in the coordinates de nes a plane, and conversely every plane has an equation of rst degree.
1. General Equation of the Plane a) with coordinates: Ax + By + Cz + D = 0 (3.372a) ~ ~rN + D = 0 b) in vector form: (3.372b) ~ (A B C ) is perpendicular to the plane. In (Fig. 3.180) the intercepts a, b, and c where the vector N ~ is called the normal vector of the plane. Its direction cosines are are shown. The vector N A cos = p 2 2 2 cos = p 2 B 2 2 cos = p 2 C 2 2 : (3.372c)
A +B +C A +B +C A +B +C If D = 0 holds, the plane goes through the origin for A = 0, or B = 0, or C = 0 the plane is parallel to the x-axis, the y-axis, or the z-axis, respectively. If A = B = 0, or A = C = 0, or B = C = 0, then the plane is parallel to the x y plane, the x z plane or the y z plane, respectively.
2. Hessian Normal Form of the Equation of the Plane a) with coordinates: x cos + y cos + z cos ; p = 0 (3.373a) 0 ~ ~rN ; p = 0 b) in vector form: (3.373b) ~ 0 is the unit normal vector of the plane and p is the distance of the plane from the origin. The where N
Hessian normal form arises from the general equation (3.372a) by multiplying by the normalizing factor (3.373c) = N1 = pA2 + 1B 2 + C 2 with N = jN~ j:
For the scalar product of two vectors see 3.5.1.5, p. 183 and in ane coordinates see 3.5.1.6, 5., p. 185 for the vector equation of the plane see 3.5.1.7, p. 188.
3.5 Vector Algebra and Analytical Geometry 215
Here the sign of must be chosen opposite to that of D.
3. Intercept Form of the Equation of the Plane With the segments a, b, c, considering them with signs depending on where the plane intersects the coordinate axes (Fig. 3.180), we have x + y + z = 1: a b c
(3.374)
4. Equation of the Plane Through Three Points If the points are P1 (x1 y1 z1 ) , P2 (x2 y2 z2) , P3 (x3 y3 z3 ) , then we have x ; x y ; y z ; z 1 1 1 a) with coordinates: x2 ; x1 y2 ; y1 z2 ; z1 = 0 (3.375a) x3 ; x1 y3 ; y1 z3 ; z1 b) in vector form: (~r ; ~r1) (~r ; ~r2) (~r ; ~r3) = 0y: (3.375b) 5. Equation of a Plane Through Two Points and Parallel to a Line
The equation of the plane passing through the two points P1 (x1 y1 z1), P2 (x2 y2 z2 ) and being parallel ~ (l m n) is the following to the line with direction vector R x ; x y ; y z ; z 1 1 1 a) with coordinates: x2 ; x1 y2 ; y1 z2 ; z1 = 0 (3.376a) l m n ~ = 0y: b) in vector form: (~r ; ~r1) (~r ; ~r2) R (3.376b)
6. Equation of a Plane Through a Point and Parallel to Two Lines
~ 1 (l1 m1 n1) and R ~ 2 (l2 m2 n2 ), then we have: If the direction vectors of the lines are R x ; x y ; y z ; z 1 1 1 a) with coordinates: l1 m1 n1 = 0 l2 m2 n2 ~ 1R ~ 2 = 0y: b) in vector form: (~r ; ~r1) R
7. Equation of a Plane Through a Point and Perpendicular to a Line
(3.377a) (3.377b)
~ (A B C ), then we have: If the point is P1 (x1 y1 z1), and the direction vector of the line is N a) with coordinates: A (x ; x1 ) + B (y ; y1) + C (z ; z1 ) = 0 (3.378a) ~ =0 b) in vector form: (~r ; ~r1) N (3.378b) 8. Distance of a Point from a Plane If we substitute the coordinates of the point P (a b c) in the Hessian normal form of the equation of the plane (3.373a) x cos + y cos + z cos ; p = 0 (3.379a) we get the distance with sign = a cos + b cos + c cos ; p (3.379b) where > 0, if P and the origin are on di erent sides of the plane in the opposite case < 0 holds. 9. Equation of a Plane Through the Intersection Line of Two Planes The equation of a plane which goes through the intersection line of the planes given by the equations A1 x + B1 y + C1z + D1 = 0 and A2x + B2 y + C2z + D2 = 0 is
a) with coordinates: A1 x + B1 y + C1z + D1 + (A2 x + B2 y + C2 z + D2 ) = 0:
(3.380a)
For the mixed product of three vectors see 3.5.1.6, 2., p. 184 For the scalar product of two vectors see 3.5.1.5, p. 183 and in ane coordinates see 3.5.1.6, p. 185 for the equation of the plane in vector form see 3.5.1.6, p. 188. y
216 3. Geometry b) in vector form: ~r N~ 1 + D1 + (~r N~ 2 + D2) = 0:
(3.380b) Here is a real parameter, so (3.380a) and (3.380b) de ne a pencil of planes. Fig. 3.181 shows a pencil of planes with three planes. If takes all the values between ;1 and +1 in (3.380a) and (3.380b), we get all the planes from the pencil. For = 1 we get the equations of the planes bisecting the angle between the given planes if their equations are in normal form.
2. Two and More Planes in Space
1. Angle between Two Planes, General Case: The angle between two planes given by the
equations A1 x + B1 y + C1z + D1 = 0 and A2x + B2 y + C2 z + D2 = 0 can be calculated by the formula cos ' = q 2 A1 A22 + B21 B2 +2 C1C22 2 : (3.381a) (A1 + B1 + C1 ) (A2 + B2 + C2 ) ~ 1 + D1 = 0 and ~r N ~ 2 + D2 = 0, then we have: If the planes are given by vector equations ~r N ~1N ~2 cos ' = N with N1 = jN~ 1j and N2 = jN~ 2j: (3.381b) N1N2 z c g N a b b a 0
y
x
z
z
0
0 y
x
Figure 3.180
Figure 3.181
y
x
Figure 3.182
2. Intersection Point of Three Planes: The coordinates of the intersection point of three planes given by the three equations A1x + B1 y + C1z + D1 = 0 A2 x + B2 y + C2z + D2 = 0, and A3x + B3 y + C3z + D3 = 0, are calculated by the formulas %x y" = ;%y z" = ;%z with x" = ;% (3.382a) A B C % D B%C A D C A B D 1 1 1 1 1 1 1 1 1 1 1 1 % = A2 B2 C2 %x = D2 B2 C2 , %y = A2 D2 C2 , %z = A2 B2 D2 . (3.382b) A3 B3 C3 D3 B3 C3 A3 D3 C3 A3 B3 D3 Three planes intersect each other at one point if % 6= 0 holds. If % = 0 holds and at least one subdeterminant of second order is non-zero, then the planes is parallel to a line if every subdeterminant of second order is zero, then the planes have a common line.
3. Conditions for Parallelism and Orthogonality of Planes: a) Conditions for Parallelism: Two planes are parallel if A1 B1 C1 ~ ~ ~ (3.383) A2 = B2 = C2 or N1 N2 = 0 holds. b) Conditions for Orthogonality: Two planes are perpendicular to each other if A1A2 + B1 B2 + C1 C2 = 0 or N~ 1N~ 2 = 0 holds. (3.384) 4. Intersection Point of Four Planes: Four planes given by the equations A1 x + B1y + C1z + D1 =
0
A2 x + B2 y + C2z + D2 = 0 A3x + B3 y + C3 z + D3 = 0, and A4x + B4 y + C4z + D4 = 0 have a
3.5 Vector Algebra and Analytical Geometry 217
common point only if for the determinant A B C D A1 B1 C1 D1 2 2 2 2 =0 = A (3.385) 3 B3 C3 D3 A4 B4 C4 D4 holds. In this case we determine the common point from three equations. The fourth equation is superuous it is a consequence of the others. 5. Distance Between Two Parallel Planes: If two planes are parallel, and they are given by the equations Ax + By + Cz + D1 = 0 and Ax + By + Cz + D2 = 0 (3.386) then their distance is 1 ; D2 j d = p jD : (3.387) A2 + B 2 + C 2
3. Equation of a Line in Space
1. Equation of a Line in Space, General Case Because a line in space can be de ned as the intersection of two planes, it can be represented by a system of two linear equations. a) In component form: A1x + B1 y + C1z + D1 = 0 A2x + B2y + C2z + D2 = 0:
(3.388a)
b) In vector form: ~ 1 + D1 = 0 ~r N ~ 2 + D2 = 0: ~r N 2. Equation of a Line in Two Projecting Planes
(3.388b)
The two equations y = kx + a z = hx + b (3.389) de ne a plane each, and these planes go through the line and are perpendicular to the x y and the x z planes resp. (Fig. 3.182). We call them projecting planes. This representation cannot be used for lines parallel to the y z plane, so in this case we have to consider other projections to other coordinate planes. z
z
z P2(x2 ,y2 ,z2)
P1(x1 ,y1 ,z1) 0
R
0
P1(x1 ,y1 ,z1) 0
y
N
P1(x1 ,y1 ,z1) y
y x
x
x
Figure 3.183
Figure 3.184
Figure 3.185
3. Equation of a Line Through a Point Parallel to a Direction Vector
The equation (or the equation system) of a line passing through a point P1 (x1 y1 z1 ) parallel to a ~ (l m n) (Fig. 3.183) has the form direction vector R
a) in component representation and in vector form: x ; x1 = y ; y1 = z ; z1 (3.390a) l
m
n
b) in parametric form and vector form:
~ = ~0 (~r ; ~r1) R
(3.390b)
218 3. Geometry ~t x = x1 + l t y = y1 + m t z = z1 + n t (3.390c) ~r = ~r1 + R (3.390d) where the numbers x1 , y1, z1 are chosen such that (3.388a) are satis ed. The representation (3.390a) follows from (3.388a) with C C A 1 1 1 1 n = A1 B1 l = B m = (3.391a) B2 C2 C2 A2 A2 B2 ~ ~ ~ or in vector form R = N1 N2: (3.391b) 4. Equation of a Line Through Two Points The equation of a line through two points P1 (x1 y1 z1 ) and P2 (x2 y2 z2 ) (Fig. 3.184) is
in component form and in vector form: a) xx ;;xx1 = yy ;; yy1 = zz ;; zz1 (3.392a)
b) (~r ; ~r1) (~r ; ~r2 ) = ~0 : (3.392b) If for instance x1 = x2, the equations in component form are x1 = x2 , yy ;; yy1 = zz ;; zz1 . If x1 = x2 2 1 2 1 2
1
2
1
2
1
and y1 = y2 are both valid, the equations in component form are x1 = x2 , y1 = y2. 5. Equation of a Line Through a Point and Perpendicular to a Plane The equation of a line passing through the point P1 (x1 y1 z1 ) perpendicular to a plane given by the equation Ax + By + ~ + D = 0 (Fig. 3.185) is Cz + D = 0 or by ~r N
in component form and in vector form: a) x ; x1 = y ; y1 = z ; z1 (3.393a)
b) (~r ; ~r1) N~ = ~0: (3.393b) A B C If for instance A = 0 holds, the equations in component form have a similar form as in the previous case.
4. Distance of a Point from a Line Given in Component Form
For the distance d of the point M (a b c) from a line given in the form (3.390a) we have: 2 2 2 d2 = (a ; x1 ) m ; (b ; y1) l] + (b ; yl21)+nm;2(+c ;n2z1 ) m] + (c ; z1 ) l ; (a ; x1 ) n] : (3.394)
5. Smallest Distance Between Two Lines Given in Component Form
If the lines are given in the form (3.390a), their distance is x ; x y ; y z ; z 1 2 1 2 1 2 m1 n1 l1 l2 m2 n2 : d = s 2 2 l1 m1 m1 n1 n1 l1 2 l2 m2 + m2 n2 + n2 l2 If the determinant in the numerator is equal to zero, the lines intersect each other.
(3.395)
6. Intersection Points of Lines and Planes
1. Equation of the Line in Component Form The intersection point of a plane given by the equation Ax + By + Cz + D = 0 and a line given by x ; x1 = y ; y1 = z ; z1 has the coordinates x" = x1 ; l
y" = y1 ; m
l z" = z1 ; n
For the product of vectors see 3.5.1.5, p. 183
m with
n
(3.396a)
3.5 Vector Algebra and Analytical Geometry 219
D = AxA1 +l +ByB1 m+ +CzC1 + n :
(3.396b)
If A l + B m + C n = 0 holds, the line is parallel to the plane. If Ax1 + By1 + Cz1 + D = 0 is also valid, the line lies in the plane. 2. Equation of the Line in Two Projecting Planes The intersection point of a plane given by the equation Ax + By + Cz + D = 0 and a line given by y = kx + a and z = hx + b has the coordinates a + C b + D , y" = kx" + a, z" = hx" + b. (3.397) x" = ; AB + Bk+Ch If A + B k + C h = 0 holds, the line is parallel to the plane. If Ba + Cb + D = 0 is also valid, the line lies in the plane. 3. Intersection Point of Two Lines If the lines are given by y = k1x + a1 z = h1x + b1 and y = k2x + a2 z = h2 x + b2 the coordinates of the intersection point, if any exists, are: a1 = b2 ; b1 y" = k1a2 ; k2a1 z" = h1b2 ; h2b1 : (3.398a) x" = ak2 ; h1 ; h2 k1 ; k2 h1 ; h2 1 ; k2 The intersection point exists only if (a1 ; a2 )(h1 ; h2 ) = (b1 ; b2 )(k1 ; k2): (3.398b) Otherwise the lines do not intersect each other.
7. Angles between Planes and Lines
1. Angle between Two Lines ; y1 = z ; z1 and x ; x2 = a) General Case: If the lines are given by the equations x ;l x1 = y m n1 l2 1 1 y ; y2 = z ; z2 or in vector form by (~r ; ~r ) R ~ 1 = ~0 and (~r ; ~r2) R ~ 2 = ~0 then for the angle 1 m n 2
2
between them we have n1 n2 cos ' = q 2 l1 l2 2+ m12m2 + or 2 (l1 + m1 + n1) (l2 + m22 + n22)
(3.399a)
~ 1R ~2 ~ ~ cos ' = R R R with R1 = jR1j and R2 = jR2j:
(3.399b)
+ Cn or sin ' = q 2 Al2 + Bm 2 (A + B + C ) ( l 2 + m 2 + n2 )
(3.402a)
~N ~ ~ ~ sin ' = R RN with R = jRj and N = jNj:
(3.402b)
1 2
b) Conditions of Parallelism: Two lines are parallel if l1 = m1 = n1 or R~ R~ = ~0: (3.400) 1 2 l2 m2 n2 c) Conditions of Orthogonality: Two lines are perpendicular to each other if l1l2 + m1 m2 + n1n2 = 0 or R~1R~2 = 0: (3.401) 2. Angle Between a Line and a Plane a) If the line and the plane are given by the equations x ;l x1 = y ;my1 = z ;n z1 and Ax + By + Cz + ~ = ~0 and ~r N ~ + D = 0, we get the angle by the formulas D = 0 or in vector form by (~r ; ~r1 ) R
220 3. Geometry b) Conditions of Parallelism: A line and a plane are parallel if ~N ~ = 0: A l + B m + C n = 0 or R c) Conditions of Orthogonality: A line and a plane are orthogonal if A = B = C or R ~ N ~ = ~0: l m n
(3.403) (3.404)
3.5.3.5 Surfaces of Second Order, Equations in Normal Form
1. Central Surfaces
We get the following equations, which are also called the normal form of the equations of surfaces of second order, from the general equations of surfaces of second order (see 3.5.3.6, 1., p. 223) by putting the center at the origin. Here the center is the midpoint of the chords passing through it. The coordinate axes are the symmetry axes of the surfaces, so the coordinate planes are also the planes of symmetry.
2. Ellipsoid
With the semi-axes a, b, c (Fig. 3.186) the equation of an ellipsoid is x2 + y2 + z2 = 1: (3.405) a2 b2 c2 We distinguish the following special cases: a) Compressed Ellipsoid of Revolution (Lens Form): a = b > c (Fig. 3.187). b) Stretched Ellipsoid of Revolution (Cigar Form): a = b < c (Fig. 3.188). c) Sphere: a = b = c so that x2 + y2 + z2 = a2 is valid. The two forms of the ellipsoid of revolution arise by rotating an ellipse in the x z plane with axes a and c around the z-axis, and we get a sphere if we rotate a circle around any axis. If a plane goes through an ellipsoid, the intersection gure is an ellipse in a special case it is a circle. The volume of the ellipsoid is V = 4abc : (3.406) 3 z
z
c a
0
b y
y x
x
Figure 3.186
Figure 3.187
3. Hyperboloid
a) Hyperboloid of One Sheet (Fig. 3.189): With a and b as real and c as imaginary semi-axes
the equation is x2 + y 2 ; z 2 = 1 a2 b2 c2
(for generator lines see p. 222): (3.407) b) Hyperboloid of Two Sheets (Fig. 3.190): With c as real and a, b as imaginary semi-axes the equation2 is 2 2 x + y ; z = ;1 : (3.408) a2 b2 c2
3.5 Vector Algebra and Analytical Geometry 221
Intersecting it by a plane parallel to the z-axis we get a hyperbola in the case of both types of hyperboloids. In the case of a hyperboloid of one sheet the intersection can also be two lines intersecting each other. The intersection gures parallel to the x y plane are ellipses in both cases. For a = b the hyperboloid can be represented by rotation of a hyperbola with semi-axes a and c around the axis 2c. This is imaginary in the case of a hyperboloid of one sheet, and real in the case of that of two sheets. z
z
z
0
y x
c
b a
y
x
y
x
Figure 3.188
Figure 3.189
4. Cone (Fig. 3.191)
Figure 3.190
If the vertex is at the origin, the equation is x2 + y 2 ; z 2 = 0 : (3.409) a2 b2 c2 As a direction curve we can consider an ellipse with semi-axes a and b, whose plane is perpendicular to the z-axis at a distance c from the origin. The cone in this representation can be considered as the asymptotic cone of the surfaces x2 + y2 ; z2 = 1 (3.410) a2 b2 c2 whose generator lines approach in nitely closely both hyperboloids at in nity (Fig. 3.192). For a = b we have a right circular cone (see 3.3.4, 9., p. 156). z
b
c
a
z
x
5. Paraboloid
y
Figure 3.191
x
y
Figure 3.192
Because a paraboloid has no center, we suppose in the following that the vertex is at the origin, the z-axis is its symmetry axis, and the x z plane and the y z plane are symmetry planes.
222 3. Geometry a) Elliptic Paraboloid (Fig. 3.193):
2 2 z = xa2 + yb2 : (3.411) The plane sections parallel to the z-axis result in parabolas as intersection gures those parallel to the x y plane result in ellipses. The volume of a paraboloid which is cut by a plane pperpendicularpto the z-axis at a distance h from the origin is given in (3.412). The parameters a = a h and b = b h are the half axis of the intersecting ellipse at height h. V = 21 abh : (3.412) The volume is half of the volume of an elliptic cylinder with the same upper surface and altitude. b) Paraboloid of Revolution: For a = b we have a paraboloid of revolution. We get it by rotating the z = x2 =a2 parabola of the x z plane around the z-axis. c) Hyperbolic Paraboloid (Fig. 3.194): 2 2 z = xa2 ; yb2 : (3.413) The intersection gures parallel to the y z plane or to the x z plane are parabolas parallel to the x y plane they are hyperbolas or two intersecting lines.
z
z
h
x
0
0 y
Figure 3.193
6. Rectilinear Generators of a Ruled Surface
y
x
Figure 3.194
These are straight lines lying completely in this surface. Examples are the generators of the surfaces of the cone and cylinder.
a) Hyperboloid of One Sheet (Fig. 3.195): x2 + y2 ; z2 = 1: a2 b2 c2
The hyperboloid of one sheet has two families of rectilinear generators with equations x + z = u 1 + y u x ; z = 1 ; y a c b a c b x + z = v 1 ; y v x ; z = 1 + y a c b a c b where u and v are arbitrary quantities.
b) Hyperbolic Paraboloid (Fig. 3.196): 2 2 z = xa2 ; yb2 :
(3.414) (3.415a) (3.415b)
(3.416)
The hyperbolic paraboloid also has two families of rectilinear generators with equations x ; y = v v x + y = z: (3.417b) x + y = u u x ; y = z (3.417a) a b a b a b a b
3.5 Vector Algebra and Analytical Geometry 223
The quantities u and v are again arbitrary values. In both cases, there are two straight lines passing through every point of the surface, one from each family. We denote only one familiy of straight lines in Fig. 3.195 and Fig. 3.196.
Figure 3.195
7. Cylinder
Figure 3.196
x2 + y2 = 1: a2 b2 2 2 b) Hyperbolic Cylinder (Fig. 3.198): xa2 ; yb2 = 1: c) Parabolic Cylinder (Fig. 3.199): y2 = 2px:
a) Elliptic Cylinder (Fig. 3.197):
z
a
0
(3.418) (3.419) (3.420) z
z
b
x
y
x
y
0 y
Figure 3.197
Figure 3.198
x
Figure 3.199
3.5.3.6 Surfaces of Second Order or Quadratic Surfaces, General Theory 1. General Equation of a Surface of Second Order 2 2 2
a11 x + a22 y + a33 z + 2a12xy + 2a23 yz + 2a31 zx + 2a14 x + 2a24 y + 2a34 z + a44 = 0: (3.421)
2. Telling the Type of Second-Order Surface from its Equation
We can determine the type of a second-order surface from its equation by the signs of its invariants %, , S , and T from Tables 3.24 and 3.25. Here we can nd the names with the normal form of the equation of the surfaces, and every equation can be transformed into a normal form. We cannot determine the coordinates of any real point from the equation of imaginary surfaces, except the vertex of the imaginary cone, and the intersection line of two imaginary planes.
224 3. Geometry
3. Invariants of a Surface of Second Order
If we substitute aik = aki, then we have a a a a a a a 11 a12 a13 a14 a11 a12 a13 22 23 24 % = aa21 (3.422a) = 21 22 23 a a a 31 32 33 34 a31 a32 a33 a41 a42 a43 a44 S = a11 + a22 + a33 (3.422c) T = a22 a33 + a33 a11 + a11 a22 ; a223 ; a231 ; a212 : During translation or rotation of the coordinate system these invariants do not change. Table 3.24 Type of surfaces of second order with 6= 0 (central surfaces)
0
S > 0 T > 01
S
and T not both
Hyperboloid of two sheets x2 + y2 ; z2 = ;1 a2 b2 c2
Imaginary ellipsoid x2 + y2 + z2 = ;1 a2 b2 c2
Hyperboloid of one sheet x2 + y2 ; z2 = 1 a2 b2 c2
Imaginary cone (with real vertex) x2 + y2 + z2 = 0 a2 b2 c2 1 For the quantities S , and T see p. 224.
(3.422d)
>0
Ellipsoid x2 + y2 + z2 = 1 a2 b2 c2
=0
(3.422b)
Cone x2 + y2 ; z2 = 0 a2 b2 c2
Table 3.25 Type of surfaces of second order with = 0 (paraboloid, cylinder and two planes)
6= 0 =0
2
< 0 (here T > 0)2
> 0 (here T < 0)
Elliptic paraboloid x2 + y2 = z a2 b2
Hyperbolic paraboloid x2 ; y2 = z a2 b2
Cylindrical surface with a second-order curve as a directrix whose type de nes di erent cylinders: For T > 0 imaginary elliptic, for T < 0 hyperbolic, and for T = 0 parabolic cylinder, if the surface does not split into two real, imaginary, or coincident planes. The condition for splitting is: a11 a12 a14 a11 a13 a14 a22 a23 a24 a21 a22 a24 + a31 a33 a34 + a32 a33 a34 = 0 a41 a42 a44 a41 a43 a44 a42 a43 a44
For the quantities % and T see p. 224.
3.6 Dierential Geometry 225
3.6 Dierential Geometry
In di erential geometry planar curves and curves and surfaces in space are discussed by the methods of di erential calculus. Therefore we suppose that the functions describing the curves and surfaces are continuous and continuously di erentiable as many times as necessary for discussion of the corresponding properties. The absence of these assumptions is allowed only at a few points of the curves and surfaces. These points are called singular points. During the discussion of geometric con gurations with their equations we distinguish properties depending on the choice of the coordinate system, such as intersection points with the coordinate axes, the slope or direction of tangent lines, maxima, minima and invariant properties independent of coordinate transformations, such as inection points, curvature, and cyclic points. There are also local properties, which are valid only for a small part of the curves and surfaces as the curvature and di erential of arc or area of surfaces, and there are properties belonging to the whole curve or surface, such as number of vertices, and arclength of a closed curve.
3.6.1 Plane Curves
3.6.1.1 Ways to Dene a Plane Curve 1. Coordinate Equations
A plane curve can be analytically de ned in the following ways.
In Cartesian coordinates: a) Implicit: F (x y ) = 0 b) Explicit: y = f (x) c) Parametric Form: x = x(t) y = y(t): In Polar Coordinates: = f ('):
(3.423) (3.424) (3.425) (3.426)
2. Positive Direction on a Curve
If a curve is given in the form (3.425), the positive direction is de ned on it in which a point P (x(t) y(t)) of the curve moves for increasing values of the parameter t. If the curve is given in the form (3.424), then the abscissa can be considered as a parameter (x = x y = f (x)), so we have the positive direction for increasing abscissa. For the form (3.426) the angle can be considered as a parameter ' (x = f (') cos ' , y = f (') sin '), so we have the positive direction for increasing ', i.e., counterclockwise. Fig.3.200a, b, c: A: x = t2 y = t3, B: y = sin x, C: = a'. y
y
0 a)
x
x
0
0
x
c)
b)
Figure 3.200
3.6.1.2 Local Elements of a Curve
Depending on whether a changing point P on the curve is given in the form (3.424), (3.425) or (3.426), its position is de ned by x, t or '. We denote a point arbitrarily close to P by N with parameter values x + dx, t + dt or ' + d'.
226 3. Geometry
1. Di erential of Arc
tan ge nt
If s denotes the length of the curve from a xed point A to the point P , we can express the in nitesimal _ increment %s =PN approximately by the di erential ds of the arclength, the dierential of arc: 8v !2 u > u > t1 + dy dx for the form (3.424), > (3.427) > < dx %s ds = > q 0 2 02 for the form (3.425), (3.428) > qx + y dt > : 2 + 02 d' for the form (3.426). (3.429) p p A: y = sin x ds = p1 + cos2 x dx . B: x = t2 y = t3 ds = t 4 + 9t2 dt. C: = a' ds = a 1 + '2 d'. y
N
P µ P
P no rm al
Figure 3.201
0
2. Tangent and Normal
Figure 3.202
ϕ
α
x
Figure 3.203
1. Tangent at a Point P to a Curve is a line in the limiting position of the secants PN for N ! P the normal is a line through P which is perpendicular to the tangent here (Fig. 3.201). 2. The Equations of the Tangent and the Normal are given in Table 3.26 for the three cases (3.423), (3.424), and (3.425). Here x, y are the coordinates of P , and X , Y are the coordinates of the points of the tangent and normal. The values of the derivatives should be calculated at the point P .
Type of equation
Table 3.26 Tangent and normal equations
Equation of the tangent
(3.423)
@F (X ; x) + @F (Y ; y) = 0 @x @y
(3.424)
dy (X ; x) Y ; y = dx
(3.425)
Y ;y = X ;x y0 x0
Equation of the normal X ;x = Y ;y
@F @F @x @y 1 Y ; y = ; dy (X ; x) dx x0(X ; x) + y0(Y ; y) = 0
Examples for equations of the tangent and normal for the following curves: A: Circle x2 + y2 = 25 at the point P (3 4): a) Equation of the tangent: 2x(X ; x) + 2y(Y ; y) = 0 or Xx + Y y = 25 considering that the point P lies on the circle: 3X + 4Y = 25. b) Equation of the normal: X2;x x = Y 2;y y or Y = xy X at the point P : Y = 43 X .
3.6 Dierential Geometry 227
B: Sine curve y = sin x at the point 0(0 0): a) Equation of the tangent: Y ; sin x = cos x(X ; x) or Y = X cos x + sin x ; x cos x at the point (0 0): Y = X . b) Equation of the normal: Y ; sin x = ; cos1 x (X ; x) or Y = ;X sec x + sin x + x sec x at the point (0 0): Y = ;X . C: Curve with x = t2 y = t3 at the point P (4 ;8) t = ;2: 3 2 a) Equation of the tangent: Y ;2 t = X ; t or Y = 3 tX ; 1 t3 at the point P : Y = ;3X + 4. 3t 2 2t 2 2 2 b) Equation of the normal: 2t (X ; t ) + 3t (Y ; t3 ) = 0 or 2X + 3tY = t2 (2 + 3t2 ) at the point P : X ; 3Y = 28. 3. Positive Direction of the Tangent and Normal of the Curve If the curve is given in one of the forms (3.424), (3.425), (3.426), the positive directions on the tangent and normal are de ned in the following way: The positive direction of the tangent is the same as on the curve at the point of contact, and we get the positive direction on the normal from the positive direction of the tangent by rotating it counterclockwise around P by an angle of 90 (Fig. 3.202). The tangent and the normal are divided into a positive and a negative half-line by the point P . 4. The Slope of the Tangent can be determined a) by the angle of slope of the tangent , between the positive directions of the axis of abscissae and the tangent, or b) if the curve is given in polar coordinates, by the angle , between the radius vector OP (OP = ) and the positive direction of the tangent (Fig. 3.203). For the angles and the following formulas are valid, where ds is calculated according to (3.427){(3.429): dy cos = dx sin = dy (3.430a) tan = dx ds ds cos = d sin = d' : tan = d (3.430b) ds ds d' sin = p cos x 2 tan = cos x, cos = p 1 2 , A: y = sin x 1 + cos x 1 + cos x 2 3 t 2 3 cos = p , sin = p 3t 2 B: x = t y = t , tan = 2 , 4 + 9 t2 4 + 9t sin = p ' 2 . tan = ', cos = p 1 2 , C: = a', 1+' 1+' 5. Segments of the Tangent and Normal, Subtangent and Subnormal (Fig. 3.204) a) In Cartesian Coordinates for the de nitions in form (3.424), (3.425):
q (segment of the tangent ), (3.431a) PT = yy0 1 + y02 q (segment of the normal ), (3.431b) PN = y 1 + y02 P 0T = yy0 (subtangent) (3.431c) P 0N = jyy0j (subnormal): (3.431d) b) In Polar Coordinates for the de nitions in form (3.426): q (segment of the polar tangent ), (3.432a) PT 0 = 0 2 + 02
228 3. Geometry q (segment of the polar normal ), (3.432b) PN 0 = 2 + 02 2 OT 0 = 0 (polar subtangent) (3.432c) ON 0 = j 0 j (polar subnormal): (3.432d) q A: y = cosh x y0 = sinh x 1 + y02 = cosh x PT = j cosh x coth xj PN = jcosh2 xj P 0T = j coth xj P 0N = j sinh x cosh q q q xj. q B: = a' 0 = a 2 + 02 = a 1 + '2 PT 0 = a' 1 + '2 PN 0 = a 1 + '2 OT 0 = 2 a' ON 0 = a. N'
y
90 0
o
T P'
Γ2
y
P N
x
β 0
T'
Figure 3.204
α1
Γ1
y
P
α2
P1 P2 x
0
x
Figure 3.205
Figure 3.206
6. Angle Between Two Curves The angle between two curves ;1 and ;2 at their intersection point P is de ned as the angle between their tangents at the point P (Fig. 3.205). By this de nition
we have reduced the calculation of the angle to the calculation of the angle between two lines with slopes ! ! df2 1 k1 = tan 1 = df (3.433a) k (3.433b) 2 = tan 2 = dx P dx P where y = f1(x) is the equation of ;1 and y = f2 (x) is the equation of ;2 , and we have to calculate derivatives at the point P . We get with the help of the formula (3.434) tan = tan(1 ; 2 ) = tan 2 ; tan 1 : 1 + tan 1 tan 2 p Determinepthe! angle between the parabolas!y = x and y = x2 at the point P (1 1): (x2 ) 2 ; tan 1 3 tan 1 = ddxx = 21 tan 2 = d dx = 2 tan = 1tan + tan tan = 4 . x=1
x=1
3. Convex and Concave Part of a Curve
1
2
If a curve is given in the explicit form y = f (x), then we can examine a small part containing the point P if the curve is concave up or down here, except of course if P is an inection point or a singular point (see 3.6.1.3, p. 231). If the second derivative f 00(x) > 0 (if it exists), then the curve is concave up, i.e., in the direction of positive y (point P2 in Fig. 3.206). If f 00(x) < 0 holds (point P1), then the curve is concave down. In the case if f 00(x) = 0 holds, we should check if it is an inection point. y = x3 (Fig. 2.15b) y00 = 6x, for x > 0 the curve is concave up, for x < 0 concave down.
4. Curvature and Radius of Curvature
1. Curvature of a Curve The curvature K of a curve at the point P is the limit of the ratio of the angle between the positive tangent directions at the points P and N (Fig. 3.207) and the arclength
3.6 Dierential Geometry 229
_ _ PN for PN ! 0: : K = _lim _ (3.435) PN !0 PN The sign of the curvature K depends on whether the curve bends toward the positive half of the normal (K > 0) or toward the negative half of it (K < 0) (see 3.6.1.1, 2., p. 227). In other words the center of curvature for K > 0 is on the positive side of the normal, for K < 0 it is on the negative side. Sometimes the curvature K is considered only as a positive quantity. Then we have to take the absolute value of the limit above. 2. Radius of Curvature of a Curve The radius of curvature R of a curve at the point P is the reciprocal value of the absolute value of the curvature: R = j1=K j: (3.436) The larger the curvature K is at a point P the smaller the radius of curvature R is. A: For a circle with radius a the curvature K = 1=a and the radius of curvature R = a are constant for every point. B: For a line we have K = 0 and R = 1. y N
0
P α
ds
3. Formulas for Curvature_and Radius of Curvature With the notation = d and PN = ds (Fig. 3.207) we have
=δ dα
α α+d
in general:
x
Figure 3.207
De nition as in (3.424): K = 2
K = d ds
ds : R = d
(3.437)
For the di erent de ning formulas of curves in 3.6.1.1, p. 225 we get di erent expressions for K and R:
d2y dx2
!233=2 dy 41 + 5 dx x0 y0 00 00 x y De nition as in (3.425): K = 3=2 2 x0 + y02 F F F Fxx Fxy Fx yx yy y F F 0 De nition as in (3.423): K = 2x y 2 3=2 Fx + Fy 2 02 00 De nition as in (3.426): K = +2 2 02;3 ( + ) =2
2 !233=2 4 dy 1 + dx 5 R = d2y dx2 x0 2 + y02 3=2 R = x0 y0 x00 y00 F 2 + F 2 3=2 x y R = F F F Fxx Fxy Fx yx yy y Fx Fy 0 2 02 3=2 R = 2( + 2+ 0 2 ;) 00 :
(3.438)
(3.439)
(3.440)
(3.441)
230 3. Geometry 6 t(4 + 9t2 )3=2 2 D: = a' K = a1 (''2 ++1)23=2 :
A: y = cosh x K = cosh1 2 x 2 C: y2 ; x2 = a2 K = (x2 +ay2)3=2
B: x = t2 y = t3 K =
5. Circle of Curvature and Center of Curvature
1. Circle of Curvature at the point P is the limiting position of the circles passing through P and two points of the curve from its neighborhood N and M , for N ! P and M ! P (Fig. 3.208). It
goes through the point of the curve and here it has the same rst and the same second derivative as the curve. Therefore it ts the curve at the point of contact especially well. It is also called the osculating circle. Its radius is the radius of curvature. It is obvious that it is the reciprocal value of the absolute value of the curvature. 2. Center of Curvature The center C of the circle of curvature is the center of curvature of the point P . It is on the concave side of the curve, and on the normal of the curve. 3. Coordinates of the Center of Curvature We can determine the coordinates (xC yC ) of the center of curvature for curves with de ning equations in 3.6.1.1, p. 225 from the following formulas. 2 !3 ! dy 41 + dy 2 5 dy 2 1 + dx dx De nition as in (3.424): xC = x ; yC = y + d2dx (3.442) d2 y y :
dx2 dx2 0 02 02 0 02 02 y ( x + y ) x ( x De nition as in (3.425): xC = x ; x0 y0 yC = y + x0 +y0y ) : x00 y00 x00 y00 2 02 ' + 0 sin ') De nition as in (3.426): xC = cos ' ; ( + 2 )(+ 2cos 02 ; 00 2 02 ' ; 0 cos ') : yC = sin ' ; ( + 2 )(+ 2sin 0 2 00 2 2 ; Fx Fx + Fy Fy Fx2 + Fy2 De nition as in (3.423): xC = x + F F F yC = y + F F F : xx xy x xx xy x Fyx Fyy Fy Fyx Fyy Fy Fx Fy 0 Fx Fy 0 These formulas can be transformed into the form xC = x ; R sin yC = y + R cos or xC = x ; R dy yC = y + R dx ds ds (Fig. 3.209), where R should be calculated as in (3.438){(3.441). P
N R C
0
y
M P(x,y) 0
Figure 3.208
C(xC ,yC) a
a
Figure 3.209
x
Figure 3.210
(3.443)
(3.444) (3.445)
(3.446) (3.447)
3.6 Dierential Geometry 231
3.6.1.3 Special Points of a Curve
Now we discuss only the points invariant during coordinate transformations. To determine maxima and minima see 6.1.5.3, p. 391.
1. Inection Points and the Rules to Determine Them
Inection points are the points of the curve where the curvature changes its sign (Fig. 3.210) while a tangent exists. The tangent line at the inection point intersects the curve, so the curve is on both sides of the line in this neighborhood. At the inection point K = 0 and R = 1 hold.
1. Explicit Form (3.424) of the Curve y = f (x) a) A Necessary Condition for the existence of an inection point is the zero value of the second
derivative f 00(x) = 0 (3.448) if it exists at the inection point (for the case of non-existant second derivative see b)). In order to determine the inection points for existing second derivative we have to consider all the roots of the equation f 00(x) = 0 with values x1 x2 : : : xi : : : xn, and substitute them into the further derivatives. If for a value xi the rst non-zero derivative has odd order, there is an inection point here. If the considered point is not an inection point, because for the rst non-disappearing derivative of k-th order, k is an even number, then for f (k)(x) < 0 the curve is concave up for f (k)(x) > 0 it is concave down. If we do not check the higher-order derivatives, for instance in the case they do not exist, see point b). b) A Sucient Condition for the existence of an inection point is the change of the sign of the second derivative f 00(x) while traversing from the left neighborhood of this point to the right, if also a tangent exists here, of course. So the question, of whether the curve has an inection point at the point with abscissa xi , can be answered by checking the sign of the second derivative traversing the considered point: If the sign changes during the traverse, there is an inection point. (Since xi is a root of the second derivative, the function has a rst derivative, and consequently the curve has a tangent.) This method can also be used in the case if y00 = 1, together with the checking of the existence of a tangent line, e.g. in the case of a vertical tangent line. 2 2 A: y = 1 +1 x2 f 00(x) = ;2 (11 ;+ 3xx2 )3 x12 = p13 f 000 (x) = 24x (11+;xx2 )4 f 000 (x12) 6= 0: ! ! 1 3 . p Inection points: A p1 3 B ; 3 4 3 4 B: y = x4 f 00(x) = 12x2 x1 = 0 f 000 (x) = 24x f 000 (x1 ) = 0 f IV (x) = 24 there is no inection point. 5 2 1 C: y = x 3 y0 = 53 x 3 y00 = 109 x; 3 for x = 0 we have y00 = 1. As the value of x changes from negative to positive, the second derivative changes its sign from \;" to \+", so the curve has an inection point at x = 0. Remark: In practice, if from the shape of the curve the existence of inection points follows, for instance between a minimum and a maximum with continuous derivatives, then we determine only the points xi and do not check the further derivatives. 2. Other De ning Forms The necessary condition (3.448) for the existence of an inection point in the case of the de ning form of the curve (3.424) will have the analytic form for the other de ning formulas as follows: x0 y0 a) De nition in parametric form as in (3.425): (3.449) x00 y00 = 0:
b) De nition in polar coordinates as in (3.426):
2 + 2 02 ; 00 = 0:
(3.450)
232 3. Geometry Fxx Fxy Fx F (x y) = 0 and Fyx Fyy Fy = 0: (3.451) Fx Fy 0 In these cases the solution system gives the possible coordinates of inection points. A: x = a t ; 21 sin t y = a 1 ; 12 cos t (curtated cycloid (Fig. 2.68b), p. 100) 1 x0 y0 a2 2 ; cos t sin t a2 x00 y00 = 4 sin t cos t = 4 (2 cos t ; 1) cos tk = 2 tk = 3 + 2k (k = 0 1 2 : : :). The curve has an in nite number of inection points for the parameter values tk . B: = p1' 2 + 2 02 ; 00 = '1 + 2'1 3 ; 4'3 3 = 4'1 3 (4'2 ; 1): The inection point is at the angle ' = 1=2 . Fxx 2 0 2x C: x2 ; y2 = a2 (hyperbola). = 0 ;2 ;2y = 8x2 ; 8y2. The equations x2 ; y2 = a2 2x ;2y 0 and 8(x2 ; y2) = 0 contradict each other, so the hyperbola has no inection point.
c) De nition in implicit form as in (3.423):
y
y
B C
A 0
a)
D
x
0
E
x
b)
Figure 3.211 Vertices are the points of the curve where the curvature has a maximum or a minimum. The ellipse haspfor instance four vertices A, B , C , D, the curve of the logarithm function has one vertex at E (1= 2 ; ln 2=2) (Fig. 3.211). The determination of vertices is reduced to the determination of the extreme values of K or, if it is simpler, the extreme values of R. For the calculation we can use the formulas from (3.438){(3.441).
2. Vertices
3. Singular Points
Singular point is a general notion for di erent special points of a curve. 1. Types of Singular Points The points a), b), etc. to j) correspond to the representation in Fig. 3.212. a) Double Point: At a double point the curve intersects itself (Fig. 3.212a). b) Isolated Point: An isolated point satis es the equation of the curve but it is separated from the curve (Fig. 3.212b). c), d) Cuspidal point: At a cuspidal point or briey a cusp the orientation of the curve changes according to the position of the tangent we distinguish a cusp of the rst kind and a cusp of the second kind (Fig.3.212c,d). e) Tacnode or point of osculation: At the tacnode the curve contacts itself (Fig. 3.212e). f) Corner point: At a corner point the curve suddenly changes its direction but in contrast to a cusp there are two di erent tangents for the two di erent branches of the curve here (Fig.3.212f). g) Terminal point: At a terminal point the curve terminates (Fig.3.212g). h) Asymptotic point: In the neighborhood of an asymptotic point the curve usually winds in and out
3.6 Dierential Geometry 233
or around in nitely many times, while it approaches itself and the point arbitrarily close (Fig.3.212h). i), j) More Singularities: It is possible that the curve has two or more such singularities at the same point (Fig. 3.212i,j).
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
Figure 3.212
2. Determination of the Tacnode, Corner, Terminal, and Asymptotic Points Singularities of these types occur only on the curves of transcendental functions (see 3.5.2.3, 5., p. 194). The corner point corresponds to a nite jump of the derivative dy=dx. Points where the function terminates correspond to the points of discontinuity of the function y = f (x) with a nite jump or to a direct termination. Asymptotic points can be determined in the easiest way in the case of curves given in polar coordinates as = f ('). If for ' ! 1 or ' ! ;1 the limit lim = 0 is equal to zero, the pole is an asymptotic point. A: The origin is a corner point for the curve y = x 1 (Fig. 6.2c) . 1 + ex B: The points (1 0) and (1 1) are points of discontinuity of the function y = 1 1 (Fig. 2.8). 1 + e x;1 C: The logarithmic spiral = aek ' (Fig. 2.75) has an asymptotic point at the origin. 3. Determination of Multiple Points (Cases from a) to e), and i), and j)) Double points, triple points, etc. are denoted by the general term multiple points. To determine them, we start with the equation of the curve of the form F (x y) = 0. A point A with coordinates (x1 y1) satisfying the three equations F = 0, Fx = 0, and Fy = 0 is a double point if at least one of the three derivatives of second order Fxx, Fxy , and Fyy does not vanish. Otherwise A is a triple point or a point with higher multiplicity. The properties of a double point depend on the sign of the Jacobian determinant Fxy : % = FFxx (3.452) yx Fyy x=x1 y=y1
Case % < 0: For % < 0 the curve intersects itself at the point A the slopes of the tangents at A are
the roots of the equation Fyy k2 + 2Fxy k + Fxx = 0: Case % > 0: For % > 0 A is an isolated point. Case % = 0: For % = 0 A is either a cusp or a tacnode the slope of the tangent is tan = ; FFxy : yy
(3.453) (3.454)
For more precise investigation about multiple points we can translate the origin to the point A, and rotate so that the x-axis becomes a tangent at A. Then from the form of the equation we can tell if it
234 3. Geometry is a cusp of rst or second order, or if it is a tacnode. A: F (x y) (x2 + y2)2 ; 2a2(x2 ; y2) = 0 (Lemniscate, Fig. 2.66, p. 99) Fx = 4x(x2 + y2 ; a2 ) Fy = 4y(x2 + y2 + a2) the equation system Fx = 0, Fy = 0 results in the three solutions (0 0), (a 0), from which only the rst one satis es the condition F = 0. Substituting (0 0) into the second derivatives we have Fxx = ;4a2 , Fxy = 0, Fyy = +4a2 % = ;16a4 < 0, i.e., at the origin the curve intersects itself the slopes of the tangents are tan = 1, their equations are y = x. B: F (x y) x3 + y3 ; x2 ; y2 = 0 Fx = x(3x ; 2), Fy = y(3y ; 2) among the points (0 0), (0 2=3), (2=3 0), and (2=3 2=3) only the rst one belongs to the curve further Fxx = ;2, Fxy = 0, Fyy = ;2, % = 4 > 0, i.e., the origin is an isolated point. C: F (x y) (y ; x2)2 ; x5 = 0. The equations Fx = 0, Fy = 0 result in only one solution (0 0), it also satis es the equation F = 0. Furthermore % = 0 and tan = 0, so the origin p is a cusp of the second kind. This can be seen from the explicit form of the equation y = x2 (1 x). y is not de ned for x < 0, while for 0 < x < 1 both values of y are positive at the origin the tangent is horizontal.
4. Algebraic Curves of Type F (x y) = 0 F (xy) Polynomial in x and y
If the equation does not contain any constant term and any rst-degree term, the origin is a double point. The corresponding tangents can be determined by making the sum of the second-degree terms equal. For the lemniscate (Fig. 2.66 p. 99) we get the equations y = x from x2 ; y2 = 0. If the equation does not contain even second-degree terms, then the origin is a triple point.
3.6.1.4 Asymptotes of Curves 1. Denition
An asymptote is a straight line to which the curve approaches while it moves away from the origin (Fig. 3.213). b) The curve can approach the line from one side a) (Fig. 3.213a), or it can intersect it again and again Figure 3.213 (Fig. 3.213b). Not every curve which goes in nitely far from the origin (in nite branch of the curve) has an asymptote. We call for instance the entire part of an improper rational expression an asymptotic approximation (see 1.1.7.2, p. 15).
2. Functions Given in Parametric Form x = x(t) y = y(t)
To determine the equation of the asymptote rst we have to know the values ti such that if t ! ti either x(t) ! 1 or y(t) ! 1 (or both) holds. We have the following cases: a) x(ti) ! 1 but y(ti) = a 6= 1 : y = a: The asymptote is a horizontal line. (3.455a) b) y(ti) ! 1 but x(ti ) = a 6= 1 : x = a: The asymptote is a vertical line: (3.455b) y(t) and b = lim y(t);kx(t)]. c) If both y(ti) and x(ti ) tend to 1, then we calculate the limits k = tlim !ti x(t) t!ti If both exist, the equation of the asymptote is y = kx + b: (3.455c) m x = cos t y = n(tan t ; t) t1 = 2 t2 = ; 2 , etc. Determine the asymptote at t1: n (sin t ; t cos t) = n , x(t1 ) = y(t1) = 1 k = lim m m t ! = 2
n m sin t ; t cos t ; 1 = ; n y = n x ; n . For the b = lim n(tan t ; t) ; m cos t = n lim cos t 2 m 2 t ! =2 t ! =2
3.6 Dierential Geometry 235
n x ; n . second asymptote, etc. we get similarly y = m 2
3. Functions Given in Explicit Form y = f (x)
The vertical asymptotes are at the points of discontinuity where the function f (x) has an in nite jump (see 2.1.5.3, p. 57) the horizontal and oblique asymptotes have the equation f (x) b = lim f (x) ; kx]: y = kx + b with k = xlim (3.456) !1 x x!1
4. Functions Given in Implicit Polynomial Form F (x y) = 0
1. To determine the horizontal and vertical asymptotes we choose the highest-degree terms with degree
m from the polynomial expression in x and y, we separate them as a function (x y) and solve the equation (x y) = 0 for x and y: (x y) = 0 yields x = '(y) y = (x): (3.457) The values y1 = a for x ! 1 give the horizontal asymptotes y = a the values x1 = b for y ! 1 the vertical ones x = b. 2. To determine the oblique asymptotes we substitute the equation of the line y = kx + b into the equation F (x y), then we order the resulting polynomial according to the powers of x: F (x kx + b) f1(k b)xm + f2 (k b)xm;1 + : (3.458) We get the parameters k and b, if they exist, from the equations f1(k b) = 0 f2(k b) = 0: (3.459) x3 + y3 ; 3axy = 0 (Cartesian folium Fig. 2.59, p. 94). From the equation F (x kx + b) (1 + k3)x3 +3(k2b ; ka)x2 + we get, according to (3.459), 1+ k3 = 0 and k2 b ; ka = 0 with the solutions k = ;1, b = ;a, so the equation of the asymptote is y = ;x ; a.
3.6.1.5 General Discussion of a Curve Given by an Equation
Curves given by their equations (3.423){(3.426) are investigated in order to know their properties and shapes.
1. Construction of a Curve Given by an Explicit Function y = f (x)
a) Determination of the domain (see 2.1.1, p. 47). b) Determination of the symmetry of the curve with respect to the origin or to the y-axis checking if the function is odd or even (see 2.1.3.3, p. 50).
c) Determination of the behavior of the function at 1 by calculating the limits x!;1 lim f (x) and lim f (x) (see 2.1.4.7, p. 53).
x!+1
d) Determination of the points of discontinuity (see 2.1.5.3, p. 57). e) Determination of the intersection points with the y-axis and with the x-axis calculating f (0) and solving the equation f (x) = 0.
f) Determination of maxima and minima and nding the intervals of monotonicity where the function is increasing or decreasing.
g) Determination of inection points and the equations of tangents at these points (see 3.6.1.3,
p. 231). With these data we can sketch the graph of the function, and if it is needed, we can calculate some substitution values to make it more appropriate. 2 Sketch the graph of the function y = 2x + 32x ; 4 : x a) The function is de ned for all x except x = 0.
236 3. Geometry b) There is no symmetry. c) For x ! ;1 we have y ! 2, and obviously y = 2 ; 0, i.e., approach from below, while x ! 1 we also have y ! 2, but y = 2 + 0, an approach from above. d) x = 0 is a point of discontinuity such that the function from left and also from right tends to ;1, because y is negative for small values of x. e) Because f (0) = 1 holds, there is no intersection point with the y-axis, and from f (x) = 2x2 + 3x ; 4 = 0 the intersection points with the x-axis are at x1 0:85 and x2 ;2:35. f) A maximum is at x = 8=3 2:66 and here y 2:56. g) An inection point is at x = 4, y = 2:5 with the slope of the tangent line tan = ;1=16. h) After sketching the graph of the function based on these data (Fig. 3.214) we can calculate the intersection point of the curve and the asymptote, which is at x = 4=3 1:33 and y = 2.
2. Construction of a Curve Given by an Implicit Function F (x y) = 0
There are no general rules for this case, because depending of the actual form of the function di erent steps can be or cannot be performed. If it is possible, the following steps are recommended: a) Determination of all the intersection points with the coordinate axes. b) Determination of the symmetry of the curve, so that we replace x by ;x and y by ;y. c) Determination of maxima and minima with respect to the x-axis and then interchanging x and y also with respect to the y-axis (see 6.1.5.3, p. 391). d) Determination of the inection points and the slopes of tangents there (see 3.6.1.3, p. 231). e) Determination of singular points (see 3.6.1.3, 3., p. 232). f) Determination of vertices (see 3.6.1.3, 2., p. 232) and the corresponding circles of curvature (see 3.6.1.2, 4., p. 228). It often happens that the curve's arc can hardly be distinguish from the circular segment of the circle of curvature on a relatively large segment. g) Determination of the equations of asymptotes (see 3.6.1.4, p. 234) and the position of the curve branches related to the asymptotes.
G2 -6 -4
-2
2
4
6 x
-2
Figure 3.214
P1 P -2 -1
2 C
C1
P P' P''
0
1
Figure 3.215
2 x
evolvent
3
G1
2
evolute
y
y 4
Figure 3.216
3.6.1.6 Evolutes and Evolvents 1. Evolute
The evolute is a second curve which is the locus of the centers of circles of curvature of the rst curve (see 3.6.1.3, 5., p. 230) at the same time it is the envelope of the normals of the rst curve (see also 3.6.1.7, p. 237). The parametric form of the evolute we can get from (3.442), (3.443), (3.444) for the center of curvature if xC and yC are considered as running coordinates. If it is possible to eliminate the parameter (x, t or ') from (3.442), (3.443), (3.444), we get the equation of the evolute in Cartesian coordinates. Determine the evolute of the parabola y = x2 (Fig. 3.215). From
3.6 Dierential Geometry 237 2 2 2 X = x ; 2x(1 +2 4x ) = ;4x3 Y = x2 + 1 +24x = 1 +26x follows with X and Y as running 2=3 coordinates the evolute Y = 1 + 3 X . 2 4
2. Evolvent or Involute
The evolvent of a curve ;2 is a curve ;1 , whose evolute is ;2. Here every normal PC of the evolvent is a _ tangent of the evolute (Fig. 3.215), and the length of arc CC 1 of the evolute is equal to the increment of the radius of curvature of the evolvent: _ (3.460) CC 1= P1C 1 ; PC: These properties show that the evolvent ;1 can be regarded as the curve traced by the end of a stretched thread unspooling from ;2 . A given evolute corresponds to a family of curves, where every curve is determined by the initial length of the thread (Fig. 3.216). We get the equation of the evolute by the integration of a system of di erential equations corresponding to its evolute. For the equation of the evolvent of the circle see 2.14.4, p. 104. The catenoid is the evolute of the tractrix the tractrix is the evolvent of the catenoid (see 2.15.1, p. 106). a
a
K a+Da a)
K
a+Da
b)
a) d) b) c)
Figure 3.217
Figure 3.218
3.6.1.7 Envelope of a Family of Curves 1. Characteristic Point
Consider the one-parameter family of curves with equation F (x y ) = 0: (3.461) Every two in nitely close curves of this family corresponding to the values of parameter and + % have points K of nearest approach . Such a point is either a point of intersection of the curves () and ( + %) or a point of the curve () whose distance from the curve ( + %) along the normal is an in nitesimal quantity of higher order than % (Fig. 3.217a,b). For % ! 0 the curve ( +%) tends to the curve (), where in some cases the point K approaches a limit position, the characteristic point.
2. Geometric Locus of the Characteristic Points of a Family of Curves
With the equation (3.461) this can be one or more curves. They are formed by the points of nearest approach or by the characteristic points of the family (Fig. 3.218a), or they form an envelope of the family, i.e., a curve which contacts tangentially every curve of the family (Fig. 3.218b). Also a combination of these two cases is possible (Fig. 3.218c,d).
3. Equation of the Envelope
The equation of the envelope can be calculated from (3.461), where can be eliminated from the following equation system: @F = 0: (3.462) F =0 @
238 3. Geometry Determine the equation of the family of straight lines arising when the ends of a line segment AB with jAB j = l are sliding along the coordinate axes (Fig. 3.219a). The equation of the family of curves is: y x l sin + l cos = 1 or F x cos + y sin ; l sin cos = 0 @F = ;x sin + y cos ; l cos2 + l sin2 = 0. @ By eliminating we have x2=3 + y2=3 = l2=3 as an envelope, which is an astroid (Fig. 3.219b, see also p. 102).
y
y A
a) 0
0
x
l B x
b)
Figure 3.219
3.6.2 Space Curves
3.6.2.1 Ways to Dene a Space Curve 1. Coordinate Equations
To de ne a space curve we have the following possibilities: a) Intersection of Two Surfaces: F (x y z) = 0 (x y z) = 0: b) Parametric Form: x = x(t) y = y(t) z = z(t) with t as an arbitrary parameter mostly we use t = x, y or z. c) Parametric Form: x = x(s) y = y(s) z = z(s) with the arc length s between a xed point A and the running point P : u Zt v u !2 dy !2 dz !2 s = t dx dt + dt + dt dt: t0
(3.463) (3.464) (3.465a) (3.465b)
2. Vector Equations
With ~r as radius vector of an arbitrary point of the curve (see 3.5.1.1, 6., p. 181) the equation (3.464) can be written in the form ~r = ~r(t) where ~r(t) = x(t)~i + y(t)~j + z(t)~k (3.466) and (3.465a) in the form ~r = ~r(s) where ~r(s) = x(s)~i + y(s)~j + z(s)~k: (3.467)
3. Positive Direction
This is the direction of increasing parameter t for a curve given in the form (3.464) and (3.466) for (3.465a) and (3.467) it is the direction in which the arclength increases.
3.6.2.2 Moving Trihedral 1. Denitions
We can de ne three lines and three planes at every point P of a space curve, apart from singular points. They intersect each other at the point P , and they are perpendicular to each other (Fig.3.220): 1. Tangent is the limiting position of the secants PN for N ! P (Fig.3.221). 2. Normal Plane is a plane perpendicular to the tangent. Every line passing through the point P and contained by this plane is called a normal of the curve at the point P . 3. Osculating Plane is the limiting position of the planes passing through three neighboring points M , P and N , for N ! P and M ! P . The tangent line is contained by the osculating plane.
3.6 Dierential Geometry 239 rectifying plane
binormal
osculating plane
b P t
N P
n
M
principal normal normal plane
tangent
Figure 3.221
Figure 3.220 4. Principal Normal is the intersection line of the normal and the osculating plane, i.e., it is the normal contained by the osculating plane. 5. Binormal is the line perpendicular to the osculating plane. 6. Rectifying Plane is the plane spanned by the tangent and binormal lines. 7. Moving Trihedral The positive directions on the lines tangent, principal normal and binormal are de ned as follows: a) On the tangent line it is given by the positive direction of the curve the unit tangent vector ~t has this direction. b) On the principal normal it is given by the sign of the curvature of the curve, and given by the unit normal vector ~n. c) On the binormal it is de ned by the unit vector ~b = ~t ~n (3.468) where the three vectors ~t, ~n, and ~b form a right-handed rectangular coordinate system, which is called the moving trihedral.
2. Position of the Curve Related to the Moving Trihedral
For the usual points of the curve the space curve is on one side of the rectifying plane at the point P , and intersects both the normal and osculating planes (Fig.3.222a). The projections of a small segment of the curve at the point P on the three planes have approximately the following shapes: 1. On the osculating plane it is similar to a quadratic parabola (Fig.3.222b). 2. On the rectifying plane it is similar to a cubic parabola (Fig.3.222c). 3. On the normal plane it is similar to a semicubical parabola (Fig.3.222d). If the curvature or the torsion of the curve are equal to zero at P or if P is a singular point, i.e., if x0 (t) = y0(t) = z0 (t) = 0 hold, then the curve may have a di erent shape. n
b
t a)
n
b
t
t b)
b
c)
Figure 3.222
n d)
240 3. Geometry
3. Equations of the Elements of the Moving Trihedral
1. The Curve is De ned in the Form (3.463) For the tangent see (3.469), for the normal plane see (3.470): Y ;y Z;z X;x X ;x Y ;y Z ;z @F @F @F @F @F = @F @F = @F @F : (3.469) @x @y @z = 0: (3.470) @y @z @z @x @x @y @ @ @ @ @ @ @ @ @ @x @y @z @y @z @x @y @z @x Here x y z are the coordinates of the point P of the curve and X Y Z are the running coordinates of the tangent or the normal plane the partial derivatives belong to the point P . 2. The Curve is De ned in the Form (3.464, 3.466) In Table 3.27 the coordinate and vector equations belonging to the point P are given with x y z and also with ~r. The running coordinates and ~ . The derivatives with respect to the radius vector of the running point are denoted by X Y Z and R the parameter t refer to the point P . 3. The Curve is De ned in the Form (3.465a, 3.467) If the parameter is the arclength s, for the tangent and binormal, and for the normal and osculating plane the same equations are valid as in case 2, we just replace t by s. The equations of the principal normal and the rectifying plane will be simpler (Table 3.28).
3.6.2.3 Curvature and Torsion 1. Curvature of a Curve
The curvature of a curve at the point P is a number which de∆t N scribes the deviation of the curve from a straight line in the very t+ ∆t close neighborhood of this point. + t The exact de nition is (Fig.3.223): ∆t t P %~t d~t K = _lim _ = ds : (3.471) Figure 3.223 PN !0 PN 1. Radius of Curvature The radius of curvature is the reciprocal value of the curvature: R = K1 : (3.472)
2. Formulas to calculate K and R a) If the curve is de ned in the form (3.465a): 2 q ~ d r K = = x002 + y002 + z002 ds2
(3.473)
where the derivatives are with respect to s. b) If the curve is de ned in the form (3.464): ! ! ! d~r 2 d2~r 2 ; d~r d2~r 2 02 02 02 002 002 002 2 2 dt (x0 x00 + y0y00 + z0 z00 )2 : K 2 = dt !23 dt dt = (x + y + z )(x (+x02y + +y02z+ )z; 02 )3 d~r dt (3.474) The derivatives are calculated here with respect to t.
3.6 Dierential Geometry 241
Table 3.27 Vector and coordinate equations of accompanying con gurations of a space curve
Vector equation r R~ = ~r + d~ dt
Coordinate equation Tangent:
X ;x = Y ;y = Z ;z x0 y0 z0
Normal plane: x0 (X ; x) + y0(Y ; y) + z0 (Z ; z) = 0 Osculating plane: X ; x Y ; y Z ; z 2 ~ ; ~r) d~r d ~r2 = 0 1 x0 y0 z0 = 0 (R dt dt x00 y00 z00 Binormal: 2~r ! d~ r d Xy0 ;z0x = Yz0 ;xy0 = Zx0;yz0 R~ = ~r + dt dt2 00 00 00 00 00 00 y z z x x y Rectifying plane: X ; x Y ; y Z ; z 2~r ! d~ r d~ r d 1 ~ x0 y0 z0 = 0 (R ; ~r) dt dt dt2 = 0 l m n with l = y0z00 ; y00z0 m = z0 x00 ; z00 x0 n = x0 y00 ; x00 y0 Principal normal: 2 ! ~R = ~r + d~r d~r d ~r2 Xy0;z0x = Yz0;x0y = Zx0;y0z dt dt dt m n n l l m
~ ; ~r) d~r = 0 (R dt
~ position vector of the accomp. con guration ~r position vector of the space curve, R 1 For the mixed product of three vectors see 3.5.1.6, p. 184 Tabelle 3.28 Vector and coordinate equations of accompanying con gurations as functions of the arclength
Element of trihedral Vector equation 2 Principal normal R~ = ~r + d ~r
Coordinate equation
X ;x = Y ;y = Z ;z x00 y00 z00 x00 (X ; x) + y00(Y ; y) + z00 (Z ; z) = 0
ds2 Rectifying plane =0 ~ position vector of the accomp. con guration ~r position vector of the space curve, R 2 ~ ; ~r) d ~r2 (R ds
242 3. Geometry 3. Helix The equations
z
x = a cos t y = a sin t z = bt (3.475) describe a so-called helix (Fig.3.224) as a right screw . If the observer is looking from the positive direction of the z-axis, which is at the same time the axis of the screw, then the screw climbs in a counter-clockwise direction. A helix winding itself in the opposite orientation is called a left screw. Determine the curvature of the helix (3.475). We replace the parameter p t by s = t a2 + b2 . Then we have x = a cos p 2s 2 , y = a sin p 2s 2 , 0 P a +b a +b t 2 + b2 P' bs a a A z = p 2 2 , and according to (3.473), K = a2 + b2 R = a . Both x a +b quantities K and R are constants. Figure 3.224 Another method, without the parameter transformation in (3.474), produces the same result.
y
2. Torsion of a Curve
The torsion of a curve at the point P is a number which describes the deviation of the curve from a plane curve in the very close neighborhood of this point. The exact de nition is (Fig.3.225): ~ ~ T = _lim %_b = ddsb : (3.476) PN !0 PN The radius of torsion is = T1 : (3.477)
∆b
b+∆b N
b P
Figure 3.225
1. Formulas for Calculating T and a) If the curve is de ned in the form (3.465a):
x0 y 0 z 0 x00 y00 z00 r d2~r d3~r = x000 y000 z000 T = 1 = R2 d~ ds ds2 ds3 (x002 + y002 + z002 ) where the derivatives are taken with respect to s. b) If the curve is de ned in the form (3.464): x0 y0 z0 x00 y00 z00 d~r d2~r d3~r 000 000 000 T = 1 = R2 dt 2 33 = R2 (x02x+ yy02 +z z02 )3 dt2!dt d~r dt
!
(3.478)
(3.479)
where R should be calculated by (3.472) and (3.473). The torsion calculated by (3.478), (3.479) can be positive or negative. In the case T > 0 an observer standing on the principal normal parallel to the binormal sees that the curve has a right turn in the case T < 0 it has a left turn. The torsion of a helix is constant. For the right screw R or the left screw L the torsion is For the mixed product of three vectors see 3.5.1.6, 2., p. 184.
3.6 Dierential Geometry 243
2 TR = a +a b
2 !2
;a sin t a cos t b ;a cos t ;a sin t 0 a sin t ;a cos t 0
(;a sin t)2 + (a cos t)2 + b2 ]3
3. Frenet Formulas
=
b a2 + b2
2 2 = a + b TL = ; 2 b 2 . b a +b
We can express the derivatives of the vectors ~t, ~n, and ~b by the Frenet formulas: ~t ~b d~b ~n d~t ~n d~n ds = R ds = ; R + ds = ; : Here R is the radius of curvature, and is the radius of torsion.
(3.480)
4. Darboux Vector
The Frenet formulas (3.480) can be represented in the clearly arranged form d~t = ~d ~t d~n = ~d ~n d~b = ~d ~b: ds ds ds Here ~d is the Darboux vector, which has the form ~d = 1~t + 1 tb ~ R :
(3.481) (3.482)
Remarks: 1. By the help of the Darboux vector the Frenet formulas can be interpreted in the sense of kinematics
(see 3.4]). 2. The modulus of the Darboux vector equals the so-called total curvator of a space curve: s 1 = R2 + 12 = j~dj: (3.483)
3.6.3 Surfaces
3.6.3.1 Ways to Dene a Surface 1. Equation of a Surface
Surfaces can be de ned in di erent ways: a) Implicit Form : F (x y z) = 0: (3.484) b) Explicit Form : z = f (x y): (3.485) c) Parametric Form : x = x(u v) y = y(u v) z = z(u v): (3.486) ~ ~ ~ ~r = ~r(u v) with ~r = x(u v)i + y(u v)j + z(u v)k: d) Vector Form : (3.487) If the parameters u and v run over all allowed values we get the coordinates and the radius vectors of all points of the surface from (3.486) and (3.487). The elimination of the parameters u and v from the parametric form (3.486) yields the implicit form (3.484). The explicit form (3.485) is a special case of the parametric form with u = x and v = y. The equation of the sphere in Cartesian coordinates, parametric form, and vector form (Fig. 3.227): x = a cos u sin v y = a sin u sin v z = a cos v (3.488b) x2 + y2 + z2 ; a2 = 0 (3.488a) ~ ~ ~ ~r = a(cos u sin vi + sin u sin vj + cos vk): (3.488c)
244 3. Geometry
2. Curvilinear Coordinates on a Surface
If a surface is given in the form (3.486) or (3.487), and we change the values of the parameter u while the other parameter v = v0 is xed, the points ~r(x y z) describe a curve ~r = ~r(u v0) on the surface. If we substitute for v di erent but xed values v = v1 v = v2 : : : v = vn one after the other, we get a family of curves on the surface. Because when moving along a curve with v = const only u is changing, we call this curve the u-line (Fig. 3.226). Analogously we get another family of curves, the v-lines, by varying v and keeping u = const xed with u1 u2 : : : un. This way we de ne a net of coordinate lines on the surface (3.486), where the two xed numbers u = ui and v = vk are the curvilinear or Gauss coordinates of the point P on the surface. If a surface is given in the form (3.485), the coordinate lines are the intersection curves of the surface with the planes x = const and y = const. With equations in implicit form F (u v) = 0 or with the parametric equations u = u(t) and v = v(t) of these coordinates, we can de ne curves on the surfaces. In the parametric equations of the sphere (3.488b,c) u means the geographical longitude of a point P , and v means its polar distance. The v lines are here the meridians APB the u lines are the parallel circles CPD (Fig. 3.227).
3.6.3.2 Tangent Plane and Surface Normal 1. Denitions
1. Tangent Plane The precise general mathematical de nition of the tangent plane is rather complicated, so we restrict our investigation to the case, when the surface is de ned by two parameters. Suppose, for a neighborhood of the point P (x y z), the mapping (u v) ! ~r(u v) is invertible, the @~r and ~r = @~r are continuous, and not parallel to each other. Then we partial derivatives ~ru = @u v @v
call P (x y z) a regular point of the surface. If P is regular, then the tangents of all curves passing through P , and having a tangent here, are in the same plane, and this plane is called the tangent plane of the surface at P . If this happens, the partial derivatives ~ru ~rv are parallel (or zero) only for certain parametrizations of the surface. If they are parallel for every parametrization, the point is called a singular point (see 3., p. 245). z A
C
u0
P
u1 u2
P v2
v 0 u
F(u,v)=0 v1
v0
x
P'
D
surface normal
N
y
ru u-line
P
rv v-line
tangent plane
B
Figure 3.226 Figure 3.227 Figure 3.228 2. Surface Normal The line perpendicular to the tangent plane going through the point P is called surface normal at the point P (Fig. 3.228). 3. Normal Vector The tangent plane is spanned by two vectors, by the tangent vectors @~r ~r = @~r ~ru = @u (3.489a) v @v of the u- and v-lines. The vector product of the tangent vectors ~ru ~rv is a vector in the direction of the
3.6 Dierential Geometry 245
surface normal. Its unit vector N~ 0 = j ~~rru ~~rrv j (3.489b) u v is called the normal vector. Its direction to one or other side of the surface depends on which variable is the rst and which one is the second coordinate among u and v. 2. Equations of the Tangent Plane and the Surface Normal (see Table 3.29) A: For the sphere with equation (3.488a) we get a) as tangent plane: 2x(X ; x) + 2y(Y ; y) + 2z(Z ; z) = 0 or xX + yY + zZ ; a2 = 0 (3.490a) b) as surface normal: X2;x x = Y 2;y y = Z 2;z z or Xx = Yy = Zz : (3.490b) B: For the sphere with equation (3.488b) we get a) as tangent plane: X cos u sin v + Y sin u sin v + Z cos v = a (3.490c) X Y Z b) as surface normal: cos u sin v = sin u sin v = cos v : (3.490d)
3. Singular Points of the Surface
If for a point with coordinates x = x1, y = y1, z = z1 of the surface with equation (3.484) all the equalities @F = 0 @F = 0 @F = 0 F (x y z) = 0 (3.491) @x @y @z are ful lled, i.e., if every rst-order derivative is zero, then the point P (x1 y1 z1) is called a singular point. All tangents going through here do not form a plane, but a cone of second order with the equation @ 2 F (X ; x )2 + @ 2 F (Y ; y )2 + @ 2 F (Z ; z )2 + 2 @ 2 F (X ; x )(Y ; y ) 1 1 1 1 1 @x2 @y2 @z2 @x@y @ 2 F (Y ; y )(Z ; z ) + 2 @ 2 F (Z ; z )(X ; x ) = 0 + 2 @y@z (3.492) 1 1 1 1 @z@x where the derivatives belong to the point P . If also the second derivatives are equal to zero, we have a more complicated type of singularity. Then the tangents form a cone of third or even higher order.
3.6.3.3 Line Elements of a Surface 1. Di erential of Arc
Consider a surface given in the form (3.486) or (3.487). Let P (u v) an arbitrary point and N (u + du v + _ dv) another one close to P , both on the surface. The arclength of the arc segment PN on the surface can be approximately calculated by the dierential of an arc or the line element of the surface with the formula ds2 = E du2 + 2F du dv + G dv2 (3.493a) where the three coecients !2 ! ! @y 2 + @z 2 F = ~r ~r = @x @x + @y @y + @z @z E = ~ru2 = @x + u v @u @u @u @u @v @u @v @u @v !2 !2 !2 @y @z @x (3.493b) G = ~rv2 = @v + @v + @v are calculated at the point P . The right-hand side (3.493a) is called the rst quadratic fundamental form of the surface.
246 3. Geometry A: For the sphere given in the form (3.488c) we have: E = a2 sin2 v
F = 0 G = a2
ds2 = a2 (sin2 v du2 + dv2):
B: For a surface given in the form (3.485) we get E = 1 + p2
F = pq
@z G = 1 + q2 with p = @x
@z : q = @y
(3.494) (3.495)
Table 3.29 Equations of the tangent plane and the surface normal
Type of equation (3.484) (3.485)
Tangent plane @F (X ; x) + @F (Y ; y) @x @y + @F ( Z ; z) = 0 @z
Z ; z = p(X ; x) + q(Y ; y) X ; x Y ; y Z ; z @x @y @z @u @u @u = 0 @x @y @z @v @v @v
Surface normal X ;x = Y ;y = Z ;z @F @F @F @x @y @z X ;x = Y ;y = Z ;z p q ;1
X ;x = Y ;y = Z ;z @y @z @z @x @x @y @u @u @u @u @u @u @y @z @z @x @x @y @v @v @v @v @v @v ~ ;~r)r~u r~v = 0 1 ~ = ~r + (r~u r~v ) ( R R (3.487) ~ ;~r)N ~ =0 ~ = ~r + N ~ or (R or R In this table x y z and ~r are the coordinates and radius vector of the points of the curve P X Y Z and R~ are the running coordinates and radius vectors of the points of the tangent @z q = @z and N~ is the normal vector. plane and surface normal furthermore p = @x @y 1 For the mixed product of three vectors see 3.5.1.6, 2., p. 184 (3.486)
2. Measurement on the Surface
1. The Arclength of a surface curve u = u(t), v = v(t) for t0 t t1 is calculated by the formula !2 !2 u Zt1 v u L = ds = tE du + 2F du dv + G dv dt: dt dt dt dt t0 t0 Zt1
r2 Pa
r1 v-line
u-line
Figure 3.229
(3.496)
2. The Angle Between Two Curves ~r1 = ~r(u1(t) v1 (t)) and ~r2 = ~r(u2(t) v2 (t)) on the surface ~r = ~r(u v) is the angle between their tangents with the direction vectors ~r_ 1 and ~r_ 2 (Fig. 3.229). It is given by the formula ~~ cos = qr_ 1 2r_ 2 2 ~r_ 1 ~r_ 2 ~ 2~ 2 ~ ~ + ~v_ 1~u_ 2) + G~v_ 1~v_ 2 (3.497) = q 2 E u_ 1 u_ 2 + F (u_ 1 v_ 22 q 2: 2 E ~u_ 1 + 2F ~u_ 1~v_ 1 + G~v_ 1 E ~u_ 2 + 2F ~u_ 2~v_ 2 + G~v_ 2
3.6 Dierential Geometry 247
Here the coecients E , F and G are calculated at the point P and ~u_ 1, ~u_ 2 , ~v_ 1 and ~v_ 2 represent the rst derivatives of u1(t), u2(t), v1 (t) and v2 (t), calculated for the value of the parameter t at the point P . If the numerator of (3.497) vanishes, the two curves are perpendicular to each other. The condition of orthogonality for the coordinate lines v = const and u = const is F = 0. 3. The Area of a Surface Patch S bounded by an arbitrary curve which is on the surface can be calculated by the double integral Z p (3.498b) with dS = EG ; F 2du dv: S = dS (3.498a) (S )
dS is called the surface element or element of area. The calculation of length, angle, and area on a surface is possible with (3.496, 3.497, 3.498a,b) if the coecients E , F , and G of the rst fundamental form are known. So the rst quadratic fundamental form de nes a metric on the surface.
3. Applicability of Surfaces by Bending
If a surface is deformed by bending, without stretching, compression or tearing, then its metric remains unchanged. In other words, the rst quadratic fundamental form is invariant under bendings. Two surfaces having the same rst quadratic fundamental form can be rolled onto each other. N
P Q
C
Γ
N
Q
P Q
Cnorm C1 N
n
N c)
b)
3.6.3.4 Curvature of a Surface
C2 N
C
Cnorm
n a)
P
a
Figure 3.230
1. Curvatures of Curves on a Surface
If di erent curves ; are drawn through a point P of the surface (Fig. 3.230), their radii of curvature at the point P are related as follows: 1. The Radius of Curvature of a curve ; at the point P is equal to the radius of curvature of a curve C , which is the intersection of the surface and the osculating plane of the curve ; at the point P (Fig. 3.230a). 2. Meusnier's Theorem For every plane section curve C of a surface (Fig. 3.230b) the radius of curvature can be calculated by the formula ~ ): = R cos(~n N (3.499) Here R is the radius of curvature of the normal section Cnorm, which goes through the same tangent ~ of the surface normal 0 is a regularization parameter . The normal equations for (4.120) are: (ATA + I)x = ATb: (4.121) The matrix of coecients of this linear equation system is positive de nite and regular for > 0, but the appropriate choice of the regularization parameter is a dicult problem (see 4.6]).
4.5 Eigenvalue Problems for Matrices 4.5.1 General Eigenvalue Problem
Let A and B be two square matrices of size (n n). Their elements can be real or complex numbers. The general eigenvalue problem is to determine the numbers and the corresponding vectors x 6= 0 satisfying the equation Ax = Bx: (4.122) The number is called an eigenvalue , the vector x an eigenvector corresponding to . An eigenvector is determined up to a constant factor, because if x is an eigenvector corresponding to , so is cx (c = constant) as well. In the special case when B = I holds, where I is the unit matrix of order n, i.e., Ax = x or (A ; I)x = 0 (4.123) the problem is called the special eigenvalue problem . We meet this form very often in practical problems, especially with a symmetric matrix A, and we will discuss it in detail in the following. More information about the general eigenvalue problem can be found in the literature (see 4.14]).
4.5.2 Special Eigenvalue Problem 4.5.2.1 Characteristic Polynomial
The eigenvalue equation (4.123) yields a homogeneous system of equations which has non-trivial solutions x 6= 0 only if det (A ; I) = 0: (4.124a)
4.5 Eigenvalue Problems for Matrices 279
By the expansion of det (A ; I) = 0 we get
a11 ; a12 a13 a1n a a22 ; a23 a2n det (A ; I) = ..21 ... ... ... ... . an1 an2 an3 ann ; = Pn() = (;1)nn + an;1n;1 + + a1 + a0 = 0:
(4.124b) So the determination of the eigenvalues is equivalent to the solution of a polynomial equation. This equation is called the characteristic equation the polynomial Pn() is the characteristic polynomial . Its roots are the eigenvalues of the matrix A. For an arbitrary square matrix A of size (n n) the following statements hold: Case 1: The matrix A(nn) has exactly n eigenvalues 1, 2, . . . , n, because a polynomial of degree n has n roots if they are considered with their multiplicity. The eigenvalues of a Hermitian matrix are real numbers, in other cases the eigenvalues can also be complex numbers. Case 2: If all the n eigenvalues are di erent, then the matrix A(nn) has exactly n linearly independent eigenvectors xi as the solutions of the equation system (4.123) with = i. Case 3: If i has multiplicity ni among the eigenvalues, and the rank of the matrix A(nn) ; iI is equal to ri , then the number of linearly independent eigenvectors corresponding to i is equal to the so-called nullity n ; ri of the matrix of coecients. The inequality 1 n ; ri ni holds, i.e., for a real or complex quadratic matrix A(nn) there exist at least one and at most n real or complex linearly independent eigenvectors. 2 ; ;3 0 2 ;3 1 1 1 3 = ;3 ; 2 + 2 = 0. A: @ 3 1 3 A det (A ; I) = 3 1 ; ;5 2 ;4 ;5 2 ;4 ; The eigenvalues are 1 = 0 2 = 1 3 = ;2. The eigenvectors are determined from the corresponding homogeneous linear equation system. 1 = 0: 2x1 ; 3x2 + x3 = 0 3x1 + x2 + 3x3 = 0 ;5x1 + 2x2 ; 4x3 = 0: We get for instance by pivoting: x1 arbitrary, x2 = 3 x1 x3 = ;2x1 +3x2 = ; 11 x1 . Choosing x1 = 10 10 10 0 10 1 the eigenvector is x1 = C1 @ 3 A, where C1 is an arbitrary constant. ;11 2 = 1: The corresponding homogeneous system 0 ;1 1yields: x3 is arbitrary, x2 = 0 x1 = 3x2 ; x3 = ;x3 . Choosing x3 = 1 the eigenvector is x2 = C2 @ 0 A , where C2 is an arbitrary constant. 1 3 = ;2: The corresponding homogeneous system yields: x2 is arbitrary, x1 = 34 x2 , x3 = ;4x1 + 0 41 3x2 = ; 7 x2 . Choosing x2 = 3 the eigenvector is x3 = C3 @ 3 A, where C3 is an arbitrary constant. 3 ;7 3 ; 0 ;1 0 3 0 ;1 1 B: @ 1 4 1 A det (A ; I) = 1 4 ; 1 = ;3 + 102 ; 32 + 32 = 0. ;1 0 3 ;1 0 3 ; The eigenvalues are 1 = 2, 2 = 3 = 4. 2 = 2: We get that x3 is arbitrary, x2 = ;x3 x1 = x3 and choosing, for instance x3 = 1 the
280 4. Linear Algebra 0 11 corresponding eigenvector is x1 = C1 @ ;1 A, where C1 is an arbitrary constant. 1
2 = 3 = 4: We get that x2 x3 are arbitrary, x1 = ;x3 . We 0 0 1have two linearly 0 ;1 1independent eigenvectors, e.g., for x2 = 1 x3 = 0 and x2 = 0, x3 = 1: x2 = C2 @ 1 A x3 = C3 @ 0 A, where C2, C3 are arbitrary constants.
0
1
4.5.2.2 Real Symmetric Matrices, Similarity Transformations
In the case of the special eigenvalue problem (4.123) for a real symmetric matrix A the following statements hold:
1. Properties Concerning the Eigenvalue Problem
1. Number of Eigenvalues The matrix A has exactly n real eigenvalues i (i = 1 2 : : : n), count-
ing them by their multiplicity.
2. Orthogonalityof the Eigenvectors The eigenvectors xi and xj corresponding to di erent eigenvalues i = 6 j are orthogonal to each other, i.e., for the scalar product of xi and xj T xi xj = (xi xj ) = 0 (4.125) is valid.
3. Matrix with an Eigenvalue of Multiplicity p For an eigenvalue which has multiplicity p ( = 1 = 2 = : : : = p), there exist p linearly independent eigenvectors x1 x2 : : : xp. Because of (4.123)
all the non-trivial linear combinations of them are also eigenvectors corresponding to . Using the Gram{Schmidt orthogonalization process we can choose p of them such that they are orthogonal to each other. Summarizing: The matrix A has exactly n real orthogonal eigenvectors. 00 1 11 A = @ 1 0 1 A det (A ; I) = ;3 + 3 + 2 = 0. The eigenvalues are 1 = 2 = ;1 and 3 = 2. 110 1 = 2 = ;1: From the corresponding homogenous equation system we get: x1 is arbitrary, x2 is arbitrary, x3 = ;x1 ; x2 . Choosing 0 1 1 rst x1 = 1, x20= 00 1then x1 = 0, x2 = 1 we get the linearly independent eigenvectors x1 = C1 @ 0 A and x2 = C2 @ 1 A, where C1 and C2 are arbitrary constants. ;1 ;1 3 = 2: We get: x01 is 1arbitrary, x2 = x1, x3 = x1 , and choosing for instance x1 = 1 we get the 1 eigenvector x3 = C3 @ 1 A, where C3 is an arbitrary constant. The matrix A is symmetric, so the 1 eigenvectors corresponding to di erent eigenvalues are orthogonal. 4. Gram{Schmidt Orthogonalization Process Let Vn be an arbitrary n-dimensional Euclidean vector space. Let the vectors x 1 x 2 : : : xn 2 Vn be linearly independent. Then there exists an orthogonal system of vectors y 1 y 2 : : : y n 2 Vn which can be obtained as follows: kX ;1 y 1 = x 1 y k = x k ; ((xy k yy i)) y i (k = 2 3 : : : n): (4.126) i=1 i i
Remarks: 1. Here (x k y i) = xTk y i is the scalar product of the vectors x k und y i. 2. Corresponding to the orthogonal system of the vectors y 1 y 2 : : : y n we get the orthonormal system
4.5 Eigenvalue Problems for Matrices 281
q x~ 1 x~ 2 : : : x~ n with x~ 1 = jjyy 1 jj x~ 2 = jjyy 2 jj : : : x~ n = jjyy njj where jjy ijj = (y i y i) is the 1 2 n Euclidean norm of the vector y i. 001 011 011 @ A @ A x 1 = 1 x 2 = 0 x 3 = @ 1 A. From here it follows: 1 1 0 001 0 21 001 0 1 1 x y 1 2 1 y 1 = @ ;1=2 A and x~ 2 = p1 @ ;1 A y 1 = x 1 = @ 1 A and x~ 1 = p2 @ 1 A y 2 = x 2 ; 6 1 y y 1 1 1= 2 0 11 0 2=3 1 1 1 y 3 = x3 ; ((yx 3 yy 1)) y 1 ; ((yx 3 yy 2 )) y 2 = @ 2=3 A and x~ 3 = p13 @ 1 A. ;1 ;2=3 1 1 2 2
2. Transformation of Principal Axes, Similarity Transformation
For every real symmetric matrix A, there is an orthogonal matrix U and a diagonal matrix D such that A = UDUT: (4.127) The diagonal elements of D are the eigenvalues of A, and the columns of U are the corresponding normed eigenvectors. From (4.127) it is obvious that D = UTAU: (4.128) Transformation (4.128) is called the transformation of principal axes . In this way A is reduced to a diagonal matrix (see also 4.1.2, 2., p. 252). If the square matrix A (not necessarily symmetric) is transformed by a square regular matrix G such a way that G;1A G = A~ (4.129) ~ are called similar and they have then it is called a similarity transformation . The matrices A and A the following properties: 1. The matrices A and A~ have the same eigenvalues, i.e., the similarity transformation does not a ect the eigenvalues. 2. If A is symmetric and G is orthogonal, then A~ is symmetric, too: A~ = GTA G with GTG = I: (4.130) Now (4.128) can be put in the form: A real symmetric matrix A can be transformed orthogonally similar to a real diagonal form D.
4.5.2.3 Transformation of Principal Axes of Quadratic Forms 1. Real Quadratic Form, Denition
A real quadratic form Q of the variables x1 , x2,. . . , xn has the form
Q=
n X n X
i=1 j =1
aij xixj = xTAx
(4.131)
where x = (x1 x2 : : : xn)T is the vector of real variables and the matrix A = (aij ) is a real symmetric matrix. The form Q is called positive denite or negative denite , if it takes only positive or only negative values respectively, and it takes the zero value only in the case x1 = x2 = : : : = xn = 0. The form Q is called positive or negative semidenite , if it takes non-zero values only with the sign according to its name, but it can take the zero value for non-zero vectors, too. A real quadratic form is called indenite if it takes both positive and negative values. According to
282 4. Linear Algebra the behavior of Q the associated real symmetric matrix A is called positive or negative de nite, or semide nite.
2. Real Positive Denite Quadratic Form, Properties
1. In a real positive de nite quadratic form Q all elements of the main diagonal of the corresponding real symmetric matrix A are positive, i.e.,
aii > 0 (i = 1 2 : : : n) (4.132) holds. (4.132) represents a very important property of positive de nite matrices. 2. A real quadratic form Q is positive de nite if and only if all eigenvalues of the corresponding matrix A are positive. 3. Suppose the rank of the matrix A corresponding to the real quadratic form Q = xTAx is equal to r. Then the quadratic form can be transformed by a linear transformation x = C~x (4.133) into a sum of pure quadratic terms, into the so-called normal form Q = x~T K~x =
r X
pix~i 2 i=1 where pi = (sign i )ki and k1, k2, . . . ,kr
(4.134) are arbitrary, previously given, positive constants.
Remark: Regardless of the non-singular transformation (4.133) that transforms the real quadratic form of rank r into the normal form (4.134), the number p of positive coecients and the number q = r ; p of negative coecients among the pi of the normal form are invariant (the inertia theorem of Sylvester). The value p is called the index of inertia of the quadratic form.
3. Generation of the Normal Form
A practical method to use the transformation (4.134) follows from the transformation of principal axes (4.128). First we perform a rotation on the coordinate system by the orthogonal matrix U, whose columns are the eigenvectors of A (i.e., the directions of the axes of the new coordinate system are the directions of the eigenvectors). Then we have the form
Q = x~T L~x =
r X i=1
ix~i 2:
(4.135)
Here L is a diagonal matrix with the eigenvalues of A in the diagonal. Then a dilatation is performed s k i by the diagonal matrix D whose diagonal elements are di = j j . The whole transformation now is i given by the matrix C = UD (4.136) and we have: Q = x~TA~x = (U D~x)TA(U D~x) = x~T(DTUTA U D)x~ = x~TDTL D~x = x~TK~x: (4.137) Remark: The transformation of principal axes of quadratic forms plays an essential role at the classication of curves and surfaces of second order (see 3.5.2.9, p. 205 and 3.5.3.6, p. 223).
4. Jordan Normal Form
Let A be an arbitrary real or complex (n n) matrix. Then there exists a non-singular matrix T such that T;1AT = J (4.138) holds, where J is called the Jordan matrix or Jordan normal form of A. The Jordan matrix has a block diagonal structure of the form (4.139), where the elemnts Jj of J are called Jordan blocks:
4.5 Eigenvalue Problems for Matrices 283
0 1 J1 B C J O B 2 C C ... C J = BBBB C: @O Jk;1 CA
(4.139)
Jk
0 1 1 B C O B 2 C C ... C J = BBBB C: @O A n;1 C n
(4.140)
They have the following structure: 1. If A has only single eigenvalues j , then Jj = j and k = n, i.e., J is a diagonal matrix (4.140). 0 1 2. If j is an eigenvalue of multiplicity pj , then there are j 1 B C one or more blocks of the form (4.141) where the sum 1 O C j B of the sizes of all such blocks is equal to pj and we have J = B . . C .. .. C (4.141) Pk p = n. The exact structure of a Jordan block de- j B B C B C j =1 j @ A 1 j pends on the structure of the elementary divisors of the O j characteristic matrix A ; I. For further information see 4.13], 19.16] vol. 1.
4.5.2.4 Suggestions for the Numerical Calculations of Eigenvalues
1. Eigenvalues can be calculated as the roots of the characteristic equation (4.124b) (see examples on p. 279). In order to do this we have to determine the coecients ai (i = 0 1 2 : : : n ; 1) of the characteristic polynomial of the matrix A. However, we should avoid this method of calculation,
because this procedure is extremely unstable, i.e., small changes in the coecients ai of the polynomial result in big changes in the roots j . 2. There are many algorithms for the solution of the eigenvalue problem of symmetric matrices. We distinguish between two types (see 4.6]): a) Transformation methods, for instance the Jacobi method, Householder tridiagonalization,QR algorithm. b) Iterative methods, for instance vector iteration, the Rayleigh{Ritz algorithm, inverse iteration, the Lanczos method, the bisection method. As an example the power method of Mises is discussed here.
3. The Power Method of Mises Assume that A is real and symmetric and has a unique dominant
eigenvalue. This iteration method determines this eigenvalue and the associated eigenvector. Let the dominant eigenvalue be denoted by 1 , that is, j1j > j2j j3j jnj: (4.142) Let x 1 x 2 : : : x n be the associated linearly independent eigenvectors. Then: 1. Ax i = ix i (i = 1 2 : : : n): (4.143) 2. Each element x 2 IRn can be expressed as a linear combination of these eigenvectors x i: x = c1 x 1 + c2x 2 + + cnx n (ci const i = 1 2 : : : n): (4.144) Multiplying both sides of (4.144) by A k times, then using (4.143) we have !k !k Ak x = c1k1 x 1 + c2k2 x 2 + + cnknx n = k1 c1 x1 + c2 2 x 2 + + cn n x n]: 1 1 (4.145) From this relation and (4.142) we see that
Ak x ;! x as k ! 1 1 k c 1 1
that is, Ak x c1k1 x 1:
This is the basis of the following iteration procedure:n Step 1: Select an arbitrary starting vector x(0) 2 IR .
(4.146)
284 4. Linear Algebra Step 2: Compute Ak x iteratively: x(k+1) = Ax(k) (k = 0 1 2 : : : x(0) is given):
From (4.147) and keeping in mind (4.146) we have: x(k) = Ak x(0) c1 k1 x 1 : Step 3: From (4.147) and (4.148) it follows that x(k+1) = Ax(k) = A(Ak x(0) ) A(Ak x(0) ) A(c1k1 x 1) = c1k1 (Ax 1 ) c1(k1 Ax 1 ) = 1(c1k1 x 1) 1x(k) therefore x(k+1) 1x(k) that is, for large values of k the consecutive vectors x(k) are 1 multiples of each other. Step 4: Relations (4.148) and (4.149) imply for x 1 and 1:
x1
x(k+1)
x(0) 1 0 0 1
(4.148)
(4.149)
x(k) x(k+1) 1 (x(k) x(k)) :
For0example, let 3:23 ;1:15 1:77 1 @ A = ;1:15 9:25 ;2:13 A 1:77 ;2:13 1:56
(4.147)
x(0)
(4.150)
011 = @ 0 A.
0 normalization x(4) x(5) normalization 3:23 14:89 88:27 1 7:58 67:75 1 ;1:15 ;18:12 ;208:03 ;2:36 ;24:93 ;256:85 ;3:79 1:77 10:93 82:00 0:93 8:24 79:37 1:17 9:964 10:177
x(1)
x(2)
x(3)
x(6)
x(7) normalization
9:66 96:40 ; 38:78 ;394:09 11:67 117:78
x(8)
x(9)
normalization 0 1 1 10:09 102:33 B 1 C ;4:09 ;41:58 ;422:49 B@ ;4:129 CA x 1 1:22 12:38 125:73 1:229 10:16 10:161 1
Remarks: 1. Since (keigenvectors are unique only up to a constant multiplier, it is preferable to normalize the ) vectors x as shown in the example.
2. The eigenvalue with the smallest absolute value and the associated eigenvector can be obtained by using the power method of Mises for A;1. If A;1 does not exist, then 0 is this eigenvalue and any vector from the null-space of A can be selected as an associated eigenvector. 3. The other eigenvalues and the associated eigenvectors of A can be obtained by repeated application of the following idea. Select a starting vector which is orthogonal to the known vector x 1 , and in this subspace 2 becomes the dominant eigenvalue that can be obtained by using the power method. In order to obtain 3, the starting vector has to be orthogonal to both x 1 and x 2 , and so on. This procedure is known as matrix deation. 4. Based on (4.147) the power method is sometimes called vector iteration.
4.5 Eigenvalue Problems for Matrices 285
4.5.3 Singular Value Decomposition
1. Singular Values and Singular Vectors Let A be a real matrix of size (m n) and its rank be equal to r. The matrices AAT and ATA have r non-zero p eigenvalues , and they are the same for both of the matrices. The positive square roots d = ( = 1 2 : : : r) of the eigenvalues of T the matrix A A are called the singular values of the matrix A. The corresponding eigenvectors u of ATA are called right singular vectors of A, the corresponding eigenvectors v of AAT left singular
vectors:
ATAu = u AATv = v ( = 1 2 : : : r):
(4.151a) The relations between the right and left singular vectors are: Au = d v ATv = d u : (4.151b) A matrix A of size (m n) with rank r has r positive singular values d ( = 1 2 : : : r). There exist r orthonormalized right singular vectors u and r orthonormalized left singular vectors v . Furthermore, there exist to the zero singular value n ; r orthonormalized right singular vectors u ( = r + 1 : : : n) and m ; r orthonormalized left singular vectors v ( = r + 1 : : : m). Consequently, a matrix of size (m n) has n right singular vectors and m left singular vectors, and two orthogonal matrices can be made from them (see 4.1.4, 9., p. 257): (4.152) U = (u1 u2 : : : un) V = (v1 v2 : : : vm):
2. Singular Value Decomposition The representation
^ T (4.153a) A = VAU
0 d1 0 0 0 0 0 1 B B C 0 d2 0 0 ... C B C B C . . . . . . . . B C . . . . B C B C 0 B C B C 0 0 d 0 0 B C r B C C with A^ = B 0 0 0 0 0 B C . B C . B C 0 0 . B C B C . . .. @ .. A 0 0
|
0 0 0
} | {z } r columns n ; r columns is called the singular value decomposition of the matrix A. The matrix A^ , as the matrix A, is of size (m n) and has only zero elements except the rst r diagonal elements a = d ( = 1 2 : : : r). The values d are the singular values of A. Remark: If we substitute AH instead of AT and consider unitary matrices U and V instead of orthogonals, then all the statements about singular value decomposition are valid also for matrices with complex elements. 3. Application Singular value decomposition can be used to determine the rank of the matrix A of size (m n) and to calculate an approximate solution of the overdetermined equation system Ax = b (see 4.4.3.1, p. 277) after the transformation according to the so-called regularization method, i.e., to solve the problem
2 X m "X n n X jjAx ; bjj2 + jjxjj2 = aik xk ; bi + x2k = min! (4.154) i=1 k=1
where > 0 is a regularization parameter.
{z
9 > > > > = r rows > > > " 9 (4.153b) > > = m ; r rows > > "
k=1
286 5. Algebra and Discrete Mathematics
5 AlgebraandDiscreteMathematics 5.1 Logic
5.1.1 Propositional Calculus 1. Propositions
A proposition is the mental reection of a fact, expressed as a sentence in a natural or arti cial language. Every proposition is considered to be true or false. This is the principle of two-valuedness (in contrast to many-valued or fuzzy logic, see 5.9.1, p. 360). \True" and \false" are called the truth value of the proposition and they are denoted by T (or 1) and F (or 0), respectively. The truth values can be considered as propositional constants.
2. Propositional Connectives
Propositional logic investigates the truth of compositions of propositions depending on the truth of the components. Only the extensions of the sentences corresponding to propositions are considered. Thus the truth of a composition depends only on that of the components and on the operations applied. So in particular, the truth of the result of the propositional operations \NOT A" (:A) (5.1) \A AND B " (A ^ B ) (5.2) \A OR B " (A _ B )
\IF A, THEN B " (A ) B )
(5.3)
(5.4)
and
\A IF AND ONLY IF B "(A , B ) (5.5) are determined by the truth of the components. Here \logical OR" always means \inclusive OR", i.e., \AND/OR". In the case of implication, for A ) B we also use the following verbal forms: A implies B B is necessary for A A is sucient for B:
3. Truth Tables
In propositional calculus, the propositions A and B are considered as variables (propositional variables) which can have only the values F and T. Then the truth tables in Table 5.1 contain the truth functions de ning the propositional operations. Table 5.1 Truth tables of propositional calculus
Negation Conjunction A :A A B A^B F T
T F
F F T T
F T F T
F F F T
Disjunction A B A_B F F T T
F T F T
4. Formulas in Propositional Calculus
F T T T
Implication A B A)B F F T T
F T F T
T T F T
Equivalence A B A,B F F T T
F T F T
T F F T
We can compose compound expressions (formulas) of propositional calculus from the propositional variables in terms of a unary operation (negation) and binary operations (conjunction, disjunction, implication and equivalence). These expressions, i.e., the formulas, are de ned in an inductive way: 1. Propositional variables and the constants T, F are formulas: (5.6) 2. If A and B are formulas, then (:A) (A ^ B ) (A _ B ) (A ) B ) (A , B ) (5.7)
5.1 Logic 287
are also formulas. To simplify formulas we omit parentheses after introducing precedence rules. In the following sequence every propositional operation binds more strongly than the next one in the sequence: : ^ _ ) ,: We often use the notation A instead of \:A" and we omit the symbol ^. By these simpli cations, for instance the formula ((A _ (:B )) ) ((A ^ B ) _ C )) can be rewritten more briey in the form: A _ B ) AB _ C:
5. Truth Functions
If we assign a truth value to every propositional variable of a formula, we call the assignment an interpretation of the propositional variables. Using the de nitions (truth tables) of propositional operations we can assign a truth value to a formula for every possible interpretation of the variables. A B C A _ B AB _ C A _ B ) AB _ C Thus for instance the formula given above determines a truth function of three variables (a F F F T F F Boolean function see 5.7.5, p. 360). F F T T T T In this way, every formula with n proposiF T F F F T tional variables determines an n-place (or nF T T F T T ary) truth function, i.e., a function which asT F F T F F signs a truth value ton every n-tuple of truth T T T T F T values. There are 22 n-ary truth functions, T T F T T T in particular these are 16 binary ones. T T T T T T
6. Elementary Laws in Propositional Calculus
Two propositional formulas A and B are said to be logically equivalent or semantically equivalent, denoted by A = B , if they determine the same truth function. Consequently, we can check the logical equivalence of propositional formulas in terms of truth tables. So we get, e.g., A_B ) AB _C = B _C , i.e., the formula A _ B ) AB _ C does not in fact depend on A, as follows from its truth table above. In particular, we have the following elementary laws of propositional calculus:
1. Associative Laws (A ^ B ) ^ C = A ^ (B ^ C ) 2. Commutative Laws A^B =B^A 3. Distributive Laws (A _ B )C = AC _ BC 4. Absorption Laws A(A _ B ) = A 5. Idempotence Laws
(5.8a)
(A _ B ) _ C = A _ ( B _ C ) :
(5.8b)
(5.9a)
A _ B = B _ A:
(5.9b)
(5.10a)
AB _ C = (A _ C )(B _ C ):
(5.10b)
(5.11a)
A _ AB = A:
(5.11b)
AA = A
(5.12a)
A _ A = A:
(5.12b)
AA = F,
(5.13a)
A _ A = T.
(5.13b)
(5.14a)
A _ B = A B:
(5.14b)
6. Excluded Middle 7. De Morgan Rules AB = A _ B
288 5. Algebra and Discrete Mathematics 8. Laws for T and F AT = A
(5.15a)
A_F=A
(5.15b)
AF = F,
(5.15c)
A _ T = T,
(5.15d)
T = F,
(5.15e)
F = T.
(5.15f)
9. Double Negation
A = A: (5.16) Using the truth tables for implication and equivalence, we get the identities (5.17a) and A , B = AB _ A B: (5.17b) A)B =A_B Therefore implication and equivalence can be expressed in terms of other propositional operations. Laws (5.17a), (5.17b) are applied to reformulate propositional formulas. The identity A _ B ) AB _ C = B _ C can be veri ed in the following way: A _ B ) AB _ C = A _ B _ AB _ C = A B _ AB _ C = AB _ AB _ C = (A _ A)B _ C = TB _ C = B _ C:
10. Further Transformations A(A _ B ) = AB
(5.18a)
A _ AB = A _ B
(5.18b)
(A _ C )(B _ C )(A _ B ) = (A _ C )(B _ C ) (5.18c) AC _ BC _ AB = AC _ BC: (5.18d) 11. NAND Function and NOR Function As we know, every propositional formula determines a truth function. We can check the following converse of this statement: Every truth function can be represented as a truth table of a suitable formula in propositional logic. Because of (5.17a) and (5.17b) we can eliminate implication and equivalence from formulas (see also 5.7, p. 342). This fact and the De Morgan rules (5.14a) and (5.14b) imply that we can express every formula, therefore every truth function, in terms of negation and disjunction only, or in terms of negation and conjunction. There are two further binary truth functions of two variables which are suitable to express all the truth functions. They are called the NAND function or ShefTable 5.2 NAND function Table 5.3 NOR function fer function (notation \ j ") and the NOR function or Peirce function (notation \ "), A B AjB A B AB with the truth tables given in Tables 5.2 and 5.3. Comparison of the truth tables F F T F F T F for these operations with the truth tables F T T F T T F T T F F of conjunction and disjunction makes the T T F T T F terminologies NAND function (NOT AND) and NOR function (NOT OR) clear.
7. Tautologies, Inferences in Mathematics
A formula in propositional calculus is said to be a tautology if the value of its truth function is identically the value T. Consequently, two formulas A and B are called logically equivalent if the formula A , B is a tautology. Laws of propositional calculus often reect inference methods used in mathematics. As an example, consider the law of contraposition, i.e., the tautology (5.19a) A ) B , B ) A: This law, which also has the form A)B=B)A (5.19b) can be interpreted in this way: To show that B is a consequence of A is the same as showing that A is a consequence of B . Indirect proof (see also 1.1.2.2, p. 5) means the following principle: To show that
5.1 Logic 289
B is a consequence of A, we suppose B to be false, and under the assumption that A is true, we derive a contradiction. This principle can be formalized in propositional calculus in several ways: A ) B = AB ) A (5.20a) or A ) B = AB ) B or (5.20b) A ) B = AB ) F.
(5.20c)
5.1.2 Formulas in Predicate Calculus
For developing the logical foundations of mathematics we need a logic which has a stronger expressive power than propositional calculus. To describe the properties of most of the objects in mathematics and the relations between these objects the predicate calculus is needed.
1. Predicates
We include the objects to be investigated into a set, i.e., into the domain X of individuums (or universe), e.g., this domain could be the set IN of the natural numbers. The properties of the individuums, as, e.g., \ n is a prime ", and the relations between individuums, e.g., \ m is smaller than n ", are considered as predicates. An n-place predicate over the domain X of individuums is an assigment P : X n ! fF,Wg, which assigns a truth value to every n-tuple of the individuums. So the predicates introduced above on natural numbers are a one-place (or unary) predicate and a two-place (or binary) predicate.
2. Quantiers
A characteristic feature of predicate logic is the use of quantiers, i.e., that of a universal quantier or \for every" quantier 8 and existential quantier or \for some" quantier 9. If P is a unary predicate, then the sentence \P (x) is true for every x in X " is denoted by 8 xP (x) and the sentence\ There exists an x in X for which P (x) is true "is denoted by 9 x P (x). Applying a quanti er to the unary predicate P , we get a sentence. If for instance IN is the domain of individuums of the natural numbers and P denotes the (unary) predicate \n is a prime", then 8 n P (n) is a false sentence and 9 n P (n) is a true sentence.
3. Formulas in Predicate Calculus
The formulas in predicate calculus are de ned in an inductive way: 1. If x1 : : : xn are individuum variables (variables running over the domain of individuum variables) and P is an n-place predicate symbol, then P (x1 : : : xn) is a formula (elementary formula): (5.21) 2. If A and B are formulas, then (:A) (A ^ B ) (A _ B ) (A ) B ) (A , B ) (8 x A) and (9 x A) (5.22) are also formulas. Considering a propositional variable to be a null-place predicate, we can consider propositional calculus as a part of predicate calculus. An occurrence of an individuum variable x is bound in a formula if x is a variable in 8 x or in 9 x or the occurrence of x is in the scope of these types of quanti ers otherwise an occurrence of x is free in this formula. A formula of predicate logic which does not contain any free occurrences of individuum variables is said to be a closed formula.
4. Interpretation of Predicate Calculus Formulas
An interpretation of predicate calculus is a pair of a set (domain of individuums) and an assignment, which assigns an n-place predicate to every n-ary predicate symbol. For every pre xed value of free variables the concept of the truth evaluation of a formula is similar to the propositional case. The truth value of a closed formula is T or F. With a formula with free variables, we can associate the values of individuums for which the truth evaluation of the formula is true these values constitute a relation (see 5.2.3, 1., p. 294) on the universe (domain of individuums). Let P denote the two-place relation on the domain IN of individuums, where IN is the set of the
290 5. Algebra and Discrete Mathematics natural numbers then P (x y) characterizes the set of all the pairs (x y) of natural numbers with x y (two-place or binary relation on IN) here x, y are free variables 8 y P (x y) characterizes the subset of IN (unary relation) consisting of the element 0 only here x is a free variable, y is a bound variable 9 x 8 y P (x y) corresponds to the sentence \ There is a smallest natural number " the truth value is true here x and y are bound variables.
5. Logically Valid Formulas
A formula is said to be logically valid (or a tautology) if it is true for every interpretation. The negation of formulas is characterized by the identities below: :8 x P (x) = 9 x :P (x) or :9 x P (x) = 8 x :P (x): (5.23) Using (5.23) the quanti ers 8 and 9 can be expressed in terms of each other: 8 x P (x) = :9 x :P (x) or 9 x P (x) = :8 x :P (x): (5.24) Further identities of the predicate calculus are: 8 x 8 y P (x y) = 8 y 8 x P (x y) (5.25) 9 x 9 y P (x y) = 9 y 9 x P (x y) (5.26) 8 x P (x) ^ 8 x Q(x) = 8 x (P (x) ^ Q(x)) (5.27) 9 x P (x) _ 9 x Q(x) = 9 x (P (x) _ Q(x)): (5.28) The following implications are also valid: 8 x P (x) _ 8 x Q(x) ) 8 x (P (x) _ Q(x)) (5.29) 9 x (P (x) ^ Q(x)) ) 9 x P (x) ^ 9 x Q(x) (5.30) 8 x (P (x) ) Q(x)) ) (8 x P (x) ) 8 x Q(x)) (5.31) 8 x (P (x) , Q(x)) ) (8 x P (x) , 8 x Q(x)) (5.32) 9 x 8 y P ( x y ) ) 8 y 9 x P ( x y ): (5.33) The converses of these implications are not valid, in particular, we have to be careful with the fact that the quanti ers 8 and 9 do not commute (the converse of the last implication is false).
6. Restricted Quantication
Often it is useful to restrict quanti cation to a subset of a given set. So we consider 8 x 2 X P (x) as a short notation of 8 x (x 2 X ) P (x)) and 9 x 2 X P (x) as a short notation of 9 x (x 2 X ^ P (x)):
(5.34) (5.35)
5.2 Set Theory
5.2.1 Concept of Set, Special Sets
The founder of set theory is Georg Cantor (1845{1918). The importance of the notion introduced by him became well known only later. Set theory has a decisive role in all branches of mathematics, and today it is an essential tool of mathematics and its applications.
1. Membership Relation
1. Sets and their Elements The fundamental notion of set theory is the membership relation. A set A is a collection of certain di erent things a (objects, ideas, etc.) that we think belong together for certain reasons. These objects are called the elements of the set. We write \a 2 A" or \a 2= A" to denote \a is an element of A" or \a is not an element of A", respectively. Sets can be given by enumerating their elements in braces, e.g., M = fa b cg or U = f1 3 5 : : :g, or by a de ning property possessed
5.2 Set Theory 291
exactly by the elements of the set. For instance the set U of the odd natural numbers is de ned and denoted by U = fx j x is an odd natural numberg. For number domains the following notation is generally used: IN = f0 1 2 : : :g set of the natural numbers, Z = f(0 1 ;1 2 ;2 : : :g ) set of the integers, p Q = p q 2 Z ^ q 6= 0 set of the rational numbers, q IR set of the real numbers, C set of the complex numbers. 2. Principle of Extensionality for Sets Two sets A and B are identical if and only if they have exactly the same elements, i.e., A = B , 8 x (x 2 A , x 2 B ): (5.36) The sets f3 1 3 7 2g and f1 2 3 7g are the same. A set contains every element only \once", even if it is enumerated several times.
2. Subsets
1. Subset If A and B are sets and 8 x (x 2 A ) x 2 B )
(5.37) holds, then A is called a subset of B , and this is denoted by A B . In other words: A is a subset of B if all elements of A also belong to B . If for A B there are some further elements in B such that they are not in A, then we call A a proper subset of B and we denote it by A B (Fig. 5.1). Obviously, every set is a subset of itself A A. Suppose A = f2 4 6 8 10g is a set of even numbers and B = f1 2 3 : : : 10g is a set of natural numbers. Since the set A does not contain odd numbers, A is a proper subset of B: 2. Empty Set or Void Set It is important and useful to introduce the notion of empty set or void set, , which has no element. Because of the principle of extensionality, there exists only one empty set. A: The set fxjx 2 IR ^ x2 + 2x + 2 = 0g is empty. B: M for every set M , i.e., the empty set is a subset of every set M . For a set A the empty set and A itself are called the trivial subsets of A. 3. Equality of Sets Two sets are equal if and only if both are subsets of each other: A = B , A B ^ B A: (5.38) This fact is very often used to prove that two sets are identical. 4. Power Set The set of all subsets A of a set M is called the power set of M and it is denoted by IP(M ), i.e., IP(M ) = fA j A M g. For the set M = fa b cg the power set is IP(M ) = f fag fbg fcg fa bg fa cg fb cg fa b cgg: It is true that: a) If a set M has m elements, its power set IP(M ) has 2m elements. b) For every set M we have M 2 IP(M ) i.e., M itself and the empty set are elements of the power set of M . 5. Cardinal number The number of elements of a nite set M is called the cardinal number of M and it is denoted by card M or sometimes by jM j. We also de ne the cardinal number of sets with in nitely many elements (see 5.2.5, p. 298).
5.2.2 Operations with Sets 1. Venn diagram
The graphical representations of sets and set operations are the so-called Venn diagrams, when we represent sets by plane gures. So, in Fig. 5.1, we represent the subset relation A B .
292 5. Algebra and Discrete Mathematics B
A
A
Figure 5.1
B
A
Figure 5.2
B
Figure 5.3
2. Union, Intersection, Complement
By set operations we form new sets from the given sets in di erent ways: 1. Union Let A and B be two sets. The union set or the union (denoted by A B ) is de ned by A B = fx j x 2 A _ x 2 B g: (5.39) We say \A union B " or \A cup B ". If A and B are given by the properties E1 and E2 respectively, the union set A B has the elements possessing at least one of these properties, i.e., the elements belonging to at least one of the sets. In Fig. 5.2 the union set is represented by the shaded region. f1 2 3g f2 3 5 6g = f1 2 3 5 6g. 2. Intersection Let A and B be two sets. The intersection set, intersection, cut or cut set (denoted by A \ B ) is de ned by A \ B = fx j x 2 A ^ x 2 B g: (5.40) We say \A intersected by B " or \A cap B ". If A and B are given by the properties E1 and E2 respectively, the intersection A \ B has the elements possessing both properties E1 and E2, i.e., the elements belonging to both sets. In Fig. 5.3 the intersection is represented by the shaded region. With the intersection of the sets of divisors T (a) and T (b) of two numbers a and b we can de ne the greatest common divisor (see 5.4.1.4, p. 323). For a = 12 and b = 18 we have T (a) = f1 2 3 4 6 12g and T (b) = f1 2 3 6 9 18 g, so T (12) \ T (18) contains the common divisors, and the greatest common divisor is g.c.d. (12 18) = 6. 3. Disjoint Sets Two sets A and B are called disjoint if they have no common element for them A\B = (5.41) holds, i.e., their intersection is the empty set. The set of odd numbers and the set of even numbers are disjoint their intersection is the empty set, i.e., fodd numbersg \ feven numbersg = : 4. Complement If we consider only the subsets of a given set M , then the complementary set or the complement CM (A) of A with respect to M contains all the elements of M not belonging to A: CM (A) = fx j x 2 M ^ x 2= Ag: (5.42) We say \complement of A with respect to M ", and M is called the fundamental set or sometimes the universal set. If the fundamental set M is obvious from the considered problem, the notation A is also used for the complementary set. In Fig. 5.4 the complement A is represented by the shaded region. M A
Figure 5.4
B
A
Figure 5.5
3. Fundamental Laws of Set Algebra
A
B
Figure 5.6
These set operations have analoguous properties to the operations in logic. The fundamental laws of set algebra are:
5.2 Set Theory 293
1. Associative Laws (A \ B ) \ C = A \ (B \ C ) 2. Commutative Laws A\B =B\A 3. Distributive Laws (A B ) \ C = (A \ C ) (B \ C ) 4. Absorption Laws A \ (A B ) = A 5. Idempotence Laws A\A=A 6. De Morgan Laws A\B =A B 7. Some Further Laws A\A= A\M =A A\ = M =
(5.43)
(A B ) C = A ( B C ) :
(5.44)
(5.45)
A B = B A:
(5.46)
(5.47)
(A \ B ) C = (A C ) \ (B C ): (5.48)
(5.49)
A (A \ B ) = A:
(5.50)
(5.51)
A A = A:
(5.52)
(5.53)
A B = A \ B:
(5.54)
(5.55)
A A = M (M fundamental set) (5.56)
(5.57)
A =A
(5.58)
(5.59)
A M =M
(5.60)
(5.61)
= M:
(5.62)
A = A: (5.63) This table can also be obtained from the fundamental laws of propositional calculus (see 5.1.1, p. 286) if we make the following substitutions: ^ by \, _ by , T by M , and F by . This coincidence is not accidental it will be discussed in 5.7, p. 342.
4. Further Set Operations
Besides the operations de ned above there are de ned some further operations between two sets A and B , the dierence set or dierence A n B the symmetric dierence A4B and the Cartesian product A B: 1. Di erence of Two Sets The set of the elements of A not belonging to B is the dierence set or dierence of A and B : A n B = fx j x 2 A ^ x 2= B g: (5.64a) If A is de ned by the property E1 and B by the property E2 , then A n B contains the elements having the property E1 but not having property E2 . In Fig. 5.5 the di erence is represented by the shaded region. f1 2 3 4g n f3 4 5g = f1 2g. 2. Symmetric Di erence of Two Sets The symmetric di erence A4B is the set of all elements belonging to exactly one of the sets A and B : A4B = fx j (x 2 A ^ x 2= B ) _ (x 2 B ^ x 2= A)g: (5.64b) It follows from the de nition that A4B = (A n B ) (B n A) = (A B ) n (A \ B ) (5.64c)
294 5. Algebra and Discrete Mathematics i.e., the symmetric di erence contains the elements which have exactly one of the de ning properties E1 (for A) and E2 (for B ). In Fig. 5.6 the symmetric di erence is represented by the shaded region. f1 2 3 4g4f3 4 5g = f1 2 5g: 3. Cartesian Product of Two Sets The Cartesian product of two sets A B is de ned by A B = f(a b) j a 2 A ^ b 2 B g: (5.65a) The elements (a b) of A B are called ordered pairs and they are characterized by (a b) = (c d) , a = c ^ b = d: (5.65b) The number of the elements of a Cartesian product of two nite sets is equal to card (A B ) = (cardA)(cardB ): (5.65c) A: For A = f1 2 3g and B = f2 3g we get A B = f(1 2) (1 3) (2 2) (2 3) (3 2) (3 3)g and B A = f(2 1) (2 2) (2 3) (3 1) (3 2) (3 3)g with cardA = 3 cardB = 2 card(A B ) = card(B A) = 6. B: Every point of the x y plane can be de ned with the Cartesian product IR IR (IR is the set of real numbers). The set of the coordinates x y is represented by IR IR, and we have: IR2 = IR IR = f(x y) j x 2 IR y 2 IRg:
4. Cartesian Product of n Sets
From n elements, by xing an order of sequence ( rst element, second element, . . . , n-th element) an ordered n-tuple is de ned. If ai 2 Ai (i = 1 2 : : : n) are the elements, the n-tuple is denoted by (a1 a2 : : : an) where ai is called the i-th component. For n = 3 4 5 we call these n-tuples triples, quadruples, and quintuples. The Cartesian product of n terms A1 A2 An is the set of all ordered n-tuples (a1 a2 : : : an) with ai 2 Ai : A1 : : : An = f(a1 : : : an) j ai 2 Ai (i = 1 : : : n)g: (5.66a) If every Ai is a nite set, the number of ordered n-tuples is card(A1 A2 An) = cardA1 cardA2 cardAn : (5.66b) Remark: The n times Cartesian product of a set A with itself is denoted by An .
5.2.3 Relations and Mappings 1. n-ary Relations
Relations de ne correspondences between the elements of one or di erent sets. An n-ary relation or n-place relation R between the sets A1 : : : An is a subset of the Cartesian product of these sets, i.e., R A1 : : : An. If the sets Ai, i = 1 : : : n, are all the same set A, then R An holds and it is called an n-ary relation in the set A.
2. Binary Relations
1. Notion of Binary Relations of a Set The two-place (binary) relations in a set have special importance (see 5.2.3, 2., p. 294). In the case of a binary relation the notation aRb is also very common instead of (a b) 2 R. As an example, we consider the divisibility relation in the set A = f1 2 3 4g, i.e., the binary relation T = f(a b) j a b 2 A ^ a is a divisor of bg (5.67a) = f(1 1) (1 2) (1 3) (1 4) (2 2) (2 4) (3 3) (4 4)g: (5.67b) 2. Arrow Diagram or Mapping Function Finite binary relations R in a set A can be represented by arrow functions or arrow diagrams or by relation matrices. The elements of A are represented as points of the plane and an arrow goes from a to b if aRb holds. Fig. 5.7 shows the arrow diagram of the relation T in A = f1 2 3 4g.
5.2 Set Theory 295
1 2 3 4 1 1 1 1 1 2 0 1 0 1 3 0 0 1 0 3 4 4 0 0 0 1 Figure 5.7 Scheme: Relation matrix 3. Relation Matrix The elements of A are used as row and column entries of a matrix (see 4.1.1, 1., p. 251). At the intersection point of the row of a 2 A with the column of b 2 B there is an entry 1 if aRb holds, otherwise there is an entry 0. The above scheme shows the relation matrix for T in A = f1 2 3 4g. 1
2
3. Relation Product, Inverse Relation
Relations are special sets, so the usual set operations (see 5.2.2, p. 291) can be performed between relations. Besides them, for binary relations, the relation product and the inverse relation also have special importance. Let R A B and S B C be two binary relations. The product R S of the relations R, S is de ned by R S = f(a c) j 9 b (b 2 B ^ aRb ^ bSc)g: (5.68) The relation product is associative, but not commutative. The inverse relation R;1 of a relation R is de ned by R;1 = f(b a) j (a b) 2 Rg: (5.69) For binary relations in a set A the following relations are valid: (R S ) T = (R T ) (S T ) (5.70) (R \ S ) T (R T ) \ (S T ) (5.71) (R S );1 = R;1 S ;1
(5.72)
(R S );1 = S ;1 R;1 :
(5.74)
(R \ S );1 = R;1 \ S ;1
4. Properties of Binary Relations
(5.73)
A binary relation in a set A can have special important properties: R is called reflexive if 8 a 2 A aRa (5.75) irreflexive if 8 a 2 A :aRa (5.76) symmetric if 8 a b 2 A (aRb ) bRa) (5.77) antisymmetric if 8 a b 2 A (aRb ^ bRa ) a = b) (5.78) transitive if 8 a b c 2 A (aRb ^ bRc ) aRc) (5.79) linear if 8 a b 2 A (aRb _ bRa): (5.80) These relations can also be described by the relation product. For instance: a binary relation is transitive if R R R holds. Especially interesting is the transitive closure tra(R) of a relation R. It is the smallest (with respect to the subset relation) transitive relation which contains R. In fact # tra(R) = Rn = R1 R2 R3 (5.81) n 1
where Rn is the n times relation product of R with itself. Let a binary relation R on the set f1 2 3 4 5g be given by its relation matrix M :
296 5. Algebra and Discrete Mathematics M 1 2 3 4 5 M2 1 2 3 4 5 M3 1 2 3 4 5 1 1 0 0 1 0 1 1 1 0 1 1 1 1 1 0 1 1 2 0 0 0 1 0 2 0 1 0 0 1 2 0 1 0 1 0 3 0 0 1 0 1 3 0 1 1 0 1 3 0 1 1 1 1 4 0 1 0 0 1 4 0 1 0 1 0 4 0 1 0 1 1 5 0 1 0 0 0 5 0 0 0 1 0 5 0 1 0 0 1 We can calculate M 2 by matrix multiplication where the values 0 and 1 are treated as truth values and instead of multiplication and addition we perform the logical operations conjunction and disjunction. So, M 2 is the relation matrix belonging to R2. Similarly, we can calculate the relation matrices of R3 R4 etc. We get the relation matrix of R R2 R3 (the matrix on the M _ M2 _ M3 1 2 3 4 5 left) if we calculate the disjunction elementwise of the matrices M M 2 and M 3 . Since the higher powers of M contains no new 1 1 1 0 1 1 1-s, this matrix already coincides with the relation matrix of 2 0 1 0 1 1 tra(R). 3 0 1 1 1 1 The relation matrix and relation product have important ap4 0 1 0 1 1 plications in search of path length in graph theory (see 5.8.2.1, 5 0 1 0 1 1 p. 351). In the case of nite binary relations, we can easily recognize the above properties from the arrow diagrams or from the relation matrices. We can recognize for instance the reexivity from \self-loops" in the arrow diagram, and from the main diagonal elements 1 in the relation matrix. Symmetry is obvious in the arrow diagram if to every arrow there belongs another one in the opposite direction, or if the relation matrix is a symmetric matrix (see 5.2.3, 2., p. 294). We can see from the arrow diagram or from the relation matrix that the divisibility T is a reexive but not symmetric relation.
5. Mappings
A mapping (or function, see 2.1.1.1, p. 47) f from a set A to a set B with the notation f : A ! B is a rule which assigns to every element a 2 A a unique element f (a) 2 B . We can consider a mapping f as a binary relation between A and B (f A B ): f A B is called a mapping from A to B , if 8 a 2 A 9 b 2 B ((a b) 2 f ) and (5.82) 8 a 2 A 8 b1 b2 2 B ((a b1) (a b2 ) 2 f ) b1 = b2 ) (5.83) hold. Here f is called a one-to-one (or injective) mapping if in addition 8 a1 a2 2 A 8 b 2 B ((a1 b) (a2 b) 2 f ) a1 = a2) (5.84) is valid. While for a mapping we supposed only that an original element has one image, injectivity means also that every image has only one original element. Here f is called a mapping from A onto B (or surjective), if 8 b 2 B 9 a 2 A ((a b) 2 f ) (5.85) holds. An injective mapping that is also surjective is called bijective. For a bijective mapping f : A ! B there is an inverse relation which is also a mapping f ;1: B ! A, the so-called inverse mapping of f . The relation product is used for composition of mappings: If f : A ! B and g: B ! C are mappings, f g is also a mapping from A to C , and is de ned by (f g)(a) = g(f (a)): (5.86) Remark: Be careful with the order of f and g in this equation (it is treated di erently in the literature!).
5.2.4 Equivalence and Order Relations
The most important classes of binary relations with respect to a set A are the equivalence and order relations.
5.2 Set Theory 297
1. Equivalence Relations
A binary relation R with respect to a set A is called an equivalence relation if R is reexive, symmetric, and transitive. For aRb we also use the notation a !R b or a ! b if the equivalence relation R is already known, and we say that a is equivalent to b (with respect to R).
Examples of Equivalence Relations: A: A = Z, m 2 IN n f0g. a !R b holds exactly if a and b have the same remainder when divided by
m (they are congruent modulo m). B: Equality relation in di erent domains, e.g., in the set Q of rational numbers: pq1 = pq2 , p1 q2 = 1 2 p2q1 , where the rst equality sign de nes an equality in Q, while the second one denotes an equality in Z. C: Similarity or congruence of geometric gures. D: Logical equivalence of expressions of propositional calculus (see 5.1.1, 6., p. 287).
2. Equivalence Classes, Partitions
1. Equivalence Classes An equivalence relation in a set A de nes a partition of A into non-empty
pairwise disjoint subsets, into equivalence classes. a]R := fb j b 2 A ^ a !R bg (5.87) is called an equivalence class of a with respect to R. For equivalence classes the following is valid: a]R 6= a !R b , a]R = b]R and a 6!R b , a]R \ b]R = : (5.88) These equivalence classes form a new set, the quotient set A=R: A=R = f a]R j a 2 Ag: (5.89) A subset Z IP(A) of the power set IP(A) is called a partition of A if # 2= Z X Y 2 Z ^ X 6= Y ) X \ Y = X = A: (5.90) X 2Z
2. Decomposition Theorem Every equivalence relation R in a set A de nes a partition Z of A,
namely Z = A=R. Conversely, every partition Z of a set A de nes an equivalence relation R in A: a !R b , 9 X 2 Z (a 2 X ^ b 2 X ): (5.91) An equivalence relation in a set A can be considered as a generalization of the equality, where \ insigni cant " properties of the elements of A are neglected, and the elements, which do not di er with respect to a certain property, belong to the same equivalence class.
3. Ordering Relations
A binary relation R in a set A is called a partial ordering if R is reexive, antisymmetric, and transitive. If in addition R is linear, then R is called a linear ordering or a chain. The set A is said to be ordered or linearly ordered by R. In a linearly ordered set any two elements are comparable. Instead of aRb we also use the notation a R b or a b, if the ordering relation R is known from the problem.
Examples of Ordering Relations: A: The sets of numbers IN, Z, Q, IR are completely ordered by the usual relation. B: The subset relation is also an ordering, but only a partial ordering. C: The lexicographical order of the English words is a chain. Remark: If Z = fA B g is a partition of Q with the property a 2 A ^ b 2 B ) a < b, then (A B ) is
called a Dedekind cut. If neither A has a greatest element nor B has a smallest element, so an irrational number is uniquely determined by this cut. Besides the nest of intervals (see 1.1.1.2, p. 2) the notion of Dedekind cuts is another way to introduce irrational numbers.
298 5. Algebra and Discrete Mathematics
4. Hasse Diagram 4 2
3
1
Figure 5.8
Finite ordered sets can be represented by the Hasse diagram: Let an ordering relation be given on a nite set A. The elements of A are represented as points of the plane, where the point b 2 A is placed above the point a 2 A if a < b holds. If there is no c 2 A for which a < c < b, we say a and b are neighbors or consecutive members. Then we connect a and b by a line segment. A Hasse diagram is a \simpli ed" arrow diagram, where all the loops, arrow-heads, and the arrows following from the transitivity of the relation are eliminated. The arrow diagram of the divisibility relation T of the set A = f1 2 3 4g is given in Fig. 5.7. T also denotes an ordering relation, which is represented by the Hasse diagram in Fig. 5.8.
5.2.5 Cardinality of Sets
In 5.2.1, p. 290 the number of elements of a nite set was called the cardinality of the set. This notion of cardinality should can also be extended to in nite sets.
1. Cardinal Numbers
Two sets A and B are called equinumerous if there is a bijective mapping between them. To every set A we assign a cardinal number jAj or card A, so that equinumerous sets have the same cardinal number. A set and its power set are never equinumerous, so no \ greatest " cardinal number exists.
2. Innite Sets
In nite sets can be characterized by the property that they have proper subsets equinumerous to the set itself. The \smallest" in nite cardinal number is the cardinal number of the set IN of the natural numbers. This is denoted by @0 (aleph 0). A set is called enumerable or countable if it is equinumerous to IN. This means that its elements can be enumerated or written as an in nite sequence a1 a2 : : :. A set is called non-countable if it is in nite but it is not equinumerous to IN. Consequently every in nite set which is not enumerable is non-countable. A: The set Z of integers and the set Q of the rational numbers are countable sets. B: The set IR of the real numbers and the set C of the complex numbers are non-countable sets. These sets are equinumerous to IP(IN), the power set of the natural numbers, and their cardinality is called the continuum.
5.3 Classical Algebraic Structures 5.3.1 Operations 1. n-ary Operations
The notion of structure has a central role in mathematics and its applications. Now we investigate algebraic structures, i.e., sets on which operations are de ned. An n-ary operation ' on a set A is a mapping ': An ! A, which assigns an element of A to every n-tuple of elements of A.
2. Properties of Binary Operations
Especially important is the case n = 2, which is called a binary operation, e.g., addition and multiplication of numbers or matrices, or union and intersection of sets. A binary operation can be considered as a mapping # : A A ! A, where instead of the notation \#(a b)" we use the inx form \a # b". A binary operation # in A is called associative if (a # b) # c = a # (b # c) (5.92) and commutative if a#b=b#a (5.93)
5.3 Classical Algebraic Structures 299
holds for every a b c 2 A. An element e 2 A is called a neutral element with respect to a binary operation # in A if a # e = e # a = a holds for every a 2 A:
3. Exterior Operations
(5.94)
Sometimes we deal with exterior operations. That are the mappings from K A to K , where K is an \exterior" and mostly already structured set (see 5.3.7, p. 316).
5.3.2 Semigroups
The most frequently occurring algebraic structures have their own names. A set H having one associative binary operation # , is called a semigroup. The notation: is H = (H #).
Examples of Semigroups: A: Number domains with respect to addition or multiplication. B: Power sets with respect to union or intersection. C: Matrices with respect to addition or multiplication. D: The set A of all \ words " (strings) over an \ alphabet " A with respect to concatenation (free semigroup). Remark: Except for multiplication of matrices and concatenation of words, all operations in these examples are also commutative in this case we talk about a commutative semigroup.
5.3.3 Groups
5.3.3.1 Denition and Basic Properties 1. Denition
A set G with a binary operation # is called a group if # is associative, # has a neutral element e, and for every element a 2 G there exists an inverse element a;1 such that a # a;1 = a;1 # a = e: (5.95) A group is a special semigroup. The neutral element of a group is unique, i.e., there exists only one. Furthermore, every element of the group has exactly one inverse. If the operation # is commutative, then the group is called an Abelian group. If the group operation is written as addition, +, then the neutral element is denoted by 0 and the inverse of an element a by ;a.
Examples of Groups: A: The number domains (except IN) with respect to addition. B: Q n f0g, IR n f0g, and C n f0g with respect to multiplication. C: SM := ff : M ! M ^ f bijectiveg with respect to composition of mappings (symmetric group). D: Consider the set Dn of all covering transformations of a regular n-gon in the plane. Here a
covering transformation is the transition between two symmetric positions of the n-gon, i.e., the moving of the n-gon into a superposable position. If we denote by d a rotation by the angle 2=n and by the reection with respect to an axis, then Dn has 2n elements: Dn = fe d d2 : : : dn;1 d : : : dn;1g: With respect to the composition of mappings Dn is a group, the dihedral group. Here the equalities dn = 2 = e and d = dn;1 hold. E: All the regular matrices (see 4.1.4, p. 254) over the real or complex numbers with respect to multiplication. Remark: Matrices have a very important role in applications, especially in representation of linear transformations. Linear transformations can be classi ed by matrix groups.
300 5. Algebra and Discrete Mathematics
2. Group Tables or Cayley's Tables
For the representation of nite groups Cayley's tables or group tables are used: The elements of the group are denoted at the row and column headings. The element a # b is the intersection of the row of the element a and the column of the element b. If M = f1 2 3g, then the symmetric group SM is also denoted by S3. S3 consists of all the bijective mappings (permutations) of the set f1 2 3g and consequently it has 3! = 6 elements (see 16.1.1, p. 745). Permutations are mostly represented in two rows, where in the rst row there are the elements of M and under each of them there is its image. So, we get the six elements of S3 as the following: " = 11 22 33 p1 = 11 23 32 p2 = 13 22 31 (5.96) p3 = 12 21 33 p4 = 12 23 31 p5 = 13 21 32 : With the successive application of these mappings (binary operations) the following group table is obtained for S3 : From the group table it can be seen that the identity per" p1 p2 p3 p4 p5 mutation " is the neutral element of the group. " " p1 p2 p3 p4 p5 In the group table every element appears exactly once in p1 p1 " p5 p4 p3 p2 (5.97) every row and in every column. p 2 p 2 p 4 " p 5 p1 p 3 It is easy to recognize the inverse of any group element in p3 p3 p5 p4 " p2 p1 the table, i.e., the inverse of p4 in S3 is the permutation p5 , p4 p4 p2 p3 p1 p5 " because at the intersection of the row of p4 with the column p5 p5 p3 p1 p2 " p4 of p5 is the neutral element ". If the group operation is commutative (Abelian group), then the table is symmetric with respect to the \main diagonal" S3 is not commutative, since, e.g., p1 p2 6= p2 p1 . The associative property cannot be recognized easily from the table.
5.3.3.2 Subgroups and Direct Products 1. Subgroups
Let G = (G #) be a group and U G. If U is also a group with respect to #, then U = (U #) is called a subgroup of G. A non-empty subset U of a group (G #) is a subgroup of G if and only if for every a b 2 U , the elements a # b and a;1 are also in U (subgroup criterion). 1. Cyclic Subgroups The group G itself and E = feg are subgroups of G, the so-called trivial subgroups. Furthermore, a subgroup corresponds to every element a 2 G, the so-called cyclic subgroup generated by a: < a > = f: : : a;2 a;1 e a a2 : : :g: (5.98) If the group operation is addition, then we write the integer multiple ka as a shorthand notation of the k times addition of a with itself instead of the power ak , as a shorthand notation of the k times operation of a by itself, i.e., < a > = f: : : (;2)a ;a 0 a 2a : : :g: (5.99) Here < a > is the smallest subgroup of G containing a. If < a > = G holds for an element a of G, then G is called cyclic. There are in nite cyclic groups, e.g., Z with respect to addition, and nite cyclic groups, e.g., the set Zm the residue class modulo m with residue class addition (see 5.4.3, 3., p. 327). If the number of elements of a nite G group is a prime, then G is always cyclic. 2. Generalization The notion of cyclic groups can be generalized as follows: If M is a non-empty subset of a group G, then the subgroup of G whose elements can be written in the form of a product of nitely many elements of M and their inverses, is denoted by < M >. The subset M is called the system of generators of < M >. If M contains only one element, then < M > is cyclic.
5.3 Classical Algebraic Structures 301
3. Order of a Group, Left and Right Cosets In group theory the number of elements of a nite group is denoted by ord G. If the cyclic subgroup < a > generated by one element a is nite, then this order is also called the order of the element a, i.e., ord < a > = ord a. If U is a subgroup of a group (G #) and a 2 G, then the subsets aU := fa # uju 2 U g and Ua := fu # aju 2 U g (5.100) of G are called left cosets and right cosets of U in G. The left or right cosets form a partition of G, respectively (see 5.2.4, 2., p. 297). All the left or right cosets of a subgroup U in a group G have the same number of elements, namely ord U . From this it follows that the number of left cosets is equal to the number of right cosets. This number is called the index of U in G. The Lagrange theorem follows from these facts. 4. Lagrange Theorem The order of a subgroup is a divisor of the order of the group. In general it is dicult to determine all the subgroups of a group. In the case of nite groups the Lagrange theorem as a necessary condition for the existence of a subgroup is useful.
2. Normal Subgroup or Invariant Subgroup
For a subgroup U , in general, aU is di erent from Ua (however jaU j = jUaj is valid). If aU = Ua for all a 2 G holds, then U is called a normal subgroup or invariant subgroup of G. These special subgroups are the basis of forming factor groups (see 5.3.3.3, 3., p. 302). In Abelian groups, obviously, every subgroup is a normal subgroup.
Examples of Subgroups and Normal Subgroups: A: IR n f0g, Q n f0g form subgroups of C n f0g with respect to multiplication. B: The even integers form a subgroup of Z with respect to addition. C: Subgroups of S3: According to the Lagrange theorem the group S3 having six elements can have subgroups only with two or three elements (besides the trivial subgroups). In fact, the group S3 has the following subgroups: E = f"g, U1 = f" p1g, U2 = f" p2g, U3 = f" p3g, U4 = f" p4 p5g, S3 .
The non-trivial subgroups U1 , U2 , U3 , and U4 are cyclic, since the numbers of their elements are primes. But the group S3 is not cyclic. The group S3 has only U4 as a normal subgroup, except the trivial normal subgroups. Anyway, every subgroup U of a group G with jU j = jGj=2 is a normal subgroup of G. Every symmetric group SIM and their subgroups are called permutation groups. D: Special subgroups of the group GL(n) of all regular matrices of type (n n) with respect to matrix multiplication: SL(n) group of all matrices A with determinant 1, O(n) group of all orthogonal matrices, SO(n) group of all orthogonal matrices with determinant 1. The group SL(n) is a normal subgroup of GL(n) (see 5.3.3.3, 3., p. 302) and SO(n) is a normal subgroup of O(n). E: As subgroups of all complex matrices of type (n n) (see 4.1.4, p. 254): U (n) group of all unitary matrices, SU (n) group of all unitary matrices with determinant 1.
3. Direct Product
1. De nition Suppose A and B are groups, whose group operation (e.g., addition or multiplication) is denoted by . In the Cartesian product (see 5.2.2, 4., p. 294) A B (5.65a) an operation # can be introduced in the following way: (a1 b1 ) # (a2 b2) = (a1 a2 b1 b2 ): (5.101a) A B becomes a group with this operation and it is called the direct product of A and B . (e e) denotes the unit element of A B , (a;1 b;1 ) is the inverse element of (a b). For nite groups A B ord (A B ) = ord A ord B (5.101b)
302 5. Algebra and Discrete Mathematics holds. The groups A0 := f(a e)ja 2 Ag and B 0 := f(e b)jb 2 B g are normal subsets of AB isomorphic to A and B , respectively. The direct product of Abelian groups is again an Abelian group. The direct product of two cyclic groups A B is cyclic if and only if the greatest common divisor of the orders of the groups is equal to 1. A: With Z2 = fe ag and Z3 = fe b b2g, the direct product Z2 Z3 = f(e e) (e b) (e b2) (a e), (a b) (a b2 )g, is a group isomorphic to Z6 (see 5.3.3.3, 2., p. 302) generated by (a b). B: On the other hand Z2 Z2 = f(e e) (e b) (a e) (a b)g is not cyclic. This group has order 4 and it is also called Klein's four-group, and it describes the covering operations of a rectangle. 2. Fundamental Theorem of Abelian Groups Because the direct product is a construction which enables us to make \larger" groups from \smaller" groups, we can reverse the question: When is it possible to consider a larger group G as a direct product of smaller groups A B , i.e., when will G be isomorphic to A B ? For Abelian groups, there exists the so-called fundamental theorem: Every nite Abelian group can be represented as direct product of cyclic groups with orders of prime powers.
5.3.3.3 Mappings Between Groups 1. Homomorphism and Isomorphism
1. Group Homomorphism Between algebraic structures we do not consider arbitrary mappings but only \structure keeping" mappings: Let G1 = (G1 #) and G2 = (G2 ) are two groups. A mapping h: G1 ! G2 is called a group homomorphism, if for all a b 2 G1 h(a # b) = h(a) h(b) (\image of product = product of images") (5.102)
is valid. As an example, consider the multiplication law for determinants (see 4.2.2, 7., p. 260): det(AB ) = (det A)(det B ): (5.103) Here on the right-hand side there is the product of non-zero numbers, on the left-hand side there is the product of regular matrices. If h: G1 ! G2 is a group homomorphism, then the set of elements of G1, whose image is the neutral element of G2, is called the kernel of h, and it is denoted by ker h. The kernel of h is a normal subgroup of G1 . 2. Group Isomorphism If a group homomorphism h is also bijective, then h is called a group isomorphism, and the groups G1 and G2 are said to be isomorphic to each other (notation: G1 ! = G2). Then ker h = E is valid. Isomorphic groups have the same structure, i.e., they di er only by the notation of their elements. The symmetric group S3 and the dihedral group D3 are isomorphic groups of order 6 and describe the covering mappings of an equilateral triangle.
2. Cayley's Theorem
The Cayley theorem says that every group can be interpreted as a permutation group (see 5.3.3.2, 2., p. 301): Every group is isomorphic to a permutation group. The permutation group P , whose elements are the permutations g (g 2 G) mapping a to (G #)g, is a subgroup of SG isomorphic to G #.
3. Homomorphism Theorem for Groups
The set of cosets of a normal subgroup N in a group G is also a group with respect to the operation aN bN = abN: (5.104) It is called the factor group of G with respect to N , and it is denoted by G=N . The following theorem gives the correspondence between homomorphic images and factor groups of a group, because of what it is called the homomorphism theorem for groups:
5.3 Classical Algebraic Structures 303
A group homomorphism h: G1 ! G2 de nes a normal subgroup of G1, namely ker h = fa 2 G1jh(a) = eg. The factor group G1= ker h is isomorphic to the homomorphic image h(G1) = fh(a)ja 2 G1 g. Conversely, every normal subgroup N of G1 de nes a homomorphic mapping natN : G1 ! G1 =N with natN (a) = aN . This mapping natN is called a natural homomorphism. Since the determinant construction det: GL(n) ! IR n f0g is a group homomorphism with kernel SL(n), SL(n) is a normal subgroup of GL(n) and (according to the homomorphism theorem): GL(n)=SL(n) is isomorphic to the multiplicative group R nf0g of real numbers (for notation see 5.3.3.2, 2., p. 301).
5.3.4 Group Representations 5.3.4.1 Denitions 1. Representation
A representation D(G) of the group G is a map (homomorphism) of G onto the group of non-singular linear transformations D on an n-dimensional (real or complex) vector space Vn: D(G) : a ! D(a) a 2 G: (5.105) The vector space Vn is called the representation space n is the dimension of the representation (see also 12.1.3, 2., p. 599). Introducing the basis feig (i = 1 2 : : : n) in Vn every vector x can be written as a linear combination of the basis vectors: n X x = xi ei x 2 Vn: (5.106) i=1
The action of the linear transformation D(a) a 2 G, on x can be de ned by the quadratic matrix (Dik (a)) (i k = 1 2 : : : n), which provides the coordinates of the transformed vector x0 within the basis ei:
x0 = D(a)x =
n X i=1
x0iei
x0i =
n X
k=1
Dik (a)xk :
(5.107)
This transformation may also be considered as a transformation of the basis feig ! fe0ig:
e0i = ei D(a) =
n X
k=1
Dki(a)ek :
Thus, every element a of the group is assigned to the representation matrix (Dik (a)): D(G) : a ! (Dik (a)) (i k = 1 2 : : : n) a 2 G: The representation matrix depends on the choice of basis.
(5.108) (5.109)
2. Faithful Representation
A representation is said to be faithful if G ! D(G) is an isomorphism, i.e., the assignment of the element of the group to the representation matrix is a one-to-one mapping.
3. Properties of the Representations
A representation has the following properties (a b 2 G I : unit operator): D(a # b) = D(a) D(b) D(a;1 ) = D;1(a) D(e) = I:
5.3.4.2 Particular Representations 1. Identity Representation
(5.110)
Any group G has a trivial one-dimensional representation (identity representation), for which every element of the group is mapped to the unit operator I : a ! I for all a 2 G. In general, in this section vectors are not printed in bold symbols.
304 5. Algebra and Discrete Mathematics
2. Adjoint Representation +
The representation D (G) is said to be adjoint to D(G) if the corresponding representation matrices are related by complex conjugation and reection in the main diagonal: D+(G) = D~ (G): (5.111)
3. Unitary Representation
For a unitary representation all representation matrices are unitary matrices: D(G) D+(G) = E where E is the unit matrix.
(5.112)
4. Equivalent Representations 0
Two representations D(G) und D (G) are said to be equivalent if for each element a of the group the corresponding representation matrices are related by the same similarity transformation with the nonsingular matrix T:
D0(a) = T;1 D(a) T Dik0 (a) =
n X
jl=1
T ;1 ij Djl(a) Tlk :
(5.113)
If such a relation does not hold two representations are called non-equivalent. The transition from D(G) to D0(G) corresponds to the transformation T : fe1 e2 : : : eng ! fe01 e02 : : : e0ng of the basis in the representation space Vn:
e0 = e T
e0i =
n X
k=1
Tkiek (i = 1 2 : : : n):
(5.114)
Any representation of a nite group is equivalent to a unitary representation.
5. Character of a Group Element
In the representation D(G) the character (a) of the group element a is de ned as the trace of the representation matrix D(a) (sum of the matrix elements on the main diagonal):
(a) = Sp (D) =
n X i=1
Dii(a):
(5.115)
The character of the unit element e is given by the dimension n of the representation: (e) = n. Since the trace of a matrix is invariant under similarity transformations, the group element a has the same character for equivalent representations. Within the shell model of atomic or nuclear physics two out of three particles with coordinates x1 x2 x3 are supposed to be in the state ' while the third particle is in the state ' (con guration 2 ). The possible occupations ' (x1)' (x2 )' (x3) = e1 ' (x1 )' (x2 )' (x3 ) = e2 ' (x1 )' (x2 )' (x3 ) = e3 form a basis fe1 e2 e3g in the three-dimensional vector space V3 for a representation of the symmetric group S3. According to (5.108) the matrix elements of the representation matrices can be found by investigating the action of the group elements (5.92) on the coordiante subscripts in the basis elements ei . For example: p1e1 = p1 ' (x1 )' (x2 )' (x3 ) = ' (x1 )' (x2 )' (x3 ) = D21 (p1)e2 p1e2 = p1 ' (x1 )' (x2 )' (x3 ) = ' (x1 )' (x2 )' (x3 ) = D12 (p1)e1 p1e3 = p1 ' (x1)' (x2 )' (x3 ) = ' (x1 )' (x2 )' (x3 ) = D33 (p1)e3 : (5.116) Altogether one nds: 01 0 01 00 1 01 00 0 11 @ A @ A D(e) = 0 1 0 D(p1) = 1 0 0 D(p2) = @ 0 1 0 A 0 01 00 10 1 0 00 01 10 1 0 10 00 01 1 (5.117) D(p3) = @ 0 0 1 A D(p4) = @ 0 0 1 A D(p5) = @ 1 0 0 A : 010 100 010
5.3 Classical Algebraic Structures 305
For the characters one has: (e) = 3 (p1 ) = (p2 ) = (p3) = 1 (p4) = (p5 ) = 0:
5.3.4.3 Direct Sum of Representations
The representations D(1) (G) D(2) (G) of dimension n1 and n2 can be composed to create a new representation D(G) of dimension n = n1 + n2 by forming the direct sum of the representation matrices: ! (1) D(a) = D(1) (a) $ D(2) (a) = D 0(a) D(2)0(a) : (5.118) The block-diagonal form of the representation matrix implies that the representation space Vn is the direct sum of two invariant subspaces Vn1 Vn2 : V n = V n 1 $ V n2 n = n1 + n 2 : (5.119) A subspace Vm (m < n) of Vn is called an invariant subspace if for any linear transformation D(a) a 2 G, every vector x 2 Vm is mapped onto an element of Vm again: (5.120) x0 = D(a)x with x x0 2 Vm : The character of the representation (5.118) is the sum of the characters of the single representations: (a) = (1) (a) + (2) (a): (5.121)
5.3.4.4 Direct Product of Representations
If ei (i = 1 2 : : : n1 ) and e0k (k = 1 2 : : : n1 ) are the basis vectors of the representation spaces Vn1 and Vn2 , respectively, then the tensor product eik = feiek g (i = 1 2 : : : n1 k = 1 2 : : : n2) (5.122) forms a basis in the product space Vn1 Vn2 of dimension n1 n2 . With the representations D(1) (G) and D(2) (G) in Vn1 and Vn2 , respectively an n1 n2 -dimensional representation D(G) in the product space can be constructed by forming the direct or (inner) Kronecker product of the representation matrices: D(G) = D(1) (G) D(2) (G) (D(G))ikjl = Dik(1) (a) Djl(2) (a) with i k = 1 2 : : : n1 j l = 1 2 : : : n2: (5.123) The character of the Kronecker product of two representations is equal to the product of the characters of the factors (1 2) (a) = (1) (a) (2) (a): (5.124)
5.3.4.5 Reducible and Irreducible Representations
If the representation space Vn possesses a subspace Vm (m < n) invariant under the group operations the representation matrices can be decomposed according to (a) 0 g m rows 1 T ;1 D(a) T = DA (5.125) D2(a) g n ; m rows by a suitable transformation T of the basis in Vn. D1(a) and D2(a) themselfes are matrix representations of a 2 G of dimension m and n ; m, respectively. A representation D(G) is called irreducible if there is no proper (non-trivial) invariant subspace in Vn. The number of non-equivalent irreducible representations of a nite group is nite. If a transformation T of a basis can be found which makes Vn to a direct sum of invariant subspaces, i.e., V n = V 1 $ $ V nj (5.126)
306 5. Algebra and Discrete Mathematics then for every a 2 G the representation matrix D(a) can be transformed into the block-diagonal form (A = 0 in (5.125)):
0 (1) D (a) ... T;1 D(a) T = D(1) (a) $ $ D(nj ) (a) = BB@
0
1 C C A:
(5.127) 0 D(nj ) (a) by a similarity transformation with T. Such a representation is called completely reducible. Remark: For the application of group theory in natural sciences a fundamental task consists in the classi cation of all non-equivalent irreducible representations of a given group. The representation of the symmetric group S3 given in (5.117) is reducible. For example, in the transformation fe1 e2 e3 g ;! fe01 = e1 + e2 + e3 e02 = e2 e03 = e3 g of the basis one obtains for the representation matrix of the permutation p3: 01 0 01 D(p3) = @ 0 0 1 A = D1A(p3) D20(p3 ) 010
with A = 00 D1(p3 ) = 1 as the identity representation of S3 and D2(p3 ) = 01 10 .
5.3.4.6 Schur's Lemma 1
If C is an operator commuting with all transformations of an irreducible representation D of a group C D(a)] = C D(a) ; D(a) C = 0 a 2 G, and the representation space Vn is an invariant subspace of C , then C is a multiple of the unit operator, i.e., a matrix (Cik ) which commutates with all matrices of an irreducible representation is a multiple of the matrix E, C = E 2 C.
5.3.4.7 Clebsch{Gordan Series
In general, the Kronecker product of two irreducible representations D(1) (G) D(2) (G) is reducible. By a suitable basis transformation in the product space D(1) (G) D(2) (G) can be decomposed into the direct sum of its irreducible parts D( ) ( = 1 2 : : : n) (Clebsch{Gordan theorem). This expansion is called the Clebsch{Gordan series:
D(1) (G) D(2) (a) =
n X
=1
$ m D( ) (G):
(5.128)
Here, m is the multiplicity with which the irreducible representation D( ) (G) occurs in the Clebsch{ Gordan series. The matrix elements of the basis transformation in the product space causing the reduction of the Kronecker product into its irreducible components are called Clebsch{Gordan coecients.
5.3.4.8 Irreducible Representations of the Symmetric Group SM
1. Symmetric Group SM
The non-equivalent irreducible representations of the symmetric group SM are characterized uniquely by the partitions of M , i.e., by the splitting of M into integers according to ] = 1 2 : : : M ] 1 + 2 + + M = M 1 2 M 0: (5.129) The graphic representation of the partitions is done by arranging boxes in Young diagrams.
5.3 Classical Algebraic Structures 307
For the group S4 one obtains ve Young dia[ 4] [ 3,1] [ 2,2] [ 2,1,1] [ 1 ] [ λ] = grams as shown in the gure. The dimension of the representation ] is given by Q ( ; + j ; i) i j i<j k
] n = M ! +k ( + k ; i)! : (5.130) i=1 i The Young diagram ~] conjugated to ] is constructed by the interchange of rows and columns. In general, the irreducible representation of SM is reducible if one restricts to one of the subgroups SM ;1 SM ;2 . In quantum mechanics for a system of identical particles the Pauli principle demands the construction of many-body wave functions that are antisymmetric with respect to the interchange of all coordinates of two arbitrary particles. Often, the wave function is given as the product of a function in space coordinates and a function in spin variables. If for such a case due to particle permutations the spatial part of the wave function transforms according to the irreducible representation ] of the symmetric group it has to be combined with a spin function transforming according to ~] in order to get a total wave function which is antisymmetric if two particles are interchanged. 4
5.3.5 Applications of Groups
In chemistry and in physics, groups are applied to describe the \symmetry" of the corresponding objects. Such objects are, for instance, molecules, crystals, solid structures or quantum mechanical systems. The basic idea of these applications is the von Neumann principle: If a system has a certain group of symmetry operations, then every physical observational quantity of this system must have the same symmetry.
5.3.5.1 Symmetry Operations, Symmetry Elements
A symmetry operation s of a space object is a mapping of the space into itself such that the length of line segments remains unchanged and the object goes into a covering position to itself. The set of xed points of the symmetry operation s is denoted by Fix s, i.e., the set of all points of space which remain unchanged for s. The set Fix s is called the symmetry element of s. We use the Schoeniess symbolism to denote the symmetry operation. Two types of symmetry operations are distinguished: Operations without a xed point and operations with at least one xed point. 1. Symmetry Operations without a Fixed Point, for which no point of the space stays unchanged, cannot occur for bounded space objects, but now we consider only such objects. A symmetry operation without a xed point is for instance a parallel translation. 2. Symmetry Operations with at least One Fixed Point are for instance rotations and reections. The following operations belong to them. a) Rotations Around an Axis by an Angle ': The axis of rotation and also the rotation itself is denoted by Cn for ' = 2=n: The axis of rotation is then said to be of n-th order. b) Reection with Respect to a Plane: Both the plane of reection and the reection itself are denoted by . If additionally there is a principal rotation axis, then we draw it perpendicularly and denote the planes of reections which are perpendicular to this axis by h (h from horizontal) and the planes of reections passing through the rotational axis are denoted by v (v from vertical) or d (d means dihedral, if certain angles are halved). c) Improper Orthogonal Mappings: An operation such that after a rotation Cn a reection h follows, is called an improper orthogonal mapping and it is denoted by Sn. Rotation and reection commute. The axis of rotation is then called an improper rotational axis of n-th order and it is also denoted by Sn. This axis is called the corresponding symmetry element, although only the symme-
308 5. Algebra and Discrete Mathematics try center stays xed under the application of the operation Sn. For n = 2, an improper orthogonal mapping is also called a point reection (see 4.3.5.1, p. 269) and it is denoted by i.
5.3.5.2 Symmetry Groups or Point Groups
For every symmetry operation S , there is an inverse operation S ;1, which reverses S \back", i.e., SS ;1 = S ;1S = : (5.131) Here denotes the identity operation, which leaves the whole space unchanged. The family of symmetry operations of a space object forms a group with respect to the successive application, which is in general a non-commutative symmetry group of the objects. The following relations hold: a) Every rotation is the product of two reections. The intersection line of the two reection planes is the rotation axis. b) For two reections and 0 0 = 0 (5.132) if and only if the corresponding reection planes are identical or they are perpendicular to each other. In the rst case the product is the identity in the second one the rotation C2. c) The product of two rotations with intersecting rotational axes is again a rotation whose axis goes through the intersection point of the given rotational axes. d) For two rotations C2 and C20 around the same axis or around axes perpendicular to each other: C2C20 = C20 C2: (5.133) The product is again a rotation. In the rst case the corresponding rotational axis is the given one, in the second one the rotational axis is perpendicular to the given ones.
5.3.5.3 Symmetry Operations with Molecules
It requires a lot of work to recognize every symmetry element of an object. In the literature, for instance in 5.9], 5.12], it is discussed in detail how to nd the symmetry groups of molecules if all the symmetry elements are known. The following notation is used for the interpretation of a molecule in space: The symbols above C in Fig. 5.9 mean that the OH group lies above the plane of the drawing, the symbol to the right-hand side of C means that the group OC2 H5 is under C. The determination of the symmetry group can be made by the following method.
1. No Rotational Axis a) If no symmetry element exists, then G = f g holds, i.e., the molecule does not have any symmetry
operation but the identity . The molecule hemiacetal (Fig.5.9) is not planar and it has four di erent atom groups. b) If is a reection or i is an inversion, then G = f g =: Cs or G = f ig = Ci hold, and with this it is isomorphic to Z2 . The molecule of tartaric acid (Fig.5.10) can be reected in the center P (inversion). H
OH CH3
C H
Figure 5.9
HO OC2H5
CO2H
C
O
O
P CO2H
C H
Figure 5.10
OH
δ
δ
H
H
Figure 5.11
2. There is Exactly One Rotational Axis C a) If the rotation can have any angle, i.e., C = C1, then the molecule is linear, and the symmetry
5.3 Classical Algebraic Structures 309
group is in nite. A: For the molecule of sodium chloride (common salt) NaCl there is no horizontal reection. The corresponding symmetry group of all the rotations around C is denoted by C1v . B: The molecule O2 has one horizontal reection. The corresponding symmetry group is generated by the rotations and by this reection, and it is denoted by D1h. b) The rotation axis is of n-th order, C = Cn , but it is not an improper rotational axis of order 2n. If there is no further symmetry element, then G is generated by a rotation d by an angle =n around Cn, i.e., G = < d > ! = Zn . In this case G is also denoted by Cn. If there is a further vertical reection v , then G = < d v > ! = Dn holds (see 5.3.3.1, p. 299), and G is denoted by Cnv . If there exists an additional horizontal reection h, then G = < d v > ! = Zn Z2 holds. G is denoted by Cnh and it is cyclic for odd n (see 5.3.3.2, p. 300). A: For hydrogen peroxide (Fig.5.11) these three cases occur in the order given above for 0 < < =2 = 0 and = =2. B: The molecule of water H2O has a rotational axis of second order and a vertical plane of reection, as symmetry elements. Consequently, the symmetry group of water is isomorphic to the group D2 , which is isomorphic to the Klein four-group V4 (see 5.3.3.2, 3., p. 301). c) The rotational axis is of order n and at the same time it is also an improper rotational axis of order 2n. We have to distinguish two cases. ) There is no further vertical reection, so G != Z2n holds, and G is denoted also by S2n. An example is the molecule of tetrahydroxy allene with formula C3 (OH)4 (Fig.5.12). ) If there is a vertical reection, then G is a group of order 4n, which is denoted by Dnh. For n = 2 we get G ! = D4, i.e., the dihedral group of order eight. An example is the allene molecule (Fig.5.13). H O C O C H
H C
H
C
C
H
O
H
H
O
H
C
C H
H
H H
Figure 5.12
Figure 5.13
Figure 5.14
3. Several Rotational Axes If there are several rotational axes, then we distinguish further cases. In particular, if several rotational axes have an order n 3, then the following groups are the corresponding symmetry groups.
a) Tetrahedral group Td : Isomorphic to S4, ord Td = 24. b) Octahedral group Oh: Isomorphic to S4 Z2, ord Oh = 48. c) Icosahedral group Ih: Ord Ih = 120. These groups are the symmetry groups of the regular polyeders discussed in 3.3.3, Table 3.7, p. 154, (Fig.3.63). The methane molecule (Fig.5.14) has the tetrahedral group Td as a symmetry group.
310 5. Algebra and Discrete Mathematics
5.3.5.4 Symmetry Groups in Crystallography
1. Lattice Structures
In crystallography the parallelepiped represents, independently of the arrangment of speci c atoms or ions, the elementary (unit) cell of the crystall lattice. It is determined by three non-coplanar basis vectors ~ai starting from one lattice point (Fig. 5.15). The in nite geometric lattice structure is created by performing all primitive translations ~tn: t~n = n1~a1 + n2~a2 + n3~a3 n = (n1 n2 n3) ni 2 Z: (5.134) Here, the coecients ni (i = 1 2 : : :) are integers. All the translations ~tn xing the space points of the lattice a L = f~tng in terms of lattice vectors form the translation group T with the group element T (~tn), the inverse element T ;1(t~n) = α a T (;t~n ), and the composition law T (t~n) # T (t~m) = T (t~n + t~m). a βγ The application of the group element T (t~n) to the position vector ~r is described by: Figure 5.15 T (t~n)~r = ~r + t~n: (5.135) 3
2
1
2. Bravais Lattices
Taking into account the possible combinations of the relative lengths of the basis vectors a~i and the pairwise related angles between them (particularly angles 90 and 120) one obtains seven di erent types of elementary cells with the corresponding lattices, the Bravais lattices (see Fig. 5.15, and Table 5.4). This classi cation can be extended by seven non-primitive elementary cells and their corresponding lattices by adding additional lattice points at the intersection points of the face or body diagonals, preserving the symmetry of the elementary cell. In this way one may distinguish one-side face-centered lattices, body-centered lattices, and all-face centered lattices. Tabelle 5.4 Primitive Bravais lattice
Elementary cell Relative lengths of basis vectors triclinic a1 = 6 a2 =6 a3 monoclinic a1 = 6 a2 =6 a3 rhombic a1 = 6 a2 =6 a3 trigonal hexagonal tetragonal cubic
a1 = a2 = a3 a1 = a2 6= a3 a1 = a2 6= a3 a1 = a2 = a3
Angles between basis vectors = 6 =6 =6 90 = = 90 = 6
= = = 90 = = < 120(6= 90) = = 90 = 120 = = = 90 = = = 90
3. Symmetry Operations in Crystal Lattice Structures
Among the symmetry operations transforming the space lattice to equivalent positions there are point group operations such as certain rotations, improper rotations, and reections in planes or points. But not all point groups are also crystallographic point groups. The requirement that the application of a group element to a lattice vector t~n leads to a lattice vector t~0n 2 L (L is the set of all lattice points) again restricts the allowed point groups P with the group elements P (R) according to: P = fR : Rt~n 2 Lg t~n 2 L: (5.136)
5.3 Classical Algebraic Structures 311
Here, R denotes a proper (R 2 SO(3)) or improper rotation operator (R = IR0 2 O(3) R0 2 SO(3) I is the inversion operator with I~r = ;~r ~r is a position vector). For example, only n-fold rotation axes with n = 1 2 3 4 or 6 are compatible with a lattice structure. Altogether, there are 32 cristallographic point groups P . The symmetry group of a space lattice may also contain operators representing simultaneous applications of rotations and primitive translations. In this way one gets gliding reections, i.e., reections in a plane and translations parallel to the plane, and screws, i.e., rotations through 2=n and translations by m~a=n (m = 1 2 : : : n ; 1 ~a are basis translations). Such operations are called non-primitive ~ (R), because they correspond to \fractional" translations. For a gliding reection R is translations V a reection and for a screw R is a proper rotation. The elements of the space group G, for which the crystal lattice is invariant is composed of elements P of the crystallographic point group P , primitive translations T (t~n) and non-primitive translations V~ (R): ~ (R) + t~n : R 2 P t~n 2 Lgg: G = ffRjV (5.137) The unit element of the space group is fej0g where e is the unit element of R. The element fejt~ng means a primitive translation, fRj0g represents a rotation or reection. Applying the group element fRj~tng to the position vector ~r one obtains: fRjt~ng~r = R~r + t~n: (5.138) Table 5.5 Bravais lattice, crystal systems, and crystallographic classes Notation: Cn { rotation about an n-fold rotation axis, Dn { dihedral group, Tn { tetrahedral group, On { octahedral group, Sn { mirror rotations with an n-fold axis.
Lattice type Crystal system Crystallographic class (holohedry) triclinic monoclinic rhombic tetragonal hexagonal trigonal cubic
Ci C2h D2h D4h D6h D3d Oh
4. Crystal Systems (Holohedry)
C1 Ci C2 Ch C2h C2v D2 D2h C4 S4 C4h D4 C4v D2d D4h C6 C3h C6h D6 C6v D3h D6h C3 S6 D3 C3v D3d T Th Td O Oh
From the 14 Bravais lattices, L = ft~ng , the 32 crystallographic point groups P = fRg and the allowed ~ (R) one can construct 230 space groups G = fRjV ~ (R) + t~ng. The point non-primitive translations V groups correspond to 32 crystallographic classes. Among the point groups there are seven groups that are not a subgroup of another point group but contain further point groups as a subgroup. Each of these seven point groups form a crystal system (holohedry). The symmetry of the seven crystal systems is reected in the symmetry of the seven Bravais lattices. The relation of the 32 crystallographic classes to the seven crystal systems is given in Table 5.5 using the notation of Schoeniess. Remark: The space group G (5.137) is the symmetry group of the \empty" lattice. The real crystal is obtained by arranging certain atoms or ions at the lattice sites. The arrangement of these crystal
312 5. Algebra and Discrete Mathematics constituents exhibits its own symmetry. Therefore, the symmetry group G0 of the real crystal possesses a lower symmetry than G (G % G0), in general.
5.3.5.5 Symmetry Groups in Quantum Mechanics
Linear coordinate transformations that leave the Hamiltonian H^ of a quantum mechanical system (see 9.2.3.5, 1., p. 538) invariant represent a symmetry group G, whose elements g commute with H^ : ^ = 0 g 2 G: g H^ ] = gH^ ; Hg (5.139) ^ The commutation property of g and H implies that in the application of the product of the operators g and H^ to a state ' the sequence of the action of the operators is arbitrary: ^ ) = H^ (g'): g(H' (5.140) ^ Hence, one has: If 'E ( = 1 2 : : : n) are the eigenstates of H with energy eigenvalue E of degeneracy n, i.e., ^ E = E'E ( = 1 2 : : : n) H' (5.141) then the transformed states g'E are also eigenstates belonging to the same eigenvalue E : ^ E = Hg' ^ E = Eg'E : gH' (5.142) The transformed states g'E can be written as a linear combination of the eigenstates 'E :
g'E =
n X
=1
D (g)'E :
(5.143)
Hence, the eigenstates 'E form the basis of an n-dimensional representation space for the representation D(G) of the symmetry group G of the Hamiltonian H^ with the representation matrices (D (g)) . This representation is irreducible if there are no \hidden" symmetries. One can state that the energy eigenstates of a quantum mechaniccal system can be labeled by the signatures of the irreducible representations of the symmetry group of the Hamiltonian. Thus, the representation theory of groups allows for qualitative statements on such patterns of the energy spectrum of a quantum mechanical system which are established by the outer or inner symmetries of the system only. Also the splitting of degenerate energy levels under the inuence of a pertubation which breaks the symmetry or the selection rules for the matrix elements of transitions between energy eigenstates follows from the investigation of representations according to which the participating states and operators transform under group operations. The application of group theory in quantum mechanics is presented extensively in the literature (see, e.g., 5.6], 5.7], 5.8], 5.9], 5.10]).
5.3.5.6 Further Applications of Group Theory in Physics
Further examples of the application of particular continuous groups in physics can only be mentioned here (see, e.g., 5.6], 5.9]). U (1): Gauge transformations in electrodynamics. SU (2): Spin and isospin multiplets in particle physics. SU (3): Classi cation of the baryons and mesons in particle physics. Many-body problem in nuclear physics. SO(3): Angular momentum algebra in quantum mechanics. Atomic and nuclear many-body problems. SO(4): Degeneracy of the hydrogen spectrum. SU (4): Wigner supermultiplets in the nuclear shell model due to the uni cation of spin and isospin degrees of freedom. Description of avor multiplets in the quark model including the charm degree of freedom.
5.3 Classical Algebraic Structures 313
SU (6): Multiplets in the quark model due to the combination of avor and spin degrees of freedom. Nuclear structure models. U (n): Shell models in atomic and nuclear physics. SU (n) SO(n): Many-body problems in nuclear physics. SU (2) U (1): Standard model of the electro weak interaction. SU (5) % SU (3) SU (2) U (1): Uni cation of fundamental interactions (GUT). Remark: The groups SU (n) and SO(n) are Lie groups, i.e. continuous groups that are not treated here (see, e.g., 5.6]).
5.3.6 Rings and Fields
In this section, we discuss algebraic structures with two binary operations.
5.3.6.1 Denitions 1. Rings
A set R with two binary operations + # is called a ring (notation: (R + #)), if (R +) is an Abelian group, (R #) is a semigroup, and the distributive laws hold: a # (b + c) = (a # b) + (a # c) (b + c) # a = (b # a) + (c # a): (5.144) If (R #) is commutative or if (R #) has a neutral element, then (R + #) is called a commutative ring or a ring with identity (ring with unit element), respectively.
2. Fields
A ring is called a eld if (R n f0g #) is an Abelian group. So, every eld is a special commutative ring with identity.
3. Field Extensions
Let K and E be two elds. If K E holds, E is called the extension eld of K .
Examples of rings and elds: A: The number domains Z, Q, IR, and C are commutative rings with identity with respect to ad-
dition and multiplication Q , IR, and C are also elds. The set of even integers is an example of a ring without identity. The set C is the extension eld of IR. B: The set Mn of all square matrices of order n with real (or complex) elements is a non-commutative ring with the identity matrix as unit element. C: The set of real polynomials p(x) = an xn + an;1xn;1 + + a1 x + a0 forms a ring with respect to the usual addition and multiplication of polynomials, the polynomial ring R x]. More generally, instead of polynomials over R, polynomial rings over arbitrary commutative rings with identity element can be considered. D: Examples of nite rings are the residue class rings Zm modulo m: Zm consists of all the classes a]m of integers having the same residue on division by m. ( a]m is the equivalence class de ned by the natural number a with respect to the relation !R introduced in 5.2.4, 1., p. 297.) The ring operations $ , & on Zm are de ned by a]m $ b]m = a + b]m and a]m & b]m = a b]m : (5.145) If the natural number m is a prime, then (Zm $ &) is a eld.
5.3.6.2 Subrings, Ideals 1. Subring
Suppose R = (R + #) is a ring and U R. If U with respect to + and # is also a ring, then U = (U + #) is called a subring of R.
314 5. Algebra and Discrete Mathematics A non-empty subset U of a ring (R + #) forms a subring of R if and only if for all a b 2 U also a +(;b) and a # b are in U .
2. Ideal
A subring I is called an ideal if for all r 2 R and a 2 I also r # a and a # r are in I . These special subrings are the basis for the formation of factor rings (see 5.3.6.3, p. 314). The trivial subrings f0g and R are always ideals of R. Fields have only trivial ideals.
3. Principal Ideal
If all the elements of an ideal can be generated by one element according to the subring criterion, then it is called a principal ideal. All ideals of Z are principal ideals. They can be written in the form mZ = fmgjg 2 Zg and we denote them by (m).
5.3.6.3 Homomorphism, Isomorphism, Homomorphism Theorem 1. Ring Homomorphism and Ring Isomorphism
a) Ring Homomorphism: Let R1 = (R1 + #) and R2 = (R2 h: R1 ! R2 is called a ring homomorphism if for all a b 2 R1 h(a + b) = h(a) + h(b) and h(a # b) = h(a) h(b) hold.
+
) be two rings. A mapping (5.146)
b) Kernel: The kernel of h is the set of elements of R1 whose image by h is the neutral element 0 of
(R2 +), and we denote it by ker h: ker h = fa 2 R1jh(a) = 0g: (5.147) Here ker h is an ideal of R1. c) Ring Isomorphism: If h is also bijective, then h is called a ring isomorphism, and the rings R1 and R2 are called isomorphic. d) Factor Ring: If I is an ideal of a ring (R + #), then the sets of cosets fa + I ja 2 Rg of I in the additive group (R +) of the ring R (see 5.3.3, 1., p. 300) form a ring with respect to the operations (a + I ) + (b + I ) = (a + b) + I and (a + I ) (b + I ) = (a # b) + I: (5.148) This ring is called the factor ring of R by I , and it is denoted by R=I . The factor ring of Z by a principal ideal (m) is the residue class ring Zm = Z=(m) (see examples of rings and elds on p. 313).
2. Homomorphism Theorem for Rings
If the notion of a normal subgroup is replaced by the notion of an ideal in the homomorphism theorem for groups, then the homomorphism theorem for rings is obtained: A ring homomorphism h: R1 ! R2 de nes an ideal of R1 , namely ker h = fa 2 R1 jh(a) = 0g. The factor ring R1 = ker h is isomorphic to the homomorphic image h(R1 ) = fh(a)ja 2 R1 g. Conversely, every ideal I of R1 de nes a homomorphic mapping natI : R1 ! R2=I with natI (a) = a + I . This mapping natI is called a natural homomorphism.
5.3.6.4 Finite Fields and Shift Registers 1. Finite Fields
Fields are in particular notable for having no zero divisors. Finite elds are even characterized by this property. Zero divisors are non-zero elements of a ring for which there is a non-zero element such that their product is zero. The residue class rings Zm mentioned in 5.3.6.1, p. 313, D have zero divisors if m is not a prime number, since in this case m = k l yields 0 = k & l (multiplication modulo m). Thus, we obtain in Zp, p prime number, (except isomorphism) all nite elds with p elements. More generally we have: For every power pn of a prime number p there is (except isomorphism) a unique eld of pn elements, and every nite elds has pn elements. The elds with pn elements are also denoted by GF (pn) (Galois eld) . Observe that for n > 1 GF (pn)
5.3 Classical Algebraic Structures 315
and Zpn are di erent. To construct nite elds with pn elements (p prime number, n > 1) polynomial rings over Zp (see 5.3.6.1, p. 313, C) and irreducible polynomials are needed: Zp x] consists of all polynomials with coecients in Zp. Addition and multiplicationof such polynomials is done by calculating with the coecients modulo p. In polynomial rings IK x] over elds IK the division algorithm (polynomial division with remainder) is valid, i.e. for f (x) g(x) 2 IK x] with deg f (x) deg g(x) there are polynomials q(x) r(x) 2 IK x] such that g(x) = q(x) f (x) + r(x) and deg r(x) deg f (x) : (5.149) This situation is described by r(x) = g(x) (mod f (x)). Repeated division with remainder yields the Euclidean algorithm for polynomial rings and the last non-zero remainder is the greatest common divisor (gcd) of the polynomials f (x) and g(x). A polynomial f (x) 2 IK x] is called irreducible if it cannot be written as a product of polynomials of lower degree. In this case (analogously to prime numbers in Z) f (x) is called a prime element of IK x]. For polynomials of second and third degree irreducibility is equivalent to the non-existence of zeroes in IK. One can show that IK x] contains irreducible polynomials of arbitrary degree. For an irreducible polynomial f (x) 2 IK x] IK x]=f (x) := fr(x) 2 IK x]j deg r(x) < deg g(x)g (5.150) is a eld with multiplication modulo f (x), i.e. g(x) # h(x) = g(x) h(x) (mod f (x)). If IK = Zp, f (x) 2 IK x] irreducible and deg f (x) = n then IK x]=f (x) is a eld with pn elements, i.e. GF (pn) = Zp x]=f (x). The multiplicative group IK = IK n f0g of a nite eld IK is cyclic, i.e. there is an element a 2 IK such that every element of IK can be expressed as a power of a. One says, that such an element a generates the multiplicative group of the eld: IK = f1 a a2 : : : aq;2 g. An irreducible polynomial f (x) 2 IK x] is called primitive, if the powers of x cover all non-zero elements of IL := IK x]=f (x), i.e. if x generates the multiplicative group of IL. With a primitive polynomial f (x) of degree n in Zp x] a \logarithm table" for GF (pn) can be produced, which simpli es calculations in this eld considerably. Construction of the eld GF (23) and its associated logarithm table. The polynomial f (x) = 1 + x + x3 is irreducible as a polynomial in Z2 x], since neither 0 nor 1 are zeroes. Therefore, we have GF 23 = Z2 x]=f (x) (5.151a) n o = a0 + a1 x + a2 x2 j a0 a1 a2 2 Z2 ^ x3 = 1 + x : (5.151b) The polynomial f (x) is even primitive and so we can establish a logarithm table. For this we associate two expressions with each polynomial a0 + a1x + a2 x2 in Z2 x]=f (x), its coecient vector a0 a1 a2 and its so-called logarithm, which is that natural number i satisfying xi = a0 + a1 x + a2x2 modulo 1 + x + x3 . We obtain for GF (8): FE CV log. Addition of eld elements (FE) in GF (8): addition of coecient vectors (CV), 1 100 0 component-wise mod 2 (in general mod p). x 010 1 Multiplication of FE in GF (8): x2 0 0 1 2 addition of logarithms (log.) mod 7 x3 1 1 0 3 (in general mod (pn ; 1)). x4 0 1 1 4 2 4 5 x 111 5 example: x3 + x4 = x6 = x;5 = x2 x6 1 0 1 6 x +x x
316 5. Algebra and Discrete Mathematics So, IL := GF (q), q = pn, can be considered as an extension eld of IK =: GF (p). According to Fermat's theorem (see 5.4.4,2., p. 331) we have for all a 2 IL the equation aq = a, i.e. every element a 2 IL is zero of a polynomial with coecients in IK, e.g. xq ; x. The minimal polynomial ma (x) 2 IK x] is characterized by having a as a zero, leading coecient 1(ma (x) is normalized) and lowest possible degree. The minimal polynomial ma (x) has the following properties: a) ma (x) is irreducible over IK. b) ma (x) divides every f (x) 2 IK x] with f (a) = 0, in particular it divides xq ; x. c) deg ma (x) n. d) If a generates IL then deg ma (x) = n. Here the prime number p can also be replaced by any power of p. Let q = pn, p prime, and gcd(n q) = 1. Then, every element a of an extension eld IL of GF (q) solving the equation xn = 1 is called n-th root of unity over GF (q). The n-th roots of unity over GF (q) form a cyclic group of order n. A generating element of this group is called primitive n-th root of unity.
2. Application to Shift Registers
Calculations with polynomials can conveniently be performed using linear feed-back shift registers (see Fig. 5.16). For a linear feed-back shift register with feed back polynomial f (x) = f0 + f1x + + -f1
-f0 s0
-fr-1
-fr-2 s1
sr-2
sr-1
Figure 5.16 fr;1xr;1 + xr we obtain from the state polynomial s(x) = s0 + s1x + + sr;1xr;1 in the next step the state polynomial s(x) x ; sr;1f (x) = s(x) x (mod f (x)). In particular, starting with s(x) = 1 we obtain after i feed-back steps the state polynomial xi (mod f (x)). Demonstration using the example of p. 315: We choose the primitive polynomial f (x) = 1+ x + x3 2 Z2 x] as feed-back polynomial. Then we obtain a shift register of length 3 with the following sequence of states: With initial state 1 0 0 =b 1 (mod f (x)) we obtain consecutively the states: 0 1 0 =b x (mod f (x)) 0 0 1 =b x2 (modf (x)) 1 1 0 =b x3 1 + x (mod f (x)) 4 2 0 1 1 =b x x + x (mod f (x)) 1 1 1 =b x5 1 + x + x2 (mod f (x)) 1 0 1 =b x6 1 + x2 (mod f (x)) 1 0 0 =b x7 1 (mod f (x)) Here the states are to be interpreted as the coecient vectors of a state polynomial s0 + s1 x + s2 x2. In general we have: a linear feed-back shift register of length r generates a sequence of states of maximal length 2r ; 1, if the feed-back polynomial is a primitive polynomial of degree r.
5.3.7 Vector Spaces 5.3.7.1 Denition
A vector space over a eld F consists of an Abelian group V = (V +) of \ vectors " written in additive form, of a eld F = (F + #) of \ scalars " and an exterior multiplication F V ! V , which assigns to every ordered pair (k v) for k 2 F and v 2 V a vector kv 2 V . These operations have the following properties: (V1) (u + v) + w = u + (v + w) for all u v w 2 V: (5.152) In this paragraph, generally, vectors are not printed in bold face.
5.3 Classical Algebraic Structures 317
There is a vector 0 2 V such that v + 0 = v for every v 2 V: To every vector v there is a vector ; v such that v + (;v) = 0: v + w = w + v for every v w 2 V: 1v = v for every v 2 V 1 denotes the unit element of F: r(sv) = (rs)v for every r s 2 F and every v 2 V: (r + s)v = rv + sv for every r s 2 F and every v 2 V: r(v + w) = rv + rw for every r 2 F and every v w 2 V: If F = IR holds, then it is called a real vector space.
(V2) (V3) (V4) (V5) (V6) (V7) (V8)
(5.153) (5.154) (5.155) (5.156) (5.157) (5.158) (5.159)
Examples of vector spaces: A: Single-column or single-row real matrices of type (n 1) and (1 n), respectively, with respect to n
matrix addition and exterior multiplication with real numbers form real vector spaces IR (the vector space of column or row vectors see also 4.1.3, p. 253). B: All real matrices of type (m n) form a real vector space. C: All real functions continuous on an interval a b] with the operations (f + g)(x) = f (x) + g(x) and (kf )(x) = k f (x) (5.160) form a real vector space. Function spaces have a fundamental role in functional analysis (1., p. 596). For further examples see 12.1.2, p. 597.
5.3.7.2 Linear Dependence
Let V be a vector space over F . The vectors v1, v2 : : :, vm 2 V are called linearly dependent if there are k1, k2 : : :, km 2 K not all of them equal to zero such that 0 = k1 v1 + k2 v2 + + kmvm holds. Otherwise they are linearly independent. Linear dependence of at least two vectors means that one of them can be expressed in terms of the other. If there is a maximal number n of linearly independent vectors in a vector space V , then the vector space V is called n-dimensional. This number n is uniquely de ned and it is called the dimension. Every n linearly independent vectors of V form a basis. If such a maximal number does not exist, then the vector space is called innite dimensional. The vector spaces in the above examples are n, m n, and in nite dimensional. In the vector space IRn, n vectors are independent if and only if the determinant of the matrix, whose columns or rows are these vectors, is not equal to zero. If fv1 v2 : : : vng form a basis of an n-dimensional vector space over F , then every vector v 2 V has a unique representation v = k1 v1 + k2 v2 + + kn vn with k1 k2 : : : kn 2 F . Every set of linearly independent vectors can be completed into a basis of the vector space.
5.3.7.3 Linear Mappings
The mappings respecting the structure of vector spaces are called linear mappings. f : V1 ! V2 is called linear if for every u v 2 V1 and every k 2 F f (u + v) = f (u) + f (v) and f (ku) = k f (u) (5.161) are valid. The linear mappings f from IRn into IRm can be given by matrices A of type (m n) by f (v) = Av.
5.3.7.4 Subspaces, Dimension Formula
1. Subspace: Let V be a vector space and U a subset of V . If U is also a vector space with respect to the operations of V , then U is called a subspace of V . A non-empty subset U of V is a subspace if and only if for every u1 u2 2 U and every k 2 F also u1 + u2 and k u1 are in U (subspace criterion).
318 5. Algebra and Discrete Mathematics 2. Kernel, Image: Let V1 , V2 be vector spaces over F . If f : V1 ! V2 is a linear mapping, then the linear subspaces kernel (notation: ker f ) and image (notation: im f ) are de ned in the following way: ker f = fv 2 V jf (v) = 0g im f = ff (v)jv 2 V g: (5.162) So, for example, the solution set of a homogeneous linear equation system Ax = 0 is the kernel of the linear mapping de ned by the coecient matrix A. 3. Dimension: The dimension dim ker f and dim im f are called the defect f and rank f , respectively. For these dimensions the equality defect f + rank f = dim V (5.163) is valid and is called the dimension formula. In particular, if the defect f = 0, i.e., ker f = f0g, then the linear mapping f is injective, and conversely. Injective linear mappings are called regular.
5.3.7.5 Euclidean Vector Spaces, Euclidean Norm
In order to be able to use notions such as length, angle, orthogonality in abstract vector spaces we introduce Euclidean vector spaces.
1. Euclidean Vector Space
Let V be a real vector space. If ': V V ! IR is a mapping with the following properties (instead of '(v w) we write v w) for every u v w 2 V and for every r 2 IR (S1) v w = w v (5.164) (S2) (u + v) w = u w + v w (5.165) (S3) r(v w) = (rv) w = v (rw) (5.166) (S4) v v > 0 if and only if v 6= 0 (5.167) then ' is called a scalar product on V . If there is a scalar product de ned on V , then we call V a Euclidean vector space. We use these properties to de ne a scalar product with similar properties on more general spaces, too (see 12.4.1.1, p. 615).
2. EuclideanpNorm The value kvk = v v denotes the Euclidean norm (length) of v. The angle between v w from V is
de ned by the formula cos = v w : (5.168) kvk kwk If v w = 0 holds, then v and w are said to be orthogonal to each other. Orthogonality of Trigonometric Functions: In the theory of Fourier series (see 7.4.1.1, p. 420), we consider functions of the form sin kx and cos kx. We can consider these functions as elements of C 0 2]. In the function space C a b] the formula
Zb f g = f (x)g(x) dx a de nes a scalar product. Since Z 2 sin kx sin lx dx = 0 (k 6= l) 0
Z 2
(5.169) (5.170)
Z 2 0
cos kx cos lx dx = 0 (k 6= l)
(5.171)
sin kx cos lx dx = 0 (5.172) the functions sin kx and cos lx for every k l 2 IN are pairwise orthogonal to each other. This orthogonality of trigonometric functions is used in the calculation of Fourier coecients in harmonic analysis 0
5.3 Classical Algebraic Structures 319
(see 7.4.1.1, p. 420).
5.3.7.6 Linear Operators in Vector Spaces
1. Notion of Linear Operators
Let V and W be two real vector spaces. A mapping a from V into W is called a linear mapping or linear transformation or linear operator (see also 12.1.5.2, p. 600) from V into W if a(u + v) = au + av for all u v 2 V (5.173) a(u) = au for all u 2 V and all real : (5.174)
A: The mapping au := R u(t) dt which transforms the space C ] of continuous real functions
into the space of real numbers is linear. In the special case when W = IR1 , as in the previous example, linear transformations are called linear functionals. B: Let V = IRn and let W be the space of all real polynomials of degree at most n ; 1. Then the mapping a(a1 a2 : : : an) := a1 + a2 x + a3 x2 + + anxn;1 is linear. In this case each n-element vector corresponds to a polynomial of degree n ; 1. C: If V = IRn and W = IRm , then all linear operators a from V into W (a : IRn ;! IRm ) can be characterized by a real matrix A = (aik ) of type (m n). The relation Ax = y corresponds to the system of linear equations (4.103a) 0 a11 a12 a1n 1 0 x1 1 0 y1 1 B B B a21 a22 a2n C x2 C y2 C B C B C B C. B C B C = B . . . . @. A @ . A @ ... C A am1 am2 amn xn ym
2. Sum and Product of two Linear Operators
Let a: V ;! W , b: V ;! W and c: W ;! U be linear operators. Then the sum a + b: V ;! W is de ned as (a + b)u = au + bu for all u 2 V and the (5.175) product ca: V ;! U is de ned as (ca)u = c(au) for all u 2 V: (5.176) Remarks: 1. If a b and c are linear, then a + b and ac are also linear operators. 2. The product (5.176) of two linear operators represents the consecutive application of these operators
a and c. 3. The product of two linear operators is usually non-commutative even if the products exist: ca 6= ac : (5.177a) We have commutability , if ca ; ac = 0 (5.177b) holds. In quantum mechanics the left-hand side of this equation ca ; ac is called the commutator. In the case (5.177a) the operators a and c do not commutate, therefore we have to be very careful about the order. As a particular example of sums and products of linear operators one may think of sums and products of the corresponding real matrices.
320 5. Algebra and Discrete Mathematics
5.4 Elementary Number Theory
Elementary number theory investigates divisibility properties of integers.
5.4.1 Divisibility
5.4.1.1 Divisibility and Elementary Divisibility Rules
1. Divisor
An integer b 2 Z is divisible by an integer a without remainder i if there is an integer q such that qa = b (5.178) holds. Here a is a divisor of b in Z, and q is the complementary divisor with respect to a b is a multiple of a. For \a divides b" we write also ajb. For \a does not divide b" we can write a=jb. The divisibility relation (5.178) is a binary relation in Z (see 5.2.3, 2., p. 294). Analogously, divisibility is de ned in the set of natural numbers.
2. Elementary Divisibility Rules
(DR1) (DR2) (DR3) (DR4) (DR5) (DR6) (DR7) (DR8) (DR9) (DR10) (DR11) (DR12)
For every a 2 Z we have 1ja aja and aj0: If ajb then (;a)jb and aj(;b): ajb and bja implies a = b or a = ;b: aj1 implies a = 1 or a = ;1: ajb and b 6= 0 imply jaj jbj: ajb implies ajzb for every z 2 Z: ajb implies azjbz for every z 2 Z: azjbz and z 6= 0 implies ajb for every z 2 Z: ajb and bjc imply ajc: ajb and cjd imply acjbd: ajb and ajc imply aj(z1b + z2c) for arbitrary z1 z2 2 Z: ajb and aj(b + c) imply ajc:
5.4.1.2 Prime Numbers
(5.179) (5.180) (5.181) (5.182) (5.183) (5.184) (5.185) (5.186) (5.187) (5.188) (5.189) (5.190)
1. Denition and Properties of Prime Numbers
A positive integer p (p > 1) is called a prime number i 1 and p are its only divisors in the set IN of positive integers. Positive integers which are not prime numbers are called composite numbers. For every integer, the smallest positive divisor di erent from 1 is a prime number. There are in nitely many prime numbers. A positive integer p (p > 1) is a prime number i for arbitrary positive integers a b, pj(ab) implies pja or pjb.
2. Sieve of Eratosthenes
By the method of the \Sieve of Eratosthenes", every prime number smaller than a given positive integer n can be determined: a) Write down the list of all positive integers from 2 to n. b) Underline 2 and delete every subsequent multiple of 2. c) If p is the rst non-deleted and non-underlined number, then underline p and delete every p-th number (beginning with 2p and counting the numbers of the original list). d) Repeat step c) for every p (p pn) and stop the algorithm. if and only if
5.4 Elementary Number Theory 321
Every underlined and non-deleted number is a prime number. In this way, all prime numbers n are obtained. The prime numbers are called prime elements of the set of integers.
3. Prime Pairs
Prime numbers with a di erence of 2 form prime pairs (twin primes). (3 5) (5 7) (11 13) (17 19) (29 31) (41 43) (59 61) (71 73) (101 103) are prime pairs.
4. Prime Triplets
Prime triplets consist of three prime numbers occuring among four consecutive odd numbers. (5 7 11) (7 11 13) (11 13 17) (13 17 19) (17 19 23) (37 41 43) are prime triplets.
5. Prime Quadruplets
If the rst two and the last two of ve consecutive odd numbers are prime pairs, then they are called a prime quadruplet. (5 7 11 13) (11 13 17 19) (101 103 107 109) (191 193 197 199) are prime quadruplets. The conjecture that there exist in nitely many prime pairs, prime triplets, and prime quadruplets, is not proved still.
6. Mersenne Primes
If 2k ; 1 k 2 IN, is a prime number, then k is also a prime number. The numbers 2p ; 1 (p prime) are called Mersenne numbers. A Mersenne prime is a Mersenne number 2p ; 1 which is itself a prime number. 2p ; 1 is a prime number for the rst ten values of p: 2, 3, 5, 13,17, 19, 31, 61, 89, 107, etc.
7. Fermat Primes
If a number 2k +1 k 2 IN, is an odd prime number, then k is a power of 2. The numbers 2k +1 k 2 IN, are called Fermat numbers. If a Fermat number is a prime number, then it is called a Fermat prime. For k = 0 1 2 3 4 the corresponding Fermat numbers 3 5 17 257 65537 are prime numbers. It is conjectured that there are no further Fermat primes.
8. Fundamental Theorem of Elementary Number Theory
Every positive integer n > 1 can be represented as a product of primes. This representation is unique except for the order of the factors. Therefore n is said to have exactly one prime factorization. 360 = 2 2 2 3 3 5 = 23 32 5. Remark: Analogously, the integers (except ;1 0 1) can be represented as products of prime elements, unique apart from the order and the sign of the factors.
9. Canonical Prime Factorization
It is usual to arrange the factors of the prime factorization of a positive integer according to their size, and to combine equal factors to powers. If every non-occurring prime is assigned exponent 0, then every positive integer is uniquely determined by the sequence of the exponents of its prime factorization. To 1 533 312 = 27 32 113 belongs the sequence of exponents (7 2 0 0 3 0 0 : : :). For a positive integer n, let p1 p2 : : : pm be the pairwise distinct primes divisors of n, and let k denote the exponent of a prime number pk in the prime factorization of n. Then
n=
m Y
k=1
pk k
(5.191a)
and this representation is called the canonical prime factorization of n. It is often denoted by Y n = pp(n) (5.191b) p
where the product applies to all prime numbers p, and where p(n) is the multiplicity of p as a divisor of n. It always means a nite product because only nitely many of the exponents p(n) di er from 0.
322 5. Algebra and Discrete Mathematics
10. Positive Divisors
If a positive integer n 1 is given by its canonical prime factorization (5.191a), then every positive divisor t of n can be written in the form
t=
m Y
k=1
pkk with k 2 f0 1 2 : : : k g for k = 1 2 : : : m:
(5.192a)
The number (n) of all positive divisors of n is (n) =
m Y
(k + 1):
k=1
(5.192b)
A: (5040) = (24 32 5 7) = (4 + 1)(2 + 1)(1 + 1)(1 + 1) = 60. B: (p1 p2 pr ) = 2r , if p1 p2 : : : pr are pairwise distinct prime numbers.
The product P (n) of all positive divisors of n is given by (5.192c) P (n) = n 21 (n) : A: P (20) = 203 = 8000. B: P (p3) = p6, if p is a prime number. C: P (pq) = p2q2, if p and q are di erent prime numbers. The sum (n) of all positive divisors of n is m k +1 Y (n) = pkp ;;1 1 : (5.192d) k=1 k A: (120) = (23 3 5) = 15 4 6 = 360. B: (p) = p + 1, if p is a prime number.
5.4.1.3 Criteria for Divisibility
1. Notation
Consider a positive integer given in decimal form: n = (ak ak;1 a2a1 a0 )10 = ak 10k + ak;110k;1 + + a2102 + a1 10 + a0: (5.193a) Then Q1 (n) = a0 + a1 + a2 + + ak (5.193b) and Q01 (n) = a0 ; a1 + a2 ; + + (;1)k ak (5.193c) are called the sum of the digits (of rst order) and the alternating sum of the digits (of rst order) of n, respectively. Furthermore, Q2 (n) = (a1a0 )10 + (a3a2 )10 + (a5a4 )10 + and (5.193d) 0 Q2 (n) = (a1a0 )10 ; (a3 a2 )10 + (a5a4 )10 ; + (5.193e) are called the sum of the digits and the alternating sum of the digits, respectively, of second order and Q3 (n) = (a2a1 a0 )10 + (a5a4 a3)10 + (a8 a7a6 )10 + (5.193f) and Q03 (n) = (a2a1 a0 )10 ; (a5 a4a3 )10 + (a8a7 a6 )10 ; + (5.193g) are called the sum of the digits and alternating sum of the digits, respectively, of third order . The number 123 456 789 has the following sum of the digits: Q1 = 9+8+7+6+5+4+3+2+1 = 45 Q01 = 9 ; 8+7 ; 6+5 ; 4+3 ; 2+1 = 5 Q2 = 89+67+45+23+1 = 225 Q02 = 89 ; 67+45 ; 23+1 = 45 Q3 = 789 + 456 + 123 = 1368 and Q03 = 789 ; 456 + 123 = 456:
5.4 Elementary Number Theory 323
2. Criteria for Divisibility
There are the following criteria for divisibility:
DC-1: DC-3: DC-5: DC-7: DC-9:
3jn , 3jQ1(n)
(5.194a)
9jn , 9jQ1(n)
(5.194c)
7jn , 7jQ03(n)
(5.194b)
11jn , 11jQ1(n)
(5.194d)
37jn , 37jQ3(n)
(5.194f)
2jn , 2ja0
(5.194h)
101jn , 101jQ2(n) (5.194g)
DC-2: DC-4: DC-6: DC-8:
5jn , 5ja0
DC-10: 2k jn , 2k j(ak;1ak;2 a1a0 )10 (5.194j)
13jn , 13jQ03(n)
(5.194e)
0
(5.194i)
0
DC-11: 5k jn , 5k j(ak;1ak;2 a1 a0)10 : (5.194k) A: a = 123 456 789 is divisible by 9 since Q1 (a) = 45 and 9j45 but it is not divisible by 7 since Q03 (a) = 456 and 7=j456. B: 91 619 is divisible by 11 since Q01(91 619) = 22 and 11j22: C: 99 994 096 is divisible by 24 since 24j4 096:
5.4.1.4 Greatest Common Divisor and Least Common Multiple 1. Greatest Common Divisor
For integers a1 a2 : : : an which are not all equal to zero, the largest number in the set of common divisors of a1 a2 : : : an is called the greatest common divisor of a1 a2 : : : an, and it is denoted by gcd(a1 a2 : : : an). If gcd(a1 a2 : : : an) = 1, then the numbers a1 a2 : : : an are called coprimes. To determine the greatest common divisor, it is sucient to consider the positive common divisors. If the canonical prime factorizations Y ai = pp(ai ) (5.195a) p
of a1 a2 : : : an are given, then
p(ai )] Y min i gcd(a1 a2 : : : an) = p : p
(5.195b)
For the numbers a1 = 15 400 = 23 52 7 11 a2 = 7 875 = 32 53 7 a3 = 3 850 = 2 52 7 11, the greatest common divisor is gcd(a1 a2 a3) = 52 7 = 175:
2. Euclidean Algorithm
The greatest common divisor of two integers a b can be determined by the Euclidean algorithm without using their prime factorization. To do this, a sequence of divisions with remainder, according to the following scheme, is performed. For a > b let a0 = a a1 = b. Then: a0 = q1 a1 + a2 0 < a 2 < a1 a1 = q2 a2 + a3 0 < a 3 < a2 ... ... ... (5.196a) an;2 = qn;1 an;1 + an 0 < an < an;1 an;1 = qn an: The division algorithm stops after a nite number of steps, since the sequence a2 a3 : : : is a strictly monotone decreasing sequence of positive integers. The last remainder an, di erent from 0 is the greatest common divisor of a0 and a1 (see rst example next page).
324 5. Algebra and Discrete Mathematics gcd(38 105) = 1, since By the recursion formula 105 = 2 38 + 29 gcd(a1 a2 : : : an) = gcd(gcd(a1 a2 : : : an;1) an) (5.196b) 38 = 1 29 + 9 the greatest common divisor of n positive integers with 29 = 3 9 + 2 n > 2 can be determined by repeated use of the Euclide9= 42+1 an algorithm. 2 = 21: gcd(150 105 56) = gcd(gcd(150 105) 56) = gcd(15 56) = 1: The Euclidean algorithm to determine the gcd (see also 1.1.1.4, 1., 55 = 1 34 + 21 p. 3) of two numbers has especially many steps, if the numbers are adja34 = 1 21 + 13 cent numbers in the sequence of Fibonacci numbers (see 5.4.1.5, p. 325). 21 = 1 13 + 8 The annexed calculation shows an example where all quotients are al13 = 1 8 + 5 ways equal to 1. 8= 15+3 5= 13+2 3. Theorem for the Euclidean Algorithm 3= 12+1 For two natural numbers a b with a > b > 0, let (a b) denote the 2= 11+1 number of divisions with remainder in the Euclidean algorithm, and let 1 = 1 1: (b) denote the number of digits of b in the decimal system. Then (a b) 5 (b): (5.197)
4. Greatest Common Divisor as a Linear Combination
It follows from the Euclidean algorithm that a2 = a0 ; q1a1 = c0 a0 + d0 a1 a3 = a1 ; q2a2 = c1 a0 + d1 a1 ... ... (5.198a) an = an;2 ; qn;1 an;1 = cn;2a0 + dn;2a1 : Here cn;2 and dn;2 are integers. Thus the gcd(a0 a1 ) can be represented as a linear combination of a0 and a1 with integer coecients: gcd(a0 a1) = cn;2a0 + dn;2a1 : (5.198b) Moreover gcd(a1 a2 : : : an) can be represented as a linear combination of a1 a2 : : : an, since: gcd(a1 a2 : : : an) = gcd(gcd(a1 a2 : : : an;1) an) = c gcd(a1 a2 : : : an;1) + dan: (5.198c) gcd(150 105 56) = gcd(gcd(150 105) 56) = gcd(15 56) = 1 with 15 = (;2) 150 + 3 105 and 1 = 15 15 + (;4) 56) thus gcd(150 105 56) = (;30) 150 + 45 105 + (;4) 56:
5. Least Common Multiple
For integers a1 a2 : : : an, among which there is no zero, the smallest number in the set of positive common multiples of a1 a2 : : : an is called the least common multiple of a1 a2 : : : an, and it is denoted by lcm(a1 a2 : : : an). If the canonical prime factorizations (5.195a) of a1 a2 : : : an are given, then:
p(ai)] Y max lcm(a1 a2 : : : an) = p i : (5.199) p
For the numbers a1 = 15 400 = 23 52 7 11 a2 = 7 875 = 32 53 7 a3 = 3 850 = 2 52 7 11 the least common multiple is lcm(a1 a2 a3) = 23 32 53 7 11 = 693 000:
6. Relation between g.c.d. and l.c.m. For arbitrary integers a b: jabj = gcd(a b) lcm(a b):
(5.200)
5.4 Elementary Number Theory 325
Therefore, the lcm(a b) can be determined with the help of the Euclidean algorithm without using the prime factorizations of a and b.
5.4.1.5 Fibonacci Numbers
1. Fibonacci Sequence
The sequence (Fn)n2IN with F1 = F2 = 1 and Fn+2 = Fn + Fn+1 (5.201) is called Fibonacci sequence. It starts with the elements 1 1 2 3 5 8 13 21 34 55 89 144 233 377 : : : The consideration of this sequence goes back to the question posed by Fibonacci in 1202: How many pairs of descendants has a pair of rabbits at the end of a year, if every pair in every month produces a new pair, which beginning with the second month itself produces new descended pairs? The answer is F14 = 377:
2. Fibonacci Recursion Formula
Besides the recursive de nition (5.201) there is an explicit formula for the Fibonacci numbers: " p n " p n ! Fn = p1 1 +2 5 ; 1 ;2 5 : (5.202) 5 Some important properties of Fibonacci numbers are the following. For m n 2 IN: (1) Fm+n = Fm;1 Fn + FmFn+1 (m > 1): (5.203a) (2) Fm jFmn: (5.203b)
(3) gcd(m n) = d implies gcd(Fm Fn) = Fd : (5.203c)
(4) gcd(Fn Fn+1) = 1:
(5) Fm jFk holds i mjk holds.
(5.203e)
(6)
(7) gcd(m n) = 1 implies Fm FnjFmn:
(5.203g)
(8)
(9) FnFn+2 ; Fn2+1 = (;1)n+1:
(5.203i)
(10) Fn2 + Fn2+1 = F2n+1 :
(11) Fn2+2 ; Fn2 = F2n+2 :
(5.203k)
n X i=1 n X i=1
(5.203d)
Fi2 = FnFn+1:
(5.203f)
Fi = Fn+2 ; 1:
(5.203h) (5.203j)
5.4.2 Linear Diophantine Equations 1. Diophantine Equations
An equation f (x1 x2 : : : xn) = b is called a Diophantine equation in n unknowns i f (x1 x2 : : : xn) is a polynomial in x1 x2 : : : xn with coecients in the set Z of integers, b is an integer constant and only integer solutions are of interest. The name \Diophantine" reminds us of the Greek mathematician Diophantus, who lived around 250 AD. In practice, Diophantine equations occur for instance, if relations between quantities are described. Until now, only general solutions of Diophantine equations of at most second degree with two variables are known. Solutions of Diophantine equations of higher degrees are only known in special cases.
2. Linear Diophantine Equations in n Unknowns
A linear Diophantine equation in n unknowns is an equation of the form a1x1 + a2 x2 + anxn = b (ai 2 Z b 2 Z) (5.204) where only integer solutions are searched for. A solution method is described in the following.
326 5. Algebra and Discrete Mathematics
3. Conditions of Solvability
If not all the coecients ai are equal to zero, then the Diophantine equation (5.204) is solvable i gcd(a1 a2 : : : an) is a divisor of b. 114x + 315y = 3 is solvable, since gcd(114 315) = 3. If a linear Diophantine equation in n unknowns (n > 1) has a solution and Z is the domain of variables, then the equation has in nitely many solutions. Then in the set of solutions there are n;1 free variables. For subsets of Z, this statement is not true.
4. Solution Method for n = 2 Let
a1x1 + a2 x2 = b (a1 a2) 6= (0 0) (5.205a) be a solvable Diophantine equation, i.e., gcd(a1 a2 )jb. To nd a special solution of the equation, the equation is divided by gcd(a1 a2) and one obtains a01 x01 + a02x02 = b0 with gcd(a01 a02 ) = 1. As described in 5.4.1, 4., p. 324, gcd(a01 a02 ) is determined to obtain nally a linear combination of a01 and a02: a01 c01 + a02 c02 = 1. Substitution in the given equation demonstrates that the ordered pair (c01 b0 c02 b0) of integers is a solution of the given Diophantine equation. 114x +315y = 6. The equation is divided by 3, since 3 = gcd(114 315). That implies 38x +105y = 2 and 38 47 + 105 (;17) = 1 (see 5.4.1, 4., p. 324). The ordered pair (47 2 (;17) 2) = (94 ;34) is a special solution of the equation 114x + 315y = 6. The family of solutions of (5.205a) can be obtained as follows: If (x01 x02) is an arbitrary special solution, which could also be obtained by trial and error, then f(x01 + t a02 x02 ; t a01 )jt 2 Zg (5.205b) is the set of all solutions. The set of solutions of the equation 114x + 315y = 6 is f(94 + 315t ;34 ; 114t)jt 2 Zg.
5. Reduction Method for n > 2
Suppose a solvable Diophantine equation a1x1 + a2 x2 + + anxn = b (5.206a) with (a1 a2 : : : an) 6= (0 0 : : : 0) and gcd(a1 a2 : : : an) = 1 is given. If gcd(a1 a2 : : : an) 6= 1, then the equation should be divided by gcd(a1 a2 : : : an). After the transformation a1x1 + a2 x2 + + an;1xn;1 = b ; anxn (5.206b) xn is considered as an integer constant and a linear Diophantine equation in n ; 1 unknowns is obtained, and it is solvable i gcd(a1 a2 : : : an;1) is a divisor of b ; anxn . The condition gcd(a1 a2 : : : an;1)jb ; anxn (5.206c) is satis ed i there are integers c cn such that: gcd(a1 a2 : : : an;1) c + ancn = b: (5.206d) This is a linear Diophantine equation in two unknowns, and it can be solved as shown in 5.4.2,4., p. 326. If its solution is determined, then it remains to solve a Diophantine equation in only n ; 1 unknowns. This procedure can be continued until a Diophantine equation in two unknowns is obtained, which can be solved with the method given in 5.4.2, 4., p. 326. Finally, the solution of the given equation is constructed from the set of solutions obtained in this way. Solve the Diophantine equation 2x + 4y + 3z = 3: (5.207a) This is solvable since gcd(2 4 3) is a divisor of 3. The Diophantine equation 2x + 4y = 3 ; 3z (5.207b)
5.4 Elementary Number Theory 327
in the unknowns x y is solvable i gcd(2 4) is a divisor of 3 ; 3z. The corresponding Diophantine equation 2z0 + 3z = 3 has the set of solutions f(;3 + 3t 3 ; 2t)jt 2 Zg. This implies, z = 3 ; 2t, and now the set of solutions ot the solvable Diophantine equation 2x + 4y = 3 ; 3(3 ; 2t) or x + 2y = ;3 + 3t (5.207c) is sought for every t 2 Z. The equation (5.207c) is solvable since gcd(1 2) = 1j(;3 + 3t). Now 1 (;1) + 2 1 = 1 and 1 (3 ; 3t) + 2 (;3 + 3t) = ;3 + 3t. The set of solution is f((3 ; 3t) + 2s (;3 + 3t) ; s)js 2 Zg . That implies x = (3 ; 3t) + 2s y = (;3 + 3t) ; s and f(3 ; 3t + 2s ;3 + 3t ; s 3 ; 2t)js t 2 Zg so obtained is the set of solutions of (5.207a).
5.4.3 Congruences and Residue Classes 1. Congruences
Let m be a positive integer m m > 1. If two integers a and b have the same remainder, when divided by m, then a and b are called congruent modulo m, denoted by a b mod m or a b(m). 3 13 mod 5, 38 13 mod 5, 3 ;2 mod 5. Remark: Obviously, a b mod m holds i m is a divisor of the di erence a ; b. Congruence modulo m is an equivalence relation (see 5.2.4, 1., p. 297) in the set of integers. Note the following properties: a a mod m for every a 2 Z (5.208a) a b mod m ) b a mod m (5.208b) a b mod m ^ b c mod m ) a c mod m: (5.208c)
2. Calculating Rules
a b mod m ^ c d mod m ) a + c b + d mod m a b mod m ^ c d mod m ) a c b d mod m a c b c mod m ^ gcd(c m) = 1 ) a b mod m a c b c mod m ^ c 6= 0 ) a b mod gcd(mc m) :
3. Residue Classes, Residue Class Ring
(5.209a) (5.209b) (5.209c) (5.209d)
Since congruence modulo m is an equivalence relation in Z, this relation induces a partition of Z into residue classes modulo m: a]m = fxjx 2 Z ^ x a mod mg: (5.210) The residue class \ a modulo m " consists of all integers having equal remainder if divided by m. Now a]m = b]m i a b mod m. There are exactly m residue classes modulo m, and normally they are represented by their smallest non-negative representatives: 0]m 1]m : : : m ; 1]m : (5.211) In the set Zm of residue classes modulo m, residue class addition and residue class multiplication are de ned by a]m $ b]m := a + b]m (5.212) a]m & b]m := a b]m : (5.213) These residue class operations are independent of the chosen representatives, i.e., a]m = a0 ]m and b]m = b0 ]m imply (5.214) a]m $ b]m = a0 ]m $ b0 ]m and a]m & b]m = a0 ]m & b0 ]m : The residue classes modulo m form a ring with unit element, with respect to residue class addition and residue class multiplication (see 5.4.3, 1., p. 327), the residue class ring modulo m. If p is a prime number, then the residue class ring modulo p is a eld (see 5.4.3, 1., p. 327).
328 5. Algebra and Discrete Mathematics
4. Residue Classes Relatively Prime to m
A residue class a]m with gcd(a m) = 1 is called a residue class relatively prime to m . If p is a prime number, then all residue classes di erent from 0]p are residue classes relatively prime to p . The residue classes relatively prime to m form an Abelian group with respect to residue class multiplication, the so-called group of residue classes relatively prime to m . The order of this group is '(m), where ' is the Euler function (see 5.4.4, 1., p. 331). A: 1]8 3]8 5]8 7]8 are residue classes relatively prime to 8. B: 1]5 2]5 3]5 4]5 are residue classes relatively prime to 5. C: '(8) = '(5) = 4 is valid.
5. Primitive Residue Classes
A residue class a]m relatively prime to m is called a primitive residue class if it has order '(m) in the group of residue classes relatively prime to m. A: 2]5 is a primitive residue class modulo 5, since ( 2]5 )2 = 4]5 ( 2]5 )3 = 3]5 ( 2]5)4 = 1]5. B: There is no primitive residue class modulo 8, since 1]8 has order 1, and 3]8 5]8 7]8 have order 2 in the group of residue classes relatively prime to m. Remark: There is a primitive residue class modulo m, i m = 2 m = 4 m = pk or m = 2pk , where p is an odd prime number and k is a positive integer. If there is a primitive residue class modulo m, then the group of residue classes relatively prime to m forms a cyclic group.
6. Linear Congruences
1. De nition If a b and m > 0 are integers, then ax b(m)
(5.215) is called a linear congruence (in the unknown x). 2. Solutions An integer x satisfying ax b(m) is a solution of this congruece. Every integer, which is congruent to x modulo m, is also a solution. If all solutions of (5.215) we searched for, then it is sucient to nd the integers pairwise incongruent modulo m which satisfy the congruence. The congruence (5.215) is solvable i gcd(a m) is a divisor of b. In this case, the number of solutions modulo m is equal to gcd(a m). In particular, if gcd(a m) = 1 holds, the congruence modulo m has a unique solution. 3. Solution Method There are di erent solution methods for linear congruences. It is possible to transform the congruence ax b(m) into the Diophantine equation ax + my = b, and to determine a special solution (x0 y0) of the Diophantine equation a0x + m0y = b0 with a0 = a=gcd(a m) m0 = m=gcd(a m) b0 = b=gcd(a m) (see 5.4.2, 1., p. 325). The congruence a0x b0(m0 ) has a unique solution since gcd(a0 m0) = 1 modulo m0, and x x0 (m0 ): (5.216a) The congruence ax b(m) has exactly gcd(a m) solutions modulo m: x0 x0 + m x0 + 2m : : : x0 + (gcd(a m) ; 1)m: (5.216b) 114x 6 mod 315 is solvable, since gcd(114 315) is a divisor of 6 there are three solutions modulo 315. 38x 2 mod 105 has a unique solution: x 94 mod 105 (see 5.4.2, 4., p. 326). 94, 199, and 304 are the solutions of 114x 3 mod 315:
7. Simultaneous Linear Congruences
If nitely many congruences x b1 (m1 ) x b2 (m2) : : : x bt (mt ) (5.217) are given, then (5.217) is called a system of simultaneous linear congruences. A result on the set of solutions is the Chinese remainder theorem: Consider a given system x b1 (m1) x b2 (m2 ) : : : x
5.4 Elementary Number Theory 329
bt (mt ), where m1 m2 : : : mt are pairwise coprime numbers. If m = m1 m2 mt a1 = mm a2 = mm : : : at = mm (5.218a) 1 2 t and xj is choosen such that aj xj bj (mj ) for j = 1 2 : : : t, then x0 = a1x1 + a2 x2 + + at xt (5.218b) is a solution of the system. The system has a unique solution modulo m, i.e., if x0 is a solution, then x00 is a solution, too, i x00 x0(m). Solve the system x 1 (2) x 2 (3) x 4 (5), where 2 3 5 are pairwise coprime numbers. Then m = 30 a1 = 15 a2 = 10 a3 = 6. The congruences 15x1 1 (2) 10x2 2 (3) 6x3 4 (5) have the special solutions x1 = 1 x2 = 2 x3 = 4. The given system has a unique solution modulo m: x 15 1 + 10 2 + 6 4 (30), i.e., x 29 (30). Remark: Systems of simultaneous linear congruences can be used to reduce the problem of solving non-linear congruences modulo m to the problem of solving congruences modulo prime number powers (see 5.4.3, 9., p. 330).
8. Quadratic Congruences
1. Quadratic Residues Modulo m One can solve every congruence ax2 + bx + c 0(m) if one can solve every congruence x2 a(m): ax2 + bx + c 0(m) , (2ax + b)2 b2 ; 4ac(m): (5.219) First quadratic residues modulo m are considered: Let m 2 IN m > 1 and a 2 Z gcd(a m) = 1. The number a is called a quadratic residue modulo m i there is an x 2 Z with x2 a(m). If the canonical prime factorization of m is given, i.e., m=
1 Y
i=1
pi i
(5.220)
then r is a quadratic residue modulo m i r is a quadratic residue modulo pi i !for i = 1 2 3 : : : . If a is a quadratic residue modulo a prime number p, then this is denoted by ap = 1 if a is a quadratic
!
non-residue modulo p, then it is denoted by ap = ;1 (Legendre symbol). The numbers 1 4 7 are quadratic residues modulo 9.
2. Properties of Quadratic Congruences ! ! (E1) p=j ab and a b(p) imply ap = pb : (E2) (E3) (E4) (E5)
(5.221a)
!
1 = 1: p ;1 ! = (;1) p;2 1 : p ! ! ! ab = a b in particular p p p ! 2 ;1 p 2 = (;1) 8 : p
(5.221b) (5.221c)
ab p
2!
!
= ap :
(5.221d) (5.221e)
330 5. Algebra and Discrete Mathematics (E6) Quadratic !reciprocity law: If p and q are distinct odd prime numbers, ! p;1 q;1 then pq pq = (;1) 2 2 : 65
(5.221f)
!
5 13 307 307 2 8 2 52 ;1 23 8 = = = = ( ; 1) = ; 307 307 307 5 13 5 13 13 13 = 2 13 ;1 ;(;1) 8 = 1: In General: A congruence x2 a(2 ), gcd(a 2) = 1, is solvable i a 1(4) for = 2 and a 1(8) for 3. If these conditions are satis ed, then modulo 2 there is one solution for = 1, there are two solutions for = 2 and four solutions for 3. A necessary condition for solvability of congruences of the general form x2 a(m) m = 2 p1 1 p2 2 pt t gcd(a m) = 1 (5.222a) is the solvability of the congruences ! ! ! a = 1 a = 1 : : : a = 1: (5.222b) a 1(4) for = 2 a 1(8) for 3 p1 p2 pt If all these conditions are satis ed, then the number of solutions is equal to 2t for = 0 and = 1, equal to 2t+1 for = 2 and equal to 2t+2 for 3.
9. Polynomial Congruences
If m1 m2 : : : mt are pairwise coprime numbers, then the congruence f (x) anxn + an;1 xn;1 + + a0 0(m1 m2 mt ) (5.223a) is equivalent to the system f (x) 0(m1) f (x) 0(m2 ) : : : f (x) 0(mt ): (5.223b) If kj is the number of solutions of f (x) 0(mj ) for j = 1 2 : : : t, then k1k2 kt is the number of solutions of f (x) 0(m1m2 mt ). This means that the solution of the congruence (5.223c) f (x) 0 (p1 1 p2 2 pt t ) where p1 p2 : : : pt are primes, can be reduced to the solution of congruences f (x) 0(p ). Moreover, these congruences can be reduced to congruences f (x) 0(p) modulo prime numbers in the following way: a) A solution of f (x) 0(p ) is a solution of f (x) 0(p), too. b) A solution x x1 (p) of f (x) 0(p) de nes a unique solution modulo p i f 0(x1) is not divisible by p: Suppose f (x1) 0(p): Let x = x1 + pt1 and determine the unique solution t01 of the linear congruence f (x1) + f 0(x )t 0(p): (5.224a) 1 1 p Substitute t1 = t01 + pt2 into x = x1 + pt1, then x = x2 + p2 t2 is obtained. Now, the solution t02 of the linear congruence f (x2) + f 0(x )t 0(p) (5.224b) 2 2 p2 has to be determined modulo p2 . By substitution of t2 = t02 + pt3 into x = x2 + p2t2 the result x = x3 + p3t3 is obtained. Continuing this process yields the solution of the congruence f (x) 0 (p ). Solve the congruence f (x) = x4 + 7x + 4 0 (27). f (x) = x4 + 7x + 4 0 (3) implies x 1 (3), i.e., x = 1 + 3t1 . Because of f 0(x) = 4x3 + 7 and 3=j f 0(1) now the solution of the congruence f (1)=3 + f 0(1) t1 4 + 11t1 0 (3) is searched for: t1 1 (3), i.e., t1 = 1 + 3t2 and x = 4 + 9t2 .
5.4 Elementary Number Theory 331
Then consider f (4)=9 + f 0(4) t2 0 (3) and the solution t2 2 (3) is obtained, i.e., t2 = 2 + 3t3 and x = 22 + 27t3. Therefore, 22 is the solution of x4 + 7x + 4 0 (27), uniquely determined modulo 27.
5.4.4 Theorems of Fermat, Euler, and Wilson 1. Euler Function
For every positive integer m with m > 0 one can determine the number of coprimes x with respect to m for 1 x m. The corresponding function ' is called the Euler function. The value of the function '(m) is the number of residue classes relatively prime to m (s. 5.4.3, 4., p. 328). For instance, '(1) = 1 '(2) = 1 '(3) = 2 '(4) = 2 '(5) = 4 '(6) = 2 '(7) = 6 '(8) = 4, etc. In general, '(p) = p ; 1 holds for every prime number p and '(p ) = p ; p ;1 for every prime number power p . If m is an arbitrary positiv integer, then '(m) can be determined in the following way: ! Y '(m) = m 1; 1 (5.225a) p pjm
where the product applies to all prime divisors p of m. '(360) = '(23 32 5) = 360 (1 ; 21 ) (1 ; 13 ) (1 ; 15 ) = 96. Furthermore X '(d) = m djm
(5.225b)
is valid. If gcd(m n) = 1 holds, then we get '(mn) = '(m)'(n). '(360) = '(23 32 5) = '(23) '(32) '(5) = 4 6 4 = 96.
2. Fermat{Euler Theorem
The Fermat{Euler theorem is one of the most important theorems of elementary number theory. If a and m are coprime positive numbers, then a'(m) 1(m): (5.226) Determine the last three digits of 999 in decimal notation. This means, determine x with x 999 (1000) and 0 x 999. Now'(1000) to Fermats theorem 9400 1 (1000). 0 =4 400, 4 and1 according 4 9 4 3 Furthermore 9 = (80 + 1) 9 0 80 1 + 1 80 1 9 = (1 + 4 80) 9 ;79 9 89 (400). From that it follows that 999 989 = (10 ; 1)89 890 100 (;1)89 + 891 101 (;1)88 + 892 102 (;1)87 = ;1 + 89 10 ; 3916 100 ;1 ; 110 + 400 = 289(1000). The decimal notation of 999 ends with the digits 289. Remark: The theorem above for m = p, i.e., '(p) = p ; 1 was proved by Fermat the general form was proved by Euler. This theorem forms the basis for encoding schemes (see 5.4.5). It contains a necessary criterion for the prime number property of a positive integer: If p is a prime, then ap;1 1(p) holds for every integer a with p =j a.
3. Wilson's Theorem
There is a further prime number criterion, called the Wilson theorem: Every prime number p satis es (p ; 1)! ;1(p). The inverse proposition is also true and therefore: The number p is a prime number i (p ; 1)! ;1(p).
5.4.5 Codes 1. RSA Codes
R. Rivest, A. Shamir and L. Adleman (see 5.15]) developed an encryption scheme for secret messages on the basis of the Euler{Fermat theorem (see 5.4.4, 2.). The scheme is called the RSA algorithm after the initials of their last names. Part of the key required for decryption can be made public without
332 5. Algebra and Discrete Mathematics endangering the con dentiality of the message for this reason, the term public key code is used in this context as well. In order to apply the RSA algorithm the recipient B chooses two very large prime numbers p and q, calculates m = pq and selects a number r relatively prime to '(m) = (p ; 1)(q ; 1) and 1 < r < '(m). B publishes the numbers m and r because they are needed for decryption. For transmitting a secret message from sender A to recipient B the text of the message must be converted rst to a string of digits that will be split into N blocks of the same length of less than 100 decimal positions. Now A calculates the remainder R of N r divided by m. N r R(m): (5.227a) Sender A calculates the number R for each of the blocks N that were derived from the original text and sends the number to B. The recipient can decipher the message R if he has a solution of the linear congruence rs 1 ('(m)). The number N is the remainder of Rs divided by m: Rs (N r )s N 1+k'(m) N (N '(m) )k N (m): (5.227b) ' ( m ) Here, the Euler{Fermat theorem with N 1(m) has been applied. Eventually, B converts the sequence of numbers into text. A recipient B who expects a secret message from sender A chooses the prime numbers p = 29 and q = 37 (actually too small for practical purposes), calculates m = 29 37 = 1073 (and '(1073) = '(29) '(37) = 1008)), and chooses r = 5 (it satis es the requirement of gcd(1008 5) = 1). B passes the values m = 1073 and r = 5 to A. A intends to send the secret message N = 8 to B. A encrypts N into R = 578 by calculating N r = 85 578 (1073), and just sends the value R = 578 to B. B solves the congruence 5 s 1 (1008), arrives at the solution s = 605, and thus determines Rs = 578605 8 = N (1073). Remark: The security of the RSA code correlates with the time needed by an unauthorized listener to factorize m. Assuming the speed of today's computers, a user of the RSA algorithm should choose the two prime numbers p and q with at least a length of 100 decimal positions in order to impose a decryption e ort of approximately 74 years on the unauthorized listener. The e ort for the authorized user, however, to determine an r relatively prime to '(pq) = (p ; 1)(q ; 1) is comparatively small.
2. International Standard Book Number (ISBN)
A simple application of the congruence of numbers is the use of control digits with the International Standard Book Number ISBN. A combination of 10 digits of the form ISBN a ; bcd ; efghi ; p: (5.228a) is assigned to a book. The digits have the following meaning: a is the group number (for example, a = 3 tells us that the book originates from Austria, Germany, or Switzerland), bcd is the publisher's number, and efghi is the title number of the book by this publisher. A control digit p will be added to detect erroneous book orders and thus help reduce expenses. The control digit p is the smallest non-negative digit that ful ls the following congruence: 10a + 9b + 8c + 7d + 6e + 5f + 4g + 3h + 2i + p 0(11): (5.228b) If the control digit p is 10, a unary symbol such as X is used (see also 5.4.5, 4., p. 333). A presented ISBN can now be checked for a match of the control digit contained in the ISBN and the control digit determined from all the other digits. In case of no match an error is certain. The ISBN control digit method permits the detection of the following errors: 1. Single digit error and 2. interchange of two digits. Statistical investigations showed that by this method more than 90% of all actual errors can be detected. All other observed error types have a relative frequency of less than 1%. In the majority of the cases the described method will detect the interchange of two digits or the interchange of two complete digit blocks.
5.4 Elementary Number Theory 333
3. Central Codes for Drugs and Medicines
In pharmacy, a similar numerical system with control digits is employed for identifying medicaments. In Germany, each medicament is assigned a seven digit control code: abcdefp: (5.229a) The last digit is the control digit p. It is the smallest, non-negative number that ful ls the congruence 2a + 3b + 4c + 5d + 6e + 7f p(11): (5.229b) Here too, the single digit error or the interchange of two digits can always be detected.
4. Account Numbers
Banks and saving banks use a uniform account number system with a maximum of 10 digits (depending on the business volume). The rst (at most four) digits serve the classi cation of the account. The remaining six digits represent the actual account number including a control digit in the last position. The individual banks and saving banks tend to apply di erent control digit methods, for example: a) The digits are multiplied alternately by 2 and by 1, beginning with the rightmost digit. A control digit p will then be added to the sum of these products such that the new total is the next number divisible by 10. Given the account number abcd efghi p with control digit p, then the congruence 2i + h + 2g + f + 2e + d + 2c + b + 2a + p 0 (mod 10): (5.230) holds. b) As in method a), however, any two-digit product is rst replaced by the sum of its two digits and then the total sum will be calculated. In case a) all errors caused by the interchange of adjacent digits and almost all single-digit errors will be detected. In case b), however, all errors caused by the change of one digit and almost all errors caused by the interchange of two adjacent digits will be discovered. Errors due to the interchange of non-adjacent digits and the change of two digits will often not be detected. The reason for not using the more powerful control digit method modulo 11 is of a non-mathematical nature. The non-numerical sign X (instead of the control digit 10 (see 5.4.5, 2., p. 332)) would require an extension of the numerical keyboard. However, renouncing those account numbers whose control digit has the value of 10 would have barred the smooth extension of the original account number in a considerable number of cases.
5. European Article Number EAN
EAN stands for European Article Number. It can be found on most articles as a bar code or as a string of 13 or 8 digits. The bar code can be read by means of a scanner at the counter. In the case of 13-digit strings the rst two digits identify the country of origin, e.g., 40 41 42 and 43 stand for Germany. The next ve digits identify the producer, the following ve digits identify a particular product. The last digit is the control digit p. This control digit will be obtained by rst multiplying all 12 digits of the string alternately by 1 and 3 starting with the left-most digit, by then totalling all values, and by nally adding a p such that the next number divisible by 10 is obtained. Given the article number abcdefghikmn p with control digit p, then the congruence a + 3b + c + 3d + e + 3f + g + 3h + i + 3k + m + 3n + p 0 (mod 10): (5.231) holds. This control digit method always permits the detection of single digit errors in the EAN and often the detection of the interchange of two adjacent digits. The interchange of two non-adjacent digits and the change of two digits will often not be detected.
334 5. Algebra and Discrete Mathematics
5.5 Cryptology
5.5.1 Problem of Cryptology
Cryptology is the science of hiding information by the transformation of data. The idea of protecting data from unauthorized access is rather old. During the 1970s together with the introduction of cryptosystems on the basis of public keys, cryptology became an independent branch of science. Today, the subject of cryptological research is how to protect data from unauthorized access and against tampering. Beside the classical military applications, the needs of the information society gain more and more in importance. Examples are the guarantee of secure message transfer via email, electronic funds transfer (home-banking), the PIN of EC-cards, etc. Today, the elds of cryptography and cryptanalysis are subsumed under the notion of cryptology. Cryptography is concerned with the development of cryptosystems whose cryptographic strengths can be assessed by applying the methods of cryptanalysis for breaking cryptosystems.
5.5.2 Cryptosystems
An abstract cryptosystem consists of the following sets: a set M of messages, a set C of ciphertexts, sets K and K 0 of keys, and sets IE and ID of functions. A message m 2 M will be encrypted into a ciphertext c 2 C by applying a function E 2 IE together with a key k 2 K , and will be transmitted via a communication channel. The recipient can reproduce the original message m from c if he knows an appropriate function D 2 ID and the corresponding key k0 2 K 0. There are two types of cryptosystems: 1. Symmetric Cryptosystems: The conventional symmetric cryptosystem uses the same key k for encryption of the message and for decryption of the ciphertext. The user has complete freedom in setting up his conventional cryptosystem. Encryption and decryption should, however, not become too complex. In any case, a trustworthy transmission between the two communication partners is mandatory. 2. Asymmetric Cryptosystems: The asymmetric cryptosystem (see 5.5.7.1, p. 338) uses two keys, one private key (to be kept secret) and a public key. The public key can be transmitted along the same path as the ciphertext. The security of the communication is warranted by the use of so-called one-way functions (see 5.5.7.2, p. 339), which makes it practically impossible for the unauthorized listener to deduce the plaintext from the ciphertext.
5.5.3 Mathematical Foundation
An alphabet A = fa0 a1 : : : an;1g is a nite non-empty totally ordered set, whose elements ai are called letters. jAj is the length of the alphabet. A sequence of letters w = a01 a02 : : : a0n of length n 2 IN and ai 2 A is called a word of length n over the alphabet A. An denotes the set of all words of length n over A. Let n m 2 IN, let A B be alphabets, and let S be a nite set. A cryptofunction is a mapping t: An S ! B m such that the mappings ts: An ! B m : w ! t(w s) are injective for all s 2 S . The functions ts and t;s 1 are called the encryption and decryption function, respectively. w is called plaintext, ts(w) is the ciphertext. Given a cryptofunction t, then the one-parameter family ftsgs2S is a cryptosystem TS . The term cryptosystem will be applied if in addition to the mapping t, the structure and the size of the set of keys is signi cant. The set S of all the keys belonging to a cryptosystem is called the key space. Then TS = fts: An ! Anjs 2 S g (5.232) is called a cryptosystem on An. If TS is a cryptosystem over An and n = 1, then ts is called a stream cipher otherwise ts is called a block cipher. Cryptofunctions of a cryptosystem over An are suited for the encryption of plaintext of any length. To this end a plaintext will be split into blocks of length n prior to applying the function to each individual
5.5 Cryptology 335
block. The last block may need padding with ller characters to obtain a block of length n. The ller characters must not distort the plaintext. There is a distinction between context-free encryption, where the ciphertext block is only a function of the corresponding plaintext block and the key, and context sensitive encryption, where the ciphertext block depends on other blocks of the message. Ideally, each ciphertext digit of a block depends on all digits of the corresponding plaintext block and all digits of the key. Small changes to the plaintext or to the key cause extended changes to the ciphertext (avalanche e ect).
5.5.4 Security of Cryptosystems
Cryptanalysis is concerned with the development of methods for deducing from the ciphertext as much information about the plaintext as possible without knowing the key. According to A. Kerkho the security of a cryptosystem rests solely in the diculty of detecting the key or, more precisely, the decryption function. The security must not be based on the assumption that the encryption algorithm is kept secret. There are di erent approaches to assess the security of a cryptosystem: 1. Absolutely Secure Cryptosystems: There is only one absolutely secure cryptosystem based on substitution ciphers, which is the one-time pad. This was proved by Shannon as part of his information theory. 2. Analytically Secure Cryptosystems: No method exists to break a cryptosystem systematically. The proof of the non-existence of such a method follows from the proof of the non-computability of a decryption function. 3. Secure Cryptosystems according to Criteria of Complexity Theory: There is no algorithm which can break a cryptosystem in polynomial time (with regard to the length of the text). 4. Practically Secure Cryptosystems: No method is known which can break the cryptosystem with available resources and with justi ed costs. Cryptanalysis often applies statistical methods such as determining the frequency of letters and words. Other methods are an exhaustive search, the trial-and-error method and a structural analysis of the cryptosystem (solving of equation systems). In order to attack a cryptosystem one can bene t from frequent aws in encryption such as using stereotype phrases, repeated transmissions of slightly modi ed text, an improper and predictable selection of keys, and the use of ller characters.
5.5.4.1 Methods of Conventional Cryptography
Besides the application of a cryptofunction it is possible to encrypt a plaintext by means of cryptological codes. A code is a bijective mapping of some subset A0 of the set of all words over an alphabet A onto the subset B 0 of the set of all words over the alphabet B . The set of all source-target pairs of such a mapping is called a code book. today evening 0815 tomorrow evening 1113 The advantage of replacing long plaintexts by short ciphertexts is contrasted with the disadvantage that the same plaintext will always be replaced by the same ciphertext. Another disadvantage of code books is the need for a complete and costly replacement of all books should the code be compromised even partially. In the following only encryption by means of cryptofunctions will be considered. Cryptofunctions have the additional advantage that they do not require any arrangement about the contents of the messages prior to their exchange. Transposition and substitution constitute conventional cryptoalgorithms. In cryptography, a transposition is a special permutation de ned over geometric patterns. The substitutions will now be discussed in detail. There is a distinction between monoalphabetic and polyalphabetic substitutions according to how many alphabets are used for presenting the ciphertext. Generally, a substitution is termed polyal-
336 5. Algebra and Discrete Mathematics phabetic even if only one alphabet is used, but the encryption of the individual plaintext letter depends on its position within the plaintext. A further, useful classi cation is the distinction between monographic and polygraphic substitutions. In the rst case, single letters will be substituted, in the latter case, strings of letters of a xed length > 1.
5.5.4.2 Linear Substitution Ciphers
Let A = fa0 a1 : : : an;1g be an alphabet and k s 2 f0 1 : : : n ; 1g with gcd(k n) = 1. The permutation tks , which maps each letter ai to tks (ai ) = aki+s, is called a linear substitution cipher. There exist n '(n) linear substitution ciphers on A. Shift ciphers are linear substituting ciphers with k = 1. The shift cipher with s = 3 was already used by Julius Caesar (100 to 44 BC) and, therefore, it is called the Caesar cipher.
5.5.4.3 Vigenere Cipher
An encryption called the Vigenere cipher is based on the periodic application of a key word whose letters are pairwise distinct. The encryption of a plaintext letter is determined by the key letter that has the same position in the key as the plaintext letter in the plaintext. This requires a key that is as long as the plaintext. Shorter keys are repeated to match the length of the plaintext. A B C D E F ... A version of the Vigenere cipher attributed to L. Carroll utilizes the so-called Vigenere tableau (see picture) for encryption and A A B C D E F ... decryption. Each row represents the cipher for the key letter B B C D E F G ... to its very left. The alphabet for the plaintext runs across the C C D E F G H ... top. The encryption step is as follows: Given a key letter D and D D E F G H I ... a plaintext letter C, then the ciphertext letter is found at the E E F G H I J ... intersection of the row labeled D and the column labeled C the F F G H I J K ... ... ... ... ... ... ... ... . . . ciphertext is F. Decryption is the inverse of this process.
Let the key be \ HUT ". Plaintext: O N C E U P O N A T I M E Key: H U T H U T H U T H U T H Ciphertext: V H V L O I V H T A C F L Formally, the Vigenere cipher can be written in the following way: let ai be the plaintext letter and aj be the corresponding key letter, then k = i + j determines the ciphertext letter ak . In the above example, the rst plaintext letter is O = a14 . The 15-th position of the key is taken by the letter H = a7 . Hence, k = i + j = 14 + 7 = 21 yields the ciphertext letter a21 = V .
5.5.4.4 Matrix Substitution
Let A = fa0 a1 : : : an;1g be an alphabet and S = (sij ) sij 2 f0 1 : : : m ; 1g, be a non-singular matrix of type (m m) with gcd(detS n)= 1. The mapping which maps the block of plaintext at(1) , at(2) : : : at(m) to the ciphertext determined by the vector (all arithmetic modulo n, vectors transposed as required) 0 0 at(1) 11T B B B B at.(2) C CC C (5.233) B A @S B @ .. C AC at(m) is called the Hill cipher. This represents a monoalphabetic matrix substitution. 0 14 8 3 1 Let the letters of the alphabet be enumerated a0 = A, a1 =B, : : : a25 = Z. For m = 3 and the plaintext AUTUMN, the strings AUT and UMN S = @ 8 5 2A: 321 correspond to the vectors (0 20 19) and (20 12 13).
5.5 Cryptology 337
Then S (0 20 19)> = (217 138 59)> (9 8 7)>(mod26) and S (20 12 13)> = (415 246 97)> (25 12 19)>(mod26). Thus, the plaintext AUTUMN is mapped to the ciphertext JIHZMT.
5.5.5 Methods of Classical Cryptanalysis
The purpose of cryptanalytical investigations is to deduce from the ciphertext an optimum of information about the corresponding plaintext without knowing the key. These analyses are of interest not only to an unauthorized \eavesdropper" but also help assess the security of cryptosystems from the user's point of view.
5.5.5.1 Statistical Analysis
Each natural language shows a typical frequency distribution of the individual letters, two-letter combinations, words, etc. For example, in English the letter e is used most frequently: Letter Relative frequency E, 12.7 % T, A, O, I, N, S, H, R 56.9 % D, L 8.3 % C, U, M, W, F, G, Y, P, B 19.9 % V, K, J, X, Q, Z 2.2 % Given suciently long ciphertexts it is possible to break a monoalphabetic, monographic substitution on the basis of the frequency distribution of letters.
5.5.5.2 Kasiski{Friedman Test
Combining the methods of Kasiski and Friedman it is possible to break the Vignere cipher. The attack bene ts from the fact that the encryption algorithm applies the key periodically. If the same string of plaintext letters is encrypted with the same portion of the key then the same string of ciphertext letters will be produced. A length > 2 of the distance of such identical strings in the ciphertext must be a multiple of the key length. In the case of several reoccurring strings of ciphertext the key length is a divisor of the greatest common divisor of all distances. This reasoning is called the Kasiski test. One should, however, be aware of erroneous conclusions due to the possibility that matches may occur accidentally. The Kasiski test permits the determination of the key length at most as a multiple of the true key length. The Friedman test yields the magnitude of the key length. Let n be the length of the ciphertext of some English plaintext encrypted by means of the Vignere method. Then the key length l is determined by 0:027n l = (n ; 1)IC ; : (5.234a) 0:038n + 0:065 Here IC denotes the coincidence index of the ciphertext. This index can be deduced from the number ni of occurrences of the letter ai (i 2 f0 1 : : : 25g) in the ciphertext: 26 P ni(ni ; 1)
i=1
IC = n(n ; 1) : (5.234b) In order to determine the key, the ciphertext of length n is split into l columns. Since the Vignere cipher produces the contents of each column by means of a shift cipher, it suces to determine the equivalence of E on a column base. Should V be the most frequent letter within a column, then the Vignere tableau points to the letter R E ... (5.234c) R: : :V
338 5. Algebra and Discrete Mathematics of the key. The methods described so far will not be successful if the Vignere cipher employs very long keys (e.g., as long as the plaintext). It is, however, possible to deduce whether the applied cipher is monoalphabetic, polyalphabetic with short period or polyalphabetic with long period.
5.5.6 One-Time Pad
The one-time pad is a substitution cipher that is considered theoretically secure. The encryption adheres to the principle of the Vignere cipher, where the key is a random string of letters as long as the plaintext. Usually, one-time pads are applied as binary Vignere ciphers: Plaintext and ciphertext are represented as binary numbers with addition modulo 2. In this particular case the cipher is involutory, which means that the twofold application of the cipher restores the original plaintext. A concrete implementation of the binary Vignere cipher is based on shift register circuits. These circuits combine switches and storage elements, whose states are 0 or 1, according to special rules.
5.5.7 Public Key Methods
Although the methods of conventional encryption can have ecient implementations with today's computers, and although only a single key is needed for bidirectional communication, there are a number of drawbacks: 1. The security of encryption solely depends on keeping the next key secret. 2. Prior to any communication, the key must be exchanged via a suciently secured channel spontaneous communication is ruled out. 3. Furthermore, no means exist to prove to a third party that a speci c message was sent by an identi ed sender.
5.5.7.1 Die{Hellman Key Exchange
The concept of encryption with public keys was developed by Die and Hellman in 1976. Each participant owns two keys: a public key that is published in a generally accessible register, and a private key that is solely known to the participant and kept absolutely secret. Methods with these properties are called asymmetric ciphers (see 5.5.2, p. 334). The public key KPi of the i-th participant controls the encryption step Ei , his private key KSi the decryption step Di. The following conditions must be ful lled: 1. Di Ei constitutes the identity. 2. Ecient implementations for Ei and Di are known. 3. The private key KSi cannot be deduced from the public key KPi with the means available in the foreseeable future. If in addition 4. also Ei Di yields the identity, then the encryption algorithm quali es as an electronic signature method with public keys. The electronic signature method permits the sender to attach a tamperproof signature to a message. If A wants to send an encrypted message m to B , then A retrieves B 0 s public key KPB from the register, applies the encryption algorithm EB , and calculates EB (m) = c. A sends the ciphertext c via the public network to B who will regain the plaintext of the message by decrypting c using his private key KSB in the decryption function DB : DB (c) = DB (EB (m)) = m. In order to prevent tampering of messages, A can electronically sign his message m to B by complying with an electronic signature method with the public key in the following way: A encrypts the message m with his private key: DA(m) = d. A attaches to d his signature \A" and encrypts the total using the public key of B : EB (DA(m), \A") = EB (d \A") = e. The text thus signed and encrypted is sent from A to B . The participant B decrypts the message with his private key and obtains DB (e) = DB (EB (d, \A")) = (d, \A"). Based on this text B can identify A as the sender and can now decrypt d using the public
5.5 Cryptology 339
key of A : EA(d) = EA(DA(m)) = m.
5.5.7.2 One-Way Function
The encryption algorithms of a method with public key must constitute a one-way function with a \trap door". A trap door in this context is some special, additional information that must be kept secret. An injective function f : X ;! Y is called a one-way function with a trap door, if the following conditions hold: 1. There is an ecient method to compute both f and f ;1. 2. The calculation of f ;1 cannot be deduced from f without the knowledge of the secret additional information. The ecient method to get f ;1 from f cannot be made without the secret additional information.
5.5.7.3 RSA Method
The RSA method described in the number theory section (see 5.4.5, 1., p. 331) is the most popular asymmetric encryption method. 1. Prerequisites: Let p and q be two large prime numbers with pq > 10200 and n = pq. The number of decimal positions of p and q should di er by a small number yet, the di erence between p and q should not be too large. Furthermore, the numbers p ; 1 and q ; 1 should contain rather big prime factors, while the greatest common divisor of p ; 1 and q ; 1 should be rather small. Let e > 1 be relatively prime to (p ; 1)(q ; 1) and let d satisfy d e 1 (mod(p ; 1)(q ; 1)). Now n and e represent the public key and d the private key.
2. Encryption Algorithm: E: f0 1 : : : n ; 1g ! f0 1 : : : n ; 1g E (x) := xe modulo n: 3. Decyphering Operations: D: f0 1 : : : n ; 1g ! f0 1 : : : n ; 1g D(x) := xd modulo n:
(5.235a)
(5.235b) Thus D(E (m)) = E (D(m) = m for message m. The function in this encryption method with n > 10200 constitutes a candidate for a one-way function with trap door (see 5.5.7.2). The required additional information is the knowledge of how to factor n. Without this knowledge it is infeasible to solve the congruence d e 1 (modulo (p ; 1)(q ; 1)). The RSA method is considered practically secure as long as the above conditions are met. A disadvantage in comparison with other methods is the relatively large key size and the fact that RSA is 1000 times slower than DES.
5.5.8 AES Algorithm (Advanced Encryption Standard)
From 1977 to 2001 the DES algorithm has served as ocial US encryption standard for con dential data (see 5.20] and also 22.7], p. 350). In 2001, after a worldwide discussion, the NIST (National Institute of Standards and Technology) has adopted a variant of the Rijndael algorithm suggested by J. Daemen and V. Rijnmen as the new ocial US encryption standard (AES). The AES algorithm consists of several rounds of substitutions and permutations. First a secret key is chosen. The length of the blocks in the plaintext to be encoded and the length of the key can be 128, 192 or 256 Bits. The encoded blocks of plaintext constitute the ciphertext, which has the same length as the plaintext. From the latter the original plaintext can be reconstructed block by block with an inverse algorithm and the key. In contrast to the encoding procedure for the decoding the subkeys generated from the key are applied in reverse order. The strength of this encoding method lies in the construction of the mappings which are applied in the separate iteration rounds. The only non-linear substitution occurs in the \SubBytes" operation. For its description the blocks to be transformed are considered as elements of a nite eld. All details of the algorithm can be found in 5.21]. Although, the AES algorithm has been laid out in the open, there
340 5. Algebra and Discrete Mathematics are no realistic possibilities of attack known to date.
5.5.9 IDEA Algorithm (International Data Encryption Algorithm)
The IDEA algorithm was developed by LAI and MASSAY and patented 1991. It is a symmetric encryption method similar to the DES algorithm and constitutes a potential successor to DES. IDEA became known as part of the reputed software package PGP (Pretty Good Privacy) for the encryption of emails. In contrast to DES not only was the algorithm published but even its basic design criteria. The objective was the use of particularly simple operations (addition modulo 2, addition modulo 216 , multiplication modulo 216+1 ). IDEA works with keys of 128 bits length. IDEA encrypts plaintext blocks of 64 bits each. The algorithm splits a block into four subblocks of 16 bits each. From the 128-bit key 52 subkeys are derived, each 16 bits long. Each of the eight encryption rounds employs six subkeys the remaining four subkeys are used in the nal transformation which constructs the resulting 64-bit ciphertext. Decryption uses the same algorithm with the subkeys in reverse order. IDEA is twice as fast as DES, its implementation in hardware, however, is more dicult. No successful attack against IDEA is known. Exhaustive attacks trying all 256 keys are infeasible considering the length of the keys.
5.6 Universal Algebra
A universal algebra consists of a set, the underlying set, and operations on this set. Simple examples are semigroups, groups, rings, and elds discussed in sections 5.3.2, p. 299 5.3.3, p. 299 and 5.3.6, p. 313. Universal algebras (mostly many-sorted, i.e., with several underlying sets) are handled especially in theoretical informatics. There they form the basis of algebraic speci cations of abstract data types and systems and of term-rewriting systems.
5.6.1 De nition
Let ) be a set of operation symbols divided into pairwise disjoint subsets )n, n 2 IN. )0 contains the constants, )n , n > 0, contain the n-ary operation symbols. The family ()n)n2IN is called the type or signature. If A is a set, and if to every n-ary operation symbol ! 2 )n an n-ary operation ! A in A is assigned, then we call A = (A f!Aj! 2 )g) an )-algebra or algebra of type (or of signature) ). If ) is nite, ) = f!1 : : : !k g, then we also write A = (A !1A : : : !kA) for A. If a ring (see 5.3.6, p. 313) is considered as an )-algebra, then ) is partitioned )0 = f!1g, )1 = f!2g, )2 = f!3 !4g, where to the operation symbols !1, !2, !3 , !4 the constant 0, taking the inverse with respect to addition, addition and multiplication are assigned. Let A and B be )-algebras. B is called an )-subalgebra of A, if B A holds and the operations !B are the restrictions of the operations !A (! 2 )) to the subset B .
5.6.2 Congruence Relations, Factor Algebras
If we want to construct factor structures for universal algebras, then we need the notion of congruence relation. A congruence relation is an equivalence relation compatible with the structure: Let A = (A f!Aj! 2 )g) be an )-algebra and R be an equivalence relation in A. R is called a congruence relation in A, if for all ! 2 )n (n 2 IN) and all ai bi 2 A with ai Rbi (i = 1 : : : n): !A(a1 : : : an) R !A(b1 : : : bn): (5.236) The set of equivalence classes (factor set) with respect to a congruence relation also form an )-algebra with respect to representative-wise calculations: Let A = (A f!Aj! 2 )g) be an )-algebra and R be a congruence relation in A. The factor set A=R (see 5.2.4, 2., p. 297) is an )-algebra A=R with the following operations !A=R (! 2 )n, n 2 IN) with !A=R( a1 ]R : : : an]R ) = !A(a1 : : : an)]R (5.237) and it is called the factor algebra of A with respect to R.
5.6 Universal Algebra 341
The congruence relations of groups and rings can be de ned by special substructures { normal subgroups (see 5.3.3.2, 2. p. 301) and ideals (see 5.3.6.2, p. 313), respectively. In general, e.g., in semigroups, such a characterization of congruence relations is not possible.
5.6.3 Homomorphism
Just as with classical algebraic structures, the homomorphism theorem gives a connection between the homomorphisms and congruence relations. Let A and B be )-algebras. A mapping h: A ! B is called a homomorphism, if for every ! 2 )n and all a1 : : : an 2 A: h(!A(a1 : : : an)) = !B (h(a1) : : : h(an)): (5.238) If, in addition, h is bijective, then h is called an isomorphism the algebras A and B are said to be isomorphic. The homomorphic image h(A) of an )-algebra A is an )-subalgebra of B . Under a homomorphism h, the decomposition of A into subsets of elements with the same image corresponds to a congruence relation which is called the kernel of h: ker h = f(a b) 2 A Ajh(a) = h(b)g: (5.239)
5.6.4 Homomorphism Theorem
Let A and B be )-algebras and h: A ! B a homomorphism. h de nes a congruence relation ker h in A. The factor algebra A= ker h is isomorphic to the homomorphic image h(A). Conversely, every congruence relation R de nes a homomorphic mapping natR : A ! A=R with natR (a) = a]R . Fig. 5.17 illustrats the homomorphism theorem. a A
h(a)
h [a]ker h
nat ker h
A/ker h
Figure 5.17
h(A)
5.6.5 Varieties
A variety V is a class of )-algebras, which is closed under forming direct products, subalgebras, and homomorphic images, i.e., these formations do not lead out of V . Here the direct products are de ned in the following way: If we consider the operations corresponding to ) componentwise on the Cartesian product of the underlying sets of )-algebras, then we again get an )algebra, the direct product of these algebras. The theorem of Birkho (see 5.6.6, p. 341) characterizes the varieties as those classes of )-algebras, which can be equationally dened.
5.6.6 Term Algebras, Free Algebras
Let ()n )n2IN be a type (signature) and X a countable set of variables. The set T(X ) of )-terms over X is de ned inductively in the following way: 1. X )0 T(X ). 2. If t1 : : : tn 2 T(X ) and ! 2 )n hold, then also !t1 : : : tn 2 T (X ) holds. The set T(X ) de ned in this way is an underlying set of an )-algebra, the term algebra T (X ) of type ) over X , with the following operations: If t1 : : : tn 2 T(X ) and ! 2 )n hold, then !T(X ) is de ned by !T(X ) (t1 : : : tn) = !t1 : : : tn: (5.240) Term algebras are the \most general" algebras in the class of all )-algebras, i.e., no \identities" are valid in term algebras. These algebras are called free algebras.
342 5. Algebra and Discrete Mathematics An identity is a pair (s(x1 : : : xn), t(x1 : : : xn)) of )-terms in the variables x1 : : : xn. An )-algebra A satises such a equation, if for every a1 : : : an 2 A we have: sA(a1 : : : an) = tA(a1 : : : an): (5.241) A class of )-algebras de ned by identities is a class of )-algebras satisfying a given set of identities. Theorem of Birkho : The classes de ned by identities are exactly the varieties. Varieties are for example the classes of all semigroups, groups, Abelian groups, and rings. But, e.g., the direct product of cyclic groups is not a cyclic group, and the direct product of elds is not a eld. Therefore cyclic groups or elds do not form a variety, and cannot be de ned by equations.
5.7 Boolean Algebras and Switch Algebra
Calculating rules, similar to the rules established in 5.2.2, 3., p. 292 for set algebra and propositional calculus (5.1.1, 6., p. 287), can be found for other objects in mathematics too. The investigation of these rules yields the notion of Boolean algebra.
5.7.1 De nition A set B , together with two binary operations u (\conjunction") and t (\disjunction"), and a unary operation (\negation"), and two distinguished (neutral) elements 0 and 1 from B , is called a Boolean algebra B = (B u t 0 1) if the following properties are valid:
(1) Associative Laws: (a u b) u c = a u (b u c) (2) Commutative Laws: aub=bua (3) Absorption Laws: a u (a t b) = a (4) Distributive Laws: (a t b) u c = (a u c) t (b u c) (5) Neutral Elements: au1=1 au0=0
(6) Complement: aua=0
(5.242)
(a t b) t c = a t (b t c):
(5.243)
(5.244)
a t b = b t a:
(5.245)
(5.246)
a t (a u b) = a:
(5.247)
(5.248)
(a u b) t c = (a t c) u (b t c): (5.249)
(5.250)
at0=a
(5.251)
(5.252)
at1=1
(5.253)
(5.254) a t a = 1: (5.255) A structure with the associative laws, commutative laws, and absorption laws is called a lattice. If the distributive laws also hold, then we call it a distributive lattice. So a Boolean algebra is a special distributive lattice. Remark: The notation used for Boolean algebras is not necessarily identical to the notation for the
5.7 Boolean Algebras and Switch Algebra 343
operations in propositional calculus.
5.7.2 Duality Principle 1. Dualizing
In the \axioms" of a Boolean algebra above we can discover the following duality: If we replace u by t, t by u, 0 by 1, and 1 by 0 in an axiom, then we always get the other axiom in the same row. The axioms in a row are dual to each other, and the substitution process is called dualization. We get the dual statement from a statement of the Boolean algebra by dualization.
2. Duality Principle for Boolean Algebras
The dual statement of a true statement for a Boolean algebra is also a true statement for the Boolean algebra, i.e., with every proved proposition, the dual proposition is also proved.
3. Properties
We get, e.g., the following properties for Boolean algebras from the axioms.
(E1) The Operations u and t are Idempotent: aua=a (5.256) (E2) De Morgan Rules: aub=atb (5.258) (E3) A further Property:
a t a = a:
(5.257)
atb=aub
(5.259)
a = a: (5.260) It is enough to prove only one of the two properties in any line above, because the other one is the dual property. The last property is self-dual.
5.7.3 Finite Boolean Algebras
All nite Boolean algebras can be described easily up to \isomorphism". Let B1, B2 be two Boolean algebras and f : B1 ! B2 a bijective mapping. f is called an isomorphism if f (a u b) = f (a) u f (b) f (a t b) = f (a) t f (b) and f (a) = f (a) (5.261) hold. Every nite Boolean algebra is isomorphic to the Boolean algebra of the power set of a nite set. In particular every nite Boolean algebra has 2n elements, and every two nite Boolean algebras with the same number of elements are isomorphic. In the following, we denote by B the Boolean algebra with two elements f0 1g with the operations u 0 1 t 0 1 ; 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 If we de ne the operations u, t, and componentwise on the n-times Cartesian product B n = f0 1g f0 1g, then B n will be a Booleannalgebra withn 0 = (0 : : : 0) and 1 = (1 : : : 1). We call B n the n times direct product of B . Because B contains 2 elements, we get all the nite Boolean algebras in this way (up to isomorphism).
5.7.4 Boolean Algebras as Orderings
We can assign an order relation to every Boolean algebra B : Here a b holds if a u b = a is valid (or equivalently, if a t b = b holds). So every nite Boolean algebra can be represented by a Hasse diagram (see 5.2.4, 4., p. 298).
344 5. Algebra and Discrete Mathematics Suppose B is the set f1 2 3 5 6 10 15 30g of the divisors of 30. We de ne the least common multiple and the greatest common divisor as binary operations, and taking the complement as unary operation. The numbers 1 and 30 correspond to the distinguished elements 0 and 1. The corresponding Hasse diagram is shown in Fig. 5.18.
5.7.5 Boolean Functions, Boolean Expressions 1. Boolean Functions
We denote by B the Boolean algebra with two elements as in 5.7.3. An n-ary Boolean function f is a mapping from B n into B . There are 22n n-ary Boolean functions. The set of all n-ary Boolean functions with the operations (f u g)(b) = f (b) u g(b)
30 6
10
15
2
3
5
1
Figure 5.18
(f t g)(b) = f (b) t g(b)
(5.262)
(5.263)
f (b) = f (b) (5.264) is a Boolean algebra. Here b always means an n-tuple of the elements of B = f0 1g, and on the righthand side of the equations the operations are performed in B . The distinguished elements 0 and 1 correspond to the functions f0 and f1 with f0(b) = 0 f1(b) = 1 for all b 2 B n: (5.265) A: In the case n = 1, i.e., for only one Boolean variable b, there are four Boolean functions: Identity f (b) = b, Negation f (b) = b, (5.266) Tautology f (b) = 1, Contradiction f (b) = 0. B: In the case n = 2, i.e., for two Boolean variables a and b, there are 16 di erent Boolean functions, among which the most important ones have their own names and notation. They are shown in Table 5.6. Table 5.6 Some Boolean functions with two variables a and b
Name of the Di erent function notation She er or NAND Peirce or NOR Antivalence or XOR Equivalence Implication
ab ajb NAND (a b) a+b ab NOR a b ab + ab a XOR b a 6 b a$b ab + ab a b a$b a+b a!b
2. Boolean Expressions
Value ! table ! for!
Di erent symbols & _1 >
!
!
a = 0 b 0
0 1
1 0
1 1
1
1
1
0
1
0
0
0
=1
+
0
1
1
0
=1
+
1
0
0
1
1
1
0
1
Boolean expressions are de ned in an inductive way: Let X = fx y z : : :g be a (countable) set of
5.7 Boolean Algebras and Switch Algebra 345 Boolean variables (which can take values only from f0 1g): 1. The constants 0 and 1 just as the Boolean variables from X are Boolean expressions. (5.267) 2. If S and T are Boolean expressions, so are T , (S u T ), and (S t T ), as well. (5.268) If a Boolean expression contains the variables x1 : : : xn, then it represents an n-ary Boolean function fT : Let b be a \valuation" of the Boolean variables x1 : : : xn, i.e., b = (b1 : : : bn) 2 B n. We assign a Boolean function to the expression T in the following way: 1. If T = 0 then fT = f0 if T = 1 then fT = f1 : (5.269a) 2. If T = xi then fT (b) = bi if T = S then fT (b) = fS (b): (5.269b) 3. If T = R u S then fT (b) = fR (b) u fS (b): (5.269c) 4. If T = R t S then fT (b) = fR (b) t fS (b): (5.269d) On the other hand, every Boolean function f can be represented by a Boolean expression T (see 5.7.6).
3. Concurrent or Semantically Equivalent Boolean Expressions
The Boolean expressions S and T are called concurrent or semantically equivalent if they represent the same Boolean function. Boolean expressions are equal if and only if they can be transformed into each other according to the axioms of a Boolean algebra. Under transformations of a Boolean expression we consider especially two aspects: Transformation in a possible \simple" form (see 5.7.7). Transformation in a \normal form".
5.7.6 Normal Forms
1. Elementary Conjunction, Elementary Disjunction
Let B = (B u t 0 1) be a Boolean algebra and fx1 : : : xng a set of Boolean variables. Every conjunction or disjunction in which every variable or its negation occurs exactly once is called an elementary conjunction or an elementary disjunction respectively (in the variables x1 : : : xn ). Let T (x1 : : : xn) be a Boolean expression. A disjunction D of elementary conjunctions with D = T is called a principal disjunctive normal form (PDNF) of T . A conjunction C of elementary disjunctions with C = T is called a principal conjunctive normal form (PCNF) of T . Part 1: In order to show that every Boolean function f can be represented as a Boolean expression, we construct the PDNF form of the function f given in the annexed table: x y z f (x y z ) The PDNF of the Boolean function f contains the elementary conjunctions x u y u z, x u y u z, x u y u z . These elementary conjunctions belong to 0 0 0 0 the valuations b of the variables where the function f has the value 1. If a 0 0 1 1 variable v has the value 1 in b, then we put v in the elementary conjunction, 0 1 0 0 otherwise we put v. 0 1 1 0 Part 2: The PDNF for the example of Part 1 is: 1 0 0 0 (x u y u z) t (x u y u z) t (x u y u z ): (5.270) 1 0 1 1 1 1 0 1 The \dual" form for PDNF is the PCNF: The elementary disjunctions be1 1 1 0 long to the valuations b of the variables for which f has the value 0. If a variable v has the value 0 in b, then we put v in the elementary disjunction, otherwise v. So the PCNF is: (x t y t z) u (x t y t z) u (x t y t z ) u (x t y t z) u (x t y t z ): (5.271) The PDNF and the PCNF of f are uniquely determined, if the ordering of the variables and the ordering of the valuations is given, e.g., if we consider the valuations as binary numbers and we arrange them in increasing order.
346 5. Algebra and Discrete Mathematics
2. Principal Normal Forms
The principal normal form of a Boolean function fT is considered as the principal normal form of the corresponding Boolean expression T . To check the equivalence of two Boolean expression by transformations often causes diculties. The principal normal forms are useful: Two Boolean expressions are semantically equivalent exactly if their corresponding uniquely determined principal normal forms are identical letter by letter. Part 3: In the considered example (see Part 1 and 2) the expressions (y u z) t (x u y u z) and (x t ((y t z) u (y t z) u (y t z ))) u (x t ((y t z) u (y t z))) are semantically equivalent because the principal disjunctive (or conjunctive) normal forms of both are the same.
5.7.7 Switch Algebra
A typical application of Boolean algebra is the simpli cation of series{parallel connections (SPC). We assign a Boolean expression to a SPC (transformation). This expression will be \simpli ed" with the transformation rules of Boolean algebra. Finally we assign a SPC to this expression (inverse transformation). As a result, we get a simpli ed SPC which produces the same behavior as the initial connection system (Fig. 5.19). A SPC has two types of contact points: the so-called \make contacts" and \break contacts", and both types have two states namely open or closed. Here we consider the usual symbolism: When the equipment is put on, the make contacts close and the break contacts open. We assign Boolean variables to the contacts of the switch equipment. SPC
electrically equivalent
transformation (modelling) Boolean expression
simplified SPC inverse transformation
simplification by Boolean algebra
simplified Boolean expression
Figure 5.19 The position \o " or \on" of the equipment corresponds to the value 0 or 1 of the Boolean variables. The contacts being switched by the same equipment are denoted by the same symbol, the Boolean variable belonging to this equipment. The contact value of a SPC is 0 or 1, according to whether the switch is electrically non-conducting or conducting. The contact value depends on the position of the contacts, so it is a Boolean function S (switch function) of the variables assigned to the switch equipment. Contacts, connections, symbols, and the corresponding Boolean expressions are represented in Fig. 5.20. The Boolean expressions, which represent switch functions of SPC, have the special property that the negation sign can occur only above variables (never over subexpressions). Simplify the SPC of Fig. 5.21. This connection corresponds to the Boolean expression S = (a u b) t (a u b u c) t (a u (b t c)) (5.272) as switch function. According to the transformation formulas of Boolean algebra we get: S = (b u (a t (a u c))) t (a u (b t c)) = (b u (a t c)) t (a u (b t c)) = (a u b) t (b u c) t (a u c) = (a u b u c) t (a u b u c) t (b u c) t (a u b u c) t (a u c) t (a u b u c) (5.273) = (a u c) t (b u c):
5.7 Boolean Algebras and Switch Algebra 347 make contact (symbol:
series connection )
a S=a
a b S=a b
)
parallel connection
break contact (symbol:
(symbol:
)
a
a
(symbol:
)
b S=a b
S=a
Figure 5.20 a a
b b
a
a
c
b
c
c
b c
Figure 5.21 Figure 5.22 Here we get a u c from (a u b u c) t (a u c) t (a u b u c), and b u c from (a u b u c) t (b u c) t (a u b u c). Finally we have the simpli ed SPC shown in Fig. 5.22. This example shows that usually it is not so easy to get the simplest Boolean expression by transformations. In the literature we can nd di erent methods for this procedure.
348 5. Algebra and Discrete Mathematics
5.8 Algorithms of Graph Theory
Graph theory is a eld in discrete mathematics having special importance for informatics, e.g., for representing data structures, nite automata, communication networks, derivatives in formal languages, etc. Besides this there are applications in physics, chemistry, electrotechnics, biology and psychology. Moreover, ows can be applied in transport networks and in network analysis in operations research and in combinatorial optimization.
5.8.1 Basic Notions and Notation 1. Undirected and Directed Graphs
A graph G is an ordered pair (V E ) of a set V of vertices and a set E of edges. There is a mapping, de ned on E , the incidence function, which uniquely assigns to every element of E an ordered or nonordered pair of (not necessarily distinct) elements of V . If a non-ordered pair is assigned then G is called an undirected graph (Fig. 5.23). If an ordered pair is assigned to every element of E , then the graph is called a directed graph (Fig. 5.24), and the elements of E are called arcs or directed edges. All other graphs are called mixed graphs. In the graphical representation, the vertices of a graph are denoted by points, the directed edges by arrows, and undirected edges by non-directed lines. v4
e'2' v3 e'1'
v5
e'3' v2
e'4' v1
v4
e'2
e'3
v3 e'1
v2
v5 e'4 v1
e7 v5
e4
v3
v1
e5 v 4 e3 e6 e 1
e2 v2
Figure 5.23 Figure 5.24 Figure 5.25 A: For the graph G in Fig. 5.25: V = fv1 v2 v3 v4 v5g E = fe1 e2 e3 e4 e5 e6 e7g, f1(e1 ) = fv1 v2g f1 (e2) = fv1 v2g f1 (e3) = (v2 v3) f1(e4 ) = (v3 v4 ) f1 (e5) = (v3 v4) f1(e6 ) = (v4 v2) f1 (e7) = (v5 v5): B: For the graph G in Fig. 5.24: V = fv1 v2 v3 v4 v5g E 0 = fe01 e02 e03 e04 g f2(e01 ) = (v2 v3) f2 (e02) = (v4 v3) f2(e03 ) = (v4 v2) f2 (e04) = (v5 v5): C: For the graph G in Fig. 5.23: V = fv1 v2 v3 v4 v5 g E 00 = fe001 e002 e003 e004 g f3(e001 ) = fv2 v3g f3 (e002 ) = fv4 v3g f3 (e003 ) = fv4 v2g f3 (e004 ) = fv5 v5g:
2. Adjacency
If (v w) 2 E , then the vertex v is said to be adjacent to the vertex w. Vertex v is called the initial point of (v w), w is called the terminal point of (v w), and v and w are called the endpoints of (v w). Adjacency in undirected graphs and the endpoints of undirected edges are de ned analogously.
3. Simple Graphs
If several edges or arcs are assigned to the same ordered or non-ordered pairs of vertices, then they are called multiple edges. An edge with identical endpoints is called a loop. Graphs without loops and multiple edges and multiple arcs, respectively, are called simple graphs.
4. Degrees of Vertices
The number of edges or arcs incident to a vertex v is called the degree dG(v) of the vertex v. Loops are counted twice. Vertices of degree zero are called isolated vertices. For every vertex v of a directed graph G, the out-degree d+G(v) and in-degree d;G(v) of v are distinguished as follows: d+G(v) = jfwj(v w) 2 E gj (5.274a) d;G(v) = jfwj(w v) 2 E gj: (5.274b)
5.8 Algorithms of Graph Theory 349
5. Special Classes of Graphs
Finite graphs have a nite set of vertices and a nite set of edges. Otherwise the graph is said to be innite. In regular graphs of degree r every vertex has degree r. An undirected simple graph with vertex set V is called a complete graph if any two di erent vertices in V are connected by an edge. A complete graph with an n-element set of vertices is denoted by Kn. If the set of vertices of an undirected simple graph G can be partitioned into two disjoint classes X and Y such that every edge of G joins a vertex of X and a vertex of Y , then G is called a bipartite graph. A bipartite graph is called a complete bipartite graph, if every vertex of X is joined by an edge with every vertex of Y . If X has n elements and Y has m elements, then the graph is denoted by Knm. Fig. 5.26 shows a complete graph with ve vertices. Fig. 5.27 shows a complete bipartite graph with a two-element set X and a three-element set Y . 1 5
2
K5
4
K2,3
x1 x2
y1 y2 y3
3
Figure 5.26 Figure 5.27 Further special classes of graphs are plane graphs, trees and transport networks. Their properties will be discussed in later paragraphs.
6. Representation of Graphs
Finite graphs can be visualized by assigning to every vertex a point in the plane and connecting two points by a directed or undirected curve, if the graph has the corresponding edge. There are examples in Fig. 5.28{5.31. Fig. 5.31 shows the Petersen graph, which is a well-known counterexample for several graph-theoretic conjectures, which could not be proved in general.
Figure 5.28
Figure 5.29
7. Isomorphism of Graphs
Figure 5.30
Figure 5.31
A graph G1 = (V1 E1) is said to be isomorphic to a graph G2 = (V2 E2) i there are bijective mappings ' from V1 onto V2 and from E1 onto E2 being compatible with the incidence function, i.e., if u v are the endpoints of an edge or u is the initial point of an arc and v is its terminal point, then '(u) and '(v) are the endpoints of an edge and '(u) is the initial point and '(v) the terminal point of an arc, respectively. Fig. 5.32 and Fig. 5.33 show two isomorphic graphs. The mapping ' with '(1) = a '(2) = b '(3) = c '(4) = d is an isomorphism. In this case, every bijective mapping of f1 2 3 4g onto fa b c dg is an isomorphism, since both graphs are complete graphs with equal number of vertices.
8. Subgraphs, Factors
If G = (V E ) is a graph, then the graph G0 = (V 0 E 0) is called a subgraph of G, if V 0 V and E 0 E . If E 0 contains exactly those edges of E which connect vertices of V 0, then G0 is called the subgraph of G induced by V 0 (induced subgraph).
350 5. Algebra and Discrete Mathematics A subgraph G0 = (V 0 E 0) of G = (V E ) with V 0 = V is called a partial graph of G. A factor F of a graph G is a regular subgraph of G containing all vertices of G. 4
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Figure 5.32
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Figure 5.33
9. Adjacency Matrix
Finite graphs can be described by matrices: Let G = (V E ) be a graph with V = fv1 v2 : : : vng and E = fe1 e2 : : : emg. Let m(vi vj ) denote the number of edges from vi to vj . For undirected graphs, loops are counted twice for directed graphs loops are counted once. The matrix A of type (n n) with A = (m(vi vj )) is called an adjacency matrix. If in addition the graph is simple, then the adjacency matrix has the following form: (vi vj ) 2 E A = (aij ) = 10 for (5.275) for (vi vj ) 62 E i.e., in the matrix A there is a 1 in the i-th row and j -th column i there is an edge from vi to vj . The adjacency matrix of undirected graphs is symmetric. A: Beside Fig. 5.34 there is the adjacency matrix A(G1) of the directed graph G1 . B: Beside Fig. 5.35 there is the adjacency matrix A(G2) of the undirected simple graph G2. v3
v4 v2 v1
00 1 0 01 A1 = BB@ 0 0 0 0 CCA
Figure 5.34
10. Incidence Matrix
0103 0100
v1 v6
v2
v5
v3 v4
00 1 0 1 0 11 B 1 0 1 0 1 0C B C B C B A2 = BB 01 10 01 10 01 10 CCC B @0 1 0 1 0 1C A 101010
Figure 5.35
For an undirected graph G = (V E ) with V = fv1 v2 : : : vng and E = fe1 e2 : : : em g, the matrix I of type (n m) given by 8 0 v is not incident with e < i j I = (bij ) = : 1 vi is incident with ej and ej is not a loop (5.276) 2 vi is incident with ej and ej is a loop is called the incidence matrix. For a directed graph G = (V E ) with V = fv1 v2 : : : vng and E = fe1 e2 : : : emg, the incidence matrix I is the matrix of type (n m), de ned by 8 0 v is not incident with e i j > < the initial point of ej and ej is not a loop I = (bij ) = > ;11 vvii isis the (5.277) terminal point of ej and ej is not a loop : ;0 v is incident to ej and ej is a loop: i
11. Weighted Graphs
If G = (V E ) is a graph and f is a mapping assigning a real number to every edge, then (V E f ) is called a weighted graph, and f (e) is the weight or length of the edge e.
5.8 Algorithms of Graph Theory 351
In applications, these weights of the edges represent costs resulting from the construction, maintenance or use of the connections.
5.8.2 Traverse of Undirected Graphs 5.8.2.1 Edge Sequences or Paths 1. Edge Sequences or Paths
In an undirected graph G = (V E ) every sequence F = (fv1 v2g fv2 v3 g : : : fvs vs+1g) of the elements of E is called an edge sequence of length s. 1 If v1 = vs+1, then the sequence is called a cycle, otherwise it is an open edge sequence. An edge sequence F is called a path i v1 v2 : : : vs are pairwise distinct vertices. A closed path is a circuit. A trail is a sequence of edges with- 5 2 out repeated edges. In the graphs in Fig. 5.36, F1 = (f1 2g f2 3g f3 5g f5 2g f2 4g) is an edge sequence of length 5, F2 = (f1 2g f2 3g f3 4g f4 2g f2 1g) is a cycle of length 5, F3 = (f2 3g f3 5g f5 2g f2 1g) is a path, F4 = 4 3 (f1 2g f2 3g f3 4g) is a path. An elementary cycle is given by F5 = Figure 5.36 (f1 2g f2 5g f5 1g).
2. Connected Graphs, Components
If there is at least one path between every pair of distinct vertices v w in a graph G, then G is said to be connected. If a graph G is not connected, it can be decomposed into components, i.e., into induced connected subgraphs with maximal number of vertices.
3. Distance Between Vertices
The distance (v w) between two vertices v w of an undirected graph is the length of a path with minimum number of edges connecting v and w. If such a path does not exist, then let (v w) = 1.
4. Problem of Shortest Paths
Let G = (V E f ) be a weighted simple graph with f (e) > 0 for every e 2 E . Determine the shortest path from v to w for two vertices v w of G, i.e., a path from v to w having minimum sum of weights of edges and arcs, respectively. There is an ecient algorithm of Dantzig to solve this problem, which is formulated for directed graphs and can be used for undirected graphs (see 5.8.6, p. 357) in a similar way. Every graph G = (V E f ) with V = fv1 v2 : : : vng has a distance matrix D of type (n n): D = (dij ) with dij = (vi vj ) (i j = 1 2 : : : n): (5.278) In the case that every edge has weight 1, i.e., the distance between v and w is equal to the minimum number of edges which have to be traversed in the graph to get from v to w, then the distance between two vertices can be determined using the adjacency matrix: Let v1 v2 : : : vn be the vertices of G. The adjacency matrix of G is A = (aij ), and the powers of the adjacency matrix with respect to the usual multiplication of matrices (see 4.1.4, 5., p. 254) are denoted by Am = (amij ) m 2 IN. There is a shortest path of length k from the vertex vi to the vertex vj (i 6= j ) i : akij 6= 0 and asij = 0 (s = 1 2 : : : k ; 1): (5.279) The weighted graph represented in Fig. 5.37 has the distance matrix D beside it. The graph represented in Fig. 5.38 has the adjacency matrix A beside it, and for m = 2 or m = 3 the matrices A2 and A3 are obtained. Shortest paths of length 2 connect the vertices 1 and 3, 1 and 4, 1 and 5, 2 and 6, 3 and 4, 3 and 5, 4 and 5. Furthermore the shortest paths between the vertices 1 and 6, 3 and 6, and nally 4 and 6 are of length 3.
352 5. Algebra and Discrete Mathematics 1
7
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00 B B B2 D = BBB 35 B @6
1 1C 1 CC 1 CC 1 CC 1A 111111
Figure 5.37
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00 1 0 0 0 01 01 1 1 1 1 01 01 5 1 1 1 11 B C B C B 1 1 1 1 1 0C 1 5 1 1 1 1C 5 9 5 5 6 1C B B B C B C B C B C 0 1 0 0 0 0 1 1 1 1 1 0 B C B C B A = BB 0 1 0 0 0 0 CC A2 = BB 1 1 1 1 1 0 CC A3 = BB 11 55 11 11 11 11 CCC : B B B @0 1 0 0 0 1C A @1 1 1 1 2 0C A @1 6 1 1 1 2C A
Figure 5.38
000010
010001
111120
5.8.2.2 Euler Trails
1. Euler Trail, Euler Graph
A trail containing every edge of a graph G is called an open or closed Euler trail of G. A connected graph containing a closed Euler trail is an Euler graph. The graph G1 (Fig. 5.39) has no Euler trail. The graph G2 (Fig. 5.40) has an Euler trail, but it is not an Euler graph. The graph G3 (Fig. 5.41) has a closed Euler trail, but it is not an Euler graph. The graph G4 (Fig. 5.42) is an Euler graph.
G1
Figure 5.39
G2
Figure 5.40
G3
Figure 5.41
G4
Figure 5.42
2. Theorem of Euler{Hierholzer A nite connected graph is an Euler graph i all vertices have positive even degrees. 3. Construction of a Closed Euler Trail
If G is an Euler graph, then choose an arbitrary vertex v1 of G and construct a trail F1 , starting at v1 , which cannot be continued. If F1 does not contain all edges of G, then construct another path F2 induced by edges not in F1 starting at a vertex v2 2 F1 , until it cannot be continued. Compose a closed trail in G using F1 and F2: Start traversing F1 at v1 until v2 is reached, then continue by traversing F2 , and nish by traversing the edges of F1 not used before. Repeating this method a closed Euler trail is obtained in nitely many steps.
4. Open Euler Trails
There is an open Euler trail in a graph G i there are exactly two vertices in G with odd degrees. Fig. 5.43 shows a graph which has no closed Euler trail, but it has an open Euler trail. The edges are consecutively enumerated with respect to an Euler trail. In Fig. 5.44 there is a graph with a closed Euler trail.
5. Chinese Postman Problem
The problem, that a postman should pass through all streets in his service area at least once and return to the initial point and use a trail as short as possible, can be formulated in graph theoretical terms as
5.8 Algorithms of Graph Theory 353 3
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Figure 5.43 Figure 5.44 Figure 5.45 follows: Let G = (V E f ) be a weighted graph with f (e) 0 for every edge e 2 E . Determine an edge sequence F with minimum total length X L = f (e): (5.280) e2F
The name of the problem refers to the Chinese mathematician Kuan, who studied this problem rst. To solve it two cases are distinguished: 1. G is an Euler graph { then every closed Euler trail is optimal { and 2. G has no closed Euler trail. An e ective algorithm solving this problem is given by Edmonds and Johnson (see 5.24]).
5.8.2.3 Hamiltonian Cycles 1. Hamiltonian Cycle
A Hamiltonian cycle is an elementary cycle in a graph covering all of the vertices. In Fig. 5.45, lines in bold face show a Hamiltonian cycle. The idea of a game to constructHamiltonian cycles in the graph of a pentagondodecaeder, goes back to Sir W. Hamilton. Remark: The problem of characterizing graphs with Hamiltonian cycles leads to one of the classical NP-complete problems. Therefore, an ecient algorithm to determine the Hamilton cycles cannot be given here.
2. Theorem of Dirac
If a simple graph G = (V E ) has at least three vertices, and dG(v) jV j=2 holds for every vertex v of G, then G has a Hamiltonian cycle. This is a sucient but not a necessary condition for the existence of Hamiltonian cycles. The following theorems with more general assumptions give only sucient but not necessary conditions for the existence of Hamilton cycles, too. Fig. 5.46 shows a graph which has a Hamiltonian cycle, but does not satisfy the assumptions of the following theorem of Ore.
3. Theorem of Ore
If a simple graph G = (V E ) has at least three vertices, and dG(v) + dG(w) jV j holds for every pair of non-adjacent vertices v w, then G contains a Hamiltonian cycle.
4. Theorem of Posa
Let G = (V E ) be a simple graph with at least three vertices. There is Figure 5.46 a Hamiltonian cycle in G if the following conditions are satis ed: 1. For 1 k < (jV j ; 1)=2, the number of vertices of degree not exceeding k is less than k. 2. If jV j is odd, then the number of vertices of degree not exceeding (jV j ; 1)=2 is less than or equal to (jV j ; 1)=2.
354 5. Algebra and Discrete Mathematics father children grandchildren
Figure 5.47
Figure 5.48
great-grandchildren
Figure 5.49
5.8.3 Trees and Spanning Trees 5.8.3.1 Trees 1. Trees
An undirected connected graph without cycles is called a tree. Every tree with at least two vertices has at least two vertices of degree 1. Every tree with n vertices has exactly n ; 1 edges. A directed graph is called a tree if G is connected and does not contain any circuit (see 5.8.6, p. 357). Fig. 5.47 and Fig. 5.48 represent two non-isomorphic trees with 14 vertices. They demonstrate the chemical structure of butane and isobutane.
2. Rooted Trees
A tree with a distinguished vertex is called a rooted tree, and the distinguished vertex is called the root. In diagrams, the root is usually on the top, and the edges are directed downwards from the root (see Fig. 5.49). Rooted trees are used to represent hierarchic structures, as for instance hierarchies in factories, family trees, grammatical structures. Fig. 5.49 shows the genealogy of a family in the form of a rooted tree. The root is the vertex assigned to the father.
3. Regular Binary Trees
If a tree has exactly one vertex of degree 2 and otherwise only vertices of degree 1 or 3, then it is called a regular binary tree. The number of vertices of a regular binary tree is odd. Regular trees with n vertices have (n + 1)=2 vertices of degree 1. The level of a vertex is its distance from the root. The maximal level occurring in a tree is the height of the tree. There are several applications of regular binary rooted trees, e.g., in informatics.
4. Ordered Binary Trees
Arithmetical expressions can be represented by binary trees. Here, the numbers and variables are assigned vertices of degree 1, the operations \+",\;", \" correspond to vertices of degree > 1, and the left and right subtree, respectively, represents the rst and second operand, respectively, which is, in general, also an expression. These trees are called ordered binary trees. The traverse of an ordered binary tree can be performed in three di erent ways, which are de ned in a recursive way (see also Fig. 5.50): Traverse the left subtree of the root (in inorder traverse), Inorder traverse : visit the root, traverse the right subtree of the root (in inorder traverse). Preorder traverse : Visit the root, traverse the left subtree (in preorder traverse), traverse the right subtree of the root (in preorder traverse). Postorder traverse : Traverse the left subtree of the root (in postorder traverse), traverse the right subtree of the root (in postorder traverse), visit the root.
5.8 Algorithms of Graph Theory 355
Using inorder traverse the order of the terms does not change in comparision with the given expression. The term obtained by postorder traverse is called postx notation PN or Polish notation. Analogously, the term obtained by preorder traverse is called prex notation or reversed Polish notation. Pre x and post x expressions uniquely describe the tree. This fact can + be used for the implementation of trees. In Fig. 5.50 the term a (b ; c)+ d is represented by a graph. Inorder . d traverse yields a b ; c + d, preorder traverse yields + a ; bcd, and postorder traversal yields abc ; d+.
5.8.3.2 Spanning Trees
−
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1. Spanning Trees
A tree, being a subgraph of an undirected graph G, and containing all vertices of G, is called a spanning tree of G. Every nite connected graph G contains a spanning tree H : If G contains a cycle, then delete an edge of this cycle. The remaining graph G1 is still connected and can be 1 transformed into a connected graph G2 by deleting a further edge of a cycle of G1, if there exists such an edge. Af2 3 4 ter nitely many steps a spanning tree of G is obtained. Fig. 5.52 shows a spanning tree H of the graph G Figure 5.51 shown in Fig. 5.51.
2. Theorem of Cayley Every complete graph with n vertices (n > 1) has exactly nn;2 spanning trees. 3. Matrix Spanning Tree Theorem
c
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Figure 5.50 1 2
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Figure 5.52
Let G = (V E ) be a graph with V = fv1 v2 : : : vng (n > 1) and E = fe1 e2 : : : emg. De ne a matrix
D = (dij ) of type (n n):
i 6= j dij = dG(vi0) for (5.281a) for i = j which is called the degree matrix. The di erence between the degree matrix and the adjacency matrix is the admittance matrix L of G: L = D ; A: (5.281b) Deleting the i-th row and the i-th column of L the matrix Li is obtained. The determinant of Li is equal to the number of spanning trees of the graph G. The adjacency matrix, the degree matrix and the admittance matrix of the graph in Fig. 5.51 are: 02 1 1 01 04 0 0 01 0 2 ;1 ;1 0 1 B C B C 1 0 2 0 0 3 0 0 A = B@ 1 2 0 1 CA D = B@ 0 0 4 0 CA L = BB@ ;;11 ;23 ;24 ;10 CCA : 0010 0001 0 0 ;1 1 Since detL3 = 5, the graph has ve spanning trees.
4. Minimal Spanning Trees
Let G = (V E f ) be a connected weighted graph. A spanning tree H of G is called a minimum spanning tree if its total length f (H ) is minimum: X f (H ) = f (e): (5.282) e2H
Minimum spanning trees are searched for, e.g., if the edge weights represent costs, and one is interested in minimum costs. A method to nd a minimum spanning tree is the Kruskal algorithm: a) Choose an edge with the least weight.
356 5. Algebra and Discrete Mathematics b) Continue, as long as it is possible, choosing a further edge having least weight and not forming a
cycle with the edges already chosen, and add such an edge to the tree. In step b) the choice of the admissible edges can be made easier by the following labeling algorithm: Let the vertices of the graph be labeled pairwise di erently. At every step, an edge can be added only in the case that it connects vertices with di erent labels. After adding an edge, the label of the endpoint with the larger label is changed to the value of the smaller endpoint label.
5.8.4 Matchings 1. Matchings
A set M of edges of a graph G is called a matching in G, i M contains no loop and two di erent edges of M do not have common endpoints. A matching M of G is called a saturated matching, if there is no matching M in G such that M M . A matching M of G is called a maximum matching, if there is no matching M in G such that jM j > jM j. If M is a matching of G such that every vertex of G is an endpoint of an edge of M , then M is called a perfect matching. 1 In the graph in Fig. 5.53 M1 = ff2 3g f5 6gg is a saturated 6 2 4 matching and M2 = ff1 2g f3 4g f5 6gg is a maximum matching which is also perfect. 5 3 Remark: In graphs with an odd number of edges there is no perfect Figure 5.53 matching.
2. Theorem of Tutte
Let q(G ; S ) denote the number of the components of G ; S with an odd number of vertices. A graph G = (V E ) has a perfect matching i jV j is even and for every subset S of the vertex set q(G ; S ) jS j. Here G ; S denotes the graph obtained from G by deleting the vertices of S and the edges incident with these vertices. Perfect machings exist for example in complete graphs with an even number of vertices, in complete bipartite graphs Knn and in arbitrary regular bipartite graphs of degree r > 0.
3. Alternating Paths
Let G be a graph with a matching M . A path W in G is called an alternating path i in W every edge e with e 2 M (or e 62 M ) is followed by an edge e0 with e0 62 M (or e 2 M ). An open alternating path is called an increasing path i none of the endpoints of the path is incident with an edge of M .
4. Theorem of Berge
A matching M in a graph G is maximum i there is no increasing alternating path in G. If W is an increasing alternating path in G with corresponding set E (W ) of traversed edges, then M 0 = (M n E (W )) (E (W ) n M ) forms a matching in G with jM 0j = jM j + 1. In the graph of Fig. 5.53 (f1 2g f2 3g f3 4g) is an increasing alternating path with respect to matching M1. Matching M2 with jM2j = jM1 j + 1 is obtained as described above.
5. Determination of Maximum Matchings
Let G be a graph with a matching M . a) First form a saturated matching M with M M . b) Chose a vertex v in G, which is not incident with an edge of M , and determine an increasing alternating path in G starting at v. c) If such a path exists, then the method described above results in a matching M 0 with jM 0 j > jM j. If there is no such path, then delete vertex v and all edges incident with v in G, and repeat step b).
5.8 Algorithms of Graph Theory 357
There is an algorithm of Edmonds, which is an e ective method to search for maximum matchings, but it is rather complicated to describe (see 5.23]).
5.8.5 Planar Graphs
Here, the considerations are restricted to undirected graphs, since a directed graph is planar i the corresponding undirected graph is a planar one.
1. Planar Graph
A graph is called a plane graph i G can be drawn in the plane with its edges intersecting only in vertices of G. A graph isomorphic with a plane graph is called a planar graph. Fig. 5.54 shows a plane graph G1 . The graph G2 in Fig. 5.55 is isomorphic to G1, it is not a plane graph but a planar graph, since it is isomorphic with G1 . Figure 5.54
Figure 5.55
3. Subdivisions
2. Non-Planar Graphs
The complete graph K5 and the complete bipartite graph K33 are non-planar graphs (see 5.8.1, 5., p. 349).
A subdivision of a graph G is obtained if vertices of degree 2 are inserted into edges of G. Every graph is a subdivision of itself. Certain subdivisions of K5 and K33 are represented in Fig. 5.56 and Fig. 5.57.
4. Kuratowski's Theorem
A graph is non-planar i it contains a subgraph which is a subdivision either of the complete bipartite graph K33 or of the complete graph K5 .
Figure 5.56
Figure 5.57
5.8.6 Paths in Directed Graphs 1. Arc Sequences
A sequence F = (e1 e2 : : : es) of arcs in a directed graph is called a chain of length s, i F does not contain any arc twice and one of the endpoints of every arc ei for i = 2 3 : : : s ; 1 is an endpoint of the arc ei;1 and the other one an endpoint of ei+1 . A chain is called a directed chain i for i = 1 2 : : : s ; 1 the terminal point of the arc ei coincides with the initial point of ei+1 . Chains or directed chains traversing every vertex at most once are called elementary chains and elementary directed chains, respectively. A closed chain is called a cycle. A closed directed path, with every vertex being the endpoint of exactly two arcs, is called a circuit. Fig. 5.58 contains examples for various kinds of arc sequences.
2. Connected and Strongly Connected Graphs
A directed graph G is called connected i for any two vertices there is a chain connecting these vertices. The graph G is said to be strongly connected i to every two vertices v w there is is assigned a directed chain connecting these vertices.
3. Algorithm of Dantzig
Let G = (V E f ) be a weighted simple directed graph with f (e) > 0 for every arc e. The following algorithm yields all vertices of G, which are connected with a xed vertex v1 by a directed chain, together with their distances from v1: a) Vertex v1 gets the label t(v1 ) = 0. Let S1 = fv1 g.
358 5. Algebra and Discrete Mathematics
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Figure 5.58 b) Denote the set of the labeled vertices by Sm. c) If Um = feje = (vi vj ) 2 E vi 2 Sm vj 62 Sm g = , then nish the algorithm. d) Otherwise choose an arc e = (x y ) with minimum t(x ) + f (e ). Label e and y . We set t(y ) = t(x ) + f (e ) and also Sm+1 = Sm fy g, and repeat b) with m := m + 1. (If all arcs have weight 1, then the length of a shortest directed chain from a vertex v to a vertex w can be found using the adjacency matrix (see 5.8.2.1, 4., p. 351)). If a vertex v of G is not labeled, then there is no div rected path from v1 to v. If v has label t(v), then t(v) is the length of such a 4 2 directed chain. A shortest directed path from v1 to 3 v 2 v v v can be found in the tree given by the labeled arcs and vertices, the distance tree with respect to v1 . 5 2 2 v 1 In Fig. 5.59, the labeled arcs and vertices repv resent the distance tree with respect to v1 in the 3 2 3 2 v graph. The lengths of the shortest directed chains 7 4 5 are: from v1 to v3 : 2 from v1 to v6 : 7 1 4 v v v from v1 to v7 : 3 from v1 to v8 : 7 v 2 2 from v1 to v9 : 3 from v1 to v14 : 8 from v1 to v2 : 4 from v1 to v5 : 8 7 v 2 1 from v1 to v10 : 5 from v1 to v12 : 9 2 from v1 to v4 : 6 from v1 to v13 : 10 from v1 to v11 : 6: v v 3 Remark: There is also a modi ed algorithm to nd the shortest directed chains in the case that Figure 5.59 G = (V E f ) has arcs with negative weights. 8
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5.8.7 Transport Networks 1. Transport Network
A connected directed graph is called a transport network if it has two labeled vertices, called the source Q and sink S which have the following properties: a) There is an arc u1 from S to Q, where u1 is the only arc with initial point S and the only arc with terminal point Q. b) Every arc ui di erent from u1 is assigned a real number c(ui) 0. This number is called its capacity. The arc u1 has capacity 1. A function ', which assigns a real number to every arc, is called a ow on G, if the equality X X '(u v) = '(v w) (5.283a) (uv)2G
(vw)2G
5.8 Algorithms of Graph Theory 359
holds for every vertex v. The sum X '(Q v)
(5.283b)
(Qv)2G
is called the intensity of the ow. A ow ' is called compatible to the capacities, if for every arc ui of G 0 '(ui) c(ui) holds. For an example of a transport network see p. 359.
2. Maximum Flow Algorithm of Ford and Fulkerson
Using the maximum ow algorithm one can recognize whether a given ow ' is maximal. Let G be a transport network and ' a ow of intensity v1 compatible with the capacities. The algorithm given below contains the following steps for labeling the vertices, and after nishing this procedure one can recognize how much the intensity of the ow could be improved depending on the chosen labeling steps. a) Label the source Q and set "(Q) = 1. b) If there is an arc ui = (x y) with labeled x and unlabeled y and '(ui) < c(ui), then label y and (x y), and set "(y) = minf"(x) c(ui) ; '(ui)g, then repeat step b), otherwise follows step c). c) If there is an arc ui = (x y) with unlabeled x and labeled y '(ui) > 0 and ui 6= u1, then we label x and (x y), substitute "(x) = minf"(y) '(ui)g and return to continue step b) if it is possible. Otherwise we nish the algorithm. If the sink S of G is labeled, then the ow in G can be improved by an amount of "(S ). If the sink is not labeled, then the ow is maximal. Maximum ow: For the graph in Fig. 5.60 the weights are written next to the edges. A ow with intensity 13, compatible to these capacities, is represented in the weighted graph in Fig. 5.61. It is a maximum ow. 8
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Figure 5.60 Transport network: A product is produced by p rms F1 F2 : : : Fp. There are q users V1 V2 : : : Vq . During a certain period there will be si units produced by Fi and tj units required by Vj . cij units can be transported from Fi to Vj during the given period. Is it possible to satisfy all the requirements during this period? The corresponding graph is shown in Fig. 5.62.
4 7
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Figure 5.61 u1 F1
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Figure 5.62
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360 5. Algebra and Discrete Mathematics
5.9 Fuzzy Logic
5.9.1 Basic Notions of Fuzzy Logic 5.9.1.1 Interpretation of Fuzzy Sets
Real situations are very often uncertain or vague in a number of ways. The word \fuzzy" also means some uncertainty, and the name of fuzzy logic is based on this meaning. Basically we distinguish two types of fuzziness: vagueness and uncertainty. There are two concepts belonging here: The theory of fuzzy sets and the theory of fuzzy measure. In the following practice-oriented introduction we discuss the notions, methods, and concepts of fuzzy sets, which are the basic mathematical tools of multi-valued logic.
1. Notions of Classical and Fuzzy Sets
The classical notion of (crisp) set is two-valued, and the classical Boolean set algebra is isomorphic to two-valued propositional logic. Let X be a fundamental set named the universe. Then for every A X there exists a function fA : X ! f0 1g (5.284a) such that it says for every x 2 X whether this element x belongs to the set A or not: fA(x) = 1 , x 2 A and fA(x) = 0 , x 62 A: (5.284b) The concept of fuzzy sets is based on the idea of considering the membership of an element of the set as a statement, the truth value of which is characterized by a value from the interval 0 1]. For mathematical modeling of a fuzzy set A we need a function whose range is the interval 0 1] instead of f0,1g, i.e.: A : X ! 0 1]: (5.285) In other words: To every element x 2 X we assign a number A(x) from the interval 0 1], which represents the grade of membership of x in A. The mapping A is called the membership function. The value of the function A(x) at the point x is called the grade of membership. The fuzzy sets A B C , etc. over X are also called fuzzy subsets of X . The set of all fuzzy sets over X is denoted by F (X ).
2. Properties of Fuzzy Sets and Further Denitions
The properties below follow directly from the de nition: (E1) Crisp sets can be interpreted as fuzzy sets with grade of membership 0 and 1. (E2) The set of the arguments x whose grade of membership is greater than zero, i.e., A(x) > 0, is called the support of the fuzzy set A: supp(A) = fx 2 X j A(x) > 0g : (5.286) The set ker(A) = fx 2 X : A(x) = 1g is called the kernel or core of A. (E3) Two fuzzy sets A and B over the universe X are equal if the values of their membership functions are equal: A = B if A(x) = B (x) holds for every x 2 X: (5.287) (E4) Discrete representation or ordered pair representation: If the universe X is nite, i.e., X = fx1 x2 : : : xng it is reasonable to de ne the membership Table 5.7 Tabular representation of a function of the fuzzy set with a table of values. The tabular fuzzy set representation of the fuzzy set A is seen in Table 5.7. x1 x2 : : : xn We can also write A(x1 ) A(x2 ) : : : A(xn )
A := A(x1)=x1 + + A(xn )=xn =
n X i=1
A(xi )=xi:
(5.288)
In (5.288) the fraction bars and addition signs have only symbolic meaning. (E5) Ultra-fuzzy set: A fuzzy set, whose membership function itself is a fuzzy set, is called, after Zadeh, an ultra-fuzzy set.
5.9 Fuzzy Logic 361
3. Fuzzy Linguistics
If we assign linguistic values, e.g., \small", \medium" or \big", to a quantity then we call it a linguistic quantity or linguistic variable. Every linguistic value can be described by a fuzzy set, for example, by the graph of a membership function (5.9.1.2) with a given support (5.286). The number of fuzzy sets (in the case of \small", \medium", \big" they are three) depends on the problem. In 5.9.1.2 the linguistic variable is denoted by x. For example, x can have linguistic values for temperature, pressure, volume, frequency, velocity, brightness, age, wearing, etc., and also medical, electrical, chemical, ecological, etc. variables. By the membership function A(x) of a linguistic variable, the membership degree of a xed (crisp) value can be determined in the fuzzy set represented by A(x). Namely, the modeling of a \high" quantity, e.g., the temperature, as a linguistic variable given by a trapezoidal membership function (Fig. 5.63) means that the given temperature belongs to the fuzzy set \high temperature" with the degree of membership (also degree of compatibility or degree of truth).
5.9.1.2 Membership Functions on the Real Line
The membership functions can be modeled by functions with values between 0 and 1. They represent the di erent grade of membership for the points of the universe being in the given set.
1. Trapezoidal Membership Functions
Trapezoidal membership functions are widespread. Piecewise (continuously di erentiable) membership functions and their special cases, e.g., the triangle shape membership functions described in the following examples, are very often used. Connecting fuzzy quantities we get smoother output functions if the fuzzy quantities were represented by continuous or piecewise continuous membership functions. A: Trapezoidal function (Fig. 5.63) corresponding to (5.289). The graph of this function turns into a triangle function m (x) if a2 = a3 = a and a1 < 80 x a1 > a < a4 . Choosing di erent > 1 > values for a1 : : : a4 we get > x ; a1 a < x < a > symmetrical or asymmetri> 1 2 > cal trapezoidal functions, a < a2 ; a1 0.5 a2 x a3 (5.289) symmetrical triangle func A(x) = > 1 b > tion (a2 = a3 = a and ja ; > a4 ; x a < x < a > a1j = ja4 ; aj) or asymmet> 3 4 > aa a a a x a4 ; a3 rical triangle function (a2 = 0 > :0 a3 = a and ja ; a1 j 6= ja4 ; x a4: Figure 5.63 aj). A
1
2
3
4
B: Membership function bounded to the left and to the right (Fig. 5.64) corresponding to (5.290): 81 µ (x) x a1 > > > 1 > a ; x 2 > a1 < x < a 2 > > < a2 ; a1 a2 x a3 (5.290) A(x) = > 0 > > x ; a3 > > > a4 ; a3 a3 < x < a4 a a a a 0 > x :1 a4 x: Figure 5.64 Α
1
2
3
4
362 5. Algebra and Discrete Mathematics C: Generalized trapezoidal function (Fig. 5.65) corresponding to (5.291). µΑ(x) 1 b3=b4
b2 b5 0
a1 a2 a3
a4 a5 a6 x
Figure 5.65
80 > > > > b2 (x ; a1 ) > > > a2 ; a1 > > > (b3 ; b2 )(x ; a2) + b > 2 > a ;a > > < b3 = b43 = 12 A(x) = > > > (b4 ; b5 )(a4 ; x) + b > 5 > a5 ; a4 > > > b5 (a6 ; x) > > > a6 ; a5 > > 0 > :
x a1 a1 < x < a2 a2 x a3 a3 < x < a4 a4 x a5 a5 < x < a6 a6 x:
80 > > < f (x) = > e;1=p(x) > :
2. Bell-Shaped Membership Functions
A: We get a class of bell-shaped, di erentiable mem-
(5.291)
xa
bership functions with the function f (x) from (5.292) if we a < x < b (5.292) choose an appropriate p(x): If p(x) = k(x ; a)(b ; x) and, e.g., k = 10 or k = 1 or 0 x b: k = 0:1, then we get a family of symmetrical curves of , ! , ! di erent width with the membership function A(x) = f (x) f a +2 b , where 1 f a +2 b is the normalizing factor (Fig. 5.66). We get the exterior curve with the value k = 10 and the interior one with k = 0:1. We get asymmetrical membership functions in 0 1] for example with p(x) = x(1 ; x)(2 ; x) or with p(x) = x(1 ; x)(x + 1) (Fig. 5.67), using appropriate normalizing factors. The factor (2 ; x) in the rst polynomial results in the shifting of the maximum to the left and it yields an asymmetrical curve shape. Similarly, the factor (x + 1) in the second polynomial results in a shifting to the right and in an asymmetric form. mA(x)
1
0.5
0
mA(x)
1 0.5
a+b 2
a
b x
0
0
0.5
1 x
Figure 5.67 Figure 5.66 B: We can get examples for a more exible class of membership functions by the formula Zx f (t(u)) du (5.293) Ft (x) = Za b f (t(u)) du a
5.9 Fuzzy Logic 363
where f is de ned by (5.292) with p(x) = (x ; a)(b ; x) and t is a transformation on a b]. If t is a smooth transformation on a b], i.e., if t is di erentiable in nitely many times in the interval a b] then Ft is also smooth, since f is smooth. If we require t to be either increasing or decreasing and to be smooth, then the transformation t allows us to change the shape of the curve of the membership function. In practice, polynomials are especially suitable for transformations. The simplest polynomial is the identity t(x) = x on the interval a b] = 0 1]. The next simplest polynomial with the given properties is t(x) = ; 32 cx3 + cx2 + 1 ; 3c x with a constant c 2 ;6 3]. The choice c = ;6 results in the polynomial of maximum curvature, its equation is q(x) = 4x3 ; 6x2 + 3x . If we choose for q0 the identity function, i.e., q0(x) = x , then we can get recursively further polynomials q by the formula qi = q qi;1 for i 2 IN. Substituting the corresponding polynomial transformations q0 q1 : : : into (5.293) for t, we get a sequence of smooth functions Fq0 Fq1 and Fq2 (Fig. 5.68), which can be considered as membership functions A(x), where Fqn converges to a line. The trapezoidal membership function can be approximated by di erentiable functions using the function Fq2 , its reection and a horizontal line (Fig. 5.69). mA(x)
mA(x)
1
1
0.5
0.5
0
0
0.25
0.5
0.75
1 x
Figure 5.68
0
0
0.25
0.5
0.75
1 x
Figure 5.69
Summary: Imprecise and non-crisp information can be described by fuzzy sets and represented by membership functions (x).
5.9.1.3 Fuzzy Sets
1. Empty and Universal Fuzzy Sets
a) Empty fuzzy set: A set A over X is called empty if A(x) = 0 8 x 2 X holds. b) Universal fuzzy set: A set is called universal if A(x) = 1 8 x 2 X holds.
2. Fuzzy Subset If B (x) A(x) 8 x 2 X , then B is called a fuzzy subset of A (we write: B A). 3. Tolerance Interval and Spread of a Fuzzy Set on the Real Line
If A is a fuzzy set on the real line, then the interval a b] = fx 2 X j A(x) = 1g (a b const a < b) (5.294) is called the tolerance interval of the fuzzy set A, and the interval c d] = cl(suppA) (c d const,c < d) is called the spread of A, where cl denotes the closure of the set. (The tolerance interval is sometimes also called the peak of set A.) The tolerance interval and the kernel coincide only if the kernel contains more then one point. A: In Fig. 5.63 a2 a3 ] is the tolerance interval, and a1 a4] is the spread. B: If a2 = a3 = a (Fig. 5.63), then we get a triangle-shaped membership function . In that case the triangular fuzzy set has no tolerance, but its kernel is the set fag. If additionally a1 = a = a4 holds, too, then we have a crisp value it is called a singleton. A singleton A has no tolerance, but ker(A) = supp(A) = fag.
4. Conversion of Fuzzy Sets on a Continuous and Discrete Universe
Let the universe be continuous, and let a fuzzy set be given on it by its membership function. Discretizing the universe, every discrete point together with its membership value determines a fuzzy singleton.
364 5. Algebra and Discrete Mathematics Conversely, a fuzzy set given on a discrete universe can be converted into a fuzzy set on the continuous universe by interpolating the membership value between the discrete points of the universe.
5. Normal and Subnormal Fuzzy Sets
If A is a fuzzy subset of X , then its height is de ned by H (A) := max f A(x)jx 2 X g: (5.295) A is called a normal fuzzy set if H (A) = 1, otherwise it is subnormal. The notions and methods represented in this paragraph are limited to normal fuzzy sets, but it easy to extend them also to subnormal fuzzy sets.
6. Cut of a Fuzzy Set
The -cut A> or the strong -cut A of a fuzzy set A are the subsets of X de ned by A> = fx 2 X j A(x) > g A = fx 2 X j A(x) g 2 (0 1]: (5.296) 0 > 0 and A = cl (A ). The -cut and strong -cut are also called -level set and strong -level set, respectively.
1. Properties a) The -cuts of fuzzy sets are crisp sets. b) The support supp(A) is a special -cut: supp(A) = A>0. c) The crisp 1-cut A 1 = fx 2 X j A(x) = 1g is called the kernel of A. 2. Representation Theorem
To every fuzzy subset A of X we can assign uniquely the families of its -cuts (A> ) 201) and its strong -cuts A 2(01]. The -cuts and strong -cuts are monotone families of subsets from X , since:
< ) A> ( A> and A ( A : (5.297a) Conversely, if there exist the monotone families (U ) 201) or (V ) 2(01] of subsets from X , then there are uniquely de ned fuzzy sets U and V such that U > = U and V = V and moreover U (x) = supf 2 0 1))jx 2 U g V (x) = supf 2 (0 1]jx 2 V g: (5.297b) 7. Similarity of the Fuzzy Sets A and B 1. The fuzzy sets A B with membership functions A B : X ! 0 1] are called fuzzy similar if for every 2 (0 1] there exist numbers i with i 2 (0 1] (i = 1 2) such that: supp(1 A) supp( B ) supp(2 B ) supp( A) : (5.298) (x) if (x) > C ( C ) represents a fuzzy set with the membership function ( C ) = 0 C otherwise and ( C ) if (x) > C represents a fuzzy set with the membership function ( C ) = 0 otherwise. 2. Theorem: Two fuzzy sets A B with membership functions A B : X ! 0 1] are fuzzy-similar if they have the same kernel: supp( A)1 = supp( B )1 (5.299a) since the kernel is equal to the 1-cut, i.e. supp( A)1 = fx 2 X j A(x) = 1g: (5.299b) 3. A B with A B : X ! 0 1] are called strongly fuzzy-similar if they have the same support and the same kernel:
5.9 Fuzzy Logic 365
supp( A)1 = supp( B )1
(5.300a)
supp( A)0 = supp( B )0:
(5.300b)
5.9.2 Aggregation of Fuzzy Sets
Fuzzy sets can be aggregated by operators. There are several di erent suggestions of how to generalize the usual set operations, such as union, intersection, and complement of fuzzy sets.
5.9.2.1 Concepts for Aggregation of Fuzzy Sets
1. Fuzzy Set Union, Fuzzy Set Intersection
The grade of membership of an arbitrary element x 2 X in the sets A B and A \ B should depend only on the grades of membership A(x) and B (x) of the element in the two fuzzy sets A and B . The union and intersection of fuzzy sets is de ned with the help of two functions (5.301) s t: 0 1] 0 1] ! 0 1] and they are de ned in the following way: AB (x) := s ( A(x) B (x)) (5.302) A\B (x) := t ( A(x) B (x)) : (5.303) The grades of membership A(x) and B (x) are mapped in a new grade of membership. The functions t and s are called the t-norm and t-conorm this last one is also called the s-norm. Interpretation: The functions AB and A\B represent the truth values of membership, which is resulted by the aggregation of the truth values of memberships A(x) and B (x).
2. Denition of the t-Norm:
The t-norm is a binary operation t in 0 1]: t: 0 1] 0 1] ! 0 1]: (5.304) It is symmetric, associative, monotone increasing, it has 0 as the zero element and 1 as the neutral element. For x y z v w 2 0 1] the following properties are valid: (E1) Commutativity: t(x y) = t(y x): (5.305a) (E2) Associativity: t(x t(y z)) = t(t(x y) z): (5.305b)
(E3) Special Operations with Neutral and Zero Elements:
t(x 1) = x and because of (E1): t(1 x) = x t(x 0) = t(0 x) = 0: (E4) Monotony: If x v and y w then t(x y) t(v w) is valid:
3. Denition of the s-Norm:
The s-norm is a binary function in 0 1]: s: 0 1] 0 1] ! 0 1]: It has the following properties: (E1) Commutativity: s(x y) = s(y x): (E2) Associativity: s(x s(y z)) = s(s(x y) z):
(E3) Special Operations with Zero and Neutral Elements: s(x 0) = s(0 x) = x s(x 1) = s(1 x) = 1:
(5.305c) (5.305d) (5.306) (5.307a) (5.307b)
(5.307c) (5.307d) With the help of these properties a class T of t-norms and a class S of s-norms can be introduced. Detailed investigations proved that the following relations hold: minfx yg t(x y) 8 t 2 T 8 x y 2 0 1] and (5.307e) (5.307f) maxfx yg s(x y) 8 s 2 S 8 x y 2 0 1]:
(E4) Monotony: If x v and y w then s(x y) s(v w) is valid:
366 5. Algebra and Discrete Mathematics
5.9.2.2 Practical Aggregator Operations of Fuzzy Sets
1. Intersection of Two Fuzzy Sets
The intersection A \ B of two fuzzy sets A and B is de ned by the minimum operation min(: :) on their membership functions A(x) and B (x). Based on the previous requirements, we get: C := A \ B and C (x) := min ( A(x) B (x)) 8 x 2 X where: (5.308a) a if a b min(a b) := b if a > b: (5.308b) The intersection operation corresponds to the AND operation of two membership functions (Fig.5.70). The membership function C (x) is de ned as the minimum value of A(x) and B (x).
2. Union of Two Fuzzy Sets
The union A B of two fuzzy sets is de ned by the maximum operation max(: :) on their membership functions A(x) and B (x). We get: C := A B and C (x) := max ( A(x) B (x)) 8 x 2 X where: (5.309a) a if a b max(a b) := b if a < b: (5.309b) The union corresponds to the logical OR operation. Fig.5.71 illustrates C (x) as the maximum value of the membership functions A(x) and B (x). The t-norm t(x y) = minfx yg and the s-norm s(x y) = maxfx yg de ne the intersection and the union of two fuzzy sets, respectively (see Fig.5.72 and Fig.5.73). µ(x)
µ(x) µΒ(x)
1
1
µΑ(x)
µΑ(x) µΒ(x)
µC(x) 0
0
x
Figure 5.70
s(x,y)
y
Figure 5.72
3. Further Aggregations
x
Figure 5.71
t(x,y)
x
µC(x)
y
x
Figure 5.73
Further aggregations are the bounded, the algebraic, and the drastic sum and also the bounded dierence, the algebraic and the drastic product (see Table 5.8). The algebraic sum, e.g., is de ned by C := A + B and C (x) := A(x) + B (x) ; A(x) B (x) for every x 2 X: (5.310a)
5.9 Fuzzy Logic 367
Author
Zadeh Lukasiewicz
Hamacher (p 0)
t-norm
Table 5.8 t- and s-norms, p 2 IR
intersection: t(x y) = minfx yg bounded di erence tb(x y) = maxf0 x + y ; 1g algebraic product ta (x y) = xy drastic product 8 < minfx yg whether x = 1 tdp(x y) = : or y = 1 0 otherwise
th(x y) = p + (1 ; p)(xyx + y ; xy) Einstein te(x y) = 1 + (1 ;xyx)(1 ; y) Frank tf (x "y) =
x py ; 1) (p > 0 p 6= 1) logp 1 + (p ;p1)( ;1 Yager (p > 0) Schweizer (p > 0) Dombi (p > 0) Weber (p ;1) Dubois (0 p 1)
tya (x y) = 1; min 1 ((1 ; x)p + (1 ; y)p)1=p ts(x y) = max(0 x;p + y;p ; 1);1=p t8do (x y) = 9 < " 1 ; x p 1 ; y !p 1=p=;1 + :1 + " x y tw (x y) = max(0 (1 + p) (x + y ; 1) ; pxy) tdu(x y) = max(xy x y p)
s-norm
union: s(x y) = maxfx yg bounded sum sb(x y) = minf1 x + yg algebraic sum sa(x y) = x + y ; xy drastic sum8 < maxfx yg whether x = 0 sds(x y) = : or y = 0 1 otherwise
; (1 ; p)xy sh(x y) = x + y1;;xy (1 ; p)xy y se(x y) = 1x++xy sf (x"y) = 1;
1 ;x p1;y ; 1) logp 1 + (p ;p1)( ;1 sya (x y) = min 1 (xp + yp)1=p ss(x y) = 1; max (0 (1 ; x);p + (1 ; y);p ; 1);1=p s8do(x y) = 1; !p 1=p9 < " x p =;1 y 1 + + : " 1;x 1;y sw (x y) = min(1 x + y + pxy) sdu(x y) = x + y ; xy ; min(x y (1 ; p)) max((1 ; x) (1 ; y) p)
Remark to Table 5.8: For the values of the t- and s-norms listed in the table, the following ordering
is valid: tdp tb te ta th t s sh sa se sb sds : (5.310b) Similarly to the union (5.309a,b), this sum also belongs to the class of s-norms. They are included in the right-hand column of Table 5.8. In Table 5.9 is given a comparision of operations in Boolean logic and fuzzy logic. Analogously to the notion of the extended sum as a union operation, the intersection can also be extended for example by the bounded, the algebraic, and the drastic product. So, e.g., the algebraic
368 5. Algebra and Discrete Mathematics product is de ned in the following way: C := A B and C (x) := A(x) B (x) for every x 2 X: (5.310c) It also belongs to the class of t-norms, similarly to the intersection (5.308a,b), and it can be found in the middle column of Table 5.8.
5.9.2.3 Compensatory Operators
Sometimes we need operators lying between the t- and the s-norms they are called compensatory operators. Examples for compensatory operators are the lambda and the gamma operator.
1. Lambda Operator
A B (x) = A(x) B (x)] + (1 ; ) A(x) + B (x) ; A(x) B (x)] with 2 0 1]: (5.311) Case = 0 : Equation (5.311) results in a form known as the algebraic sum (Table 5.8, s-norms) it belongs to the OR operators. Case = 1: Equation (5.311) results in the form known as the algebraic product (Table 5.8, t-norms) it belongs to the AND operators.
2. Gamma Operator
AB (x) = A(x) B (x)]1; 1 ; (1 ; A(x)) (1 ; B (x))] with 2 0 1]: Case = 1: Equation (5.312) results in the representation of the algebraic sum. Case = 0: Equation (5.312) results in the representation of the algebraic product. The application of the gamma operator on fuzzy sets of any numbers is given by "Y
1; " Y
n n (x) = i(x) 1 ; (1 ; i(x)) i=1
and with weights i:
(x) =
"Y n
i=1
i(x)i
i=1
1 ; "
1;
n Y i=1
(1 ; i(x))i
5.9.2.4 Extension Principle
with x 2 X
n X i=1
(5.312)
(5.313)
i = 1 2 0 1]: (5.314)
In the previous paragraph, we discussed the possibilities of generalizing the basic set operations for fuzzy sets. Now, we want to extend the notion of mapping on fuzzy domains. The basis of the concept is the acceptance grade of vague statements. The classical mapping : X n ! Y assigns a crisp function value (x1 : : : xn) 2 Y to the point (x1 : : : xn) 2 X n. This mapping can be extended for fuzzy variables as follows: The fuzzy mapping is ^ : F (X )n ! F (Y ), which assigns a fuzzy function value ^( 1 : : : n) to the fuzzy vector variables (x1 : : : xn) given by the membership functions ( 1 : : : n) 2 F (X )n.
5.9.2.5 Fuzzy Complement
A function c : 0 1] ! 0 1] is called a complement function if the following properties are ful lled for 8 x y 2 0 1]: (EK1) Boundary Conditions: c(0) = 1 and c(1) = 0: (5.315a) (EK2) Monotony: x < y ) c(x) c(y): (5.315b) (EK3) Involutivity: c(c(x)) = x: (5.315c) (5.315d) (EK4) Continuity: c(x) should be continuous for every x 2 0 1]: A: The most often used complement function is (continuous and involutive): c(x) := 1 ; x: (5.316) B: Other continuous and involutive complements are the Sugeno complement c (x) := (1 ; x)(1 + x);1 with 2 (;1 1) and the Yager complement cp(x) := (1 ; xp)1=p with p 2 (0 1).
5.9 Fuzzy Logic 369
Table 5.9 Comparison of operations in Boolean logic and in fuzzy logic
Operator Boolean logic AND C =A^B OR C =A_B NOT C = :A
Fuzzy logic ( A B 2 0 1])
A\B = min( A B ) AB = max( A B ) CA = 1 ; A ( CA as complement of A)
5.9.3 Fuzzy-Valued Relations 5.9.3.1 Fuzzy Relations
1. Modeling Fuzzy-Valued Relations
Uncertain or fuzzy-valued relations, as e.g. \approximately equal", \practically larger than", or \practically smaller than", etc., have 2an important role in practical applications. A relation between numbers is interpreted as a subsets of IR . So, the equality \=" is de ned as the set n o A = (x y) 2 IR2 jx = y (5.317) 2 i.e., by a straight line y = x in IR . Modeling the relation \approximately equal" denoted by R1, we can use a fuzzy subset on IR2, the kernel of which is A, and we require that the membership function should decrease and tend to zero getting far from the line A. A linear decreasing membership function can be modeled by R1 (x y) = maxf0 1 ; ajx ; yjg with a 2 IR a > 0: (5.318) For modeling the relation R2 \practically larger than", it is useful to start with the crisp relation \ ". The corresponding set of values is given by n o (x y) 2 IR2jx y : (5.319) It describes the crisp domain above the line x = y. The modi er \practically" means that a thin zone under the half-space in (5.319) is still acceptable with some grade. So, the model of R2 is f0 1 ; ajx ; yjg for y < x R2 (x y) = max (5.320) 1 for y x with a 2 IR a > 0: If the value of one of the variables is xed, e.g., y = y0, then R2 can be interpreted as a region with uncertain boundaries for the other variable. Handling the uncertain boundaries by fuzzy relations has practical importance in fuzzy optimization, qualitative data analysis and pattern classi cation. The foregoing discussion shows that the concept of fuzzy relations, i.e., fuzzy relations between several objects, can be described by fuzzy sets. In the following, we discuss the basic properties of binary relations over universe which consists of ordered pairs.
2. Cartesian Product
Let X and Y be two universes. Their \cross product" X Y , or Cartesian product, is a universe G: G = X Y = f(x y)jx 2 X ^ y 2 Y g: (5.321) Then, a fuzzy set on G is a fuzzy relation, analogously to classical set theory, if it consists of the valued pair of universes X and Y . A fuzzy relation R in G is a fuzzy subset R 2 F (G), where F (G) denotes the set of all the fuzzy sets over X Y . R can be given by a membership function R(x y) which assigns a membership degree R (x y) from 0 1] to every element of (x y) 2 G.
3. Properties of Fuzzy-Valued Relations
(E1) Since the fuzzy relations are special fuzzy sets, all propositions stated for fuzzy sets will also be valid for fuzzy relations. (E2) All aggregations de ned for fuzzy sets can be de ned also for fuzzy relations they yield a fuzzy
370 5. Algebra and Discrete Mathematics relation again. (E3) The notion of -cut de ned above can be transmitted without diculties to fuzzy relations. (E4) The 0-cut (the closure of the support) of a fuzzy relation R 2 F (G) is a usual relation on G: (E5) We denote the membership value by R(x y), i.e., the degree by which the relation R between the pair (x y) holds. The value R(x y) = 1 means that R holds perfectly for the pair (x y), and the value R (x y) = 0 means that R does not at all hold for the pair (x y). (E6) Let R 2 F (G) be a fuzzy relation. Then the fuzzy relation S := R;1, the inverse of R, is de ned by S (x y) = R(y x) for every (x y) 2 G: (5.322) The inverse relation R2;1 means \practically smaller than" (see 5.9.3.1, 1., p. 369) the union R1 R2;1 can be determined as \practically smaller or approximately equal".
4. n-Fold Cartesian Product
Let n be the number of universal sets. Their cross product is an n-fold Cartesian product. A fuzzy set on an n-fold Cartesian product represents an n-fold fuzzy relation. Consequences: The fuzzy sets, considered until now, are unary fuzzy relations, i.e., in the sense of the analysis they are curves above a universal set. A binary fuzzy relation can be considered as a surface over the universal set G. A binary fuzzy relation on a nite discrete support can be represented by a fuzzy relation matrix. Colour-ripe grade relation: The well-known correspondence between the colour x and the ripe grade y of a friut is modeled in the form of a binary relation matrix with elements f0 1g. The possible colours are X = fgreen, yellow, redg and the ripe grades are Y = funripe, half-ripe, ripeg. The relation matrix (5.323) belongs to the table: 01 0 01 unripe half-ripe ripe green 1 0 0 R = @0 1 0A: (5.323) yellow 0 1 0 001 red 0 0 1 Interpretation of this relation matrix: IF a fruit is green, THEN it is unripe. IF a fruit is yellow, THEN it is half-ripe. IF a fruit is red, THEN it is ripe. Green is uniquely assigned to unripe, yellow to half-ripe and red to ripe. If we want to formalize that a green fruit can be considered half-ripe in a certain percentage, then we can get the following table with discrete membership values: R (green, unripe) = 1:0, R (green, half-ripe) = 0:5, The relation matrix with R 2 0 1] R (green, ripe) = 0:0, R (yellow, unripe) = 0:25, is: 0 1:0 0:5 0:0 1 R (yellow, half-ripe) = 1:0, R (yellow, ripe) = 0:25, @ 0:25 1:0 0:25 A : R = (5.324) R (red, unripe) = 0:0, R (red, half-ripe) = 0:5, 0:0 0:5 1:0 R (red, ripe) = 1:0.
5. Rules of Calculations
The AND-type aggregation of fuzzy sets, e.g. 1 : X ! 0 1] and 2 : Y ! 0 1] given on di erent universes is formulated by the min operation as follows: R(x y) = min( 1(x) 2(y)) or ( 1 2)(x y) = min( 1(x) 2 (y)) with (5.325a) 1 2 : G ! 0 1] where G = X Y: (5.325b) The result of this aggregation is a fuzzy relation R on the cross product set (Cartesian product universe of fuzzy sets) G with (x y) 2 G. If X and Y are discrete nite sets and so 1(x) 2(y) can be represented as vectors, then we get: 1 2 = 1 T2 and R;1 (x y) := R(y x) 8 (x y) 2 G: (5.326) The aggregation operator does not denote here the usual matrix product. The product is calculated here by the componentwise min operation and addition by the componentwise max operation.
5.9 Fuzzy Logic 371
The validity grade of an inverse relation R;1 for the pair (x y) is always equal to the validity grade of R for the pair (y x). If the fuzzy relations are given on the same Cartesian product universe, then the rules of their aggregations can be given as follows: Let R1 R2 : X Y ! 0 1] be binary fuzzy relations. The evaluation rule of their AND-type aggregation uses the min operator, namely for 8(x y) 2 G: R1\R2 (x y) = min( R1 (x y) R2 (x y)): (5.327) A corresponding evaluation rule for the OR-type aggregation is given by the max operation: (5.328) R1R2 (x y) = max( R1 (x y) R2 (x y)):
5.9.3.2 Fuzzy Product Relation R 1. Composition or Product Relation
S
Suppose R 2 F (X Y ) and S 2 F (Y Z ) are two relations, and it is additionally assumed that R S 2 F (G) with G X Z . Then the composition or the fuzzy product relation R S is: RS (x z) := supy2Y fmin( R(x y) S (y z))g 8 (x z) 2 X Z: (5.329) If a matrix representation is used for a nite universal set analogously to (5.324), then the composition R S is motivated as follows: Let X = fx1 : : : xng Y = fy1 : : : ymg Z = fz1 : : : zl g and R 2 F (X Y ) S 2 F (Y Z ) and let the matrix representations R S be in the form R = (rij ) and S = (sjk ) for i = 1 : : : n j = 1 : : : m k = 1 : : : l, where rij = R(xi yj ) and sjk = S (yj zk ): (5.330) If the composition T = R S has the matrix representation tik , then tik = sup minfrij sjk g: (5.331) j The nal result is not a usual matrix product, since instead of the summation operation there is the least upper bound (supremum) operation and instead of the product we have the minimum operator. With the representations for rij and sjk and with (5.329), the inverse relation R;1 (rij )T, can also be computed taking into consideration that R;1 can be represented by the transpose matrix, i.e., R;1 = (rij )T. Interpretation: Let R be a relation from X to Y and S be a relation from Y to Z . Then the following compositions are possible: a) If the composition R S of R and S is de ned as a max-min product, then the resulted fuzzy composition is called a max-min composition. The symbol sup stands for supremum and denotes the largest value, if no maximum exists. b) If the product composition is de ned as with the usual matrix multiplication, then we get the maxprod composition. c) For max-average composition, \multiplication" is replaced by the average.
2. Rules of Composition
The following rules are valid for the composition of fuzzy relations R S T 2 F (G):
(E1) Associative Law:
(R S ) T = R (S T ):
(5.332)
(E2) Distributive Law for Composition with Respect to the Union: R (S T ) = (R S ) (R T ): (5.333) (E3) Distributive Law in a Weaker Form for Composition with Respect to Intersection: R (S \ T ) (R S ) \ (R T ): (5.334) (E4) Inverse Operations: (R S );1 = S ;1 R;1 (R S );1 = R;1 S ;1 and (R \ S );1 = R;1 \ S ;1: (5.335)
372 5. Algebra and Discrete Mathematics (E5) Complement and Inverse:
;1 ;1 ;1 C R = R RC = R;1 :
(5.336)
(E6) Monotonic Properties: R S ) R T S T und T R T S: (5.337) A: Equation (5.329) for the product relation R S is de ned by the min operation as we have done for intersection formation. In general, any t{norm can be used instead of the min operation. B: The -cuts with respect to the union, intersection, and complement are: (A B )> = A> B > , (A \ B )> = A> \ B > , (AC )> = A 1; = fx 2 X j A(x) 1 ; g. Corresponding statements are valid for strong -cuts.
3. Fuzzy Logical Inferences
It is possible to make a fuzzy inference, e.g., with the IF THEN rule by the composition rule 2 = 1 R. The detailed formulation for the conclusion 2 is given by 2(y) = maxx2X min( 1(x) R (x y)) (5.338) with y 2 Y 1 : X ! 0 1] 2 : Y ! 0 1] R : G ! 0 1] und G = X Y .
5.9.4 Fuzzy Inference (Approximate Reasoning)
Fuzzy inference is an application of fuzzy relations with the goal of getting fuzzy logical conclusions with respect to vague information (see 5.9.6.3, p. 375). Vague information means here fuzzy information but not uncertain information. Fuzzy inference, also called implication, contains one or more rules, a fact and a consequence. Fuzzy inference, which is called by Zadeh, approximate reasoning, cannot be described by classical logic.
1. Fuzzy Implication, IF THEN Rule
The fuzzy implication contains one IF THEN rule in the simplest case. The IF part is called the premise and it represents the condition. The THEN part is the conclusion. Evaluation happens by 2 = 1 R and (5.338). Interpretation: 2 is the fuzzy inference image of 1 under the fuzzy relation R, i.e., a calculation prescription for the IF THEN rule or for a group of rules.
2. Generalized Fuzzy Inference Scheme
The rule IF A1 AND A2 AND A3 : : : AND An THEN B with Ai : i : Xi ! 0 1] (i = 1 2 : : : n) and the membership function of the conclusion B : : Y ! 0 1] is described by an (n + 1)-valued relation R: X1 X2 Xn Y ! 0 1]: (5.339a) For the actual input with crisp values x01 x02 : : : x0n the rule (5.339a) de nes the actual fuzzy output by B0 (y) = R(x01 x02 : : : x0n y) = min( 1(x01 ) 2(x02 ) : : : n(x0n ) B (y)) where y 2 Y: (5.339b) Remark: The quantity min( 1(x01 ) 2 (x02) : : : n(x0n)) is called the degree of fulllment, and the quantities f 1(x01 ) 2(x02) : : : n(x0n)g represent the fuzzy-valued input quantities. Forming the fuzzy relations for a connection between the quantities \medium" pressure and \high" temperature (Fig. 5.74): ~1(p T ) = 1(p) 8 T 2 X2 with 1 : X1 ! 0 1] is a cylindrical extension (Fig. 5.74c) of the fuzzy set medium pressure (Fig. 5.74a). Analogously, ~2 (p T ) = 2(T ) 8 p 2 X1 with 2 : X2 ! 0 1] is a cylindrical extension (Fig. 5.74d) of the fuzzy set high temperature (Fig. 5.74b), where ~1 ~2 : G = X1 X2 ! 0 1]. Fig. 5.75a shows the graphic result of the formation of fuzzy relations: In Fig. 5.75b the result of the composition medium pressure AND high temperature with the min operator R(p T ) = min( 1(p), 2(T )) is represented, and (Fig. 5.75b) shows the result of the composition OR with the max operator R(p T ) = max( 1(p) 2(T )).
5.9 Fuzzy Logic 373 m1(p) 1
m2(T) 1
medium
5 p pressure
0 a)
mR(p,.) 1
c)
0
10
b)
50 100 T temperature
0
mR(.,T) 1
T
5
high
10 p
d)
0
T 100
10 p
5
Figure 5.74 mR(p,T) 1
a)
0
T
5
T
mR(p,T) 1
10 p
b)
0
10 p
5
Figure 5.75
5.9.5 Defuzzi cation Methods
Often we have to get a crisp set from a fuzzy-valued set. This process is called defuzzication. There are di erent methods to do this. 1. Maximum-Criterion Method An arbitrary value 2 Y is selected from the domain where the fuzzy set Output x1 :::xn has the maximal membership degree. 2. Mean-of-Maximum Method (MOM) The output value is the mean value of the maximal membership values: Z sup Output x1 :::xn := Output y dy fy 2 Y j x1:::xn (y) x1:::xn (y ) 8 y 2 Y g (5.340) :(5.341) MOM = Zy2sup( x1:::xn ) dy Output i.e., the set Y is an interval, which should not be empty and y2sup( x1 :::xn ) it is characterized by (5.340), from which we get (5.341).
3. Center of Gravity Method (COG)
In the center of gravity method, we take the abscissa value of the center of gravity of a surface with a ctitious homogeneous density of value 1.
Z ysup
(y)y dy COG = Zyinfysup : (y) dy yinf
(5.342)
374 5. Algebra and Discrete Mathematics 4. Parametrized Center of Gravity Method (PCOG) The parametrized method works with the exponent 2 IR. From (5.343) it follows for = 1 PCOG = COG and for ! 0, PCOG = MOM. 5. Generalized Center of Gravity Method (GCOG) The exponent is considered as a function of y in the PCOG method. Then (5.344) follows obviously. The GCOG method is a generalization of the PCOG method, where (y) can be changed by the special weight depending itself on y.
6. Center of Area (COA) Method
We calculate a line parallel to the ordinate axis so that the area under the membership function is the same on the left- and on the right-hand side of it.
Z ysup
(y) y dy y inf Z PCOG = ysup : (y) dy y
(5.343)
inf
Z ysup
(y)(y) y dy y inf Z GCOG = ysup (y) : (5.344) (y) dy y inf
Z yinf
(y) dy =
Z PB
Z ysup
(y) dy: (5.345)
Z ysup
(y) dy = (y) dy: (5.346) 7. Parametrized Center of Area (PCOA) Method yinf PF 8. Method of the Largest Area (LA) The signi cant subset is selected and one of the methods de ned above, e.g., the method of center of gravity (COG) or center of area (COA) is used for this subset.
5.9.6 Knowledge-Based Fuzzy Systems
There are several application possibilities of multi-valued fuzzy logic, based on the unit interval, both in technical and non-technical life. The general concept is that we fuzzify quantities and distinguish marks, we aggregate them in an appropriate knowledge base with operators, and if necessary, we defuzzify the possibly fuzzy result set.
5.9.6.1 Method of Mamdani
The following steps are applied for a fuzzy control process: 1. Rule Base Suppose, for example, for the i-th rule Ri : If e is E i AND e_ is %E i THEN u is U i: (5.347) Here e characterizes the error, e_ the change of the error and u the change of the (not fuzzy valued) output value. Every quantity is de ned on its domain E %E and U . Let the entire domain be E %E U . The error and the change of the error will be fuzzi ed on this domain, i.e., they will be represented by fuzzy sets, where linguistic description is used. 2. Fuzzifying Algorithm In general, the error e and its change e_ are not fuzzy-valued, so they must be fuzzi ed by a linguistic description. The fuzzy values will be compared with the premisses of the IF THEN rule from the rule base. From this it follows, which rules are active and how large are their weights. 3. Aggregation Module The active rules with their di erent weights will be combined with an algebraic operation and applied to the defuzzi cation. 4. Decision Module In the defuzzi cation process a crisp value should be given for the control quantity. With a defuzzi cation operation, a non-fuzzy-valued quantity is determined from the set of possible values, i.e., a crisp quantity. This quantity expresses how the control parameters of the system should be set up to keep the deviation minimal. Fuzzy control means that the steps from 1. to 4. are repeated until the goal, the smallest deviation e and its change e_, is reached.
5.9 Fuzzy Logic 375
5.9.6.2 Method of Sugeno
The Sugeno method is also used for planning of a fuzzy control process. It di ers from the Mamdani concept in the rule base and in the defuzzi cation method. It has the following steps: 1. Rule Base: The rule base consists of rules of the following form: Ri : IF x1 is Ai1 AND : : : AND xk is Aik THEN ui = pi0 + pi1x1 + pi2 x2 + + pik xk : (5.348) The notations mean: Aj : fuzzy sets, which can be determined by membership functions xj : crisp input values as, e.g., the error e and the change of the error e_, which tell us something about the dynamics of the system pij : weights of xj (j = 1 2 : : : k) ui: the output value belonging to the i-th rule (i = 1 2 : : : n). 2. Fuzzifying Algorithm: A i 2 0 1] is calculated for every rule Ri. 3. Decision Module: A non-fuzzy-valued quantity is calculated from the weighted mean of ui, where the weights are i from the fuzzy cation:
u=
n X i=1
iui
n !;1 X i :
(5.349)
i=1
Here u is a crisp value. The defuzzi cation of the Mamdani method does not work here. The problem is to get the weight parameters pij available. These parameters can be determined by a mechanical learning method, e.g., by an arti cial neuronetwork (ANN).
5.9.6.3 Cognitive Systems
To clarify the method, the following known example will be investigated with the Mamdami method: The regulation of a pendulum that is perpendicular to its moving base (Fig. 5.76). The aim of the control process is to keep a pendulum in balance so that the pendulum rod should stand vertical, i.e., the angular displacement from the vertical direction and the angular velocity should be zero. It must be done by a force F acting at the lower end of the pendulum. This force is the control quantity. The model is based on the activity of a human \control expert" (cognitive problem). The expert formulates its knowledge in linguistic rules. Linguistic rules consist, in general, of a premisse, i.e., a speci cation of the measured values, and a conclusion which gives the appropriate control value. For every set of values X1 X2 : : : Xn for the measured values and Y for the control quantity the appropriate linguistic terms are de ned as \approximately zero", \small positive", etc. Here \approximately zero" with respect to the measured value 1 can have a di erent meaning as for the measured value 2.
Inverse Pendulum on a Moving Base (Fig. 5.76) 1. Modeling For the set X1 (values of angle) and analogously for the input
quantity X2 (values of the angular velocity) the seven linguistic terms, negative large (nl), negative medium (nm), negative small (ns), zero (z), positive small (ps), positive medium (pm) and positive large (pl) are chosen. For the mathematical modeling, a fuzzy set must be assigned by graphs to every one of these linguistic terms (Fig. 5.75), as was shown for fuzzy inference (see 5.9.4, p. 372).
2. Determination of the Domain of Values Values of angles: !(;90 < ! < 90): X1 := ;90 90]. Values of angular velocity: !_ (;45 s;1 !_ 45 s;1): X2 := ;45 s;1 45 s;1 ]. Values of force F : (;10 N F 10 N): Y := ;10N 10 N].
q F
Figure 5.76
376 5. Algebra and Discrete Mathematics The partitioning of the input quantities X1 and X2 and the output quantity Y is represented graphically in Fig. 5.77. Usually, the initial values are actual measured values, e.g., ! = 36 !_ = ;2:25 s;1 . .
m(q) 1
nl nm
a)
45
m(q) 1
pm pl
q
nl
b)
m(F) 1
.
22.5
pl q
nl
c)
5
pl
F
Figure 5.77 3. Choice of Rules Considering the following table, there are 49 possible rules (7 7) but there are only 19 important in practice, and we discuss the following two, R1 and R2, from them. R1: If ! is positive small (ps) and !_ zero (z), then F is positive small n (ps). For theodegree of fulllment (also called the weight of the rules) of the premise with = min (1) (!) (1)(!_ ) = minf0:4 0:8g = 0:4 we get the output set (5.350) by an -cut, hence the output fuzzy set is positive small (ps) in the height = 0:4 (Fig. 5.78c). Table: Rule base with 19 practically meaning82 ful rules > 0y !_ n! nl nm ns z ps pm pl > 5y > > nl ps pl 1y4 > < 0:4 nm pm Output (R1) ( y ) = (5.350) ns nm ns ps 36;2:25 > > 2 ; 2y 4 < y 5 z nl nm ns z ps pm pl > 5 > ps ns ps pm > > otherwise: pm nm :0 pl nl ns 82 > y ; 1 2:5 y < 4 > _ R2: If ! is positive medium (pm) and ! is > 5 > > zero (z), then F is positive medium (pm). 4y6 > < 0:6 For the performance score Output (R2) n (2) of the(2) premise o ( y ) = (5.351) 36 ; 2 : 25 > we get = min (!) !_ = > 3 ; 25 y 6 < y 7:5 > > minf0:6 0:8g = 0:6, the output set (5.351) > > otherwise: analogously to rule R1 (Fig. 5.78f). :0
4. Decision Logic The evaluation of rule R1 with the min operation results in the fuzzy set in Figs. 5.78a{c. The corresponding evaluation for the rule R2 is shown in Figs. 5.78d{f. The control quantity is calculated nally by a defuzzi cation method from the fuzzy proposition set (Fig. 5.78g). We obtain the fuzzy set (Fig. 5.78g) by using the max operation if we take into account the fuzzy sets (Fig. 5.78c) and (Fig. 5.78f). a) Evaluation of the fuzzy set obtained in this way, which is aggregated by operators (see max-min composition 5.9.3.2, 1., p. 371). The decision logic yields: n n (1) oo (n) Output (5.352) x1 :::xn : Y ! 0 1] y ! maxr2f1:::kg min ilr (x1 ) : : : ilr (xn ) ir (y ) :
5.9 Fuzzy Logic 377
b) For the function graph of the fuzzy set we
82 > > for > 5y > > > 0:4 for > > > > < 2 y ; 1 for 5 Output ( y ) = 36;2:25 > > 0:6 for > > > > > 3 ; 52 y for > > > :0 for
0y 0 holds, we can calculate the derivative y0 starting with the function ln y(x), whose derivative (considering the chain rule) is: d(ln y(x)) = 1 y0 : (6.11) dx y(x) From this rule y) (6.12) y0 = y(x) d(ln dx follows. Remark 1: With the help of logarithmic di erentiation it is possible to simplify some di erentiation
6.1 Dierentiation of Functions of One Variable 383
problems, and there are functions such that this is the only way to calculate the derivative, for instance, when the function has the form y = u(x)v(x) with u(x) > 0: (6.13) The logarithmic di erentiation of this equality follows from the formula (6.12) ! uv ) = y d(v ln u) = uv v0 ln u + vu0 : y0 = y d (ln (6.14) dx dx u 0 y = (2x + 1)3x ln y = 3x ln(2x + 1) yy = 3 ln(2x + 1) + 23xx+ 21 y0 = 3 (2x + 1)3x ln(2x + 1) + 2x2+x 1 . Remark 2: Logarithmic di erentiation is often used when we have to di erentiate a product of several functions. p A: y = x3 e4x sin x ln y = 21 (3 ln x + 4x + ln sin x) y0 = 1 3 + 4 + cos x y0 = 1 px3 e4x sin x 3 + 4 + cot x. y 2 x sin x 2 x 0 y 1 0 B: y = u v ln y = ln u + ln v y = u u + v1 v0. From this identity it follows that y0 = (u v)0 = v u0 + u v0, so we get the formula for the derivative of a product (6.7a) (under the assumption u v > 0). 0 0 C: y = uv ln y = ln u ; ln v yy = u1 u0 ; v1 v0. From this identity it follows that y0 = uv = u0 ; uv0 = v u0 ; u v0 , which is the formula for the derivative of a quotient (6.8) (under the assumption v v2 v2 u v > 0).
8. Derivative of the Inverse Function
If y = '(x) is the inverse function of the original function y = f (x), then both representations y = f (x) and x = '(y) are equivalent. For every corresponding value of x and y such that f is di erentiable with respect to x, and ' is di erentiable with respect to y, e.g., none of the derivatives is equal to zero, between the derivatives of f and its inverse function ' we have the following relation: dy = 1 : f 0(x) = '01(y) or dx (6.15) dx dy The function y = f (x) = arcsin x for ;1 < x < 1 is equivalent to the function x = '(y) = sin y for ;=2 < y < =2. From (6.15) it follows that (arcsin x)0 = (sin1 y)0 = cos1 y = q 1 2 = p 1 2 , because cos y 6= 0 for ;=2 < y < =2. 1;x 1 ; sin y
9. Derivative of an Implicit Function
Suppose the function y = f (x) is given in implicit form by the equation F (x y) = 0. Considering the rules of di erentiation for functions of several variables (see 6.2, p. 392) calculating the derivative with respect to x we get @F + @F y0 = 0 and so y0 = ; Fx (6.16) @x @y Fy if the partial derivative Fy di ers from zero.
384 6. Dierentiation 2 2 The equation x2 + y2 = 1 of an ellipse with semi-axes a and b can be written in the form F (x y) = a b x2 + y2 ; 1 = 0. For the slope of the tangent line at the point of the ellipse (x y) we get according to a2 b2 (6.16) 2 . y0 = ; 2ax2 2by2 = ; ab 2 xy :
10. Derivative of a Function Given in Parametric Form
If a function y = f (x) is given in parametric form x = x(t) y = y(t), then the derivative y0 can be calculated by the formula dy 0 y_ (6.17) dx = f (x) = x_ dx with the help of the derivatives y_ (t) = dy dt and x_ (t) = dt with respect to the variable t, if of course x_ (t) 6= 0 holds. Polar Coordinate Representation: If a function is given with polar coordinates (see 3.5.2.2, 3., p. 191) = ('), then the parametric representation is x = (') cos ' y = (') sin ' (6.18) with the angle ' as a parameter. For the slope y0 of the tangent of the curve (see 3.6.1.2, 2., p. 227 or 6.1.1, 2., p. 379) we get from (6.17) sin ' + cos ' where _ = d : y0 = __ cos (6.19) ' ; sin ' d'
Remarks: 1. The derivatives x_ y_ are the components of the tangent vector at the point (x(t) y(t)) of the curve. 2. It is often useful to consider the complex relation: x(t) + i y(t) = z(t)
x_ (t) + i y_ (t) = z_ (t): (6.20) i( !t + ) 2 . The tangent Circular Movement: z(t) = rei!t (r ! const) z_ (t) = ri!ei!t = r!e vector runs ahead by a phase-shift =2 with respect to the position vector. given direction MN . To
11. Graphical Di erentiation
If a di erentiable function y = f (x) is represented by its curve ; in the Cartesian coordinate system in an interval a < x < b, then the curve ;0 of its derivative can be constructed approximately. The construction of a tangent estimated by eye is pretty inaccurate. However, if the direction of the tangent MN (Fig. 6.4) is given, then we can determine the point of contact A more precisely.
1. Construction of the Point of Contact of a Tangent
N1
N N2 P
A R2 R Q 1
M2 We draw two secants M1N 1 and M2 N 2 parallel to the direction MN of the tangent so that the curve is intersected in points being not far from M M1 each other. Then we determine the midpoints of the secants, and draw a straight line through them. This line PQ intersects the curve at the point A, which is approximately the point, where the tangent has the Figure 6.4 check the accuracy, we draw a third line close to and parallel to the rst two lines, and the line PQ should intersect it at the midpoint.
2. Construction of the Derivative Curve a) Choose some directions l1 l2 : : : ln which could be the directions of some tangents of the curve
6.1 Dierentiation of Functions of One Variable 385
y = f (x) in the considered interval as in Fig. 6.5, and determine the corresponding points of contact A1 A2 : : : An, where the tangents themselves must not be constructed. y b) Choose a point P , a \pole", on the negative side of A3 A4 A5 l1 A2 the x-axis, where the longer the segment PO = a, the A6 l2 atter the curve is. A1 l3 c) Draw the lines through the pole P parallel to the dil4 G rections l1 l2 : : : ln, and denote their intersection points l5 with the y-axis by B1 B2 : : : Bn. l6 B1 C1 d) Construct the horizontal lines B1C1 B2C2 : : : BnCn B2 C2 through the points B1 B2 : : : Bn to the intersection B3 C3 points C1 C2 : : : Cn with the orthogonal lines from the C4 B4 D1 D2 P points A1 A2 : : : An. D3 D D5 x 0 B 5 4 e) Connect the points C1 C2 : : : Cn with the help of a C5 curved ruler. The resulting curve satis es the equation D6 y = af 0(x). If the segment a is chosen so that it correG' sponds to the unit length on the y-axis, then the curve B6 we get is the curve of the derivative. Otherwise, we have C6 to multiply the ordinates of C1 C2 : : : Cn by the factor 1=a. The points D1 D2 : : : Dn given in Fig. 6.5 are on the correctly scaled curve ;0 of the derivative. Figure 6.5
6.1.3 Derivatives of Higher Order
6.1.3.1 Denition of Derivatives of Higher Order !
d dy , is called the second derivative of the The derivative of y0 = f 0(x), which means (y0)0 or dx dx 2y 2 f (x) d 00 00 function y = f (x) and it is denoted by y y dx2 f (x) or d dx 2 . Higher derivatives can be de ned analogously. The notation for the n-th derivative of the function y = f (x) is: dny = f (n) (x) = dnf (x) n = 0 1 : : : y(0) (x) = f (0) (x) = f (x) : y(n) = dx (6.21) n dxn
6.1.3.2 Derivatives of Higher Order of some Elementary Functions The n-th derivatives of the simplest functions are collected in Table 6.3.
6.1.3.3 Leibniz's Formula
To calculate the n-th-order derivative of a product of two functions, the Leibniz formula can be used: Dn(uv) = u Dnv + 1!n Du Dn;1v + n(n2!; 1) D2u Dn;2v + : (n ; m + 1) Dmu Dn;mv + + Dnu v see also p. 385: (6.22) + n(n ; 1) : :m ! n d ( n ) Here, we use the notation D = dx . If D0u is replaced by u and D0v by v, then we get the formula (6.23) whose structure corresponds to the binomial formula (see p. 12): n n! X DmuDn;mv : (6.23) Dn(uv) = m=0 m
386 6. Dierentiation Expression Constant function Constant multiple Sum Product of two functions Product of n functions Quotient Chain rule for two functions Chain rule for three functions Power Logarithmic di erentiation Di erentiation of the inverse function Implicit di erentiation
Table 6.2 Di erentiation rules
Formula for the derivative c0 = 0 (c const) (cu)0 = cu0 (c const) (u v)0 = u0 v0 (uv)0 = u0v + uv0 n (u1u2 un)0 = P u1 u0i un i=1 u 0 vu0 ; uv0 (v 6= 0) v = v2 dv y = u(v(x)): y0 = du dv dx dv dw y = u(v(w(x))): y0 = du dv dw dx (u )0 = u ;1u0 ( 2 IR 6= 0) 0 0 1 u specially : u = ; u2 (u 6= 0)
d(ln y(x)) = 1 y0 =) y0 = y d(ln y) dx y dx 0 ! special : (uv )0 = uv v0 ln u + vu (u > 0) u
' inverse function of f i.e. y = f (x) () x = '(y) : dy = 1 f 0(x) = '01(y) or dx dx dy 0 F (x y) = 0: Fx + Fy y = 0 or ! @F F 6= 0 y0 = ; FFx Fx = @F F y= y @x @y y
Derivative in parameter form
x = x(t) y = y(t) (t parameter): ! dy = y_ dy y0 = dx x_ = dx x_ dt y_ = dt
Derivative in polar coordinates
(') cos ' r = r('): xy = = (') sin ' dy = _ sin ' + cos ' y0 = dx _ cos ' ; r sin '
(angle ' as parameter) ! dx _ = d'
A: (x2 cos ax)(50) : If v = x2 u = cos ax are substituted, then we have u(k) = ak cos ax + k 2 0 v = 2x v00 = 2 v000 = v(4) = = 0 . Except the rst three cases, summandsare equal to all50the 50 49 49 (50) 2 50 zero, so (uv) = x a cos ax + 50 2 + 1 2xa cos ax + 49 2 + 1 2 2a48 cos ax + 48 2
6.1 Dierentiation of Functions of One Variable 387
= a48 (2450 ; a2 x2 ) cos ax ; 100ax sin ax] . ! ! ! ! B: (x3 ex)(6) = 60 x3 ex + 61 3x2 ex + 62 6xex + 63 6ex. Table 6.3 Derivatives of higher order of some elementary functions
Function xm ln x loga x ekx ax akx sin x cos x sin kx cos kx sinh x cosh x
n-th-order derivative
m(m ; 1)(m ; 2) : : : (m ; n + 1)xm;n (for integer m and n > m the n-th derivative is 0) (;1)n;1(n ; 1)! x1n (;1)n;1 (n ; 1)! 1n ln a x knekx (ln a)nax (k ln a)n akx sin(x + n 2) cos(x + n ) 2 n k sin(kx + n 2) n kn cos(kx + 2 ) sinh x for even n, cosh x for odd n cosh x for even n, sinh x for odd n
6.1.3.4 Higher Derivatives of Functions Given in Parametric Form
If a function y = f (x) is given in the parametric form x = x(t) y = y(t), then its higher derivatives (y00 y000, etc.) can be calculated by the following formulas, where y_ (t) = dy x_ (t) = dx dt dt y(t) = 2 2 d y x = d x , etc., denote the derivatives with respect to the parameter t: dt2 dt2 d2y x_ y ; y_ x d3y = x_ 2 y___ ; 3x_ xy + 3y_ x2 ; x_ y_ x___ : : : (x_ (t) 6= 0): (6.24) dx2 = x_ 3 dx3 x_ 5
6.1.3.5 Derivatives of Higher Order of the Inverse Function
If y = '(x) is the inverse function of the original function y = f (x), then both representations y = f (x) and x = '(y) are equivalent. Supposing '0(y) 6= 0 holds, the relation (6.15) is valid for the derivatives of the function f and its inverse function '. For higher derivatives (y00 y000, etc.) we get
d2y = ; '00 (y) dx2 '0(y)]3
d3y = 3 '00(y)]2 ; '0(y)'000(y) : : : : dx3 '0(y)]5
(6.25)
388 6. Dierentiation
6.1.4 Fundamental Theorems of Dierential Calculus 6.1.4.1 Monotonicity
If a function f (x) is de ned and continuous in a connected interval, and if it is di erentiable at every interior point of this interval, then the relations f 0(x) 0 for a monotone increasing function (6.26a) f 0(x) 0 for a monotone decreasing function (6.26b) are necessary and sucient. If the function is strictly monotone increasing or decreasing, then the derivative function f 0(x) must not be identically zero on any subinterval of the given interval. In Fig. 6.6b this condition is not ful lled on the segment BC . The geometrical meaning of monotonicity is that the curve of an increasing function never falls for increasing values of the argument, i.e., it either rises or runs horizontally (Fig. 6.6a). Therefore the tangent line at any point of the curve forms an acute angle with the positive x-axis or it is parallel to it. For monotonically decreasing functions (Fig. 6.6b) analogous statements are valid. If the function is strictly monotone, then the tangent can be parallel to the x-axis only at some single points, e.g., at the point A in Fig. 6.6a, i.e., not on a subinterval such as BC in Fig. 6.6b. y
y B
A a a)
0
y
x
C a
b) 0
Figure 6.6
6.1.4.2 Fermat's Theorem
A
c2 x
x
0 c1
B
Figure 6.7
If a function y = f (x) is de ned on a connected interval, and it has a maximum or a minimum value at an interior point x = c of this interval (Fig. 6.7), i.e., if for every x in this interval f (c) > f (x) (6.27a) or f (c) < f (x) (6.27b) holds, and if the derivative exists at the point c, then the derivative must be equal to zero there: f 0(c) = 0: (6.27c) The geometrical meaning of the Fermat theorem is that if a function satis es the assumptions of the theorem, then its curve has tangents parallel to the x-axis at A and B (Fig. 6.7). The Fermat theorem gives only a necessary condition for the existence of a maximum or minimum value at a point. From Fig. 6.6a it is obvious that having a zero derivative is not sucient to give an extreme value: At the point A, f 0(x) = 0 holds, but there is no maximum or minimum here. To have an extreme value di erentiability is not a necessary condition. The function in Fig. 6.8d has a maximum at e, but the derivative does not exist here.
6.1.4.3 Rolle's Theorem
If a function y = f (x) is continuous on the closed interval a b], and di erentiable on the open interval (a b), and f (a) = 0 f (b) = 0 (a < b) (6.28a) hold, then there exists at least one point c between a and b such that f 0(c) = 0 (a < c < b) (6.28b) holds. The geometrical meaning of Rolle's theorem is that if the graph of a function y = f (x) which is continuous on the interval (a b) intersects the x-axis at two points A and B , and it has a non-vertical
6.1 Dierentiation of Functions of One Variable 389
tangent at every point, then there is at least one point C between A and B such that the tangent is parallel to the x-axis here (Fig. 6.8a). It is possible, that there are several such points in this intery
y
y C
c
E
a 0 d)
e
C A d B
D a 0 A a)
y
b B x
a 0 A c b)
0
e B d
E
x
b x c)
b x
Figure 6.8 val, e.g., the points C , D, and E in Fig. 6.8b. The properties of continuity and di erentiability are important in the theorem: in Fig. 6.8c the function is not continuous at x = d, and in Fig. 6.8d the function is not di erentiable at x = e. In both cases f 0(x) 6= 0 holds everywhere where the derivative exists.
6.1.4.4 Mean Value Theorem of Di erential Calculus
If a function y = f (x) is continuous on the closed interval a b], di erentiable on the open interval (a b) and it has a non-vertical tangent at every point, then there exists at least one point c between a and b such that f (b) ; f (a) = f 0(c) (a < c < b) B y (6.29a) b;a holds. If we substitute b = a + h, and ! means a number between C 0 and 1, then the theorem can be written in the form f (a + h) = f (a) + h f 0(a + ! h) (0 < ! < 1): (6.29b) 1. Geometrical Meaning The geometrical meaning of the A theorem is that if a function y = f (x) satis es the conditions of the theorem, then its graph has at least one point C between A 0 a c b x and B such that the tangent line at this point is parallel to the line segment between A and B (Fig. 6.9). There can be several such points (Fig. 6.8b). Figure 6.9 The properties of continuity and di erentiability are important in the theorem, as can be observed in Fig. 6.8c,d. 2. Applications The mean value theorem has several useful applications. A: This theorem can be used to prove some inequalities in the form jf (b) ; f (a)j < K jb ; aj (6.30) where K is an upper bound of jf 0(x)j for every x in the interval a b]. B: How accurate is the value of f () = 1 +1 2 if is replaced by the approximate value = 3:14? We have: jf () ; f (")j = 2c 2 2 j ; " j 0:053 0:0016 = 0:000 085, which means 1 +1 2 is (1 + c ) between 0:092 084 0:000 085.
6.1.4.5 Taylor's Theorem of Functions of One Variable
If a function y = f (x) is continuously di erentiable (it has continuous derivatives) n ; 1 times on the interval a a + h], and if also the n-th derivative exists in the interior of the interval, then the Taylor
390 6. Dierentiation formula or Taylor expansion is 2 n;1 n f (a + h) = f (a) + 1!h f 0(a) + h2! f 00(a) + + (nh; 1)! f (n;1) (a) + hn! f (n) (a + ! h)
(6.31)
with 0 < ! < 1. The quantity h can be positive or negative. The mean value theorem (6.29b) is a special case of the Taylor formula for n = 1.
6.1.4.6 Generalized Mean Value Theorem of Di erential Calculus (Cauchy's Theorem)
If two functions y = f (x) and y = '(x) are continuous on the closed interval a b] and they are differentiable at least in the interior of the interval, and '0(x) is never equal to zero in this interval, then there exists at least one value c between a and b such that f (b) ; f (a) = f 0(c) (a < c < b): (6.32) '(b) ; '(a) '0(c) The geometrical meaning of the generalized mean value theorem corresponds to that of the rst mean value theorem. Supposing, e.g., that the curve in Fig.6.9 is given in parametric form x = '(t) y = f (t), where the points A and B belong to the parameter values t = a and t = b respectively. Then for the point C f (a) = f 0(c) tan = 'f ((bb)) ; (6.33) ; '(a) '0 (c) is valid. For '(x) = x the generalized mean value theorem is simpli ed into the rst mean value theorem.
6.1.5 Determination of the Extreme Values and Inection Points 6.1.5.1 Maxima and Minima
The substitution value f (x0) of a function f (x) is called the relative maximum (M ) or relative minimum (m) if one of the inequalities f (x0 + h) < f (x0) (for maximum) (6.34a) f (x0 + h) > f (x0) (for minimum) (6.34b) holds for arbitrary positive or negative values of h small enough. At a relative maximum the value f (x0 ) is greater than the substitution values in the neighborhood, and similarly, at a minimum it is smaller. The relative maxima and minima are called relative or local extrema. The greatest or the smallest value of a function in an interval is called the global or absolute maximum or global or absolute minimum in this interval.
6.1.5.2 Necessary Conditions for the Existence of a Relative Extreme Value
A function can have a relative maximum or minimum only at the points where its derivative is equal to zero or does not exist. That is: At the points of the graph of the function corresponding to the relative extrema the tangent line is whether parallel to the x-axis (Fig. 6.10a) or parallel to the y-axis (Fig. 6.10b) or does not exist (Fig. 6.10c). Anyway, these are not sucient conditions, e.g., at the points A B C in Fig. 6.11 these conditions are obviously ful lled, but there are no extreme values of the function. If a continuous function has relative extreme values, then maxima and minima follow alternately, that means, between two neighboring maxima there is a minimum, and conversely.
2. Method of Sign Change
For values x; and x+ , which are slightly smaller and greater than xi , and for which between xi and x; and x+ no more roots or points of discontinuity of f 0(x) exist, we check the sign of f 0(x). When during the transition from f 0 (x;) to f 0(x+ ) the sign of f 0(x) changes from \+" to \;", then there is a relative
6.1 Dierentiation of Functions of One Variable 391 y
y
x
0 a)
y
M
M
m
0
x
0 b)
M
x
c)
m
m
Figure 6.10
6.1.5.3 Relative Extreme Values of a Differentiable Explicit Function y = f (x)
y
1. Determine the Points of Extreme Values 0
C
Since f (x) = 0 is a necessary condition where the derivative exists, after determining the derivative f 0(x), rst we calculate all the real roots x1 x2 : : : xi : : : xn of the x equation f 0 (x) = 0. Then we check each of them, e.g., xi 0 Figure 6.11 with the following method. maximum of the function f (x) at x = xi (Fig. 6.12a) if it changes from \;" to \+", then there is a relative minimum there (Fig. 6.12b). If the derivative does not change its sign (Fig. 6.12c,d), then there is no extremum at x = xi , but it has an inection point with a tangent parallel to the x-axis. A
B
3. Method of Higher Derivatives
If a function has higher derivatives at x = xi, then we can substitute, e.g., the root xi into the second derivative f 00(x). If f 00(xi ) < 0 holds, then there is a relative maximum at xi , and if f 00(xi) > 0 holds, a relative minimum. If f 00 (xi) = 0 holds, then xi must be substituted into the third derivative f 000 (x). If f 000 (xi) 6= 0 holds, then there is no extremum at x = xi but an inection point. If still f 000(xi ) = 0 holds, then we substitute it into the forth derivative, etc. If the rst non-zero derivative at x = xi is an even one, then f (x) has an extremum here: If the derivative is positive, then there is minimum, if it is negative, then there is a maximum. If the rst non-zero derivative is an odd one, then there is no extremum there (actually, there is an inection point). y
y
=0 −
+ 0 ~ x a)
x1
−
~ ~ ~ x x 0 x b)
y
y =0 + ~ x1 x x
+ 0 ~ x c)
=0
+
−
=0 −
x1
~ x x
0 ~ x d)
x1
~ ~x x
Figure 6.12
4. Further Conditions for Extreme Points and Determination of Inection Points
If a continuous function is increasing below x0 and decreasing after, then it has a maximum there if it is decreasing below and increasing after, then it has a minimum there. Checking the sign change of the derivative is a useful method even if the derivative does not exist at certain points as in Fig. 6.10b,c and Fig. 6.11. If the rst derivative exists at a point where the function has an inection point, then the rst derivative has an extremum there. So, to nd the inection points with the help of derivatives, we have to do the same investigation for the derivative function as we have done for the original function to nd its extrema. Remark: For non-continuous functions, and sometimes also for certain di erentiable functions the
392 6. Dierentiation determination of extrema needs individual ideas. It is possible that a function has an extremum so that the rst derivative exists and it is equal to zero, but the second derivative does not exist, and the rst one has in nitely many roots in an arbitrary neighborhood of the considered point, so it is meaningless to say it changes its sign there. For instance f (x) = x2 (2 + sin (1=x)) for x 6= 0 and f (0) = 0.
6.1.5.4 Determination of Absolute Extrema
The considered interval of the independent variable is divided into subintervals such that in these intervals the function has a continuous derivative. The absolute extreme values are among the relative extreme values, or at the endpoints of the subintervals, if their endpoints belong to them. For noncontinuous functions or for non-closed intervals it is possible that no maximum or minimum exists on the considered interval.
Examples of 2the Determination of Extrema: A: y = e;x , interval ;1 +1]. Greatest value at x = 0, smallest at the endpoints (Fig. 6.13a). B: y = x3 ; x2 , interval ;1 +2]. Greatest value at x = +2, smallest at x = ;1, at the ends of the interval (Fig. 6.13b). C: y = 1 x1 , interval ;3 +3], x =6 0. There is no maximum or minimum. Relative minimum 1+e at x = ;3, relative maximum at x = 3. If we de ne y = 1 for x = 0, then there will be an absolute maximum at x = 0 (Fig. 6.13c). D: y = 2 ; x 23 , interval ;1 +1]. Greatest value at x = 0 (Fig. 6.13d, the derivative is not nite).
6.1.5.5 Determination of the Extrema of Implicit Functions
If the function is given in the implicit form F (x y) = 0, and the function F itself and also its partial derivatives Fx Fy are continuous, then its maxima and minima can be determined in the following way: 1. Solution of the Equation System F (x y) = 0 Fx (x y) = 0 and substitution of the resulting values (x1 y1) (x2 y2) : : : (xi yi) : : : in Fy and Fxx. 2. Sign Comparison for Fy and Fxx at the Point (xi yi): When they have di erent signs, the function y = f (x) has a minimum at xi when Fy and Fxx have the same sign, then it has a maximum at xi . If either Fy or Fxx vanishes at (xi yi), then we need further and rather complicated investigation. y
y -x 2
3
m1
M
−1 0 a)
+1
y= 2
0 x
m
b)
x
2/3
y=2−x
1/x
M1
m1 −3
y
1 1+e
−1
m2
y
M
2
y=x −x
y=e
0
M2 3 x
c)
M
m1 −1
m2 0 d)
1
x
Figure 6.13
6.2 Dierentiation of Functions of Several Variables
6.2.1 Partial Derivatives
6.2.1.1 Partial Derivative of a Function
The partial derivative of a function u = f (x1 x2 : : : xi : : : xn) with respect to one of its n variables, e.g., with respect to x1 is de ned by @u f (x1 + %x1 x2 x3 : : : xn) ; f (x1 x2 x3 : : : xn) (6.35) x1 !0 @x1 = lim % x1
6.2 Dierentiation of Functions of Several Variables 393
so only one of the n variables is changing, the other n ; 1 are considered as constants. The symbols @u u0 @f f 0 . A function of n variables can have n rst-order partial for the partial derivatives are @x x @x x @u @u @u @u derivatives: @x @x @x : : : @x . The calculation of the partial derivatives can be done following 1 2 3 n the same rules we have for the functions of one variable. 2 2xy @u = x2 @u = ; x2 y . u = xzy @u = @x z @y z @z z2
6.2.1.2 Geometrical Meaning for Functions of Two Variables
If a function u = f (x y) is represented as a surface in a Cartesian coordinate system, and this surface is intersected through its point P by a plane parallel to the x u plane (Fig. 6.14), then we have @u = tan (6.36a) @x where is the angle between the positive x-axis and the tangent line of the intersection curve at P , which is the same as the angle between the positive x-axis and the perpendicular projection of the tangent line into the x u plane. Here, is measured starting at the x-axis, and the positive direction is counterclockwise if we are looking toward the positive half of the y-axis. Analogously to , is de ned with a plane parallel to the y u plane: @u = tan : (6.36b) @y The derivative with respect to a given direction, the so-called directional derivative, and derivative with respect to volume, will be discussed in vector analysis (see 13.2.1, p. 649 and p. 650). u
u u=f(x,y) Du
P b
du
0
y
y
P dx
a x
x
Figure 6.14
dy
Figure 6.15
6.2.1.3 Di erentials of x and f (x)
1. The Di erential dx of an Independent Variable x is equal to the increment %x, i.e., dx = %x for an arbitrary value of %x.
(6.37a)
2. The Di erential dy of a Function y = f (x) of One Variable x
is de ned for a given value of x and for a given value of the di erential dx as the product dy = f 0(x) dx:
(6.37b)
394 6. Dierentiation
3. The Increment y from x to x + x of One Variable x is the di erence %y = f (x + %x) ; f (x):
4. Geometrical Meaning of the Di erential
(6.37c)
If the function is represented by a curve in a Cartesian coordinate system, then dy is the increment of the ordinate of the tangent line for the change of x by a given increment dx (Fig. 6.1).
6.2.1.4 Basic Properties of the Di erential
1. Invariance
Independently of whether x is an independent variable or a function of a further variable t dy = f 0(x) dx is valid.
(6.38)
2. Order of Magnitude
If dx is an arbitrarily small value, then dy and %y = y(x + %x) ; y(x) are also arbitrarily small, but %y = 1. Consequently, the di erence between them is also arbitrarily equivalent amounts, i.e., lim x!0 dy small, but of higher order than dx dy and %x (except if dy = 0 holds). Therefore, we get the relation (6.39) lim %y = 1 %y dy = f 0(x) dx x!0 dy which allows us to reduce the calculation of a small increment to the calculation of its di erential. This formula is frequently used for approximate calculations (see 6.1.4.4, p. 389 and 16.4.2.1, 2., p. 795).
6.2.1.5 Partial Di erential
For a function of several variables u = f (x y : : :) we can form the partial di erential with respect to one of its variables, e.g., with respect to x, which is de ned by the equality dxu = dxf = @u (6.40) @x dx:
6.2.2 Total Dierential and Dierentials of Higher Order
6.2.2.1 Notion of Total Di erential of a Function of Several Variables (Complete Di erential) 1. Di erentiability
The function of several variable u = f (x1 x2 : : : xi : : : xn) is said to be di erentiable at the point P0(x10 x20 : : : xi0 : : : xn0 ) if at a transition to an arbitrarily close point P (x10 + dx1 x20 + dx2 : : : xi0 +dxi : : : xn0 + dxn) with the arbitrarily small quantities dx1 dx2 : : : dxi : : : dxn the complete increment %u = f (x10 + dx1 x20 + dx2 : : : xi0 + dxi : : : xn0 + dxn) ;f (x10 x20 : : : xi0 : : : xn0 ) (6.41a) of the function di ers from the sum of the partial di erentials of all variables (6.41b) ( @u dx1 + @u dx2 + : : : + @u dxn )x10x20 :::xn0 @x1 @x2 @xn by an arbitrarily small amount in higher order than the distance q P0P = dx21 + dx22 + : : : + dx2n : (6.41c) A continuous function of several variables is di erentiable at a point if its partial derivatives, as functions of several variables, are continuous in a neighborhood of this point. This is a sucient but not a
6.2 Dierentiation of Functions of Several Variables 395
necessary condition, while the simple existence of the partial derivatives at the considered point is not sucient even for the continuity of the function.
2. Total Di erential
If u is a di erentiable function, then the sum (6.41b) @u dx + @u dx + : : : + @u dx du = @x 1 @x2 2 @xn n 1 is called the total dierential of the function. With the n-dimensional vectors ! @u @u : : : @u T (6.42b) dr = (dx1 dx2 : : : dxn)T grad u = @x @xn 1 @x2 the total di erential can be expressed as the scalar product du = (grad u)T dr: In (6.42b), there is the gradient, de ned in 13.2.2, p. 650, for n independent variables.
(6.42a) (6.42c) (6.42d)
3. Geometrical Representation
The geometrical meaning of the total di erential of a function of two variables u = f (x y), represented in a Cartesian coordinate system as a surface (Fig. 6.15), is that du is the same as the increment of the applicate (see 3.5.3.1, 2., p. 208) of the tangent plane (at the same point) if dx and dy are the increments of x and y. From the Taylor formula (see 6.2.2.3, 1., p. 396) it follows for functions of two variables that @f f (x y) = f (x0 y0) + @f (6.43a) @x (x0 y0)(x ; x0 ) + @y (x0 y0)(y ; y0) + R1 : Ignoring the remainder R1, we have that @f u = f (x0 y0) + @f (6.43b) @x (x0 y0)(x ; x0 ) + @y (x0 y0)(y ; y0) gives the equation of the tangent plane of the surface u = f (x y) at the point P0(x0 y0 u0).
4. The Fundamental Property of the Total Di erential
is the invariance with respect to the variables as formulated in (6.38) for the one-variable case.
5. Application in Error Calculations
In error calculations we use the total di erential du for an estimation of the error %u (see (6.41a)) (see, e.g., 16.4.1.3, 5., p. 792). From the Taylor formula (see 6.2.2.3, 1., p. 396) we have j%uj = jdu + R1 j jduj + jR1 j jduj (6.44) i.e., the absolute error j%uj can be replaced by jduj as a rst approximation. It follows that du is a linear approximation for %u.
6.2.2.2 Derivatives and Di erentials of Higher Order
1. Partial Derivatives of Second Order, Schwarz's Exchange Theorem
The second-order partial derivative of a function u = f (x1 , x2 , : : : , xi, : : : , xn ) can be calculated @ 2 u @ 2 u : : : or with respect to another with respect to the same variable as the rst one was, i.e., @x 2 @x2 1 2 2u 2u 2u @ @ @ variable, i.e. @x @x @x @x @x @x : : : . In this second case we talk about mixed derivatives. If 1 2 2 3 3 1 at the considered point the mixed derivatives are continuous, then @2u = @2u (6.45) @x1 @x2 @x2 @x1
396 6. Dierentiation holds for given x1 and x2 independently of the order of sequence of the di erentiation (Schwarz's exchange theorem). 3 3 Partial derivatives of higher order such as, e.g., @ u3 @ u 2 : : : are de ned analogously. @x @x@y
2. Second-Order Di erential of a Function of One Variable u = f (x)
The second-order di erential of a function y = f (x) of one variable, denoted by the symbols d2 y d2f (x), is the di erential of the rst di erential: d2y = d(dy) = f 00 (x)dx2. These symbols are appropriate only if x is an independent variable, and they are not appropriate if x is given, e.g., in the form x = z(v). Differentials of higher order are de ned analogously. If the variables x1 x2 : : : xi : : : xn are themselves functions of other variables, then we get more complicated formulas (see 6.2.4, p. 399).
3. Total Di erential of Second Order of a Function of Two Variables u = f (x y) 2 @ 2 u dx dy + @ 2 u dy2 d2u = d(du) = @@xu2 dx2 + 2 @x@y @y2
or symbolically ! @ dx + @ dy 2 u: d2u = @x @y
4. Total Di erential of!n-th Order of a Function of Two Variables n
(6.46a) (6.46b)
@ dx + @ dy u: dnu = @x @y
(6.47)
dnu = @x@ dx1 + @x@ dx2 + : : : + @x@ dxn u: 1 2 n
(6.48)
5. Total Di erential of n-th Order of a Function of Several Variables !n
6.2.2.3 Taylor's Theorem for Functions of Several Variables 1. Taylor's Formula for Functions of Two Variables a) First Form of Representation:
x y) @f (x y) f (x y) = f (a b) + @f (@x (xy)=(ab) (x ; a) + @y (xy)=(ab) (y ; b) 2 (x y) 2 2 + 2 @ f (x y ) + 2!1 @ f@x ( x ; a ) 2 @x@y (xy)=(ab) (x ; a)(y ; b) (xy)=(ab) 2 1 1 2 + @ f (x2 y) (6.49a) @y (xy)=(ab) (y ; b) + 3! f: : :g + + n! f: : :g + Rn : Here (a b) is the center of expansion and Rn is the remainder. Sometimes we write, e.g., instead of @f (x y) @f @x (xy)=(x0 y0 ) the shorter expression @x (x0 y0) . The terms of higher order in (6.49a) can be represented in a clear way with the help of operators: @ + (y ; b) @ f (x y) f (x y) = f (a b) + 1!1 (x ; a) @x (xy)=(ab) @y 2 @ + (y ; b) @ f (x y) + 2!1 (x ; a) @x (xy)=(ab) @y 1 1 (6.49b) + 3! f: : :g3f (x y)(xy)=(ab) + + n! f: : :gnf (x y)(xy)=(ab) + Rn :
6.2 Dierentiation of Functions of Several Variables 397
This symbolic form means that after using the binomial theorem the powers of the di erential operators @ @ @x and @y represent the higher-order derivatives of the function f (x y). Then the derivatives must be taken at the point (a b).
b) Second Form of the Representation:
! ! @ h + @ k f (x y) + 1 @ h + @ k 2 f (x y) f (x + h y + k) = f (x y) + 1!1 @x @y 2! @x @y !3 ! @ h + @ k f (x y) + + 1 @ h + @ k n f (x y) + R : (6.49c) + 3!1 @x n @y n! @x @y c) Remainder: The expression for the remainder is ! @ h + @ k n+1 f (x + !h y + !k) (0 < ! < 1): Rn = (n +1 1)! @x (6.49d) @y
2. Taylor Formula for Functions of m Variables
The analogous representation with di erential operators is f (x + h y +k : : : t + l) ! n X @ h + @ k + + @ l i f (x y : : : t) + R = f (x y : : : t) + i1! @x n @y @t i=1 where the remainder can be calculated by the expression ! @ h + @ k + + @ l n+1f (x + ! h y + ! k : : : t + ! l) Rn = (n +1 1)! @x @y @t (0 < ! < 1):
(6.50a)
(6.50b)
6.2.3 Rules of Dierentiation for Functions of Several Variables 6.2.3.1 Di erentiation of Composite Functions
1. Composite Function of One Independent Variable u = f (x1 x2 : : : xn ) x1 = x1 ( ) x2 = x2 ( ) : : : du = @u dx1 + @u dx2 + : : : + @u dxn : d @x1 d @x2 d @xn d
xn = xn( )
(6.51a) (6.51b)
2. Composite Function of Several Independent Variables u = f (x1 x2 : : : xn) x1 = x1 ( : : : ) x2 = x2 ( : : : ) : : : 9 @u = @u @x1 + @u @x2 + + @u @xn > > @ @x1 @ @x2 @ @xn @ > > @u = @u @x1 + @u @x2 + + @u @xn > > = @ @x1 @ @x2 @ @xn @ > ... = ... + ... + ... + ... > > @u = @u @x1 + @u @x2 + + @u @xn : > > > @ @x @ @x @ @x @ " 1
2
n
xn = xn( : : : )
(6.52a)
(6.52b)
398 6. Dierentiation
6.2.3.2 Di erentiation of Implicit Functions
1. A Function y = f (x) of One Variable is given by the equation
F (x y) = 0: (6.53a) Di erentiating (6.53a) with respect to x with the help of (6.51b) we get and y0 = ; FFx (Fy 6= 0): (6.53c) Fx + Fy y0 = 0 (6.53b) y Di erentiation of (6.53b) yields in the same way Fxx + 2Fxy y0 + Fyy (y0)2 + Fy y00 = 0 (6.53d) so considering (6.53b) we have 2 2 y00 = 2FxFy Fxy ; (F(Fy ) )F3 xx ; (Fx) Fyy : (6.53e) y In an analogous way we can calculate the third derivative Fxxx + 3Fxxy y0 + 3Fxyy (y0)2 + Fyyy (y0)3 + 3Fxy y00 + 3Fyy y0y00 + Fy y000 = 0 (6.53f) from which y000 can be expressed. 2. A Function u = f (x1 x2 : : : xi : : : xn) of Several Variables is given by the equation F (x1 x2 : : : xi : : : xn u) = 0: (6.54a) The partial derivatives @u Fx1 @u Fx2 @u Fxn (6.54b) @x1 = ; Fu @x2 = ; Fu : : : @xn = ; Fu can be calculated similarly as we have shown above but we use the formulas (6.52b). The higher-order derivatives can be calculated in the same way. 3. Two Functions y = f (x) and z = '(x) of One Variable are given by the system of equations F (x y z) = 0 and (x y z) = 0: (6.55a) Then di erentiation of (6.55a) according to (6.51b) results in Fx + Fy y0 + Fz z0 = 0 x + y y0 + z z0 = 0 (6.55b) ; z Fx z0 = Fxy ; Fy x : y0 = FFz x ; (6.55c) Fy z ; Fz y y z Fz y The second derivatives y00 and z00 are calculated in the same way by di erentiation of (6.55b) considering y0 and z0 . 4. n Functions of One Variable Let the functions y1 = f (x) y2 = '(x) : : : yn = (x) be given by a system F (x y1 y2 : : : yn) = 0 (x y1 y2 : : : yn) = 0 : : : " (x y1 y2 : : : yn) = 0 (6.56a) of n equations. Di erentiation of (6.56a) using (6.51b) results in 9 Fx + Fy1 y10 + Fy2 y20 + + Fyn yn0 = 0 > > > x + y1 y10 + y2 y20 + + yn yn0 = 0 = (6.56b) ... + ... + ... + ... + ... = 0 > > > "x + "y1 y10 + "y2 y20 + + "yn yn0 = 0: " Solving (6.56b) we get the derivatives y10 y20 : : : yn0 , we are looking for. In the same way we can calculate the higher-order derivatives.
6.2 Dierentiation of Functions of Several Variables 399
5. Two Functions u = f (xy) v = '(x y) of Two Variables are given by the system of
equations F (x y u v) = 0 and (x y u v) = 0: (6.57a) Then di erentiation of (6.57a) with respect to x and y with the help of (6.52b) results in 9 @F + @F @u + @F @v = 0 9 > @F + @F @u + @F @v = 0 > > > = = @y @u @y @v @y @x @u @x @v @x (6.57b) (6.57c) > @ @ @u @ @v @ + @ @u + @ @v = 0 > > + + = 0: > " " @y @u @y @v @y @x @u @x @v @x @v @v @u Solving the system (6.57b) for @x @x and the system (6.57c) for @u @y @y give the rst-order partial derivatives. The higher-order derivatives should be calculated in the same way. 6. n Functions of m Variables Given by a System of n Equations The rst-order and higher-order partial derivatives can be calculated in the same way as we did in the previous cases.
6.2.4 Substitution of Variables in Dierential Expressions and Coordinate Transformations 6.2.4.1 Function of One Variable
Suppose, given a function and a functional relation containing the independent variable, the function, and its derivatives: ! dy d2y d3 y : : : : y = f (x) (6.58a) H = F x y dx (6.58b) dx2 dx3 If the variables are substituted, then the derivatives can be calculated in the following way: Case 1a: The variable x is replaced by the variable t, and they have the relation x = '(t): (6.59a) Then we have ( ) 2 dy = 1 dy d2y = 1 0 ' (t) d y2 ; '00(t) dy (6.59b) 0 2 0 3 dx ' (t) dt dx ' (t)] dt dt ( ) 3 2 d3y = 1 0 ' (t)]2 d y3 ; 3 '0(t) '00 (t) d y2 + 3 '00(t)]2 ; '0(t) '000(t)] dy : : : : (6.59c) 3 0 5 dx ' (t)] dt dt dx Case 1b: If the relation between the variables is not explicit but it is given in implicit form (x t) = 0 (6.60) 2 y d3 y dy d 0 then the derivatives dx dx2 dx3 are calculated by the same formulas, but the derivatives ' (t) '00(t) '000(t) must be calculated according to the rules for implicit functions. In this case it can happen that the relation (6.58b) contains the variable x. To eliminate x, the relation (6.60) is used. Case 2: If the function y is replaced by a function u, and the relation between them is y = '(u) (6.61a) then the calculation of the derivatives can be performed using the following formulas: ! dy = '0(u) du d2y = '0(u) d2u + '00 (u) du 2 (6.61b) dx dx dx2 dx2 dx ! d3y = '0(u) d3u + 3'00(u) du d2u + '000(u) du 3 : : : : (6.61c) dx3 dx3 dx dx2 dx
400 6. Dierentiation Case 3: The variables x and y are replaced by the new variables t and u, and the relations between them are given by x = '(t u) y = (t u): (6.62a) For the calculation of the derivatives the following formulas are used: @ @ du dy = @t + @u dt (6.62b) dx @' + @' du @t @u dt 2 2 @ @ du 3 @ @ du 3 ! d2y = d dy = d 66 @t + @u dt 77 = 1 d 66 @t + @u dt 77 (6.62c) dx2 dx dx dx 4 @' + @' du 5 @' + @' du dt 4 @' + @' du 5 @t @u dt @t @u dt @t @u dt ! 1 d A = 1 B dA ; A dB (6.62d) B dt B B 3 dt dt with A = @ + @ du @t @u dt
(6.62e)
@' du B = @' @t + @u dt :
d3y can be done in an analogous way. The determination of the third derivative dx 3 For the transformation from Cartesian coordinates into polar coordinates according to x = cos ' y = sin ' the rst and second derivatives should be calculated as follows: 2 2 02 00 dy = 0 sin ' + cos ' (6.63b) d y2 = 0 + 2 ; 3 : 0 dx cos ' ; sin ' dx ( cos ' ; sin ')
(6.62f)
(6.63a) (6.63c)
6.2.4.2 Function of Two Variables
Suppose given a function and a functional relation containing the independent variables, the function and its partial derivatives: ! = f (x y) (6.64a)
! @! @ 2 ! @ 2 ! @ 2 ! : : : : H = F x y ! @! @x @y @x2 @x@y @y2
If x and y are replaced by the new variables u and v given by the relations x = '(u v) y = (u v) then the rst-order partial derivatives can be expressed from the system of equations @! = @! @' + @! @ @! = @! @' + @! @ @u @x @u @y @u @v @x @v @y @v with the new functions A B C , and D of the new variables u and v @! = A @! + B @! @! = C @! + D @! : @x @u @v @y @u @v
(6.64b) (6.65a) (6.65b) (6.65c)
6.2 Dierentiation of Functions of Several Variables 401
The second-order partial derivatives are calculated with the same formulas, only we do not use ! in @! them but its partial derivatives @! @x and @y , e.g.,
! ! ! @ 2 ! = @ @! = @ A @! + B @! = A A @ 2 ! + B @ 2 ! + @A @! + @B @! @x2 @x @x @x @u @v @u2 @u@v @u @u @u @v ! 2 2 +B A @ ! + B @ !2 + @A @! + @B @! : (6.66) @u@v @v @v @u @v @v The higher partial derivatives can be calculated in the same way. Express the Laplace operator (see 13.2.6.5, p. 657) in polar coordinates (see 3.5.2.1, 2., p. 190): 2 2 x = cos ' y = sin ': (6.67b) %! = @ !2 + @ !2 (6.67a) @x @y The calculations are: @! = @! cos ' + @! sin ' @! = ; @! sin ' + @! cos ' @ @x @y @' @x @y @! = cos ' @! ; sin ' @! @! = sin ' @! + cos ' @! @x @ @' @y @ @' ! ! 2 @ ! = cos ' @ cos ' @! ; sin ' @! ; sin ' @ cos ' @! ; sin ' @! : @x2 @ @ @' @' @ @' 2 Similarly, @@y!2 is calculated, so nally: 2 2 %! = @@ !2 + 12 @@'!2 + 1 @! (6.67c) @ : Remark: If functions of more than two variables should be substituted, then similar substitution formulas can be derived.
6.2.5 Extreme Values of Functions of Several Variables 6.2.5.1 Denition
A function u = f (x1 x2 : : : xi : : : xn) has a relative extreme value at a point P0 (x10 x20 : : : xi0 : : : xn0 ), if there is a number such that for every point belonging to the domain x10 ; < x1 < x10 + x20 ; < x2 < x20 + : : : xn0 ; < xn < xn0 + and to the domain of the function but di erent from P0 , then for a maximum the inequality f (x1 x2 : : : xn) < f (x10 x20 : : : xn0 ) (6.68a) holds, and for a minimum the inequality f (x1 x2 : : : xn) > f (x10 x20 : : : xn0 ) (6.68b) holds. Using the terminology of several dimensional spaces (see 2.18.1, p. 117) a function has a relative maximum or a relative minimum at a point if it is greater or smaller there than at the neighboring points.
6.2.5.2 Geometric Representation
In the case of a function of two variables, represented in a Cartesian coordinate system as a surface (see 2.18.1.2, p. 117), the relative extreme value geometrically means that the applicate (see 3.5.3.1, 2., p. 208) of the surface in the point A is greater or smaller than the applicate of the surface in any other point in a suciently small neighborhood of A (Fig. 6.16).
402 6. Dierentiation u
u
u
P0
P0 P0
0 x0 a) x
y0
0 x0
y A
b) x
0 x0
y y0
A
c) x
y y0
A
Figure 6.16 If the surface has a relative extremum at the point P0 which is an interior point of its domain, and if the surface has a tangent plane at this point, then the tangent plane is parallel to the x y plane (Fig. 6.16a,b). This property is necessary but not sucient for a maximum or minimum at a point P0. For example Fig. 6.16c shows a surface having a horizontal tangent plane at P0 , but there is a saddle point here and not an extremum.
6.2.5.3 Determination of Extreme Values of Functions of Two Variables
If u = f (x y) is given, then we solve the system of equations fx = 0 fy = 0. The resulting pairs of values (x1 y1) (x2 y2) : : : can be substituted into the second derivatives 2 @2f C = @2f : A = @@xf2 B = @x@y (6.69) @y2 Depending on the expression A B 2 2 % = B (6.70) C = AC ; B = fxxfyy ; (fxy ) ]x=xiy=yi (i = 1 2 : : :) it can be decided whether an extreme value exists: 1. In the case % > 0 the function f (x y) has an extreme value at (xi yi), and for fxx < 0 it is a maximum, for fxx > 0 it is a minimum (sucient condition). 2. In the case % < 0 the function f (x y) does not have an extremum. 3. In the case % = 0, we need further investigation.
6.2.5.4 Determination of the Extreme Values of a Function of n Variables
If u = f (x1 x2 : : : xn) is given, then rst we nd a solution (x10 x20 : : : xn0 ) of the system of the n equations fx1 = 0 fx2 = 0 : : : fxn = 0 (6.71) because it is a necessary condition for an extreme value. We prepare a matrix of the second-order par2 tial derivatives such that aij = @ f . Then we substitute a solution of the system of equations into @xi @xj the terms, and we prepare the sequence of left upper subdeterminants (a11, a11 a22 ; a12 a21 : : :). Then we have the following cases: 1. The signs of the subdeterminants follow the rule ; + ; + : : :, then there is a maximum there. 2. The signs of the subdeterminants follow the rule + + + + : : :, then there is a minimum there. 3. There are some zero values among the subdeterminants, but the signs of the non-zero subdeterminants coincide with the signs of the corresponding positions of one of the rst two cases. Then further investigation is required: Usually we check the values of the function in a close neighborhood of x10 x20 : : : xn0.
6.2 Dierentiation of Functions of Several Variables 403
4. The signs of the subdeterminants does not follow the rules given in cases 1. and 2.: There is no extremum at that point. The case of two variables is of course a special case of the case of n variables, (see 6.4]).
6.2.5.5 Solution of Approximation Problems
Several di erent approximation problems can be solved with the help of the determination of the extreme values of functions of several variables, e.g., tting problems or mean squares problems.
Problems to solve: Determination of Fourier coecients (see 7.4.1.2, p. 421, 19.6.4.1, p. 927). Determination of the coecients and parameters of the approximation function (see 19.6.2, p. 919 ). Determination of an approximate solution of an overdetermined linear system of equations (see p. 893, 19.2.1.3). Remark: For these problems the following notations are equivalent: Gaussian least squares method (see, e.g., 19.6.2, p. 919). Least squares method (see 19.6.2.2, p. 921). Approximation in mean square (continuous and discrete) (see, e.g., 19.6.2, p.919). Calculus of observations (or tting) (see 19.6.2, p. 919) and regression (see 16.3.4.2, 1., p. 780).
6.2.5.6 Extreme Value Problem with Side Conditions
Suppose we have to determine the extreme values of a function u = f (x1 x2 : : : xn) of n variables with the side conditions '(x1 x2 : : : xn) = 0 (x1 x2 : : : xn ) = 0 : : : (x1 x2 : : : xn) = 0: (6.72a) Because of the conditions, the variables are not independent, and if the number of conditions is k, obviously k < n must hold. One possibility to determine the extreme values of u is to express k variables with the others from the system of equations of the conditions, to substitute them into the original function, then the result is an extreme value problem without conditions for n ; k variables. The other way is the Lagrange multiplier method. We introduce k unde ned multipliers : : : , and we compose the Lagrange function (Lagrangian) of n + k variables x1 x2 : : : xn : : : : (x1 x2 : : : xn : : : ) = f (x1 x2 : : : xn) + '(x1 x2 : : : xn) + (x1 x2 : : : xn) + + (x1 x2 : : : xn): (6.72b) An extremum of the function can be only at a point (x10 x20 : : : xn0 0 0 : : : 0) which is the solution of the system of n + k equations (6.71) ' = 0 = 0 : : : = 0 x1 = 0 x2 = 0 : : : xn = 0 (6.72c) with unknowns x1 x2 : : : xn : : : . Since the equations of the side conditions are ful lled, the extreme value of will be an extreme value also for f . So we have to look for the extremum points of f among the solutions x10 x20 : : : xn0 of the system of equations (6.72b). To determine whether there are really extreme values at these points ful lling the necessary conditions requires further investigations, for which the general rules are fairly complicated. Usually we use some appropriate and individual calculations depending on the function f to prove if there is an extremum there, or not. The extreme value of the function u = f (x y) with the side condition '(x y) = 0 will be determined from the three equations @ f (x y) + '(x y)] = 0 @ f (x y) + '(x y)] = 0: '(x y) = 0 @x (6.73) @y There are three unknowns, x y .
404 7. Innite Series
7 InniteSeries
7.1 Sequences of Numbers
7.1.1 Properties of Sequences of Numbers 7.1.1.1 Denition of Sequence of Numbers
An innite sequence of numbers is an in nite system of numbers a1 a2 : : : an : : : or briey fak g with k = 1 2 : : : (7.1) arranged in a given order. The numbers of the sequence of numbers are called the terms of the sequence. Among the terms of a sequence of numbers the same numbers can occur several times. A sequence is considered to be de ned if the law of formation, i.e., a rule is given, by which any term of the sequence can be uniquely determined. Mostly there is a formula for the general term an.
Examples of Sequences of Numbers: A: an = n: 1 2 3 4 5 : : : . n;1 C: an = 3 ; 12 : 3 ; 32 , 34 , ; 38 , 163 : : : .
B: an = 4 + 3(n ; 1): 4 7 10 13 16 : : : . D: an = (;1)n+1: 1 ;1 1 ;1 1 : : : .
E: an = 3 ; 2n1;2 : 1 2 2 12 2 34 2 78 : : : (read 2 34 = 114 ) . n;1 F: an = 3 1 ; 1 10; 2 for odd n and
3 3 n an = 3 31 + 23 10; 2 + 1 for even n: 3 4 3:3 3:4 3:33 3:34 3:333 3:334 : : : . G: an = n1 : 1 12 13 14 15 : : : . H: an = (;1)n+1n: 1 ;2 3 ;4 5 ;6 : : : . I: an = ; n +2 1 for odd n and an = 0 for even n: ;1 0 ;2 0 ;3 0 ;4 0 : : : . J: an = 3 ; n 1; 3 for odd n and an = 13 ; n 1; 2 for even n: 1 11 2 12 2 12 12 12 2 34 12 43 : : : . 22 22 2
7.1.1.2 Monotone Sequences of Numbers
A sequence a1 a2 : : : an : : : is monotone increasing if a1 a2 a3 an (7.2) is valid and it is monotone decreasing if a1 a2 a3 an (7.3) is valid. We talk about a strictly monotone increasing sequence or strictly monotone decreasing sequence, if equality never holds in (7.2) or (7.3).
Examples of Monotone Sequences of Numbers: A: Among the sequences from A to J the sequences A, B, E are strictly monotone increasing. B: The sequence G is strictly monotone decreasing.
7.1.1.3 Bounded Sequences
A sequence is called bounded if for all terms janj < K (7.4) is valid for a certain K > 0. If such a K (bound) does not exist, then the sequence is unbounded.
7.1 Sequences of Numbers 405
Among the sequences from A to J the sequences C with K = 4, D with K = 2, E with K = 3, F with K = 5, G with K = 2 and J with K = 13 are bounded.
7.1.2 Limits of Sequences of Numbers 1. Limit of a Sequence of Numbers
An in nite sequence of numbers (7.1) has a limit A if for an unlimited increase of the index n the di erence an ; A becomes arbitrarily small. Precisely de ned this means: For an arbitrarily small " > 0 there exists an index n0 (") such that for every n > n0 jan ; Aj < ": (7.5a) The sequence has the limit +1 (;1), if for arbitrary K > 0 there exists an index n0 (K ) such that for every n > n0 an > K (an < ;K ): (7.5b)
2. Convergence of a Sequence of Numbers
If a sequence of numbers fang satis es (7.5a), then we say it converges to A. This is denoted by an = A or an ! A: (7.6) nlim !1 Among the sequences from A to J on the previous page, C with A = 0, E with A = 3, F with A = 3 13 , G with A = 0 are convergent.
3. Divergence of a Sequence of Numbers
Non-convergent sequences of numbers are called divergent. We talk about proper divergence in the case of (7.5b), i.e., if as n exceeds any value, an exceeds any large positive number K (K > 0) so that it never goes below, or if as n exceeds any value, an goes below any negative number ;K (K > 0) with arbitrarily large magnitude and never increases above it, i.e., if it has the limit 1. We use the notation: a = 1 (an > K 8n > n0 ) or n!;1 lim an = ;1 (an < ;K 8n > n0): (7.7) nlim !1 n Otherwise the sequence is called improperly divergent.
Examples of Divergent Sequences of Numbers: A: Among the sequences from A to J on the previous page, A and B tend +1, they are properly divergent. B: Among the sequences D is improperly divergent.
4. Theorems for Limits of Sequences
a) If the sequences fang and fbng are convergent, then (a + bn) = nlim a + nlim b nlim !1 n !1 n !1 n
(7.8) (a b ) = (nlim a )( lim b ) (7.9) nlim !1 n n !1 n n!1 n hold, and if bn 6= 0 for every n, and nlim b 6= 0, then !1 n lim a n an = n!1 (7.10) nlim !1 bn b nlim !1 n is valid. If nlim b = 0 and fang is bounded, then nlim (a b ) = 0 even if fang does not have any nite !1 n !1 n n limit. b) If nlim a = nlim b = A is valid and at least, from an index n1 and beyond, the inequality an !1 n !1 n cn bn holds, then we also have cn = A: (7.11) nlim !1 c) A monotone and bounded sequence has a nite limit. If a monotone increasing sequence a1 a2 a3 : : : is bounded above, i.e., an K1 for all n, then it is convergent, and its limit is equal to
406 7. Innite Series its least upper bound which is the smallest possible value for K1. If a monotone decreasing sequence a1 a2 a3 : : : is bounded below, i.e., an K2 , then it is convergent, and its limit is equal to its greatest lower bound which is the largest possible value for K2 .
7.2 Number Series
7.2.1 General Convergence Theorems
7.2.1.1 Convergence and Divergence of Innite Series 1. Innite Series and its Sum
From the terms ak of an innite sequence fak g (see 7.1.1.1, p. 404) the formal expression
a1 + a2 + + an + =
1 X
k=1
ak
(7.12)
can be composed and this is called an innite series. The nite sums
S1 = a1 S2 = a1 + a2 : : : Sn = are called partial sums.
n X
k=1
ak
(7.13)
2. Convergent and Divergent Series
A series (7.12) is called convergent if the sequence of partial sums fSng is convergent. The limit
S = nlim Sn = !1
1 X
ak
(7.14)
k=1 is called the sum, and ak
is called the general term of the series. If the limit (7.14) does not exist or it is equal to 1, then we call the series divergent. In this case the partial sums are not bounded or they oscillate. So the determination of the convergence of an in nite series is reduced to the determination of the limit of a sequence fSng. A: The geometric series (see 1.2.3, p. 19) 1 + 21 + 14 + 18 + + 21n + (7.15) is convergent. B: The harmonic series (7.16) and the series (7.17) and (7.18) 1+1+1++1+ (7.17) 1 + 21 + 13 + + n1 + (7.16) 1 ; 1 + 1 ; + (;1)n;1 + (7.18) are divergent. For the series (7.16) and (7.17) nlim S = 1 holds, (7.18) oscillates. n !1
3. Remainder
1 X
The remainder of a convergent series S = ak is the di erence between its sum S and the partial sum k=1 Sn :
Rn = S ; Sn =
1 X
k=n+1
ak = an+1 + an+2 + :
(7.19)
7.2.1.2 General Theorems about the Convergence of Series
1. Leaving out the Initial Terms If we leave out nitely many initial terms of a series or introduce nitely many further terms into it at the begin or we change the order of nitely many terms, then its
7.2 Number Series 407
convergence behavior does not change. Exchange of the order of nitely many terms does not a ect the sum if it exists. 2. Multiplication of Terms If all the terms of a convergent series are multiplied by the same factor c, then the convergence of the series does not change its sum is multiplied by the factor c. 3. Termwise Addition or Subtraction If we add or subtract two convergent series
a1 + a2 + + an + =
1 X
k=1
ak = S1 (7.20a) b1 + b2 + + bn + =
1 X
k=1
bk = S2 (7.20b)
term by term, then the result is a convergent series, and its sum is (a1 b1 ) + (a2 b2) + + (an bn) + = S1 S2: (7.20c) 4. Necessary Criterion for the Convergence of a Series The sequence of terms of a convergent series is a null sequence: a = 0: (7.21) nlim !1 n This is only a necessary but not sucient condition. For the harmonic series (7.16) nlim a = 0 holds, but nlim S = 1. !1 n !1 n
7.2.2 Convergence Criteria for Series with Positive Terms 7.2.2.1 Comparison Criterion Suppose we have two series
a1 + a2 + + an + =
1 X
k=1
ak
(7.22a)
b1 + b2 + + bn + =
1 X
k=1
bk
(7.22b)
with only positive terms (an > 0, bn > 0). If an bn holds from a certain n, then the convergence of the series (7.22a) yields the convergence of the series (7.22b), and the divergence of the series (7.22b) yields the divergence of the series (7.22a). A: Comparing the terms of the series 1 + 12 + 13 + + 1n + (7.23a) 2 3 n with the geometric series (7.15), the convergence of the series (7.23a) follows. From n = 2 the terms of the series (7.23a) are smaller than the terms of the convergent series (7.15): 1 1 (7.23b) nn < 2n;1 (n 2): B: From the comparison of the terms of the series 1 + p1 + p1 + + p1n + (7.24a) 2 3 with the terms of the harmonic series (7.16) follows the divergence of the series (7.24a). For n > 1 the terms of the series (7.24a) are greater than those of the divergent series (7.16): p1n > n1 (n > 1): (7.24b)
7.2.2.2 D'Alembert's Ratio Test If for the series 1 X a1 + a2 + + an + = ak k=1
(7.25a)
408 7. Innite Series all the ratios aan+1 are smaller than a number q < 1 from a certain n onwards, then the series is convern gent: an+1 < q < 1: (7.25b) an If these ratios are greater than a number Q > 1 from a certain n onwards, then the series is divergent. From the two previous statements it follows that if the limit an+1 = (7.25c) nlim !1 an exists, then for < 1 the series is convergent, for > 1 it is divergent. In the case = 1 the ratio test gives no information whether the series is convergent or not. A : The series 21 + 222 + 233 + + 2nn + (7.26a) is convergent, because n + 1 n 1+ 1 1 n = < 1 holds: = nlim : = nlim (7.26b) !1 !1 2n+1 2n 2 2 B : For the series 2 + 34 + 49 + + n n+2 1 + (7.27a) the ratio test does not give any result about whether the series is convergent or not, because ! n + 2 : n + 1 = 1: = nlim (7.27b) !1 (n + 1)2 n2
7.2.2.3 Root Test of Cauchy If for a series 1 X a1 + a2 + + an + = ak k=1
from a certain n onwards for every value pn an pn a < q < 1
(7.28a)
(7.28b) n holds, then the series is convergent. If from a certain n every value pn an is greater than a number Q where Q > 1 holds, then the series is divergent. From the previous pn a = statements it follows that if (7.28c) n nlim !1 exists, in the case < 1 the series is convergent, in the case > 1 it is divergent. For = 1 with this test we cannot tell anything about the convergence behavior of the series. 4 9 n n2 The series 21 + 23 + 34 + + n + (7.29a) 1 + is convergent because
= nlim !1
s n
0 1n C 1 1 n n2 = lim B B C A = e < 1 holds. n!1 @ 1 n+1 1+ n
7.2.2.4 Integral Test of Cauchy
(7.29b)
1. Convergence If a series has the general term an = f (n), and f (x) is a monotone decreasing function such that the improper integral
7.2 Number Series 409
Z1 c
f (x) dx
(see 8.2.3.2, 1., p. 454)
(7.30)
exists (it is convergent), then the series is convergent. 2. Divergence If the above integral (7.30) is divergent, then the series with the general term an = f (n) is divergent, too. The lower limit c of the integral is almost arbitrary but it must be chosen so that for c < x < 1 the function f (x) should be monotone decreasing. The series (7.27a) is divergent because
Z 1 x+1 1 1 = 1: f (x) = x x+2 1 dx = ln x ; (7.31) x2 xc c
7.2.3 Absolute and Conditional Convergence 7.2.3.1 Denition Along with the series
a1 + a2 + + an + =
1 X
k=1
ak
(7.32a)
whose terms can have di erent signs, we consider also the series
ja1j + ja2j + + janj + =
1 X
k=1
jak j
(7.32b)
whose terms are the absolute values of the terms of the original sequence (7.32a). If the series (7.32b) is convergent, then the original one (7.32a) is convergent, too. (This statement is valid also for series with complex terms.) In this case, the series (7.32a) is called absolutely convergent. If the series (7.32b) is divergent, then the series (7.32a) can be either divergent or convergent. In the second case, the series (7.32a) is called conditionally convergent. A : The series sin2 + sin222 + + sin2nn + (7.33a) where is an arbitrarily constant number, is absolutely convergent, because the series of absolute values sin n with terms n is convergent. This is obvious by comparing it with the geometric series (7.15): sin n 2 1 n n : (7.33b) 2 2 B : The series 1 ; 12 + 31 ; + (;1)n;1 n1 + (7.34) is conditionally convergent, because it is convergent according to (7.36b), and the series made of the absolute values of the terms is the divergent harmonic series (7.16) whose general term is n1 = janj.
7.2.3.2 Properties of Absolutely Convergent Series
1. Exchange of Terms
a) The terms of an absolutely convergent series can be exchanged with each other arbitrarily (even
in nitely many of them) and the sum does not change. b) Exchanging an in nite number of terms of a conditionally convergent series can change the sum and even the convergence behavior. Theorem of Riemann: The terms of a conditionally convergent series can be rearranged so that the sum will be equal to any given value, even to 1.
410 7. Innite Series
2. Addition and Subtraction
Absolutely convergent series can be added and subtracted term-by-term the result is absolutely convergent.
3. Multiplication
Multiplying a sum by a sum, the result is a sum of the products where every term of the rst factor is multiplied by every term of the second one. These two-term products can be arranged in di erent ways. The most common way for this arrangement is made as if the series were power series, i.e.: (a1 + a2 + + an + )(b1 + b2 + + bn + ) = a|{z} 1 b1 + |a2 b1 + {z a1 b2} + a| 3b1 + a{z2 b2 + a1 b3} + + a| nb1 + an;1 b{z2 + + a1 bn} + : (7.35a) If two series are absolutely convergent, then X their product is absolutely convergent, so it has the same X sum in any arrangement. If an = Sa and bn = Sb hold, then the sum of the product is S = SaSb : (7.35b) 1 1 P P If two series a1 + a2 + + an + = an and b1 + b2 + + bn + = bn are convergent, and n=1 n=1 at least one of them is absolutely convergent, then their product is also convergent, but not necessarily absolutely convergent.
7.2.3.3 Alternating Series
1. Leibniz Alternating Series Test (Theorem of Leibniz)
For an alternating series a1 ; a2 + a3 ; an (7.36a) where an are positive numbers, a sucient condition of convergence is if the following two relations hold: 1: nlim a = 0 and 2: a1 > a2 > a3 > > an > : (7.36b) !1 n The series (7.34) is convergent because of this criterion.
2. Estimation of the Remainder of an Alternating Series
If we consider the rst n terms of an alternating series, then the remainder Rn has the same sign as the rst omitted term an+1, and the absolute value of Rn is smaller than jan+1j: sign Rn = sign(an+1) with Rn = S ; Sn (7.37a) jS ; Snj < jan+1j: (7.37b) Considering the series 1 1 ; 21 + 13 ; 41 + n1 = ln 2 (7.38a) the remainder is j ln 2 ; Snj < n + 1 : (7.38b)
7.2.4 Some Special Series
7.2.4.1 The Values of Some Important Number Series 1 + 1!1 + 2!1 + 3!1 + + n1! + = e
1 ; 1!1 + 2!1 ; 3!1 + n1! = 1e 1 ; 1 + 1 ; 1 + 1 = ln 2 2 3 4 n 1 1 1 1 + 2 + 4 + 8 + + 21n + = 2
(7.39) (7.40) (7.41) (7.42)
7.2 Number Series 411
1 ; 1 + 1 ; 1 + 1n = 2 2 4 8 2 3 1 1 1 1 1 1 ; 3 + 5 ; 7 + 9 ; 2n ; 1 = 4
1 1 1 1 1 2 + 2 3 + 3 4 + + n(n + 1) + = 1 1 1 1 1 1 1 3 + 3 5 + 5 7 + + (2n ; 1)(2n + 1) + = 2 1 + 1 + 1 ++ 1 + = 3 13 24 35 (n ; 1)(n + 1) 4 1 + 1 + 1 ++ 1 + = 1 ; 3 5 7 9 11 13 (4n ; 1)(4n + 1) 2 8 1 + 1 ++ 1 1 123 234 n(n + 1)(n + 2) + = 4 1 1 1 1 1 2 : : : l + 2 3 : : : (l + 1) + + n : : : (n + l ; 1) + = (l ; 1)(l ; 1)! 2 1 + 12 + 12 + 12 + + 12 + = 2 3 4 n 6 2 1 1 1 1 1; 2 + 2 ; 2 + 2 = 2 3 4 n 12 1 + 1 + 1 + + 1 + = 2 12 32 52 (2n + 1)2 8 4 1 + 214 + 314 + 414 + + n14 + = 90 4 1 ; 214 + 314 ; n14 = 7720 1 1 1 1 4 14 + 34 + 54 + + (2n + 1)4 + = 96 2k 2 k ;1 1 + 12k + 12k + 12k + + 12k + = 2 Bk 2 3 4 n (2k)! 2k 2k;1 1 ; 12k + 12k ; 12k + 12k = (2 ; 1) Bk 2 3 4 n (2k)! 2k 2k 1 + 312k + 512k + 712k + + (2n ;1 1)2k + = 2(2 (2k;)! 1) Bk 2k+1 1 ; 321k+1 + 521k+1 ; 721k+1 + (2n ;11)2k+1 = 22k+2(2k)! Ek :y y
Bk are the Bernoulli numbers Ek are the Euler numbers
(7.43) (7.44) (7.45) (7.46) (7.47) (7.48) (7.49) (7.50) (7.51) (7.52) (7.53) (7.54) (7.55) (7.56) (7.57) (7.58) (7.59) (7.60)
412 7. Innite Series
7.2.4.2 Bernoulli and Euler Numbers
1. First De nition of the Bernoulli Numbers The Bernoulli numbers Bk occur in the power series expansion of some special functions, e.g., in the trigonometric functions tan x, cot x and csc x, also in the hyperbolic functions tanh x, coth x, and cosech x. The Bernoulli numbers Bk can be de ned as follows x x x2 x4 x2n n+1 (7.61) ex ; 1 = 1 ; 2 + B1 2! ; B2 4! + (;1) Bn (2n)! (jxj < 2) and they can be calculated by the coecient comparison method with respect to the powers of x. Their values are given in Table 7.1.
k 1 2 3
Bk 1 6 1 30 1 42
Table 7.1 The rst Bernoulli numbers
k 4 5 6
Bk
1 30 5 66 691 2 730
k 7 8 9
Bk
7 6 3 617 510 43 867 798
k 10 11
Bk
174 611 330 854 513 138
2. Second De nition of Bernoulli Numbers Some authors de ne the Bernoulli numbers in the following way: x x x2 x2n (7.62) ex ; 1 = 1 + B1 1! + B2 2! + + B2n (2n)! + (jxj < 2): So we get the recursive formula Bk+1 = (B + 1)k+1 (k = 1 2 3 : : :) (7.63) where after the application of the binomial theorem (see 1.1.6.4, 1., p. 12) we have to replace B by B , i.e., the exponent becomes the index. The rst few numbers are: 1 B = 1 B1 = ; 12 B2 = 16 B4 = ; 30 6 42 1 5 691 B8 = ; 30 B10 = 66 B12 = ; 2730 B14 = 76 (7.64) B16 = ; 3617 B3 = B5 = B7 = = 0: 510 : : : The following relation is valid: (7.65) Bk = (;1)k+1B2k (k = 1 2 3 : : :): 3. First De nition of Euler Numbers The Euler numbers Ek occur in the power series expansion of some special functions, e.g., in the functions sec x and sech x. The Euler numbers Ek can be de ned as follows 2 4 2n sec x = 1 + E1 x + E2 x + + En x + (jxj < ) (7.66) 2! 4! (2n)! 2 and they can be calculated by coecient comparison with respect to the powers of x. Their values are given in Table 7.2.
7.2 Number Series 413
4. Second De nition of Euler Numbers Analogously to (7.63) the Euler numbers can be de ned with the recursive formula (E + 1)k + (E ; 1)k = 0 (k = 1 2 3 : : :) (7.67) where after the application of the binomial theorem we have to replace E by E . For the rst values we get: E2 = ;1 E4 = 5 E6 = ;61 E8 = 1 385 E10 = ;50 521 E12 = 2 702 765 E14 = ;199 360 981 (7.68) E16 = 19 391 512 145 : : : E1 = E3 = E5 = = 0: The following relation is valid: Ek = (;1)k E2k (k = 1 2 3 : : :): (7.69)
k 1 2 3 4
Table 7.2 First Euler numbers
Ek
k
Ek
1 5 61 1 385
5 6 7
50 521 2 702 765 199 360 981
5. Relation Between the Euler and Bernoulli Numbers The relation between the Euler and Bernoulli numbers is: 2k+1 2k+1 E2k = 24k + 1 Bk ; 14
(k = 1 2 : : :) :
(7.70)
7.2.5 Estimation of the Remainder 7.2.5.1 Estimation with Majorant
In order to determine how well the n-th partial sum approximates the sum of the series, the absolute value of the remainder
X 1
X 1 ak jak j k=n+1 k=n+1
jS ; Snj = jRnj =
1 X ak must be estimated. For this estimation we use a majorant for jak j, usually a k=1 k=n+1 geometric series or another series which is easy to sum or estimate. 1 X Estimate the remainder of the series e = n1! . For the ratio aam+1 of two subsequent terms of this m n=0 a m ! 1 m+1 series with m n + 1 we have: a = (m + 1)! = m + 1 n +1 2 = q < 1. So the remainder m Rn = (n +1 1)! + (n +1 2)! + (n +1 3)! + can be majorized by the geometric series (7.15) with the quotient q = 1 and with the initial term a = 1 , and it yields: n+2 (n + 1)! 1 n + 2 1 n + 2 a (7.72) Rn < 1 ; q = (n + 1)! n + 1 < n! n2 + 2n = n 1n! :
of the series
1 X
(7.71)
414 7. Innite Series
7.2.5.2 Alternating Convergent Series
For a convergent alternating series, whose terms tend to zero with monotone decreasing absolute values, the easy estimation for the remainder is (see 7.2.3.3, 1., p. 410): jRnj = jS ; Snj < jan+1j : (7.73)
7.2.5.3 Special Series
For some special series, e.g., Taylor series, there are special formulas to estimate the remainder (see 7.3.3.3, p. 417).
7.3 Function Series
7.3.1 De nitions
1. Function Series is a series whose terms are functions of the same variable x: 1 X f1(x) + f2(x) + + fn(x) + = fk (x): k=1
(7.74)
2. Partial Sum Sn(x) is the sum of the rst n terms of the series (7.74): Sn(x) = f1 (x) + f2 (x) + + fn(x): (7.75) 3. Domain of Convergence of a function series (7.74) is the set of values of x = a for which all the functions fn(x) are de ned and the series of constant terms
f1(a) + f2(a) + + fn(a) + =
1 X
k=1
fk (a)
(7.76)
is convergent, i.e., for which the limit of the partial sums Sn(a) exists:
S (a) = nlim nlim !1 n !1
n X
k=1
fk (a) = S (a):
(7.77)
4. The Sum of the Series (7.74) is the function S (x), and we say that the series converges to the function S (x).
5. Remainder Rn(x) is the di erence between the sum S (x) of a convergent function series and its partial sum Sn(x): Rn(x) = S (x) ; Sn(x) = fn+1(x) + fn+2(x) + + fn+m(x) + :
(7.78)
7.3.2 Uniform Convergence
7.3.2.1 Denition, Weierstrass Theorem
According to the de nition of the limit of a sequence of numbers (see 7.1.2, p. 405 and 7.2.1.1, 2., p. 406) the series (7.74) converges at a point x to S (x) if for an arbitrary " > 0 there is an integer N such that jS (x) ; Sn(x)j < " holds for every n > N . For function series we distinguish between two cases: 1. Uniformly Convergent Series If there is a number N such that for every x in the domain of convergence of the series (7.74), jS (x) ; Sn(x)j < " holds for every n > N , then the series is called uniformly convergent on this domain. 2. Non-Uniform Convergence of Series If there is no such number N which is good for every value of x in the domain of convergence, i.e., there are such values of " for which there is at least one x in the domain of convergence such that jS (x) ; Sn(x)j > " holds for arbitrarily large values of n, then the series is non-uniformly convergent. 2 n (7.79a) A : The series 1 + 1!x + x2! + + xn! +
7.3 Function Series 415
with the sum ex (see Table 21.5, p. 1017) is convergent for every value of x. The convergence is uniform for every bounded domain of x, and for every jxj < a using the remainder of the Maclaurin formula (see 7.3.3.3, 2., p. 418) the inequality n+1 n+1 jS (x) ; Sn(x)j = (nx+ 1)! ex < (na+ 1)! ea (0 < ! < 1) (7.79b) is valid. Because (n + 1)! increases more quick than an+1 , the expression on the right-hand side of the inequality, which is independent of x, for suciently large n will be less than ". The series is not uniformly on the whole numerical axis: For any large n there will be a value of x such that n+1 convergent x ex is greater than a previously given ". (n + 1)!
B : The series x + x(1 ; x) + x(1 ; x)2 + + x(1 ; x)n + (7.80a) is convergent for every x in 0 1], because corresponding to the d'Alembert ratio test (see 7.2.2.2, p. 407) an+1 = j1 ; xj < 1 holds for 0 < x 1 (for x = 0 S = 0 holds): = nlim (7.80b) !1 an The convergence is non-uniform, because S (x) ; Sn(x) = x (1 ; x)n+1 + (1 ; x)n+2 + ] = (1 ; x)n+1 (7.80c) is valid and for every n there is an x such that (1 ; x)n+1 is close enough to 1, i.e., it is not smaller than ". In the interval a x 1 with 0 < a < 1 the series is uniformly convergent. 3. Weierstrass Criterion for Uniform Convergence The series f1(x) + f2(x) + + fn(x) + (7.81a) is uniformly convergent in a given domain if there is a convergent series of constant positive terms c1 + c2 + + cn + (7.81b) such that for every x in this domain the inequality jfn(x)j cn (7.81c) is valid. We call (7.81c) a majorant of the series (7.81a).
7.3.2.2 Properties of Uniformly Convergent Series
1. Continuity If the functions f1(x) f2 (x) fn(x) are continuous in a domain and the series f1(x)+ f2 (x)+ + fn (x)+ is uniformly convergent in this domain, then the sum S (x) is continuous
in the same domain. If the series is not uniformly convergent in a domain, then the sum S (x) may have discontinuities in this domain. A: The sum of the series (7.80a) is discontinuous: S (x) = 0 for x = 0 and S (x) = 1 for x > 0. B: The sum of the series (7.79a) is a continuous function: The series is uniformly convergent for any nite domain it cannot have any discontinuity at any nite x. 2. Integration and Di erentiation of Uniformly Convergent Series In the domain a b] of uniform convergence it is allowed to integrate the series term-by-term. It is also allowed to di erentiate term-by-term if the result is a uniformly convergent series. That is:
Zx X 1
x0 n=1 1 X
n=1
fn(t) dt =
1 Zx X
n=1x0
fn(t) dt for x0 x 2 a b]
!0 X 1 fn(x) = fn0 (x) n=1
for x 2 a b]:
(7.82a) (7.82b)
416 7. Innite Series
7.3.3 Power series
7.3.3.1 Denition, Convergence
1. De nition The most important function series are the power series of the form 1 X a0 + a1x + a2x2 + + anxn + = anxn or n=0
a0 + a1(x ; x0 ) + a2 (x ; x0 )2 + + an(x ; x0 )n + =
X 1
n=0
an(x ; x0)n
(7.83a) (7.83b)
where the coecients ai and the centre of expansion x0 are constant numbers.
2. Absolute Convergence and Radius of Convergence A power series is convergent either only for x = x0 or for all values of x or there is a number r > 0, the radius of convergence, such that the series is absolutely convergent for jx ; x0 j < r and divergent for jx ; x0 j > r (Fig. 7.1). The radius of convergence can be calculated by the formulas an or r = lim q 1 r = nlim !1 n!1 n
an+1
janj
domain of convergence
x0−r
x0+r
x0
Figure 7.1
(7.84)
if the limits exist. If these limits do not exist, then we have to take the limit superior (lim) instead of the usual limit (see 7.6], Vol. I). At the endpoints x = +r and x = ;r for the series (7.83a) and x = x0 + r and x = x0 ; r for the series (7.83b) the series can be either convergent or divergent.
3. Uniform Convergence A power series is uniformly convergent on every subinterval jx ; x0 j
r0 < r of the domain of convergence (theorem of Abel). 2 n n + 1 = 1 i.e., r = 1: For the series 1 + x1 + x2 + + xn + we get 1r = nlim (7.85) !1 n Consequently the series is absolutely convergent in ;1 < x < +1, for x = ;1 it is conditionally convergent (see series (7.34) on p. 409) and for x = 1 it is divergent (see the harmonic series (7.16) on p. 406). According to the theorem of Abel the series is uniformly convergent in every interval ;r1 +r1], where r1 is an arbitrary number between 0 and 1.
7.3.3.2 Calculations with Power Series
1. Sum and Product Convergent power series can be added, multiplied, and multiplied by a constant factor term-by-term inside of their common domain of convergence. The product of two power series is ! X ! 1 1 X anxn bnxn = a0b0 + (a0 b1 + a1 b0)x + (a0 b2 + a1b1 + a2 b0 )x2 n=0
n=0
+(a0b3 + a1 b2 + a2b1 + a3b0 )x3 + :
2. First Terms of Some Powers of Power Series: S = a + bx + cx2 + dx3 + ex4 + fx5 +
S 2 = a2 + 2abx + (b2 + 2ac)x2 + 2(ad + bc)x3 + (c2 + 2ae + 2bd)x4 +2(af + be + cd)x5 + " ! 2! p 1 b3 x3 S = S 12 = a 12 1 + 12 ab x + 12 ac ; 18 ab 2 x2 + 21 da ; 14 abc2 + 16 a3
(7.86) (7.87) (7.88)
7.3 Function Series 417
!
2 2 4 + 1 e ; 1 bd2 ; 1 c2 + 3 b 3c ; 5 b 4 x4 + 2 a 4 a 8 a 16 a 128 a " ! 2 3! p1 = S ; 12 = a; 12 1 ; 12 ab x + 38 ab 2 ; 12 ac x2 + 34 abc2 ; 12 ad ; 165 ab 3 x3 S !
3 c2 ; 1 e ; 15 b2 c + 35 b4 x4 + + 34 bd + 2 2 3 4 a 8 a 2 a 16 a 128 a " ! ! 1 = S ;1 = a;1 1 ; b x + b2 ; c x2 + 2bc ; d ; b3 x3 2 2 3 S a a a a a a
2 2 c b4 ! 2 bd c e b + a2 + a2 ; a ; 3 a3 + a4 x4 + " ! ! 1 = S ;2 = a;2 1 ; 2 b x + 3 b2 ; 2 c x2 + 6 bc ; 2 d ; 4 b3 x3 2 2 2 3 S a a a a a a
2 2c 4! bd c e b b + 6 a2 + 3 a2 ; 2 a ; 12 a3 + 5 a4 x4 + :
(7.89)
(7.90)
(7.91)
(7.92)
3. Quotient of Two Power Series 1 X
an xn
1x + 2x2 + = a0 1 + ( ; )x + ( ; + 2 ; )x2 = ab 0 11 + 1 1 2 1 1 1 2 + 1x + 2x2 + b0 0 bn xn n=0 +(3 ; 2 1 ; 1 2 ; 3 ; 1 3 + 1 12 + 212 )x3 + ]: (7.93) We get this formula by considering the quotient as a series with unknown coecients, and after multiplying by the numerator we get the unknown coecients by coecient comparison. 4. Inverse of a Power Series If the series y = f (x) = ax + bx2 + cx3 + dx4 + ex5 + fx6 + (a 6= 0) (7.94a) is given, then its inverse is the series x = '(y) = Ay + By2 + Cy3 + Dy4 + Ey5 + Fy6 + : (7.94b) Taking powers of y and comparing the coecients we get A = a1 B = ; ab3 C = a15 (2b2 ; ac) D = a17 (5abc ; a2 d ; 5b3 ) E = a19 (6a2bd + 3a2 c2 + 14b4 ; a3e ; 21ab2 c) (7.94c) F = a111 (7a3be + 7a3 cd + 84ab3 c ; a4 f ; 28a2b2 d ; 28a2bc2 ; 42b5 ): The convergence of the inverse series must be checked in every case individually. n=0 1 X
7.3.3.3 Taylor Series Expansion, Maclaurin Series
There is a collection of power series expansions of the most important elementary functions in Table 21.5 (see p. 1015). Usually, we get them by Taylor expansion.
1. Taylor Series of Functions of One Variable
If a function f (x) has all derivatives at x = a, then it can often be represented with the Taylor formula as a power series (see 6.1.4.5, p. 390).
418 7. Innite Series a) First Form of the Representation: 2 n f (x) = f (a) + x ; a f 0(a) + (x ; a) f 00(a) + + (x ; a) f (n) (a) + :
(7.95a) 1! 2! n! This representation (7.95a) is correct only for the x values for which the remainder Rn = f (x) ; Sn tends to zero if n ! 1. This notion of the remainder is not identical to the notion of the remainder given in 7.3.1, p. 414 in general, only in the case if the expressions (7.95b) can be used. There are the following formulas for the remainder: n+1 Rn = (x(n;+a)1)! f (n+1)( ) (a < < x or x < < a) (Lagrange formula) (7.95b)
Zx Rn = n1! (x ; t)nf (n+1) (t) dt a
(Integral formula):
(7.95c)
b) Second Form of the Representation:
2 n f (a + h) = f (a) + 1!h f 0(a) + h2! f 00(a) + + hn! f (n) (a) + : The expressions for the remainder are: n+1 Rn = (nh+ 1)! f (n+1) (a + !h) (0 < ! < 1) Zh Rn = n1! (h ; t)nf (n+1) (a + t) dt: 0
(7.96a) (7.96b) (7.96c)
2. Maclaurin Series
The power series expansion of a function f (x) is called a Maclaurin series if it is a special case of the Taylor series with a = 0. It has the form 2 n f (x) = f (0) + 1!x f 0(0) + x2! f 00(0) + + xn! f (n)(0) + (7.97a) with the remainder n+1 Rn = (nx+ 1)! f (n+1) (!x) (0 < ! < 1) (7.97b)
Zx Rn = n1! (x ; t)nf (n+1) (t) dt: 0
(7.97c)
The convergence of the Taylor series and Maclaurin series can be proven either by examining the remainder Rn or determining the radius of convergence (see 7.3.3.1, p. 416). In the second case it can happen that although the series is convergent, the sum S (x) is not equal to f (x). For instance for the 1 function f (x) = e; x2 for x 6= 0, and f (0) = 0 the Maclaurin series is the identically zero series.
7.3.4 Approximation Formulas
Considering only a neighborhood small enough of the centre of the expansion, we can introduce rational approximation formulas for several functions with the help of the Taylor expansion. The rst few terms of some functions are shown in Table 7.3. The data about accuracy are given by estimating the remainder. Further possibilities for approximate representation of functions, e.g., by interpolation and tting polynomials or spline functions, can be found in 19.6, p. 917 and 19.7, p. 931.
7.3 Function Series 419
Table 7.3 Approximation formulas for some frequently used functions
Tolerance interval for x with an error of 0:1% 1% 10% from to from to from to 3 x ; 0 : 077 0 : 077 ; 0 : 245 0 : 245 ; 0 : 786 0 : 786 sin x x ; 6 ;4:4 4:4 ;14:0 14:0 ;45:0 45:0 3 5 ; 0 : 580 0 : 580 ; 1:005 ;1:632 1:632 x x sin x x ; 6 + 120 ;33:2 33:2 ;157:005 :6 57:6 ;93:5 93:5 2 0:045 ;0:141 0:141 ;0:415 0:415 cos x 1 ; x2 ;;02:045 :6 2:6 ;8:1 8:1 ;25:8 25:8 2 4 ; 0 : 386 0 : 386 ; 0 : 662 0 : 662 ; 1:036 x x cos x 1 ; 2 + 24 ;22:1 22:1 ;37:9 37:9 ;159:036 :3 59:3 3 0:054 ;0:172 0:172 ;0:517 0:517 tan x x + x3 ;;03:054 :1 3:1 ;9:8 9:8 ;29:6 29:6 3 x 2 ; 0 : 293 0 : 293 ; 0 : 519 0 : 519 ; 0:895 tan x x + 3 + 15 x5 ;16:8 16:8 ;29:7 29:7 ;051:895 :3 51:3 2 p2 a + x a + 2xa ; 8xa3 ;0:085a2 0:093a2 ;0:247a2 0:328a2 ;0:607a2 1:545a2 ! Approximate formula
2 = 12 a + a a+ x p 21 a1 ; 2xa3 a +x 1 1 x a + x a ; a2 ex 1 + x
ln(1 + x) x
Next term
+ 38xa5
2
+ xa3 2 + x2 2 ; x2 2
;0:051a2 0:052a2 ;0:157a2 0:166a2 ;0:488a2 0:530a2 ;0:031a 0:031a ;0:099a 0:099a ;0:301a 0:301a ;0:045
0:045
;0:134
0:148
;0:375
0:502
;0:002
0:002
;0:020
0:020
;0:176
0:230
7.3.5 Asymptotic Power Series
Even divergent series can be useful for calculation of substitution values of functions. In the following we consider some asymptotic power series with respect to x1 to calculate the values of functions for large values of jxj.
7.3.5.1 Asymptotic Behavior
Two functions f (x) and g(x), de ned for x0 < x < 1, are called asymptotically equal for x ! 1 if f (x) = 1 (7.98a) or f (x) = g(x) + o(g(x)) for x ! 1 (7.98b) xlim !1 g (x) hold. Here, o(g(x)) is the Landau symbol \little o" (see 2.1.4.9, p. 56). If (7.98a) or (7.98b) is ful lled, then we write also f (x) ! g(x).
420 7. Innite Series
p A: x2 + 1 ! x:
B: e x1 ! 1:
7.3.5.2 Asymptotic Power Series
C: 4x33x++x 2+ 2 ! 43x2 :
1. Notion of Asymptotic Series
a A series P1 =0 x is called an asymptotic power series of the function f (x) de ned for x > x0 if n X f (x) = xa + O xn1+1 (7.99) =0 1 holds for every n = 0 1 2 : : : . Here, O xn+1 is the Landau symbol \big O". For (7.99) we also write 1 X f (x) xa . =0
2. Properties of Asymptotic Power Series
a) Uniqueness: If for a function f (x) the asymptotic power series exists, then it is unique, but a func-
tion is not uniquely determined by an asymptotic power series. b) Convergence: Convergence is not required from an asymptotic power series. 1 X A: e x1 !1x is an asymptotic series, which is convergent for every x with jxj > x0 (x0 > 0). =0 Z 1 ;xt B: The integral f (x) = 0 1e + t dt (x > 0) is convergent for x > 0 and repeated partial integration results in the representation f (x) = 1 ; 1!2 + 2!3 ; 3!4 + (;1)n;1 (n ;n 1)! + Rn(x) x x x x x Z 1 e;xt n! Z 1 e;xt dt = n! we get with Rn (x) = (;1)n xnn! dt . Because of j R ( x ) j n xn 0 xn+1 0 (1 + t)n+1 1 Rn(x) = O xn+1 , and with this estimation Z 1 e;xt 1 X ! : dt
(;1) x +1 (7.100) 1+t 0
=0
The asymptotic power series (7.100) is divergent for every x, because the absolute value of the quotient 1 of the (n + 1)-th and of the n-th terms has the value n + x . However, this divergent series can be used for a reasonably good approximation of f (x). For instance, for x = 10 with the partial sums S4(10) Z 1 e;10t and S5(10) we get the estimation 0:09152 < 1 + t dt < 0:09164. 0
7.4 Fourier Series
7.4.1 Trigonometric Sum and Fourier Series 7.4.1.1 Basic Notions
1. Fourier Representation of Periodic Functions
Often it is necessary or useful to represent a given periodic function f (x) with period T exactly or approximatively by a sum of trigonometric functions of the form sn(x) = a20 + a1 cos !x + a2 cos 2!x + + an cos n!x +b1 sin !x + b2 sin 2!x + + bn sin n!x: (7.101)
7.4 Fourier Series 421
This is called the Fourier expansion. Here the frequency is ! = 2 . In the case T = 2 we have ! = 1. T We can get the best approximation of f (x) in the sense given on 7.4.1.2, p. 421 by an approximation function sn(x), where the coecients ak and bk (k = 0 1 2 : : : n) are the Fourier coecients of the given function. They are determined with the Euler formulas T= xZ0 +T ZT Z2 ak = T2 f (x) cos k!x dx = T2 f (x) cos k!x dx = T2 f (x) + f (;x)] cos k!x dx (7.102a)
and
x0
0
0
T= xZ0 +T ZT Z2 bk = T2 f (x) sin k!x dx = T2 f (x) sin k!x dx = T2 f (x) ; f (;x)] sin k!x dx x0
0
0
(7.102b)
or approximatively with the help of methods of harmonic analysis (see 19.6.4, p. 927).
2. Fourier Series
If there are x values such that the sequence of functions sn(x) tends to a limit s(x) for n ! 1, then the given function has a convergent Fourier series for these x values. This can be written in the form s(x) = a20 + a1 cos !x + a2 cos 2!x + + an cos n!x + +b1 sin !x + b2 sin 2!x + + bn sin n!x + (7.103a) and also in the form s(x) = a20 + A1 sin(!x + '1) + A2 sin(2!x + '2) + + An sin(n!x + 'n) + (7.103b) where in theqsecond case: Ak = ak 2 + bk 2 tan 'k = ab k : (7.103c) k
3. Complex Representation of the Fourier Series In many cases the complex form is very useful:
s(x) =
+ X1
ck eik!x
81 > > a0 for k = 0 , > T > Z < 21 1 ;ik!x ck = T f (x)e dx = > (ak ; ibk ) for k > 0 , 2 > 0 > > : 1 (a;k + ib;k ) for k < 0 . 2 k=;1
(7.104a) (7.104b)
7.4.1.2 Most Important Properties of the Fourier Series 1. Least Mean Squares Error of a Function
If a function f (x) is approximated by a trigonometric sum n n X X sn(x) = a20 + ak cos k!x + bk sin k!x (7.105a) k=1 k=1 also called the Fourier sum, then the mean square error (see 19.6.2.1, p. 919, and 19.6.4.1, 2., p. 928) ZT (7.105b) F = T1 f (x) ; sn(x)]2 dx 0
422 7. Innite Series is smallest if we de ne ak and bk as the Fourier coecients (7.102a,b) of the given function.
2. Convergence of a Function in the Mean, Parseval Equation The Fourier series converges in mean to the given function, i.e.,
ZT 0
f (x) ; sn(x)]2 dx ! 0 for n ! 1
(7.106a)
holds, if the function is bounded and in the interval 0 < x < T it is piecewise continuous. A consequence of the convergence in the mean is the Parseval equation: 1 2 ZT f (x)]2 dx = a20 + X 2 2 (7.106b) T0 2 k=1(ak + bk ):
3. Dirichlet Conditions
If the function f (x) satis es the Dirichlet conditions, i.e., a) the interval of de nition can be decomposed into a nite number of intervals where the function f (x) is continuous and monotone, and b) at every point of discontinuity of f (x) the values f (x + 0) and f (x ; 0) are de ned, then the Fourier series of this function is convergent. At the points where f (x) is continuous the sum is equal to f (x), at the points of discontinuity the sum is equal to f (x ; 0) + f (x + 0) . 2
4. Asymptotic Behavior of the Fourier Coe cients
If a periodic function f (x) and its derivatives up to k-th order are continuous, then for n ! 1 both expressions annk+1 and bn nk+1 tend to zero. f(x)
f(x)
f(x)
0 0
T 2
T
T 2
T
x
x
Figure 7.2
Figure 7.3
0 T 2
T
x
Figure 7.4
7.4.2 Determination of Coecients for Symmetric Functions 7.4.2.1 Di erent Kinds of Symmetries
1. Symmetry of the First Kind If f (x) is an even function, i.e., if f (x) = f (;x) (Fig. 7.2), then
for the coecients we have ZT=2 ak = T4 f (x) cos k 2Tx dx 0
bk = 0
(k = 0 1 2 : : :) :
(7.107)
2. Symmetry of the Second Kind If f (x) is an odd function, i.e., if f (x) = ;f (;x) (Fig. 7.3), then for the coecients we have ZT=2 (7.108) ak = 0 bk = T4 f (x) sin k 2Tx dx (k = 0 1 2 : : :): 0
7.4 Fourier Series 423
3. Symmetry of the Third Kind If f (x + T=2) = ;f (x) holds (Fig. 7.4), then the coecients
are
T= Z2 a2k+1 = T4 f (x) cos(2k + 1) 2Tx dx
b2k+1 = T4
a2k = 0
0 T= Z2 0
(7.109a)
f (x) sin(2k + 1) 2Tx dx b2k = 0 (k = 0 1 2 : : :):
(7.109b)
4. Symmetry of the Fourth Kind If the function f (x) is odd and also the symmetry of the third kind is satis ed (Fig. 7.5a), then the coecients are T= Z4 ak = b2k = 0 b2k+1 = T8 f (x) sin(2k + 1) 2Tx dx (k = 0 1 2 : : :):
(7.110)
0
If the function f (x) is even and also the symmetry of the third kind is satis ed (Fig.7.5b), then the coecients are T= Z4 bk = a2k = 0 a2k+1 = T8 f (x) cos(2k + 1) 2Tx dx (k = 0 1 2 : : :): (7.111) 0 f(x)
f(x)
0 T T 4 2
T
0 T T 4 2
x
a)
x
T
b)
Figure 7.5 f(x) f(x)
0
f(x)
f(x) l
2l f1(x)
Figure 7.6
0
l
2l f2(x)
x
x
Figure 7.7
7.4.2.2 Forms of the Expansion into a Fourier Series
Every function f (x), satisfying the Dirichlet conditions in an interval 0 x l (see 7.4.1.2, 3., p. 422), can be expanded in this interval into a convergent series of the following forms: 2x 2x 1: f1(x) = a20 + a1 cos 2x l + a2 cos 2 l + + an cos n l + 2x 2x (7.112a) + b1 sin 2x l + b2 sin 2 l + + bn sin n l + :
424 7. Innite Series The period of the function f1(x) is T = l in the interval 0 < x < l the function f1 (x) coincides with the function f (x) (Fig. 7.6). At the points of discontinuity the substitution values are f (x) = 1 f (x ; 0) + f (x + 0)]. The coecients of the expansion are determined with the Euler formulas 2 (7.102a,b) for ! = 2 . l 2: f2(x) = a20 + a1 cos xl + a2 cos 2 xl + + an cos n xl + : (7.112b) The period of the function f2 (x) is T = 2l in the interval 0 x l the function f2 (x) has a symmetry of the rst kind and it coincides with the function f (x) (Fig. 7.7). The coecients of the expansion of f2 (x) are determined by the formulas for the case of symmetry of the rst kind with T = 2l. 3: f3(x) = b1 sin xl + b2 sin 2 xl + + bn sin n xl + : (7.112c) The period of the function f3 (x) is T = 2l in the interval 0 < x < l the function f3 (x) has a symmetry of the second kind and it coincides with the function f (x) (Fig. 7.8). The coecients of the expansion are determined by the formulas for the case of symmetry of the second kind with T = 2l.
0
f(x)
y l
2l
2
f(x)
x
0
T T 4 2
Tx
f3(x)
Figure 7.8
Figure 7.9
7.4.3 Determination of the Fourier Coecients with Numerical Methods
If the periodic function f (x) is a complicated one or in the interval 0 x < T its values are known only for a discrete system xk = kT N with k = 0 1 2 : : : N ; 1, then we have to approximate the Fourier coecients. Furthermore, e.g., also the number of measurements N can be a very large number. In these cases we use the methods of numerical harmonic analysis (see 19.6.4, p. 927).
7.4.4 Fourier Series and Fourier Integrals 1. Fourier integral
If the function f (x) satis es the Dirichlet conditions (see 7.4.1.2, 3., p. 422) in an arbitrarily nite interval and, moreover, the integral
+ Z1
;1
jf (x)j dx is convergent (see 8.2.3.2, 1., p. 454), then the following
formula holds (Fourier integral): + + Z1 Z1 Z1 +Z 1 f (x) = 21 ei!x d! f (t)e;i!t dt = 1 d! f (t) cos !(t ; x) dt: ;1 ;1 ;1 0 At the points of discontinuity we substitute f (x) = 12 f (x ; 0) + f (x + 0)]:
(7.113a) (7.113b)
7.4 Fourier Series 425
2. Limiting Case of a Non-Periodic Function
The formula (7.113a) can be regarded as the expansion of a non-periodic function f (x) into a trigonometric series in the interval (;l +l) for l ! 1. With Fourier series expansion a periodic function with period T is represented as the sum of harmonic vibrations with frequency !n = n 2 with n = 1 2 : : : and with amplitude An. This representation is T based on a discrete frequency spectrum. With the Fourier integral the non-periodic function f (x) is represented as the sum of in nitely many harmonic vibrations with continuously varying frequency !. The Fourier integral gives an expansion of the function f (x) into a continuous frequency spectrum. Here the frequency ! corresponds to the density g(!) of the spectrum: + Z1 g(!) = 21 f (t)e;i!t dt: (7.113c) ;1 The Fourier integral has a simpler form if f (x) is either a) even or b) odd: Z1 Z1 (7.114a) a) f (x) = 2 cos !x d! f (t) cos !t dt 0 0
Z1
Z1
0
0
b) f (x) = 2 sin !x d! f (t) sin !t dt:
(7.114b)
The density of the spectrum of the even function f (x) = e;jxj and the representation of this function are Z1 Z 1 cos !x g(!) = 2 e;t cos !t dt = 2 !2 1+ 1 (7.115a) and e;jxj = 2 d!: (7.115b) 0 0 !2 + 1
7.4.5 Remarks on the Table of Some Fourier Expansions
In Table 21.6 there are given the Fourier expansions of some simple functions, which are de ned in a certain interval and then they are periodically extended. The shapes of the curves of the expanded functions are graphically represented.
1. Application of Coordinate Transformations
Many of the simplest periodic functions can be reduced to a function represented in Table 21.6 when we either change the scale (unit of measure) of the coordinate axis or we translate the origin. A function f (x) = f (;x) de ned by the relations 8 T > < 2 for 0 < x < 4 , y=> (7.116a) : 0 for T < x < T 4 2 (Fig. 7.9), can be transformed into the form 5 given in Table 21.6, if we substitute a = 1 and we introduce the new variables Y = y ; 1 and X = 2Tx + 2 . By the substitution of the variables in series 5, because sin(2n + 1) 2Tx + 2 = (;1)n cos(2n + 1) 2Tx we get for the function (7.116a) the expression (7.116b) y = 1 + 4 cos 2Tx ; 13 cos 3 2Tx + 15 cos 5 2Tx ; :
426 7. Innite Series
2. Using the Series Expansion of Complex Functions
Many of the formulas given in Table 21.6 for the expansion of functions into trigonometric series can be derived from power series expansion of functions of a complex variable. The expansion of the function 1 = 1 + z + z2 + (jzj < 1) (7.117) 1;z yields for z = aei' (7.118) after separating the real and imaginary parts a cos ' 1 + a cos ' + a2 cos 2' + + an cos n' + = 1 ;12;a cos ' + a2 a sin ' (7.119) a sin ' + a2 sin 2' + + an sin n' + = 1 ; 2a cos ' + a2 for jaj < 1:
427
8 IntegralCalculus
1. Integral Calculus and Inde nite Integrals Integration represents the inverse operation of 0
di erentiation in the following sense: While di erentiation calculates the derivative function f (x) of a given function f (x), integration determines a function whose derivative f 0(x) is previously given. This process does not have a unique result, so we get the notion of an indenite integral. 2. De nite Integral If we start with the graphical problem of the integral calculus, to determine the area between the curve of y = f (x) and the x-axis, and for this purpose we approximate it with thin rectangles (Fig. 8.1), then we get the notion of the denite integral. 3. Connection Between De nite and Inde nite Integrals The relation between these two types of integral is the fundamental theorem of calculus (see 8.2.1.2, 1., p. 442).
8.1 Indenite Integrals
8.1.1 Primitive Function or Antiderivative 1. Denition
Consider a function y = f (x) given on an interval a b]. F (x) is called a primitive function or antiderivative of f (x) if F (x) is di erentiable everywhere on a b] and its derivative is f (x): F 0(x) = f (x): (8.1) Because under di erentiation an additive constant disappears, a function has in nitely many primitive functions, if it has any. The di erence of two primitive function is a constant. So, the graphs of all primitive functions F1(x) F2 (x) : : : Fn(x) can be got by parallel translation of a particular primitive function in the direction of the ordinate axis (Fig. 8.2). y
y
f(x)
y=f(x)
0 a=x0 x1 x2
xk-1 xk xk xk+1
xn-1 xn=b x
Figure 8.1 y
x
0 y
F1(x)
y=F(x)
F2(x) F3(x) 0
x
Figure 8.2
2. Existence
x
0
Figure 8.3
Every function continuous on a connected interval has a primitive function on this interval. If there are some discontinuities, then we decompose the interval into subintervals in which the original function is continuous (Fig. 8.3). The given function y = f (x) is in the upper part of the gure the function
428 8. Integral Calculus y = F (x) in the lower part is a primitive function of it on the considered intervals.
8.1.1.1 Indenite Integrals
The indenite integral of a given function f (x) is the set of primitive functions Z F (x) + C = f (x) dx:
Z
(8.2)
The function f (x) under the integral sign is called the integrand , x is the integration variable , and C is the integration constant . It is also a usual notation, especially in physics, to put the di erential dx right after the integral sign and so before the function f (x). Table 8.1 Basic integrals
Z Z Z Z Z Z
Powers xn+1
xn dx = n + 1 (n 6= ;1) dx = ln jxj x
Trigonometric functions
sin x dx = ; cos x cos x dx = sin x tan x dx = ; ln j cos xj
cot x dx = ln j sin xj dx2 = tan x Z cos x dx sin2 x = ; cot x
Z
Z
Fractional rational functions
Z Z Z Z Z Z
Exponential functions ex dx = ex x ax dx = lna a
Hyperbolic functions
sinh x dx = cosh x cosh x dx = sinh x tanh x dx = ln j cosh xj
coth x dx = ln j sinh xj dx2 = tanh x Z cosh x dx sinh2 x = ; coth x
Z
Z
Irrational functions
dx 2 = 1 arctan x p 2dx 2 = arcsin xa 2 a a ;x Z a +x a Z p 2 2 a + x dx 1 x 1 dx x p a2 ; x2 = a Artanh a = 2a ln a ; x a2 + x2 = Arsinh a = ln x + x + a (for jxj < a) Z Z dx 1 x 1 ln x ; a p dx = Arcosh x = ln x + px2 ; a2 = ; Arcoth = 2 2 a a a 2a x + a x ;a x2 ; a2 (for jxj > a)
8.1.1.2 Integrals of Elementary Functions
1. Basic Integrals
The integration of elementary functions in analytic form is reduced to a sequence of basic integrals. These basic integrals can be got from the derivatives of well-known elementary functions, since inde nite integration means the determination of a primitive function F (x) of the function f (x). The collection of integrals given in Table 8.1 comes from reversing the di erentiation formulas in Table 6.1 (Derivatives of elementary functions). The integration constant C is omitted.
8.1 Indenite Integrals 429
2. General Case
For the solution of integration problems, we try to reduce the given integral by algebraic and trigonometric transformations, or by using the integration rules to basic integrals. The integration methods given in section 8.1.2 make it possible in many cases to integrate those functions which have an elementary primitive function. The results of some integrations are collected in Table 21.7 (Inde nite integrals). The following remarks are very useful in integration: a) The integration constant is mostly omitted. Exceptions are some integrals, which in di erent forms can be represented with di erent arbitrary constants. b) If in the primitive function there is an expression containing ln f (x), then we have to consider always ln jf (x)j instead of it. c) If the primitive function is given by a power series, then the function cannot be integrated in an elementary fashion. A wide collection of integrals and their solutions are given in 8.1] and 8.2].
8.1.2 Rules of Integration
The integral of an integrand of arbitrary elementary functions is not usually an elementary function. In some special cases we can use some tricks, and by practice we can gain some knowledge of how to integrate. Today we leave the calculation of integrals mostly to computers. The most important rules of integration, which are nally discussed here, are collected in Table 8.2.
1. Integrand with a Constant Factor
A constant factor in the integrand can be factored out in front of the integral sign (constant multiple rule ): Z Z f (x) dx = f (x) dx: (8.3)
2. Integration of a Sum or Di erence
The integral of a sum or di erence can be reduced to the integration of the separate terms if we can tell their integrals separately (sum rule ): Z Z Z Z (u + v ; w) dx = u dx + v dx ; w dx: (8.4) The variables u v w are functions of x. Z Z 5 3 2 (x + 3)2 (x2 + 1) dx = (x4 + 6x3 + 10x2 + 6x + 9) dx = x5 + 32 x4 + 10 3 x + 3 x + 9x + C .
3. Transformation of the Integrand
The integration of a complicated integrand can sometimes be reduced to a simpler integral by algebraic or trigonometric transformations. Z Z sin 2x cos x dx = 21 (sin 3x + sin x) dx.
4.Z Linear Transformation in the Argument
If f (x) dx = F (x) is known, e.g., from an integral table, then we get:
Z
Z
f (ax) dx = a1 F (ax) + C
(8.5a)
Z
f (x + b) dx = F (x + b) + C
f (ax + b) dx = a1 F (ax + b) + C: Z Z A: sin ax dx = ; a1 cos ax + C . B: eax+b dx = a1 eax+b + C .
(8.5b) (8.5c)
430 8. Integral Calculus C:
Z
dx = arctan(x + a) + C . 1 + (x + a)2 Table 8.2 Important rules of calculation of inde nite integrals
Rule
Formula for integration
Integration constant
f (x) dx = F (x) + C (C const) F 0(x) = dF dx = fZ(x) Z f (x) dx = f (x) dx ( const) Z Z Z u(x) v(x)] dx = u(x) dx v(x) dx Z Z u(x)v0(x) dx = u(x)v(x) ; u0(x)v(x) dx
Integration and di erentiation Constant multiple rule Sum rule Partial integration Substitution rule
Special form of the integrand Integration of the inverse function
Z
x = u(t) or t = v(x) uZ und v are inverse functions of each other : Z f (x) dx = f (u(t))u0(t) dt or Z Z f (x) dx = vf0((uu((tt)))) dt Z 0 1: f (x) dx = ln jf (x)j + C (logarithmic integration) Z f (x) 2: f 0 (x)f (x) dx = 1 f 2 (x) + C 2 uZ und v are inverse functions of each other : u(x) dxZ = xu(x) ; F (u(x)) + C1 with F (x) = v(x) dx + C2 (C1 C2 const)
5. Power and Logarithmic Integration
If the integrand has the form of a fraction such that in the numerator we have the derivative of the denominator, then the integral is the logarithm of the absolute value of the denominator: Z f 0(x) Z d f (x) (8.6) f (x) dx = f (x) = ln jf (x)j + C: Z A: x2 2+x3+x 3; 5 dx = ln jx2 + 3x ; 5j + C . If the integrand is a product of a power of a function multiplied by the derivative of the function, and the power is not equal to ;1, then Z Z +1 f 0(x)f (x) dx = f (x)d f (x) = f +(1x) + C ( const 6= ;1): Z B: (x2 +2x3+x ;3 5)3 dx = (;2)(x2 +1 3x ; 5)2 + C .
8.1 Indenite Integrals 431
6. Substitution Method
If x = u(t) where t = v(x) is the inverse function of x = u(t), then according to the chain rule of di erentiation we get Z Z Z Z f (x) dx = f (u(t)) u0(t) dt or f (x) dx = vf0((uu((tt)))) dt: (8.7) Z ex ; 1 1 A: ex + 1 dx. Substituting x = ln t (t > 0) dx dt = t , then taking the decomposition into partial we get: Z ex ; fractions 1 dx = Z t ; 1 dt = Z 2 ; 1 dt = 2 ln(ex + 1) ; x + C . ex + 1 t+1 t t+1 t Z x dx dt = 2x then we get Z x dx = Z dt = 1 ln(1+ x2 )+ C . B: 1 + x2 . Substituting 1+ x2 = t dx 1 + x2 2t 2
7. Partial Integration
Reversing the rule for the di erentiation of a product we get Z Z u(x)v0(x) dx = u(x) v(x) ; u0(x) v(x) dx (8.8) where u(x) and v(x) have continuous derivatives. Z The integral x ex dx can be calculated by partial integration where we choose u = x and v0 = ex,
Z
Z
so we get u0 = 1 and v = ex: xex dx = x ex ; ex dx = (x ; 1)ex + C .
8. Non-Elementary Integrals
Integrals of elementary functions are not always elementary functions. These integrals are calculated mostly in the following three ways, where the primitive function will be approximated by a given accuracy: 1. Table of Values The integrals which have a particular theoretical or practical importance but cannot be expressed by elementary functions can be given by a table of values. (Of course, the table lists values of one particular primitive function.) Such special functions usually have special names. Examples are: A: Logarithmic integral (see 8.2.5, 3., p. 460): Z x dx = Li (x): (8.9) 0 ln x B: Elliptic integral of the rst kind (see 8.1.4.3, p. 437): Z sin ' dx q = F (k '): (8.10) 2 0 (1 ; x )(1 ; k2x2 ) C: Error function (see 8.2.5, 5., p. 461): Zx p2 0 e;t2 dt = erf(x): (8.11) 2. Integration by Series Expansion We take the series expansion of the integrand, and if it is uniformly convergent, then it can be integrated term-by-term. Z A: sinx x dx (see also Sine integral p. 460). Z x B: ex dx (see also Exponential integral p. 461).
432 8. Integral Calculus 3. Graphical integration is the third approximation method, which is discussed in 8.2.1.4, 5., p. 446.
8.1.3 Integration of Rational Functions
Integrals of rational functions can always be expressed by elementary functions.
8.1.3.1 Integrals of Integer Rational Functions (Polynomials)
Integrals of integer rational functions are calculated directly by term-by-term integration: Z ( anxn + an;1xn;1 + + a1 x + a0) dx = an xn+1 + an;1 xn + + a1 x2 + a0 x + C: n+1 n 2
(8.12)
8.1.3.2 Integrals of Fractional Rational Functions
Z The integrand of an integral of a fractional rational function P (x) dx, where P (x) and Q(x) are Q(x) polynomials with degree m and n, respectively, can be transformed algebraically into a form which is easy to integrate. We perform the following steps: 1. We simplify the fraction by the greatest common divisor, so P (x) and Q(x) have no common factor. 2. We separate the integer rational part of the expression. If m n holds, then we divide P (x) by Q(x). Then we have to integrate a polynomial and a proper fraction. 3. We decompose the denominator Q(x) into linear and quadratic factors (see 1.6.3.2, p. 44): Q(x) = an(x ; )k (x ; )l (x2 + px + q)r (x2 + p0 x + q0)s (8.13a) 2 02 with p4 ; q < 0 p4 ; q0 < 0 : : : : (8.13b) 4. We factor out the constant coecient an in front of the integral sign. 5. We decompose the fraction into a sum of partial fractions: The proper fraction we get after the division, which can nolonger be simpli ed and whose denominator is decomposed into a product of irreducible factors, can be decomposed into a sum of partial fractions (see 1.1.7.3, p. 15), which are easy to integrate.
8.1.3.3 Four Cases of Partial Fraction Decomposition 1. Case: All Roots of the Denominator are Real and Single Q(x) = (x ; )(x ; ) (x ; ) a) We form the decomposition: P (x) = A + B + + L Q(x) x ; x ; x;
(8.14a) (8.14b)
B = QP0(()) : : : L = QP0(()) : (8.14c) b) The numbers A B C : : : L can also be calculated by the method of undetermined coecients (see 1.1.7.3, 1., p. 16). c) We integrate by the formula Z A dx (8.14d) x ; = A ln(x ; ): with A = QP0(())
8.1 Indenite Integrals 433
I= = ; 32 , Z I=
Z (2x + 3) dx 2x + 3 A + B + C , A = P (0) = 2x + 3 : = 3 2 x +x ; 2x x(x ; 1)(x + 2) x x ; 1 x + 2 Q0(0) 3x2 + 2x ; 2 x=0 2 x + 3 5 2 x + 3 1 B = 3x2 + 2x ; 2 = , C = 3x2 + 2x ; 2 = ;6 , x=;2 !x=1 3 5 1 3 ; 2 + 3 + ; 6 dx = ; 3 ln x + 5 ln(x ; 1) ; 1 ln(x + 2) + C = ln C (x ; 1)5=3 . 1 x x;1 x+2 2 3 6 x3=2 (x + 2)1=6
2. Case: All Roots of the Denominator are Real, Some of them with a Higher Multiplicity Q(x) = (x ; )l(x ; )m : a) We form the decomposition: P (x) = A1 + A2 + + Al Q(x) (x ; ) (x ; )2 (x ; )l B B Bm + : + (x ;1 ) + (x ;2 )2 + + (x ; )m
(8.15a)
(8.15b)
b) We calculate the constants A1 A2 : : : Al B1 B2 : : : Bm : : : by the method of undetermined coefcients (see 1.1.7.3, 1., p. 16). c) We integrate by the rule Z A1 dx Z Ak dx Ak = A1 ln (x ; ) =; (k > 1): (8.15c) x; (x ; )k (k ; 1)(x ; )k;1 Z 3 1 x3 + 1 A B1 B2 B3 I = x(xx ;+ 1) 3 dx : x(x ; 1)3 = x + x ; 1 + (x ; 1)2 + (x ; 1)3 . The method of undetermined coecients yields A + B1 = 1 ;3A ; 2B1 + B2 = 0, 3A + B1 ; B2 + B3 = 0 ;A = 1 A = ;1 B1 = 2 B2 =" 1 B3 = 2. The result of the integration
is Z 1 2 1 2 I = ; x + x ; 1 + (x ; 1)2 + (x ; 1)3 dx = ; ln x + 2 ln(x ; 1) ; x ;1 1 ; (x ;1 1)2 + C 2 = ln (x ; 1) ; x 2 + C . x (x ; 1)
3. Case: Some Roots of the Denominator are Single Complex
Suppose all coecients of the denominator Q(x) are real. Then, with a single complex root of Q(x) its conjugate complex number is a root too and we can compose them into a quadratic polynomial. Q(x) = (x ; )l(x ; )m : : : (x2 + px + q)(x2 + p0x + q0) : : : (8.16a) 2 02 with p4 < q p4 < q0 : : : because the quadratic polynomials have no real zeros. a) We form the decomposition: P (x) = A1 + A2 + + Al + B1 + B2 + + Bm Q(x) x ; (x ; )2 (x ; )l x ; (x ; ) 2 (x ; )m Ex + F Cx + D + x2 + px + q + x2 + p0x + q0 + :
(8.16b)
(8.16c)
434 8. Integral Calculus b) We calculate the constants by the method of undetermined coecients (see 1.1.7.3, 1., p. 16). +D c) We integrate the expression x2Cx + px + q by the formula Z (Cx + D) dx C D ; Cp=2 2 q qx + p=2 (8.16d) x2 + px + q = 2 ln(x + px + q) + q ; p2 =4 arctan q ; p2=4 : Z +D I = x34+dx4x : x3 +4 4x = Ax + Cx x2 + 4 . The method of undetermined coecients yields the equations Z 1 A + Cx = 0 D = 0 4A 1= 4 A = 1 C = ;1 D =C0.1x I = x ; x2 + 4 dx = ln x ; 2 ln(x2 + 4) + ln C1 = ln p 2 , where in this particular case the x +4 term arctan is missing.
4. Case: Some Roots of the Denominator are Complex with a Higher Multiplicity
Q(x) = (x ; )k (x ; )l : : : (x2 + px + q)m(x2 + p0x + q0)n : : : : (8.17a) a) We form the decomposition: P (x) A1 A2 Ak B1 B2 Bl Q(x) = x ; + (x ; )2 + + (x ; )k + x ; + (x ; )2 + + (x ; )l + D1 + C2 x + D2 + + Cmx + Dm + xC2 1+x px + q (x2 + px + q)2 (x2 + px + q)m E E Fn 1 x + F1 2 x + F2 + x2 + p0x + q0 + (x2 + p0x + q0)2 + + (x2E+nxp0+ (8.17b) x + q 0 )n : b) We calculate the constants by the method of undetermined coecients. c) We integrate the expression (xC2 m+xpx++Dqm)m for m > 1 in the following steps: ) We transform the numerator into the form (8.17c) Cmx + Dm = C2m (2x + p) + Dm ; C2mp : ) We decompose the integrand into the sum of two summands, where the rst one can be integrated directly: Z Cm (2x + p) dx Cm 1 (8.17d) 2 (x2 + px + q)m = ; 2(m ; 1) (x2 + px + q)m;1 : ) The second one will be integrated by the following recursion formula, not considering its coecient: Z dx x + p=2 (x2 + px + q)m = 2(m ; 1) (q ; p2 =4) (x2 + px + q)m;1 Z m;3 dx + 2(m ;21) (8.17e) 2 (q ; p =4) (x2 + px + q)m;1 : Z 2 2x + 13 2x2 + 2x + 13 A C1 x + D1 C2x + D2 I = (x2x; + 2)(x2 + 1)2 dx : (x ; 2)(x2 + 1)2 = x ; 2 + x2 + 1 + (x2 + 1)2 . The method of undetermined coecients results in the following system of equations: A + C1 = 0 ;2C1 + D1 = 0 2A + C1 ; 2D1 + C2 = 2 ;2C1 + D1 ; 2C2 + D2 = 2, A ; 2D1 ; 2D2 = 13 the coecients are A = 1 C1 = ;1 !D1 = ;2 C2 = ;3 D2 = ;4, Z I = x ;1 2 ; xx2 ++21 ; (x32x++1)4 2 dx.
8.1 Indenite Integrals 435
Z Z dx = 2x + 1 2dx = 2x + 1 arctan x, and According to (8.17e) we get 2 2 (x + 1) 2(x + 1) 2 x + 1 2(x + 1) 2 2 nally the result is I = 2(3x;2 +4x1) + 12 ln (xx2;+2)1 ; 4 arctan x + C . Table 8.3 Substitutions for integration of irrational functions I
Integral
Substitution
! ax + b dx cx + e ! r r Z n ax + b dx R x cx ++ eb m ax cx + e Z
R x
r
r
n
ax + b = t cx + e r ax + b = t cx + e where r is the lowest common multiple of the numbers m n : : : . One of the three Euler substitutions : p 2 p ax + bx + c = t ; ax p 2 p ax + bx + c = xt + c n
r
Z p R x ax2 + bx + c dx: 1. For a > 0 y 2. For c > 0 3. If the the polynomial ax2 + bx + c
has di erent real roots: p 2 ax2 + bx + c = a(x ; )(x ; ) ax + bx + c = t(x ; ) The symbol R denotes a rational function of the expressions in parentheses. The numbers n m : : : are integers. y If a < 0, and the polynomial p ax2 + bx + c has complex roots, then the integrand is not de ned for any value of x, since ax2 + bx + c is imaginary for every real value of x. In this case the integral is meaningless.
8.1.4 Integration of Irrational Functions
8.1.4.1 Substitution to Reduce to Integration of Rational Functions
Irrational functions cannot always be integrated in an elementary way. Table 21.7 contains a wide collection of integrals of irrational functions. In the simplest cases we can introduce substitutions, as in Table 8.3, such that the integral can be reduced to an integral of a rational function. Z p The integral R (x ax2 + bx + c) dx can be reduced to one of the following three forms
Z Z
p
R (x x2 + 2) dx
p
(8.18a)
Z
p
R (x x2 ; 2) dx
(8.18b)
(8.18c) R (x 2 ; x2 ) dx because the quadratic polynomial ax2 + bx + c can always be written as the sum or as the di erence of two complete squares. Then, we can use the substitutions given in Table 8.4. " 2 " 2 A: 4x2 +16x +17 = 4 x2 + 4x + 4 + 41 = 4 (x + 2)2 + 12 = 4 x21 + 12 with x1 = x +2.
436 8. Integral Calculus
p !2 2 p !2 B: x2 + 3x + 1 = x2 + 3x + 49 ; 54 = x + 32 ; 25 = x21 ; 25 with x1 = x + 32 . C: ;x2 + 2x = 1 ; x2 + 2x ; 1 = 12 ; (x ; 1)2 = 12 ; x21 with x1 = x ; 1. Tabelle 8.4 Substitutions for integration of irrational functions II
Integral
Z p 2 + 2 dx R x x Z p 2 ; 2 dx R x x Z p R x 2 ; x2 dx
Substitution x = sinh t or x = tan t x = cosh t or x = sec t x = sin t or x = cos t
8.1.4.2 Integration of Binomial Integrands
An expression of the form xm (a + bxn )p (8.19) is called a binomial integrand, where a and b are arbitrary real numbers, and m n p are arbitrary positive orZ negative rational numbers. The theorem of Chebyshev tells that the integral xm (a + bxn )p dx (8.20) can be expressed by elementary functions only in the following three cases: Case 1: If p is an integer, then the expression (a + bxn )p can be expanded by the binomial theorem, so the integrand after eliminating the parentheses will be a sum of terms in the form cxk , which are easy to integrate. Case 2: If m n+ 1 is an integer, then the integral (8.20) can be reduced to the integral of a rational p function by substituting t = r a + bxn , where r is the denominator of the fraction p. Case 3: If m n+ 1 + p is an integer, then the integral (8.20) can be reduced to the integral of a rational s n function by substituting t = r a +xnbx , where r is the denominator of the fraction p. Z q3 1 + p4 x Z px dx = x;1=2 1 + x1=4 1=3 dx m = ; 21 n = 14 p = 13 m n+ 1 = 2 (Case 2): A: Z q3 1 + p4 x Z q3 p4 3 4 2 3 3 px dx = 12 (t6 ; t3 ) dt Substitution t = 1 + x x = (t ; 1) dx = 12t (t ; 1) dt = 37 t4 (4t3 ; 7) + C . Z Z 3 . B: p4 x1 +dxx3 = x3 (1 + x3 );1=4 : m = 3 n = 3 p = ; 41 m n+ 1 = 34 m n+ 1 + p = 13 12 Because none of the three conditions is ful lled, the integral is not an elementary function.
8.1 Indenite Integrals 437
8.1.4.3 Elliptic Integrals
1. Indenite Elliptic Integrals
Elliptic integrals are integrals of the form Z Z q p R (x ax3 + bx2 + cx + e) dx R (x ax4 + bx3 + cx2 + ex + f ) dx: (8.21) Usually they cannot be expressed by elementary functions if it is still possible, the integral is called pseudoelliptic. The name of this type of integral originates from the fact that the rst application of them was to calculate the perimeter of the ellipse (see 8.2.2.2, 2., p. 449). The inverses of elliptic integrals are the elliptic functions (see 14.6.1, p. 702). Integrals of the types (8.21), which are not integrable in elementary terms, can be reduced by a sequence of transformations into elementary functions and integrals of the following three types (see 21.1], 21.2], 21.6]): Z Z (1 ; k2t2 ) dt dt q q (0 < k < 1) (8.22a) (0 < k < 1) (8.22b) (1 ; t2 )(1 ; k2 t2) (1 ; t2 )(1 ; k2 t2)
Z
q dt (0 < k < 1): (8.22c) (1 + nt2 ) (1 ; t2)(1 ; k2t2 ) Concerning the parameter n in (8.22c) one has to distinguish certain cases (see 14.1]). By the substitution t = sin ' 0 < ' < 2 the integrals (8.22a,b,c) can be transformed into the Legendre form : Z q d' 2 : Elliptic Integral of the First Kind: (8.23a) 1 ; k2 sin '
Elliptic Integral of the Second Kind: Elliptic Integral of the Third Kind:
Zq
Z
1 ; k2 sin2 ' d':
d' q : (1 + n sin2 ') 1 ; k2 sin2 '
(8.23b) (8.23c)
2. Denite Elliptic Integrals
De nite integrals with zero as the lower bound corresponding to the inde nite elliptic integrals are denoted by Z' Z' q q d 2 = F (k ') (8.24a) 1 ; k2 sin2 d = E (k ') (8.24b) 1 ; k2 sin 0 0
Z'
d q = +(n k ') (for all three integrals 0 < k < 1 holds). (8.24c) (1 + n sin2 ) 1 ; k2 sin2 We call these integrals incomplete elliptic integrals of the rst, second, and third kind for ' = 2 . The rst two integrals are called complete elliptic integrals, and we denote them by 0
Z2 K = F k 2 = q d 2 1 ; k2 sin 0
(8.25a)
438 8. Integral Calculus Z2 q E = E k 2 = 1 ; k2 sin2 d: 0
(8.25b)
Tables 21.9.1, 2, 3 contain the values for incomplete and complete elliptic integrals of the rst and
second kind F E and also K and E . The calculation of the perimeter of the ellipse leads to a complete elliptic integral of the second kind as a function of the numerical eccentricity e (see 8.2.2.2, 2., p. 449). For a = 1:5 b = 1 it follows that e = 0:74: Since e = k = 0:74 holds, we get from Table 21.9.3: sin = 0:74, i.e., = 47 and E (k 2 ) = E (0:74) = 1:33. It follows that U = 4aE (0:74) = 4aE ( = 47) = 4 1:33a = 7:98. Calculation with the approximation formula (3.326c) yields 7:93:
8.1.5 Integration of Trigonometric Functions 8.1.5.1 Substitution
With the substitution t2 t = tan x2 i.e. dx = 12+dtt2 sin x = 1 +2t t2 cos x = 11 ; (8.26) + t2 an integral of the form Z R (sin x cos x) dx (8.27) can be transformed into an integral of a rational function, where R denotes a rational function of its arguments. Z 1 + sin x Z 1 + 2t 2 2 2 1 t2 1 1Z 1 + t 1 + t! sin x(1 + cos x) dx = 2t 1 + 1 ; t2 dt = 2 t + 2 + t dt = 4 + t + 2 ln t + C 1 + t2 1 + t2 x 2 tan = 4 2 + tan x2 + 12 ln tan x2 + C . In some special cases we can apply simpler substitutions. If the integrand in (8.27) contains only odd powers of the functions sin x and cos x, then by the substitution t = tan x a rational function can be obtained in a simpler way.
8.1.5.2 Z Simplied Methods Case 1: Case 2:
Z Z
R (sin x) cos x dx:
Substitution t = sin x
cos x dx = dt :
(8.28)
R (cos x) sin x dx:
Substitution t = cos x
sin x dx = ;dt:
(8.29)
Case 3: sinn x dx : a) n = 2m + 1, odd: Z
Z
(8.30a)
Z
sinn x dx = (1 ; cos2 x)m sin x dx = ; (1 ; t2 )m dt with t = cos x: b) n = 2m, even: m Z Z
Z sinn x dx = 12 (1 ; cos 2x) dx = 2m1+1 (1 ; cos t)m dt with t = 2x:
(8.30b) (8.30c)
8.1 Indenite Integrals 439
We halve the power in this way. After removing the parentheses in (1 ; cos t)m we integrate term-byterm. Z Case 4: cosn x dx : (8.31a) a) n =Z 2m + 1, odd:Z Z cosn x dx = (1 ; sin2 x)m cos x dx = (1 ; t2 )m dt with t = sin x: (8.31b) b) n = 2m, even: m Z Z
Z cosn x dx = 21 (1 + cos 2x) dx = 2m1+1 (1 + cos t)m dt with t = 2x: (8.31c) We halve the power in this way. After removing the parentheses we integrate term-by-term. Z Case 5: sinn x cosm x dx : (8.32a) a) One Zof the numbers m orZn is odd: We reduce it to the cases Z 1 or 2. A: sin2 x cos5 x dx = sin2 x (1 ; sin2 x)2 cos x dx = t2(1 ; t2 )2 dt with t = sin x. Z Z dt x pt with t = cos x: B: psin cos x dx = ; b) The numbers m and n are both even: we reduce it to the cases 3 or 4 by halving the powers using the trigonometric formulas 2x 2x : sin x cos x = sin22x sin2 x = 1 ; cos cos2 x = 1 + cos (8.32b) 2 2 Z Z Z Z sin2 x cos4 x dx = (sin x cos x)2 cos2 x dx = 81 sin2 2x(1 + cos 2x) dx = 18 sin2 2x cos 2x dx + Z 1 1 3 1 1 16 (1 ; Zcos 4x) dx = 48Z sin 2x + 16 x ; 64 sin 4x +Z C . Z Case 6: tann x dx = tann;2 x(sec2 x ; 1) dx = tann;2 x (tan x)0 dx ; tann;2 x dx n;1 x Z n;2 = tan (8.33a) n ; 1 ; tan x dx: By repeating this process we decrease the power and depending on whether n is even or odd we nally get the integral Z Z dx = x or tan x dx = ; ln cos x (8.33b) respectively. Z Case 7: cotn x dx: (8.34) The solution is similar to case 6. Remark: Table 21.7, p. 1023 contains several integrals with trigonometric functions.
8.1.6 Integration of Further Transcendental Functions 8.1.6.1 Integrals with Exponential Functions
Integrals with exponential functions can be reduced to integrals of rational functions if it is given in the form Z R (emx enx : : : epx) dx (8.35a)
440 8. Integral Calculus where m n : : : p are rational numbers. We need two substitutions to calculate the integral: 1. Substitution of t = ex results in an integral Z 1 m n p (8.35b) t R (t t : : : t ) dt:
p 2. Substitution of z = r t, where r is the loweest common multiple of the denominators of the fractions m n : : : p, results in an integral of a rational function.
8.1.6.2 Integrals with Hyperbolic Functions
Integrals with hyperbolic functions, i.e., containing the functions sinh x cosh x tanh x and coth x in the integrand, can be calculated as integrals with exponential functions, if the hyperbolic Z functions are replaced by the corresponding exponential functions. The most often occurring cases sinhn x dx
Z
Z
coshn x dx sinhn x coshm x dx can be integrated in a similar way to the trigonometric functions (see 8.1.5, . p. 438).
8.1.6.3 Application of Integration by Parts
If the integrand is a logarithm, inverse trigonometric function, inverse hyperbolic function or a product of xm with ln x, eax, sin ax or cos ax or their inverses, then the solution can be got by a single or repeated integration by parts. In some cases the repeated partial integration results in an integral of the same type as the original integral. In this case we have to solve anZ algebraic equation Z with respect to this expression. We can calculate in this way, e.g., the integrals eax cos bx dx eax sin bx dx, where we need integration by parts twice. We choose the same type of function for the factor u in both steps, either the exponential or the trigonometric function. Z Z We also use integration by parts if we have integrals in the forms P (x)eax dx P (x) sin bx dx and
Z
P (x) cos bx dx, where P (x) is a polynomial. (Choosing u = P (x) the degree of the polynomial will be decreased at every step.)
8.1.6.4 Integrals of Transcendental Functions
The Table 21.7, p. 1023, contains many integrals of transcendental functions.
8.2 Denite Integrals
8.2.1 Basic Notions, Rules and Theorems
8.2.1.1 Denition and Existence of the Denite Integral
1. Denition of the Denite Integral
The de nite integral of a bounded function y = f (x) de ned on a nite closed interval a b] is a number, which is de ned as a limit of a sum, where either a < b can hold (case A) or a > b can hold (case B). In a generalization of the notion of the de nite integral (see 8.2.3, p. 453) we will consider functions de ned on an arbitrary connected domain of the real line, e.g., on an open or half-open interval, on a half-axis or on the whole numerical axis, or on a domain which is only piecewise connected, i.e., everywhere, except nitely many points. These types of integrals belong to improper integrals (see 8.2.3, 1., p. 453).
2. Denite Integral as the Limit of a Sum
We get the limit, leading to the notion of the de nite integral, by the following procedure (see Fig. 8.1, 427):
8.2 Denite Integrals 441
1. Step: The interval a b] is decomposed into n subintervals by the choice of n ; 1 arbitrary points x1 x2 : : : xn;1 so that one of the following cases occurs: a = x0 < x1 < x2 < < xi < < xn;1 < xn = b a = x0 > x1 > x2 > > xi > > xn;1 > xn = b
(case A) or
(8.36a)
(case B):
(8.36b)
2. Step: A point i is chosen in the inside or on the boundary of each subinterval as in Fig. 8.4: xi;1 i xi (in case A) or xi;1 i xi (in case B): (8.36c) xl-1 xl xn-1 xn=b ξl ξn ∆x2 ∆x1 ∆x0
x1 x2 x3 ξ1 ξ2 ξ3 ∆xn-1 ∆xl-1
a=x0
b=xn xn-1 ξn
xl
ξl
∆xn-1
xl-1
x3
x2 x1 ξ3 ξ2
ξ1
x0=a
8
8 8
-
∆xl-1
(A)
8
∆x0 ∆x1 ∆x2
-
(B)
Figure 8.4
3. Step: The value f ( i) of the function f (x) at the chosen point is multiplied by the corresponding di erence %xi;1 = xi ; xi;1, i.e., by the length of the subinterval taken with a positive sign in case A and taken with negative sign in case B. This step is represented in Fig. 8.1, p. 427 for the case A. 4. Step: Then all the n products f ( i) %xi;1 are added. 5. Step: The limit of the obtained integral approximation sum or Riemann sum n X i=1
f ( i) %xi;1
(8.37)
is calculated if the length of each subinterval %xi;1 tends to zero and consequently their number n tends to 1. Based on this, we can also denote %xi;1 as an in nitesimal quantity. If this limit exists independently of the choice of the numbers xi and i, then it is called the denite Riemann integral of the considered function on the given interval. We write
Zb a
f (x) dx = xlim i;1 !0 n!1
n X i=1
f ( i) %xi;1:
(8.38)
The endpoints of the interval are called limits of integration and the interval a b] is the integration interval a is the lower limit, b is the upper limit of integration x is called the integration variable and f (x) is called the integrand.
3. Existence of the Denite Integral
The de nite integral of a continuous function on a b] is always de ned, i.e., the limit (8.38) always exists and is independent of the choice of the numbers xi and i. Also for a bounded function having only a nite number of discontinuities on the interval a b] the de nite integral exists. The function whose de nite integral exists on a given interval is called an integrable function on this interval.
8.2.1.2 Properties of Denite Integrals
The most important properties of de nite integrals explained in the following are enumerated in Table 8.5, p. 443.
442 8. Integral Calculus
1. Fundamental Theorem of Integral Calculus
If the integrand f (x) is continuous on the interval a b], and F (x) is a primitive function, then
Zb a
Zb b f (x) dx = F 0(x) dx = F (x)a = F (b) ; F (a)
(8.39)
a
holds, i.e., the calculation of a de nite integral is reduced to the calculation of the corresponding indefinite integral, to the determination of the antiderivative: Z F (x) = f (x) dx + C: (8.40) Remark: There are integrable functions which do not have any primitive function, but we will see that, if a function is continuous, it has a primitive function.
2. Geometric Interpretation and Rule of Signs
1. Area under a Curve For all x in a b] let f (x) 0. Then the sum (8.37) can be considered as the total area of the rectangles (Fig. 8.1), p. 427, which approximate the area under the curve y = f (x).
Therefore the limit of this sum and together with it the de nite integral is equal to the area of the region A, which is bounded by the curve y = f (x), the x-axis, and the parallel lines x = a and x = b:
Zb A = f (x) dx (a < b and f (x) 0 for a x b):
(8.41)
a
2. Sign Rule If a function y = f (x) is piecewise positive or negative in the integration interval (Fig. 8.5), then the integrals over the corresponding subintervals, that is, the area parts, have positive or negative values, so the integration over the total interval yields the sum of signed areas. In Fig. 8.5a{d four cases are represented with the di erent possibilities of the sign of the area. y
y
y
y
a 0
0
a b x f(x)>0, a0, a>b
c) Figure 8.5
f(x) b (see 8.2.1.1, 2., p. 441) Zb a
f (x) dx = (b ; a)f ( )
(8.48)
is valid. The geometric meaning of this theorem is that between the points a and b there exists at least one point
8.2 Denite Integrals 445
such that the area of the gure ABCD is equal to the area of the rectangle AB 0C 0D in Fig. 8.8. The value Zb m = b ;1 a f (x) dx (8.49) a is called the mean value or the arithmetic average of the function f (x) in the interval a b] .
2. Generalized Mean Value Theorem If the functions f (x) and '(x) are continuous on the closed interval a b], and '(x) does not change its sign in this interval, then there exists at least one point such that Zb a
Zb f (x)'(x) dx = f ( ) '(x) dx
(a < < b)
a
is valid.
(8.50)
5. Estimation of the Denite Integral The value of a de nite integral lies between the values of the products of the in mum m and the supremum M of the function on the interval a b] multiplied by the length of the interval:
Zb
m(b ; a) f (x) dx M (b ; a):
y M
(8.51)
a
If f is continuous, then m is the minimum and M is the maximum of the function. It is easy to recognize the geometrical interpretation of this theorem in Fig. 8.9.
m 0 a
bx
Figure 8.9
8.2.1.4 Evaluation of the Denite Integral 1. Principal Method
The principal method of calculating a de nite integral is based on the fundamental theorem of integral calculus, i.e., the calculation of the inde nite integral (see 8.2.1.2, 1., p. 442), e.g., using Table 21.7. Before substituting the limits we have to check if we have an improper integral. Nowadays we have computer algebra systems to determine analytically inde nite and de nite integrals (see Chapter 20).
2. Transformation of Denite Integrals
In many cases, de nite integrals can be calculated by appropriate transformations, with the help of the substitution method or partial integration. Z ap A: Use the substitution method for I = 0 a2 ; x2 dx. First we substitute: x = '(t) = a sin t t = (x) = arcsin xa (0) = 0 (a) = 2 . We get: Z arcsin 1 q Z =2 Z =2 1 Z ap (1 + cos 2t) dt. a2 ; x2 dx = a2 1 ; sin2 t cos t dt = a2 cos2 t dt = a2 I= arcsin 0 0 0 0 2 z With the further substitution t = '(z) = 2 z = (t) = 2t (0) = 0 2 = we get: 2 2Z 2 2 2 I = a2 tj02 + a4 cos z dz = a4 + a4 sin zj0 = a4 . 0
B: Method of partial integration:
Z1 0
1 Z 1 x ex dx = xex 0 ; ex dx = e ; (e ; 1) = 1. 0
446 8. Integral Calculus
3. Method for Calculation of More Di cult Integrals
If the determination of an inde nite integral is too dicult and complicated, or it is not possible to express it in terms of elementary functions, then there are still some further ideas to determine the value of the integral in several cases. Here, we mention integration of functions with complex variables (see the examples on p. 694{697) or the theorem about the di erentiation of an integral with respect to a parameter (see 8.2.4, p. 459): d Zb f (x t) dx = Zb @ f (x t) dx: (8.52) dt a @t a Z1 Z1 t I = xln;x1 dx. Introducing the parameter t: F (t) = xln;x1 dx F (0) = 0 F (1) = I . 0 Z 1 @ " xt ; 1 Z 1 xt ln x 0 Z 1 1 d F Using (8.52) for F (t): d t = @ t ln x dx = dx = xt dx = t +1 1 xt+1 0 = t +1 1 . 0 0 ln x 0 Zt t Integration: F (t) ; F (0) = t d+t1 = ln(t + 1)0 = ln(t + 1). Result: I = F (1) = ln 2. 0
4. Integration by Series Expansion
If the integrand f (x) can be expanded into a uniformly convergent series f (x) = '1(x) + '2(x) + + 'n(x) + in the integration interval a b], then the integral can be written in the form Z Z Z Z f (x) dx = '1(x) dx + '2 (x) dx + + 'n(x) dx + : In this way the de nite integral can be represented as a convergent numerical series:
Zb a
Zb Zb Zb f (x) dx = '1(x) dx + '2 (x) dx + + 'n(x) dx + : a
a
a
(8.53) (8.54) (8.55)
When the functions 'k (x) are easy to integrate, if, e.g., f (x) can be expanded in a power series, which Zb is uniformly convergent in the interval a b], then the integral f (x) dx can be calculated to arbitrary a accuracy. Z 1=2 2 2 Calculate the integral I = e;x dx with an accuracy of 0:0001. The series e;x2 = 1 ; x1! + 0 x4 ; x6 + x8 ; is uniformly convergent in any nite interval according to the Abel theorem (see 2! 3! 4! Z ! 2 4 6 8 2 7.3.3.1, p. 416), so e;x dx = x 1 ; 1!x 3 + 2!x 5 ; 3!x 7 + 4!x 9 ; holds. With this result it Z 1=2 2 follows that I = e;x dx = 12 1 ; 22 11! 3 + 24 12! 5 ; 26 13! 7 + 28 14! 9 ; 1 10 1 + 1 ; . To achieve the accuracy 0:0001 for the calculation of the + 160 ; 2688 = 12 1 ; 12 55296 integral it is enough to consider the rst four terms, according to the theorem of Leibniz about alternating series (see 7.2.3.3, 1., p. 410): Z 1=2 2 I 12 (1 ; 0:08333 + 0:00625 ; 0:00037) = 21 0:92255 = 0:46127 e;x dx = 0:4613. 0
5. Graphical Integration
Graphical integration is a graphical method to integrate a function y = f (x) which is given by a curve Zb AB (Fig. 8.10), i.e., to calculate graphically the integral f (x) dx, the area of the region M0 ABN : a
8.2 Denite Integrals 447 Mn
y
B
An
P
M3 A3 A2 M2 A1 A M1 0 M0 x1 x2 x3 x1/ x3/ x5/ 2
2
N xn x
2
Figure 8.10
1. The interval M0 N is divided by the points
x1=2 x1 x3=2 x2 : : : xn;1 xn;1=2 into 2n equal parts, where the result is more accurate if there are more points of division. 2. At the points of division x1=2 x3=2 : : : xn;1=2 we draw vertical lines intersecting the curve. The ordinate values of the segments are denoted on the y-axis by OA1 OA2 : : : OAn: 3. A segment OP of arbitrary length is placed on the negative x-axis, and P is connected with the points A1 A2 : : : An. 4. Through the point M0 a line segment is drawn parallel to PA1 to the intersection point with the line x = x1 this is the segment M0M1 . Through the point M1 the segment M1 M2 is drawn parallel to PA2 to the intersection with the line x = x2 , etc., until the last point Mn is reached with the abscissa xn . The integral is numerically equal to the product of the length of OP and the length of NMn : Zb f (x) dx = OP NMn : (8.56) a
By a suitable choice of the arbitrary segment OP the shape of our result can be inuenced the smaller Zb the graph we want, the longer we should choose the segment OP . If OP = 1, then f (x) dx = NMn , a and the broken line M0 M1 M2 : : : Mn represents approximately Z the graph of a primitive function of f (x), i.e., one of the functions given by the inde nite integral f (x) dx.
6. Planimeter and Integraph
A planimeter is a tool to nd the area bounded by a closed plane curve, thus also to compute a de nite integral of a function Z Z y = f (xZ) given by the curve. Special types of planimeters can evaluate not only y dx, but also y2 dx and y3 dx. An integraph is a device which can be used to draw the graph of a primitive function Y = the graph of a function y = f (x) is given (see 19.29]).
Zx a
f (t) dt if
7. Numerical Integration
If the integrand of a de nite integral is too complicated, or the corresponding inde nite integral cannot be expressed by elementary functions, or we have the values of the function only at discrete points, e.g., from a table of values, then we use the so-called quadrature formulas or other methods of numerical mathematics (see 19.3.1, p. 898).
8.2.2 Application of De nite Integrals
8.2.2.1 General Principles for Application of the Denite Integral
1. We decompose the quantity we want to calculate into a large number of very small quantities, i.e.,
into in nitesimal quantities: A = a1 + a2 + + an: (8.57) 2. We replace every one of these in nitesimal quantities ai by a quantity a~i, which di ers only very slightly in value from ai , but which can be integrated by known formulas. Here the error i = ai ; a~i should be an in nitesimal quantity of higher order than ai and a~i. 3. We express a~i by a variable x and a function f (x) so that a~i has the form f (xi ) %xi. 4. We evaluate the desired quantity as the limit of the sum
448 8. Integral Calculus A = nlim !1
n X i=1
a~i = nlim !1
n X i=1
Zb f (xi)%xi = f (x) dx a
(8.58)
H
hi
where %xi 0 holds for every i. The lower and upper limit for x is denoted by a and b. We evaluate the volume V of a pyramid with base area S and height H (Fig. 8.11a-c): a) We decompose the required volume V by plane sections into frustums (Fig. 8.11a): V = v1 + v2 + + vn. b) We replace every frustum by a prism, whose volume is v~i, with the same height and with a base area of the top base of the frustum (Fig. 8.11b). The di erence of their volumes is an in nitesimal quantity of higher order than vi . c) We represent the volume v~i in the form v~i = Si %hi , where hi (Fig. 8.11c) is the distance of the top surface from the vertex of the pyramid. Since Si : S = h2i : H 2 2 Si i we can write: v~i = Sh Dhi H 2 %hi . d) We calculate the limit of the sum S c) b) a) ZH 2 n n Sh2 X X i %h = Sh dh = SH : V = lim v ~ = lim i i 2 n!1 i=1 n!1 i=1 H Figure 8.11 H2 3 0
8.2.2.2 Applications in Geometry 1. Area of Planar Figures
1. Area of a Curvilinear Trapezoid Between B and C (Fig. 8.12a) if the curve is given by an equation in explicit form (y = f (x) and a x b) or in parametric form (x = x(t) y = y(t) t1 t t2 ): Zb Zt2 SABCD = f (x) dx = y(t)x0(t) dt: a t1
(8.59a)
2. Area of a Curvilinear Trapezoid Between G and H (Fig. 8.12b) if the curve is given by an equation in explicit form (x = g(y) and y ) or in parametric form (x = x(t) y = y(t) t1 t t2 ): Z Zt2 SEFGH = g(y) dy = x(t)y0(t) dt: (8.59b) t1 3. Area of a Curvilinear Sector (Fig. 8.12c), bounded by a curve between K and L, which is given by an equation in polar coordinates ( = ('), '1 ' '2): Z2 SOKL = 12 2 d': '
'1
(8.59c)
Areas of more complicated gures can be calculated by partition of the area into simple parts, or by line integrals (see 8.3, p. 462) or by double integrals (see 8.4.1, p. 471).
8.2 Denite Integrals 449 y
C (t2) B
(t1)
A 0 a
a)
y
β dy α
D dx b x
y (t2)
F (t1)
E
G dϕ
H
0
x
b)
2. Arclength of Plane Curves
L
K
0 ϕ ϕ 1 2
c)
x
Figure 8.12
1. Arclength of a Curve Between Two Points (I) A and B , given in explicit form (y = f (x) or x = g(y)) or in parametric form (x = x(t), y = y(t)) (Fig. 8.13a) can be calculated by the integrals: Zb q Z q Zt2 q L _ = 1 + f 0(x)]2 dx = g0(y)]2 + 1 dy = x0 (t)]2 + y0(t)]2 dt: (8.60a) AB
a
With the di erential of the arclength dl we get
Z L = dl
with
t1
dl2 = dx2 + dy2:
(8.60b)
The perimeter of the ellipse WithZ theqsubstitutions x = x(t) = a sin t y = Z t2with q the help of (8.60a): t2 p 2 2 2 2 y(t) = b cos t we get L _ = a ; (a ; b ) sin t dt = a 1 ; e2 sin2 t dt, where e = a2 ; b2 =a AB t1 t1 is the numerical eccentricity of the ellipse. Since x = 0 y = b and x = a y = 0, the limits of the integral in the rst quadrant are L _ = AB Z =2 q 2 2 4a 1 ; e sin t dt = a E (k 2 ) with k = e. The value of the integral E (k 2 ) is given in Ta0 ble 21.9 (see example on p. 438). 2. Arclength of a Curve Between Two Points (II) C and D, given in polar coordinates ( = (')) (Fig. 8.13b):
! u Z'2v u d 2 d': L _ = t 2 + d' CD ' 1
(8.60c)
Z L = dl with dl2 = 2 d'2 + d 2 :
(8.60d)
With the di erential of the arclength dl we get
3. Surface Area of a Body of Revolution (see also First Guldin Rule, p. 453)
1. The area of the surface of a body given by rotating the graph of the function y = f (x) around the x-axis (Fig. 8.14a) is: ! u Zb Zb v u dy 2 dx: (8.61a) S = 2 y dl = 2 y(x)t1 + dx a a
450 8. Integral Calculus y B
b
dl dx
A
a
D
dp
dy
j pd
dj
dl C
0
a
a)
0 j j 1 2
b x
x
b) Figure 8.13
y
y
dl y x
x
dl
dy
b
α
0
β
a
a)
dx
b)
0
x
Figure 8.14 2. The area of the surface of a body given by rotating x = f (y) around the y-axis (Fig. 8.14b) is:
v !2 u u S = 2 x dl = 2 x(y)t1 + dx dy dy:
(8.61b)
Zb V = y2 dx:
(8.62a)
Z
Z
To calculate the area of more complicated surfaces see the application of double integrals in 8.4.1.3, p. 474 and the application of surface integrals of the rst kind, 8.5.1.3, p. 482. General formulas for the calculation of surface areas with double integrals are given in Table 8.9 (Application of double integrals), p. 475. 4. Volume (see also Second Guldin Rule, p. 453) 1. The volume of a rotationally symmetric body given by a rotation around the x-axis (Fig. 8.14a) is: a
2. The volume of a rotationally symmetric body given by a rotation around the y-axis (Fig. 8.14b) is:
Zb V = x2 dy: a
(8.62b)
3. The volume of a body, whose section perpendicular to the x-axis (Fig. 8.15) has an area given by the function S = f (x), is:
Zb V = f (x) dx: a
(8.63)
8.2 Denite Integrals 451
General formulas to calculate volumes by multiple integrals are given in Table 8.9 (Applications of double integrals, see p. 475) and Table 8.11 (Applications of triple integrals, see p. 480). z S=f(x) 0
x
x y
0 h1 h2
y y1(x)
y2(x)
x
Figure 8.15
Figure 8.16
8.2.2.3 Applications in Mechanics and Physics
1. Distance Traveled by a Point
The distance traveled by a moving point during the time from t0 until T with a time-dependent velocity v = f (t) is
ZT s = v dt: t0
(8.64)
2. Work
To determine the work in moving a body in a force eld we suppose that the direction of the eld and the direction of the movement are constant and coincident. We de ne the x-axis in this direction. If the magnitude of the force F~ is changing, i.e., jF~ j = f (x), then we get for the work W necessary to move the body from the point x = a to the point x = b along the x-axis:
Zb W = f (x) dx: a
(8.65)
In the general case, when the direction of the force eld and the direction of the movement are not coincident, we calculate the work as a line integral (see (8.130), p. 469) of the scalar product of the force and the variation of the position vector at every point of ~r along the given path.
3. Pressure
In a uid at rest with a density % we distinguish between gravitational pressure and lateral pressure. This second one is exerted by the uid on one side of a vertical plate immersed in the uid. Both depend on the depth. 1. Gravitational Pressure The gravitational pressure ph at depth h is: ph = % g h (8.66) where g is the gravitational acceleration. 2. Lateral Pressure The lateral pressure ps, e.g., on the cover of a lateral opening of a container of some uid with the di erence of depth h1 ; h2 (Fig. 8.16) is:
Zh2 ps = % g x y2 (x) ; y1(x)] dx: h1
The left and the right boundary of the cover is given by the functions y1(x) and y2(x).
(8.67)
452 8. Integral Calculus
4. Moments of Inertia
1. Moment of Inertia of an Arc The moment of inertia of a homogeneous curve segment y = f (x) with constant density % in the interval a b] with respect to the y-axis (Fig. 8.17a) is: Zb Zb q Iy = % x2 dl = % x2 1 + (y0)2 dx: a
(8.68)
a
If the density is a function %(x), then its analytic expression is in the integrand. y
y
b x
y
dl
a
x
0
a)
x
b)
0
a
dx
b
x
Figure 8.17 2. Moment of Inertia of a Planar Figure The moment of inertia of a planar gure with a homogeneous density % with respect to the y-axis, where y is the length of the cut parallel to the y-axis (Fig. 8.17b), is:
Zb Iy = % x2 y dx:
(8.69)
a
(See also Table 8.4.2.3, (Applications of line integral), p. 479.) If the density is position dependent, then its analytic expression must be in the integrand. y
y L
yC
yC
C
A
0 a
b x
xC
a)
0
y
(x) y=f 1 C
B A
a
(x) y=f 2
b x
xC
b)
yC
y
B
B A
0 c)
(x) y=f 1 C
yC A
C xC
x
0
a
B
(x) y=f 2 xC
b x
d)
Figure 8.18
5. Center of Gravity, Guldin Rules
1. Center of Gravity of an Arc Segment The center of gravity C of an arc segment of a homogeneous plane curve y = f (x) in the interval a b] with a length L (Fig. 8.18a) considering (8.60a), p. 449, has the coordinates:
Zb q x 1 + y02 dx
xC = a
Zb q y 1 + y02 dx
yC = a
: (8.70) L 2. Center of Gravity of a Closed Curve The center of gravity C of a closed curve y = f (x) (Fig. 8.18b) with the equations y1 = f1 (x) for the upper part and y2 = f2(x) for the lower part, and L
8.2 Denite Integrals 453
with a length L has the coordinates:
Zb q q x( 1 + (y10 )2 + 1 + (y20 )2) dx
xC = a
Zb q q (y1 1 + (y10 )2 + y2 1 + (y20 )2 ) dx
yC = a : (8.71) L L 3. First Guldin Rule Suppose a plane curve segment is rotated around an axis which lies in the plane of the curve and does not intersect the curve. We choose it as the x-axis. The surface area Srot of the body generated by the rotated curve segment is the product of the perimeter of the circle drawn by the centre of gravity at a distance yC from the axis of rotation, i.e., 2yC , and the length of the curve segment L: Srot = L 2yC : (8.72) 4. Center of Gravity of a Trapezoid The center of gravity C of a homogeneous trapezoid bounded above by a curve segment between the points of the curve A and B (Fig. 8.18c), with an area S of the trapezoid, and with the equation y = f (x) of the curve segment AB , has the coordinates: Zb Zb 1 y 2 dx x y dx 2 xC = a S yC = a S : (8.73) 5. Center of Gravity of an Arbitrary Planar Figure The center of gravity C of an arbitrary planar gure (Fig. 8.18d) with area S , bounded above and below by the curve segments with the equations y1 = f1 (x) and y2 = f2(x), has the coordinates Zb 1 Zb (y2 ; y2) dx x(y1 ; y2) dx 2a 1 2 xC = a y : (8.74) C= S S Formulas to calculate the center of gravity with multiple integrals are given in Table 8.9 (Application of double integrals, p. 475) and in Table 8.11 (Application of triple integrals, p. 480). 6. Second Guldin Rule Suppose a plane gure is rotated around an axis which is in the plane of the gure and does not intersect it. We choose it as the x-axis. The volume V of the body generated by the rotated gure is equal to the product of the perimeter of the circle drawn by the center of gravity under the rotation, i.e., 2yC , and the area of the gure S : Vrot = S 2yC : (8.75)
8.2.3 Improper Integrals, Stieltjes and Lebesgue Integrals 8.2.3.1 Generalization of the Notion of the Integral
The notion of the de nite integral (see 8.2.1.1, p. 440), as a Riemann integral (see 8.2.1.1, 2., p. 441), was introduced under the assumptions that the function f (x) is bounded, and the interval a b] is closed and nite. Both assumptions can be relaxed in the generalizations of the Riemann integral. In the following we mention a few of them.
1. Improper Integrals
These are the generalization of the integral to unbounded functions and to unbounded intervals. We discuss integrals with innite integration limits and integrals with unbounded integrands in the next paragraph.
2. Stieltjes Integral for Functions of One Variable
We start from two nite functions f (x) and g(x) de ned on the nite interval a b]. We make a partition of the interval into subintervals, just as with the Riemann integral, but instead of the Riemann sum (8.37) we compose the sum
454 8. Integral Calculus n X i=1
f ( i) g(xi) ; g(xi;1)]:
(8.76)
If the limit of (8.76) exists, when the length of the subintervals tends to zero, and it is independent of the choice of the points xi and i, then this limit is called a denite Stieltjes integral (see 8.8]). For g(x) = x the Stieltjes integral becomes the Riemann integral.
3. Lebesgue Integral
Another generalization of the integral notion is connected with measure theory (see 12.9, p. 635), where the measure of a set, measure spaces, and measurable functions are introduced. In functional analysis the Lebesgue integral is de ned (see 12.9.3.2, p. 637) based on these notions (see 8.6]). The generalization with comparison to the Riemann integral is, e.g., the domain of integration can be a rather general subset of IRn and it is partitioned into measurable subsets. There are di erent notations for the generalizations of the integrals (see 8.8]).
8.2.3.2 Integrals with Innite Integration Limits
1. Denitions
a) If the integration domain is the closed half-axis a +1), and if the integrand is de ned there, then the integral is by de nition + Z1
a
ZB f (x) dx = Blim f (x) dx: !1
(8.77)
a
If the limit exists, then the integral is called a convergent improper integral. If the limit does not exist, then the improper integral (8.77) is divergent. b) If the domain of a function is the closed half-axis (;1 b] or the whole real axis (;1 +1), then we de ne analogously the improper integrals
Zb
;1
Zb f (x) dx = A!;1 lim f (x) dx A
(8.78a)
+ Z1
;1
ZB f (x) dx = Alim !;1 f (x) dx: B!1 A
(8.78b)
c) At the limits of (8.78b) the numbers A and B tend to in nity independently of each other. If the limit (8.78b) does not exist, but the limit lim A!1
Z+A f (x) dx
;A
(8.78c)
exists, then this limit (8.78c) is called the principal value of the improper integral, or Cauchy's principal value. Remark: An obviously necessary but not sucient condition for the convergence of the integral (8.77) is xlim f (x) = 0. !1
2. Geometrical Meaning of Integrals with Innite Limits
The integrals (8.77), (8.78a) and (8.78b) give the area of the gures represented in Fig. 8.19. Z1 Z B dx A: 1 dx = lim ln B = 1 (divergent). B !1 !1 x x = Blim 1 1 1 Z 1 dx Z 1B dx ; B = 2 (convergent). B: 2 x2 = Blim = Blim !1 2 x2 !1 2 Z +1 dx Z B dx ; ; = (convergent). C: ;1 1 + x2 = Alim = lim arctan B ; arctan A ] = 2 !;1 A!;1 2 2 B!+1 A 1 + x B ! +1
8.2 Denite Integrals 455 y +
a
x −
y
x
b 0
b y − 0 x
x
+ + + ++ a − x − − − − b − f(x)dx − f(x)dx a b) −
+ + +
b0
x
8
x 0
x
−
y
+
+
8
a)
0 y
+ c) −
y 0
− − − f(x)dx
8
y 0
+
+
+ + −− x
−
8
0
a
0
y
y
Figure 8.19
3. Su cient Criteria for Convergence
If the direct calculation of the limits (8.77), (8.78a) and (8.78b) is complicated, or if only the convergence or divergence of an improper integral is the question, then one of the following sucient criteria can be used. Here, only the integral (8.77) is considered. The integral (8.78a) can be transformed into (8.77) by substitution of x by ;x:
Za
f (x) dx =
;1
+ Z1
;a
f (;x) dx:
(8.79)
The integral (8.78b) can be decomposed into the sum of two integrals of type (8.77) and (8.78a): + Z1
Zc
f (x) dx =
;1
f (x) dx +
;1
+ Z1
c
f (x) dx
(8.80)
where c is an arbitrary number. Criterion 1: If f (x) is integrable on any nite subinterval of a 1) and if the integral + Z1
a
j f (x) j dx
(8.81)
exists, then there exists also the integral (8.77). The integral (8.77) is in this case said to be absolutely convergent, and the function f (x) is absolute integrable on the half-axis a +1). Criterion 2: If for the functions f (x) and '(x) f (x) > 0 '(x) > 0 and f (x) '(x) for a x < +1 (8.82a) hold, then from the convergence of the integral + Z1
a
'(x) dx
(8.82b)
the convergence of the integral
+ Z1
a
f (x) dx
(8.82c)
456 8. Integral Calculus follows, and conversely, from the divergence of the integral (8.82c) the divergence of the integral (8.82b) follows. Criterion 3: If we substitute '(x) = x1 (8.83a) and we consider that for a > 0 > 1 the integral + Z 1 dx 1 = ( ; 1) (8.83b) x a ;1 (a > 0 > 1) a is convergent and has the value of the right-hand side, and the integral of the left-hand side is divergent for 1, then we can deduce a further convergence criterion from the second one: If f (x) in a x < 1 is a positive function, and there exists a number > 1 such that for x large enough f (x) x < k < 1 (k > 0 const) (8.83c) holds, then the integral (8.77) is convergent if f (x) is positive and there exists a number 1 such that f (x) x > c > 0 (c > 0 const) (8.83d) holds from a certain point, then the integral (8.77) is divergent. Z +1 x3=2 dx 1 , then we get x3=2 x1=2 = x2 ! 1. The integral is . If we substitute = 2 1+x 2 1 + x2 1 + x2 0 divergent.
4. Relations Between Improper Integrals and Innite Series
If x1 x2 : : : xn : : : is an arbitrary, unlimited increasing in nite sequence, i.e., if a < x1 < x2 < < xn < with n!lim x =1 (8.84a) +1 n and if the function f (x) is positive for a x < 1, then the problem of convergence of the integral (8.77) can be reduced to the problem of convergence of the series
Zx1 a
Zx2 Zxn f (x) dx + f (x) dx + + f (x) dx + : xn;1
x1
(8.84b)
If the series (8.84b) is convergent, then the integral (8.77) is also convergent, and it is equal to the sum of the series (8.84b). If the series (8.84b) is divergent, then the integral (8.77) is also divergent. So the convergence criteria for series can be used for improper integrals, and conversely, in the integral criterion for series (see 7.2.2.4, p. 408) we use the improper integrals to investigate the convergence of in nite series.
8.2.3.3 Integrals with Unbounded Integrand 1. Denitions
1. Right Open or Closed Interval For a function f (x), which has a domain open on the right a b) or a domain closed on the right a b], but at the point b it has the limit x!lim f (x) = 1, we have the b;0 de nition of the improper integral in both cases: Zb a
f (x) dx = "lim !0
Zb;" f (x) dx: a
(8.85)
If this limit exists and is nite, then the improper integral (8.85) exists, and we call it a convergent improper integral. If the limit does not exist or it is not nite, then we call the integral a divergent improper integral.
8.2 Denite Integrals 457
2. Left Open or Closed Interval For a function f (x), which has a domain open on the left (a b] or a domain closed on the left a b], but at the point a it has the limit x!lim f (x) = 1, we de ne the a+0 improper integral analogously to (8.85). That is: Zb a
f (x) dx = "lim !0
Zb
a+"
f (x) dx:
(8.86)
3. Two Half-Open Contiguous Intervals For a function f (x), which is de ned on the interval
a b] except at an interior point c with a < c < b, i.e., for a function f (x) de ned on the half-open intervals a c) and (c b], or is de ned on the interval a b], but at the interior point c it has an in nite limit at least from one side x!lim f (x) = 1 or x!lim f (x) = 1, the de nition of the improper integral c+0 c;0 is:
Zb a
f (x) dx = "lim !0
Zc;" Zb f (x) dx + lim f (x) dx: !0 a
(8.87a)
c+
Here the numbers " and tend to zero independently of each other. If the limit (8.87a) does not exist, but the limit
8 Zc;" 9 Zb < = lim f (x) dx + f (x) dx" "!0 : a c+"
(8.87b)
does, then we call the limit (8.87b) the principal value of the improper integral or Cauchy's principal value.
2. Geometrical Meaning
The geometrical meaning of the integral of discontinuous functions (8.85), (8.86), and (8.87a) is to nd the area of the gures bounded, e.g., from one side by a vertical asymptote as represented in Fig.8.20. y
+ 0a
bx
a) s. (8.85)
y
y
+ b 0 a − x
+ b 0 a − x
y
y
0
a
−
b) s. (8.86) Figure 8.20
b
x
a 0 −
y
+ c
bx 0 a
+ c b − x
c) s. (8.87a)
Z b dx p : Case (8.86), singular point at x = 0. Z b dx Z 0b dxx p p p p p = "lim = "lim (2 b ; 2 ") = 2 b (convergent). ! 0 ! 0 " x 0 x Z =2 B: 0 tan x dx : Case (8.85), singular point at x = 2 .
Z =2 Z =2;" tan x dx = "lim tan x dx = lim ln cos 0 ; ln cos !0 0 "!0 2 ; " = 1 (divergent). 0
A:
458 8. Integral Calculus Z 8 dx p3 : Case (8.87a), singular point at x = 0. Z ;81 dxx Z ;" dx Z 8 dx 3 (4 ; 2=3 ) = 9 (convergent). p p p !0 32 ("2=3 ; 1) + lim = lim + lim 3 3 !0 2 x "!0 ;1 x !0 3 x = "lim 2 ;1 Z 2 2x dx D: ;2 x2 ; 1 : Case (8.87a), singular point at x = 1 . Z 2 2x dx Z ;1;" Z 1; Z2 = "lim + lim + lim 2 ! 0 ! 0 ! 0 ;2 x ; 1 ;2 1+ !0 ;1+ 2 ; 1);1;" + = lim ln(1 + 2" + "2 ; 1) ; ln 3] + = 1 (divergent). = "lim ln( x ;2 !0 "!0
C:
3. The Application of the Fundamental Theorem of Integral Calculus
1. Warning The calculation of improper integrals of type (8.87a) with the mechanical use of the formula
Zb a
b f (x) dx = F (x)a with F 0(x) = f (x)
(8.88)
(see 8.2.1.1, p. 440) usually results in mistakes if the singular points in the interval a b] are not taken into consideration. E: Using formally the fundamental theorem we get for the example D Z 2 2x dx 2 = ln(x2 ; 1);2 = ln 3 ; ln 3 = 0 ;2 x2 ; 1 though this integral is divergent. 2. General Rule The fundamental theorem of integral calculus can be used for (8.87a) only if the primitive function of f (x) can be de ned to be continuous at the singular point. F: In the example D the function ln(x2 ; 1) is discontinuous at x = 1, so the conditions are not ful lled. Consider the example C. The function y = 32 x2=3 is such a primitive function of p31x on the intervals a 0) and (0 b] which can be de ned continuously at x = 0, so the fundamental theorem can be used in the example C: Z 8 dx 3 8 3 p3 x = 2 x2=3 ;1 = 2 (82=3 ; (;1)2=3 ) = 92 : ;1
4. Su cient Conditions for the Convergence of an Improper Integral with UnboundedZ Integrand Z
1. If the integral
b
a
b
jf (x)j dx exists, then the integral a f (x) dx also exists. In this case we call it an
absolutely convergent integral and the function f (x) is an absolutely integrable function on the considered interval.
2. If the function f (x) is positive in the interval a b), and there is a number < 1 such that for the
values of x close enough to b f (x) (b ; x) < 1 (8.89a) holds, then the integral (8.87a) is convergent. But, if the function f (x) is positive in the interval a b), and there is a number > 1 such that for the values of x close enough to b f (x) (b ; x) > c > 0 (c const) (8.89b) holds, then the integral (8.87a) is divergent.
8.2 Denite Integrals 459
8.2.4 Parametric Integrals
8.2.4.1 Denition of Parametric Integrals The de nite integral
Zb a
f (x y) dx = F (y)
(8.90)
is a function of the variable y considered here as a parameter. In several cases the function F (y) is no longer elementary, even if f (x y) is an elementary function of x and y. The integral (8.90) can be an ordinary integral, or an improper integral with in nite limits or unbounded integrand f (x y). For theoretical discussions about the convergence of improper integrals depending on a parameter see, e.g., 8.3]. Gamma Function or Euler Integral of the Second Kind (see 8.2.5, 6., p. 461):
Z1 ; (y) = xy;1 e;x dx
(convergent for y > 0):
0
(8.91)
8.2.4.2 Di erentiation Under the Symbol of Integration
1. Theorem If the function (8.90) is de ned in the interval c y e, and the function f (x y) is continuous on the rectangle a x b c y e and it has here a partial derivative with respect to y, then for arbitrary y in the interval c e]: d Zb f (x y) dx = Zb @f (x y) dx: (8.92) dy a @y a This is called dierentiation under the symbol of integration. ! Z1 Z1 Z 1 x dx 1 2 For arbitrary y > 0: d arctan x dx = @ arctan x dx = ; = ln y 2 . 2 2 dy 0 y y 2 1+y 0 @y 0 x +y Z1 2 2 ! 1 1 1 y d 1 1 y y2 x Checking: arctan dx = arctan + y ln y y 2 1 + y2 dy arctan y + 2 y ln 1 + y2 = 2 ln 1 + y2 . 0 For y = 0 the condition of continuity is not ful lled, and there exists no derivative. 2. Generalization for Limits of Integration Depending on Parameters The formula (8.92) can be generalized, if with the same assumptions we made for (8.92) the functions (y) and (y) are de ned in the interval c e], they are continuous and di erentiable there, and the curves x = (y) x = (y) do not leave the rectangle a x b c y e: (y) (y) d Z f (x y) dx = Z @f (x y) dx + 0(y) f ( (y) y) ; 0(y) f ((y) y) : dy (y) @y (y)
(8.93)
8.2.4.3 Integration Under the Symbol of Integration
If the function f (x y) is continuous on the rectangle a x b c y e, then the function (8.90) is de ned in the interval c e], and
3 3 Ze 2Zb Zb 2Ze 4 f (x y) dx5 dy = 4 f (x y) dy5 dx c
a
a
c
(8.94)
is valid. This is called integration under the symbol of integration. A: Integration of the function f (x y) = xy on the rectangle 0 x 1, a y b. The function xy is discontinuous at x = 0, y = 0, for a > 0 it is continuous. So we can change the order of integration:
460 8. Integral Calculus
Z 1 "Z b Zb xy dx dy = xy dy dx: On the left-hand side we get 1 dy = ln 1 + b , on the right1+a a 0 0 a a +y Z 1 xb ; xa hand side dx. The inde nite integral cannot be expressed by elementary functions. Anyway, 0 ln x the de nite integral is known, so we get: Z 1 xb ; xa +b dx = ln 11 + a (0 < a < b). 0 ln x 2 2 B: Integration of the function f (x y) = (xy2 +; yx2)2 over the rectangle 0 x 1, 0 y 1. The function is discontinuous at the point (0 0), so the formula (8.94) cannot be used. Checking it we get: Z 1 dy Z 1 y 2 ; x2 x=1 1 dx = 2 x 2 x=0 = 1 2 = arctan y0 = 2 2 2 2 x +y 1+y 4 0 1+y 0 (x + y ) Z 1 y 2 ; x2 Z 1 dx y=1 y 1 ; x2 + 1 = ; arctan x10 = ; 4 . (x2 + y2)2 dy = x2 + y2 y=0 = ; x2 + 1 Z b Z 1
0
0
8.2.5 Integration by Series Expansion, Special Non-Elementary Functions
It is not always possible to express an integral by elementary functions, even if the integrand is an elementary function. In many cases we can express these non-elementary integrals by series expansions. If the integrand can be expanded into a uniformlyZ convergent series in the interval a b], then we get x also a uniformly convergent series for the integral f (t) dt if we integrate it term by term. a
1. Integral Sine (jxj < 1, see also 14.4.3.2, 2., p. 696) Zx sin t ; Z1 sin t dt dt = t 2 x t 0 3 5 = x ; x + x ;++
Si (x) =
3 3!
5 5!
(;1)nx2n+1 (2n + 1) (2n + 1)! + :
(8.95)
2. Integral Cosine (0 < x < 1) Z1 Zx Ci (x) = ; cos t dt = C + ln x ; 1 ; cos t dt t x t 0 2 4 n x2n = C + ln x ; 2x 2! + 4x 4! ; + + 2(;n 1) (2n)! + with Z1 C = ; e;t ln t dt = 0:577 215 665 : : : 0
(Euler constant):
(8.96a) (8.96b)
3. Integral Logarithm (0 < x < 1, for 1 < x < 1 as Cauchy Principal Value) Li (x) =
Zx dt (ln x)2 + + (ln x)n + : = C + ln j ln x j + ln j x j + ln t 2 2! n n! 0
(8.97)
8.2 Denite Integrals 461
4. Exponential Integral (;1 < x < 0, for 0 < x < 1 as Cauchy Principal Value) Ei (x) =
Zx et x2 + + xn + : dt = C + ln j x j + x + t 2 2! n n! ;1
(8.98a)
Ei (ln x) = Li (x):
(8.98b)
5. Gauss Error Integral and Error Function
The Gauss error integral is de ned for the domain jxj < 1 and it is denoted by . The following de nitions and relations are valid: Zx t2 (x) = 1 (8.99b) (x) = p1 e; 2 dt (8.99a) xlim !1 2 ;1
Zx t2 (8.99c) 0(x) = p1 e; 2 dt = (x) ; 21 : 2 0 The function (x) is the distribution function of the standard normal distribution (see 16.2.4.2, p. 759) and its values are tabulated in Table 21.17, p. 1091. The error function erf (x), often used in statistics (see also 16.2.4.2, p. 759), has a strong relation with the Gauss error integral: Zx p erf (x) = 1 (8.100b) erf (x) = p2 e;t2 dt = 20 (x 2) (8.100a) xlim !1 0 ! 3 5 1)n x2n+1 + erf (x) = p2 x ; 1!x 3 + 2!x 5 ; + + n(; (8.100c) ! (2n + 1) Zx 0
erf (t) dt = x erf (x) + p1 e;x2 ; 1 (8.100d)
6. Gamma Function and Factorial
1. De nition The gamma function, the Euler integral of the second kind (8.91), is an extension of the notion of factorial for arbitrary numbers x, even complex numbers, except zero and the negative integers. The curve of the function ; (x) is represented in Fig. 8.21. Its values are given in Table 21.10, p. 1063. It can be de ned in two ways:
Z1
; (x) = e
;t
0
tx;1 dt
(x > 0)
(8.101a)
nx n! ; (x) = nlim !1 x(x + 1)(x + 2) : : : (x + n) (x 6= 0 ;1 ;2 : : :):
(8.101b)
2. Properties of the Gamma Function ; (x + 1) = x; (x)
(8.102a)
d erf (x) = p2 e;x2 : dx
(8.100e)
G(x) 5 4 3 2 1 -5 -4 -3 -2 -1 0 1 2 3 4 x -1 -2 -3 -4 -5
Figure 8.21
; (n + 1) = n! (n = 0 1 2 : : :)
(8.102b)
462 8. Integral Calculus Z1 p ; (x) ; (1 ; x) = sinx (x 6= 0 1 2 : : :) (8.102c) ; 21 = 2 e;t2 dt = (8.102d) 0 1 (2n)!p ; n + 2 = n!22n (n = 0 1 2 : : :) (8.102e) p n 2n ; ;n + 21 = (;1)(2nn!2)! (n = 0 1 2 : : :): (8.102f) The same relations are valid for complex arguments z, but only if Re (z) > 0 holds. 3. Generalization of the Notion of Factorial The notion of factorial, de ned until now only for positive integers n (see 1.1.6.4, 3., p. 13), leads to the function x! = ; (x + 1) (8.103a) as its extension for arbitrary real numbers. The following equalities are valid: For positive integers x: x! = 1 2 3 x (8.103b) for x = 0: 0! = ; (1) = 1 (8.103c) p for negative integers x: x! = 1 (8.103d) for x = 1 : 1 ! = ; 3 = (8.103e) 2 2 2 2 p p for x = ; 12 : ; 12 ! = ; 12 = (8.103f) for x = ; 32 : ; 32 ! = ; ; 12 = ;2 : (8.103g) An approximate determination of a factorial can be performed for large numbers (> 10), also for fractions n with the Stirling formula: n p n! ne 2n 1 + 121n + 2881n2 + (8.103h) 1 p ln(n!) n + 2 ln n ; n + ln 2: (8.103i)
7. Elliptic Integrals
For the complete elliptic integrals (see 8.1.4.3, 2., p. 437) the following series expansions are valid:
" 2
2 2 Z2 K = p d 2 2 = 2 1 + 12 k2 + 12 34 k4 + 12 34 56 k6 + k2 < 1 (8.104) 1 ; k sin 0 "
2 2 2 4 2 6 1 ; k2 sin2 d = 2 1 ; 12 k1 ; 12 34 k3 ; 12 34 56 k5 ; 0 k2 < 1: (8.105) The numerical values of the elliptic integrals are given in Table 21.9, p. 1061. E=
Z2 q
8.3 Line Integrals
The notion of the integral can be generalized in di erent ways. While the domain of an ordinary de nite integral is an interval on the numerical axis, for a line integral, the domain of integration is a segment of a planar or space curve. The curve, i.e., the path of integration can also be closed it is called also circuit integral and it gives the circulation of the function along the curve. We distinguish line integrals of the rst type, of the second type, or of general type.
8.3 Line Integrals 463
The line integral of the rst type or integral over an arc is the de nite integralZ f (x y) ds (8.106)
An B
Pn An-1
∆s i-1 A i ∆s 2 A Ai-1Pi P3 3 A2 P2 A1 P1 A0 A
∆s0 ∆ s
1
(C )
y -1
8.3.1.1 Denitions
∆sn
8.3.1 Line Integrals of the First Type
where u = f (x y) is a function of two variables de ned on a con_ nected domain and the integration is performed over an arc C AB of a plane curve given by its equation. The considered arc is in the x same domain, and we call it the path of integration. The numerical 0 value of the line integral of the rst type can be determined in the following way (Fig. 8.22): Figure 8.22 _ 1. We decompose the recti able arc segment AB into n elementary parts by points A1 A2 : : : An;1 chosen arbitrarily, starting at the initial point A A0 and nishing at the endpoint B An. _ 2. We choose arbitrary points Pi inside or at the end of the elementary arcs Ai;1Ai , with coordinates i and i . _ 3. We multiply the values of the function f ( i i) at the chosen points with the arclength Ai;1 Ai= %si;1 which should be taken positive. (Since the arc is recti cable, %si;1 is nite.) 4. We add the n products f ( i i)%si;1 . 5. We evaluate the limit of the sum n X i=1
f ( i i)%si;1
(8.107a)
as the arclength of every elementary curve segment %si;1 tends to zero, while n obviously tends to 1. If the limit of (8.107a) exists and is independent of the choice of the points Ai and Pi, then this limit is called the line integral of the rst type, and we write
Z
(C )
f (x y) ds = lim si !0
n X
n!1 i=1
f ( i i)%si;1:
(8.107b)
We can de ne analogously the line integral of the rst type for a function u = f (x y z) of three variables, whose path of integration is a curve segment of a space curve:
Z
(C )
f (x y z) ds = lim si ! 0
n X
n!1 i=1
f ( i i i)%si;1:
8.3.1.2 Existence Theorem
(8.107c)
The line integral of the rst type (8.107b) or (8.107c) exists if the function f (x y) or f (x y z) and also the curve along the arc segment C are continuous, and the curve has a tangent which varies continuously. In other words: The above limits exist and are independent of the choice of Ai and Pi. In this case, the functions f (x y) or f (x y z) are said to be integrable along the curve C .
8.3.1.3 Evaluation of the Line Integral of the First Type
We calculate the line integral of the rst type by reducing it to a de nite integral.
464 8. Integral Calculus
1. The Equation of the Path of Integration is Given in Parametric Form If the de ning equations of the path are x = x(t) and y = y(t), then
Z
(C )
ZT q f (x y) ds = f x(t) y(t)] x0(t)]2 + y0(t)]2 dt
(8.108a)
t0
holds, and in the case of a space curve x = x(t) y = y(t), and z = z(t)
Z
(C )
ZT q f (x y z) ds = f x(t) y(t) z(t)] x0 (t)]2 + y0(t)]2 + z0 (t)]2 dt t0
(8.108b)
where t0 is the value of the parameter t at the point A and T is the parameter value at B . The points A and B are chosen so that t0 < T holds.
2. The Equation of the Path of Integration is Given in Explicit Form We substitute t = x and get from (8.108a) for the planar case
Z
(C )
Zb q f (x y) ds = f x y(x)] 1 + y0(x)]2 dx
(8.109a)
a
and from (8.108b) for the three dimensional case
Z
(C )
Zb q f (x y z) ds = f x y(x) z(x)] 1 + y0(x)]2 + z0 (x)]2 dx: a
(8.109b)
Here a and b are the abscissae of the points A and B , where the relation a < b must be ful lled. We can consider x as a parameter if every point corresponds to exactly one point on the projection of the curve segment C onto the x-axis, i.e., every point of the curve is uniquely determined by the value of its abscissa. If this condition does not hold, we have to partition the curve segment into subsegments having this property. The line integral along the whole segment is equal to the sum of the line integrals along the subsegments.
8.3.1.4 Application of the Line Integral of the First Type
Some applications of the line integral of the rst type are given in Table 8.6. The curve elements needed for the calculations of the line integrals are given for di erent coordinate systems in Table 8.7.
8.3.2 Line Integrals of the Second Type 8.3.2.1 Denitions
A line integral of the second type or an integral over a projection (onto the x-, y- or z-axis) is the de nite integral
Z
(C )
f (x y) dx
(8.110a)
or
Z
(C )
f (x y z) dx
(8.110b)
where f (x y) or f (x y z) are two or three variable functions de ned on a connected domain, and we _ integrate over a projection of a plane or space curve C AB (given by its equation) onto the x-, y-, or z-axis. The path of integration is in the same domain. We get the line integral of the second type similarly to the line integral of the rst type, but in the third step the values of the function f ( i i) or f ( i i i) are not multiplied by the arclength of the _ elementary curve segments Ai;1Ai , but by its projections onto a coordinate axis (Fig. 8.23).
8.3 Line Integrals 465
Table 8.6 Line integrals of the rst type Length of a curve segment C L = Mass of an inhomogeneous curve segment C
Z
(C )
M=
Z
ds
% ds (% = f (x y z) density function) Z Z Z Center of gravity coordinates xC = L1 x%ds yC = L1 y%ds zC = L1 z%ds (C ) Z (C ) Z (C ) Moments of inertia of a plane 2 %ds I = 2 %ds I = x y x y curve in the x y plane (C ) (C ) Z Z 2 2 Ix = (y + z )%ds Iy = (x2 + z2 )%ds Moments of inertia of a (C ) (C ) space curve with respect Z to the coordinate axes Iz = (x2 + y2)%ds (C )
(C )
In the case of homogeneous curves we substitute % = 1. Table 8.7 Curve elements Cartesian coordinates x y = y(x) Plane curve in Polar coordinates ' = (') the x y plane Parametric form in Cartesian coordinates x = x(t) y = y(t) Space curve Parametric form in Cartesian coordinates x = x(t) y = y(t) z = z(t)
1. Projection onto the x-Axis
_ With Prx Ai;1 Ai= xi ; xi;1 = %xi;1 we get
Z
(C )
Z
(C )
f (x y) dx = xlim i;1 !0
n X
n!1 i=1
f (x y z) dx = xlim i;1 !0 n!1
f ( i i) %xi;1
n X i=1
q ds = 1 + y0(x)]2dx q ds = 2 (') + 0 (')]2d' q ds = x0 (t)]2 + y0(t)]2 dt q ds = x0 (t)]2 + y0(t)]2 + z0 (t)]2 dt
(8.111) (8.112a)
(C )
f (x y) dy = ylim i;1 !0
n X
n!1 i=1
Ai
yi yi-1
B
Ai-1Pi(xi ,hi ) A
f ( i i i) %xi;1 :
(8.112b)
f ( i i) %yi;1
0
xi-1 xi
Figure 8.23
2. Projection onto the y-Axis Z
y
(8.113a)
x
466 8. Integral Calculus Z (C )
f (x y z) dy = ylim i;1 !0
n X
n!1 i=1
f ( i i i) %yi;1:
(8.113b)
3. Projection onto the z-Axis Z
(C )
f (x y z) dz = zlim i;1 !0
n X
n!1 i=1
f ( i i i) %zi;1 :
(8.114)
8.3.2.2 Existence Theorem
The line integral of the second type in the form (8.112a), (8.113a), (8.112b), (8.113b) or (8.114) exists if the function f (x y) or f (x y z) and also the curve are continuous along the arc segment C , and the curve has a continuously varying tangent there.
8.3.2.3 Calculation of the Line Integral of the Second Type
We reduce the calculation of the line integrals of the second type to the calculation of de nite integrals.
1. The Path of Integration is Given in Parametric Form
With the parametric equations of the path of integration x = x(t) y = y(t) and (for a space curve) z = z(t) we get the following formulas:
(8.115)
ZT f (x y) dx = f x(t) y(t)]x0(t) dt: (8.116a) (C ) t0 Z ZT For (8.113a) f (x y) dy = f x(t) y(t)]y0(t) dt: (8.116b) (C ) t0 Z ZT For (8.112b) f (x y z) dx = f x(t) y(t) z(t)]x0 (t) dt: (8.116c) (C ) t0 Z ZT For (8.113b) f (x y z) dy = f x(t) y(t) z(t)]y0(t) dt: (8.116d) (C ) t0 Z ZT For (8.114) f (x y z) dz = f x(t) y(t) z(t)]z0 (t) dt: (8.116e) (C ) t0 Here, t0 and T are the values of the parameter t for the initial point A and the endpoint B of the arc segment. In contrast to the line integral of the rst type, here we do not require the inequality t0 < T . Remark: If we reverse the path of the integral, i.e., interchange the points A and B , the sign of the integral changes. For (8.112a)
Z
2. The Path of Integration is Given in Explicit Form In the case of a plane or space curve with the equations y = y(x) or y = y(x) z = z(x)
(8.117)
8.3 Line Integrals 467
as the path of integration, with the abscissae a and b of the points A and B , where the condition a < b is no longer necessary, the abscissa x takes the place of the parameter t in the formulas (8.112a) { (8.114).
8.3.3 Line Integrals of General Type 8.3.3.1 Denition
A line integral of general type is the sum of the integrals of the second type along all the projections of a curve. If two functions P (x y) and Q(x y) of two variables, or three functions P (x y z), Q(x y z), and R(x y z) of three variables, are given along the given curve segment C , and the corresponding line integrals of the second type exist, then the following formulas are valid for a planar or for a space curve.
1. Planar Curve Z (C )
(P dx + Q dy) =
2. Space Curve Z (C )
Z
(C )
P dx +
(P dx + Q dy + R dz) =
Z (C )
Z
(C )
Q dy:
P dx +
Z (C )
(8.118a)
Q dy +
Z (C )
R dz:
(8.118b)
The vector representation of the line integral of general type and an application of it in mechanics will be discussed in the chapter about vector analysis (see 13.3.1.1, p. 660).
8.3.3.2 Properties of the Line Integral of General Type
1. The Decomposition of the Path of the Integral
_ by a point M , which is on the curve, and it can even be outside of AB (Fig. Z8.24), results in theZdecomposition of theZ integral into two parts: (P dx + Q dy) = (P dx + Q dy) + (P dx + Q dy): (8.119) _ _ _ AB AM MB
B M a) A A
M B
b)
Figure 8.24
2. The Reverse of the Sense of the Path of Integration
changes Z the sign of the integral: Z (P dx + Q dy) = ; (P dx + Q dy): _ _ AB BA
(8.120)
3. Dependence on the Path
In general, the value of the line integral is dependent not only on the initial and endpoints but also on the path of integration (Fig. 8.25): Z Z (P dx + Q dy) 6= (P dx + Q dy): (8.121) _ _ AMB ADB
M B
A D
Figure 8.25 Z A: I = (xy dx + yz dy + zx dz), where C is one turn of the helix x = a cos t y = a sin t z = bt (C )
(see Helix on p. 242) from t0 to T = 2: Z 2 2 I = (;a3 sin2 t cos t + a2 bt sin t cos t + ab2 t cos t) dt = ; a2 b . 0 Similar formulas are valid for the three-variable case.
468 8. Integral Calculus Z
y2 dx + (xy ; x2 ) dy], where C is the arc of the parabola y2 = 9x between the points (C ) Z 3 "2 3 4 ! 3. 3+ y ; y A(0 0) and B (1 3): I = y dy = 6 20 9 9 81
B: I =
0
8.3.3.3 Integral Along a Closed Curve
1. Notion of the Integral Along a Closed Curve A circuit integral or the circulation along a
curve is a line integral along a closed path of integration C , i.e., the initial point A and the end point B are identical. We use the notation:
I
(C )
(P dx + Q dy)
or
I
(C )
(P dx + Q dy + R dz):
(8.122)
In general, this integral di ers from zero. But it is equal to zero if the conditions (8.127) are satis ed, or if the integration is performed in a conservative eld (see 13.3.1.6, p. 662). (See also zero-valued circulation, 13.3.1.6, p. 662.) 2. The Calculation of the Area of a Plane Figure is a typical example of the application of the integral along a closed curve in the form I S = 12 (x dy ; y dx) (8.123) (C ) where C is the boundary curve of the plane gure. The integral is positive if the path is oriented counterclockwise.
8.3.4 Independence of the Line Integral of the Path of Integration
The condition for independence of a line integral of the path of integration is also called integrability of the total dierential.
8.3.4.1 Two-Dimensional Case If the line integral
Z
P (x y) dx + Q(x y) dy] (8.124) with continuous functions P and Q de ned on a simple connected domain depends only on the initial point A and the endpoint B of the path of integration, and does not depend on the curve connecting these points, i.e., for arbitrary A and B and arbitrary paths of integration ACB and ADB (Fig. 8.25) the equality Z Z (P dx + Q dy) = (P dx + Q dy) (8.125) _ _ ACB ADB
holds, then it is a necessary and sucient condition for the existence of a function U (x y) of two variables, whose total di erential is the integrand of the line integral: @U (8.126b) P dx + Q dy = d U (8.126a) i.e., P = @U @x Q = @y : The function U (x y) is a primitive function of the total di erential (8.126a). In physics, the primitive
8.3 Line Integrals 469
function U (x y) means the potential in a vector eld (see 13.3.1.6, 4., p. 663).
8.3.4.2 Existence of a Primitive Function
A necessary and sucient criterion for the existence of the primitive function, the integrability condition for the expression P dx + Q dy, is the equality of the partial derivatives @P = @Q (8.127) @y @x where also the continuity of the partial derivatives is required.
8.3.4.3 Three-Dimensional Case
The condition of independence of the line integral Z P (x y z) dx + Q(x y z) dy + R(x y z) dz] (8.128) of the path of integration analogously to the two-dimensional case is the existence of a primitive function U (x y z) for which P dx + Q dy + R dz = d U (8.129a) holds, i.e., @U @U P = @U (8.129b) @x Q = @y R = @z :
The integrability condition is now that the three equalities for the partial derivatives @Q = @R @R = @P @P = @Q (8.129c) @z @y @x @z @y @x should be simultaneously satis ed, provided that the partial derivatives are continuous. The work W (see also 8.2.2.3, 2., p. 451) is de ned as the scalar product of force F~ (~r) and displacement ~s. In a conservative eld the work depends only on the place ~r, but not on the velocity ~v. With ~ = dx~ex + dy~ey + dz~ez the relations (8.129a), (8.129b) are F~ = P~ex + Q~ey + R~ez = gradV and ds satis ed for the potential V (~r), and (8.129c) is valid. Independently of the path between the points P1 and P2 we get: Z P2 Z ~ = P2 P dx + Q dy + R dz] = V (P2) ; V (P1): W = F~ (~r) ds (8.130) P1
P1
z
y P(x, y)
L
A(x0, y0)
K
0
x
Figure 8.26
P(x, y, z) A(x0, y0, z0)
0
K
L y
x
Figure 8.27
8.3.4.4 Determination of the Primitive Function
1. Two-Dimensional Case
If the integrability condition (8.127) is satis ed, then along an arbitrary path of integration connecting an arbitrary xed point A(x0 y0) with the variable point P (x y) (Fig.8.26) and passing through the
470 8. Integral Calculus domain where (8.127) is valid, the primitive function U (x y) is equal to the line integral Z U = (P dx + Q dy): (8.131) _ AP In practice, it is convenient to choose a path of integration parallel to the coordinate axes, i.e., one of the segments AKP or ALP , if they are inside the domain where (8.127) is valid. With these we have two formulas for the calculation of the primitive function U (x y) and the total di erential P dx + Q dy: Z Z Zx Zy U = U (x0 y0) + + = C + P ( y0) d + Q(x ) d (8.132a) x0 y0 AK KP Z Z Zy Zx U = U (x0 y0) + + = C + Q(x0 ) d + P ( y) d : (8.132b) y0 x0 AL LP Here C is an arbitrary constant.
2. Three-Dimensional Case (Fig. 8.27)
If the condition (8.129c) is satis ed, the primitive function can be calculated for the path of integration AKLP with the formulas Z Z Z U = U (x0 y0 z0) + + + AK
KL
Zy
LP
Zz P ( y0 z0) d + Q(x z0) d + R(x y ) d + C (C arbitrary constant): (8.133) x0 y0 z0 For the other ve possibilities of a path of integration with the segments being parallel to the coordinate axes we get ve further formulas. @Q y2 ; x2 A: P dx+Q dy = ; x2y+dxy2 + x2x+dyy2 . The condition (8.129c) is satis ed: @P @y = @x = (x2 + y2)2 . Application of the formula (8.132b) and the substitution of x0 = 0 y0 = 1 (x0 = 0 y0 = 0 may not be chosen since the functions P and Q are discontinuous at the point (0 0)) results in U = Z y 0 d Z x ;y d + 2 + y2 + U (0 1) = ; arctan xy + C = arctan xy + C1. 1 02 + 2 0 B: P dx + Q dy + R dz = z x12 y ; x2 +1 z2 dx + xyz 2 dy + x2 +x z2 ; xy1 dz. The relations (8.129c) are satis ed. Application of the formula (8.133)! and substitution of x0 = 1 y0 = 1 z0 = 1 Zx Zy Zz x 1 z z result in U = 0 d + 0 d + x2 + 2 ; xy d + C = arctan x ; xy + C . =
Zx
1
1
0
8.3.4.5 Zero-Valued Integral Along a Closed Curve
The integral along a closed curve, i.e., the line integral P dx + Q dy is equal to zero, if the relation (8.127) is satis ed, and if there is no point inside the curve where even one of the functions P Q @P @y @Q or @x is discontinuous or not de ned. Remark: The value of the integral can be equal to zero also without this conditions, but then we get this value only after performing the corresponding calculations.
8.4 Multiple Integrals 471
8.4 Multiple Integrals
The notion of the integral can be extended to higher dimensions. If the domain of integration is a region in the plane or on a surface in space, then the integral is called a surface integral, if the domain is a part of space, then it is called a volume integral. For the di erent special applications we use di erent notations.
8.4.1 Double Integrals
8.4.1.1 Notion of the Double Integral
1. Denition
The double integral of a function of two variables u = f (x y) over a planar domain S is denoted by Z ZZ f (x y) dS = f (x y) dy dx: (8.134) S
S
It is a number, if it exists, and it is de ned in the following way (Fig. 8.28): 1. We consider a partition of the domain S into n elementary domain. 2. We chose an arbitrary point Pi(xi yi) in the interior or on the boundary of every elementary domain. 3. We multiply the value of the function u = f (xi yi) at this point by the area %Si of the corresponding elementary domain. 4. We add these products f (xi yi)%Si. 5. We calculate the limit of the sum n X f (xi yi)%Si (8.135a) i=1
as the diameter of the elementary domains tends to zero, consequently %Si tends to zero, and so n tends to 1. (The diameter of a set of points is the supremum of the distances between the points of the set.) The requirement %S tends to zero is not enough, because, e.g., in the case of a rectangle the area can be close to zero also if only one side is small and the other is not, so the considered points could be far from each other. z
y
u=f(x,y)
Pi(xi ,yi) S
DSi 0 S x
0
x
DSi
y Pi(xi ,yi)
Figure 8.28 Figure 8.29 If this limit exists independently of the partition of the domain S into elementary domains and also of the choice of the points Pi(xi yi), then we call it the double integral of the function u = f (x y) over the domain S , the domain of integration, and we write:
Z S
f (x y) dS = lim Si !0
n X
n!1 i=1
2. Existence Theorem
f (xi yi) %Si:
(8.135b)
If the function f (x y) is continuous on the domain of integration including the boundary, then the double integral (8.135b) exists. (This condition is sucient but not necessary.)
472 8. Integral Calculus
3. Geometrical Meaning
The geometrical meaning of the double integral is the volume of a solid whose base is the domain in the x y plane, whose side is a cylindrical surface with generators parallel to the z-axis, and it is bounded above by the surface de ned by u = f (x y) (Fig. 8.29). Every term f (xi yi)%Si of the sum (8.135b) corresponds to an elementary cell of a prism with base %Si and with altitude f (xi yi). The sign of the volume is positive or negative, according to whether the considered part of the surface u = f (x y) is above or under the x y plane. If the surface intersects the x y plane, then the volume is the algebraic sum of the positive and negative parts. If the value of the function is identically 1 (f (x y) 1), then the volume has the numerical value of the area of the domain S in the x y plane.
8.4.1.2 Evaluation of the Double Integral
The evaluation of the double integral is reduced to the evaluation of a repeated integral, i.e., to the evaluation of two consecutive integrals.
1. Evaluation in Cartesian Coordinates
If the double integral exists, then we can consider any type of partition of the domain of integration, such as a partition into rectangles. We divide the domain of integration into in nitesimal rectangles by coordinate lines (Fig. 8.30a). Then we calculate the sum of all di erentials f (x y)dS starting with all the rectangles along every vertical stripe, then along every horizontal stripe. (The interior sum is an integral approximation sum with respect to the variable y, the exterior one with respect to x.) If the integrand is continuous, then this repeated integral is equal to the double integral on this domain. The analytic notation is:
Z S
2 3 Zb 6 'Z2 (x) Zb 'Z2(x) f (x y) dS = 4 f (x y) dy75 dx = f (x y) dy dx: a '1 (x)
(8.136a)
a '1 (x)
_ Here y = '2(x) and y = '1(x) are the equations of the upper and lower boundary curves (AB )above _ and (AB )below of the surface patch S . Here a and b are the abscissae of the points of the curves to the very left and to the very right. The elementary area in Cartesian coordinates is dS = dx dy: (8.136b) (The area of the rectangle is %x%y independently of the value of x.) For the rst integration x is y (x)
ϕ2
dy
a)
B
(x) 1 y=ϕ
A a
dx
b x
β dy α 0
y
) (y D
ψ1
x=
ψ C x=
dx
4
(y) 2
x
n
B
0
m ϕ2
dρ
ρdϕ
0
y=
y
ϕ1
A
b) Figure 8.30 Figure 8.31 Figure 8.32 handled as a constant. The square brackets in (8.136a) can be omitted, since according to the notation the interior integral is referred to the interior integration variable, the exterior integral is referred to the second variable. In (8.136a) the di erential signs dx and dy are at the end of the integrand. It is also usual to put these signs right after the corresponding integral signs, in front of the integrand. We can perform the summation in reversed order, too, (Fig. 8.30b). If the integrand is continuous, 0
2
x
8.4 Multiple Integrals 473
then it results also in the double integral:
Z S
Z Z2(y) f (x y) dx dy:
f (x y) dS =
(8.136c)
1 (y)
Z A = xy2 dS , where S is the surface patch between the parabola y = x2 and the line y = 2x in S Z2 Z 2 Z 2x Z2 3 2x Fig. 8.31. A = 0 x2 xy2 dy dx = 0 x dx y3 2 = 31 0 (8x4 ; x7 ) dx = 325 or x p Z 4 Z py Z2 Z4 2 y 2! x 1 2 2 A= xy dx dy = y dy 2 = 2 y2 y ; y4 dy = 32 5. 0 y=2 0 0 y=2
2. Evaluation in Polar Coordinates
The integration domain is divided by coordinate lines into elementary parts bounded by the arcs of two concentric circles and two segments of rays issuing from the pole (Fig. 8.32). The area of the elementary domain in polar coordinates has the form d d' = dS: (8.137a) (The area of an elementary part determined by the same % and %' is obviously smaller being close to the origin, and larger far from it.) With an integrand given in polar coordinates w = f ( ') we perform a summation rst along each sector, then with respect to all sectors:
Z S
f ( ') dS =
Z'2 Z2 (') f ( ') d d'
(8.137b)
'1 1 (')
where = 1 (') and = 2 (') are the equations of the interior and the exterior boundary curves _ _ AmB and AnB of the surface S and '1 and '2 are the in mum and supremum of the polar angles of the points of the domain. The reverse order of integration is seldom used. Z A = sin2 ' dS , where S is a half-circle = 3 cos ' (Fig. 8.33): S
A=
Z =2 Z 3 cos ' 0
0
2 sin2 ' d d' =
Z =2 0
Z =2 3 3 cos ' sin2 ' d' 3 =9 sin2 ' cos3 ' d' = 56 . 0 0
3. Evaluation with Arbitrary Curvilinear Coordinates u and v
The coordinates are de ned by the relations x = x(u v) y = y(u v) (8.138) (see 3.6.3.1, p. 244). The domain of integration is partitioned by coordinate lines u = const and v = const into in nitesimal surface elements (Fig. 8.34) and the integrand is expressed by the coordinates u and v. We perform the summation along one strip, e.g., along v = const, then over all strips:
Z S
f (u v) dS =
Zu2 vZ2 (u) f (u v)jDj dv du:
u1 v1 (u)
(8.139)
_ _ Here v = v1(u) and v = v2(u) are the equations of the boundary curves AmB and AnB of the surface S . We denote by u1 and u2 the in mum and supremum of the values of u of the points belonging to the
474 8. Integral Calculus surface S . jDj denotes the absolute value of the Jacobian determinant (functional determinant) @x @x D ( x y ) @v : D = D(u v) = @u (8.140a) @y @y @u @v The area of the elementary domain in curvilinear coordinates can be easily expressed: jDj dv du = dS: (8.140b) y v=con s
0
u=const.
A x (3, 0)
a
t. u=u1
m dS
a x
u2
u=
B
-a
n
-a
Figure 8.33 Figure 8.34 Figure 8.35 The formula (8.137b) is a special case of (8.139) for the polar coordinates x = cos ' y = sin '. The functional determinant here is D = . We choose curvilinear coordinates so that the limits of integration in the formula (8.139) are as simple as possible, and also the integrand is not very complicated. Z Calculate A = f (x y) dS for the case when S is the interior of an asteroid (see 2.13.4, p. 102), S
with x = a3 cos3 t y = a sin3 t (Fig. 8.35). First we introduce the curvilinear coordinates x = u cos3 v, y = u sin v whose coordinate lines u = c1 represents a family of similar asteroids with equations x = c1 cos3 t and y = c1 sin3 t. The coordinate lines v = c2 are rays with the equations y = kx, where 3 k = tan c32 holds. We2 get cos v sin v = 3u sin2 v cos2 v, A = Z a Z 2 f (x(u v) y(u v)) 3u sin2 v cos2 v dv du. D = sin3 vv ;33uu cos 2 sin v cos v 0
8.4.1.3 Applications of the Double Integral
0
Some applications of the double integral are collected in Table 8.9. The required areas of elementary domains in Cartesian and polar coordinates are given in Table 8.8 Tabelle 8.8 Plane elements of area
Coordinates
Element of area
Cartesian coordinates x y
dS = dy dx
Polar coordinates '
dS = d d'
Arbitrary curvilinear coordinates u v dS = jDj du dv (D Jacobian determinant)
8.4 Multiple Integrals 475
Table 8.9 Applications of the double integral
General formula Cartesian coordinates 1. Area of a plane gure: S=
Z
dS
S
2. Surface: Z dS SO = cos S
=
ZZ
dy dx
Polar coordinates =
! ! u ZZ v u @z 2+ @z 2 dy dx = ZZ = t1 + @x @y
3. Volume of a cylinder: Z V = z dS S
ZZ
=
ZZ
z dy dx
=
ZZ
d d'
v !2 !2 u u t 2 + 2 @z + @z d d' @ @' z d d'
4. Moment of inertia of a plane gure, with respect to the x-axis: Z Ix = y2 dS S
=
ZZ
y2 dy dx
=
ZZ
3 sin2 ' d d'
5. Moment of inertia of a plane gure, with respect to the pole 0: Z I0 = 2 dS S
=
ZZ
(x2 + y2) dy dx
=
6. Mass of a plane gure with the density function %: Z M = % dS S
=
ZZ
% dy dx
=
ZZ ZZ
3 d d'
% d d'
7. Coordinates of the center of gravity of a homogeneous plane gure: Z
xC = S
Z yC = S
x dS S y dS S
ZZ
= ZZ
ZZ = ZZ
x dy dx dy dx y dy dx dy dx
ZZ
=
ZZ =
2 cos ' d d' ZZ d d' 2 sin ' d d' ZZ d d'
476 8. Integral Calculus
8.4.2 Triple Integrals
The triple integral is an extension of the notion of the integral into three-dimensional domains. We also call it volume integral.
8.4.2.1 Notion of the Triple Integral 1. Denition
We de ne the triple integral of a function f (x y z) of three variables over a three-dimensional domain V analogously to the deZZnition Z Z of the double integral. We write: f (x y z) dV = f (x y z) dz dy dx: (8.141) V
V
The volume V (Fig. 8.36) is partitioned into elementary volumes %Vi. Then we form the products f (xi yi zi)%Vi, where the point Pi(xi yi zi) is inside the elementary volume or it is on the boundary. The triple integral is the limit of the sum of these products with all the elementary volumes in which the volume V is partitioned, then the diameter of every elementary volume tends to zero, i.e., their number tends to 1. The triple integral exists only if the limit is independent of the partition into elementary volumes and the choice of the points Pi(xi yi zi). Then we have:
Z
V
f (x y z) dV = lim Vi !0
n X
n!1 i=1
f (xi yi zi ) %Vi:
(8.142)
2. Existence Theorem
The existence theorem for the triple integral is a perfect analogue of the existence theorem for the double integral. z z
z=ψ 2(x, y)
y
dz
V
Γ z=ψ 1(x, y)
Pi
0
DVi
dx b
0 x
a
y
y=ϕ1(x)
C x
Figure 8.36
y=ϕ2(x)
dy
Figure 8.37
8.4.2.2 Evaluation of the Triple Integral
The evaluation of triple integrals is reduced to repeated evaluation of three ordinary integrals. If the triple integral exists, then we can consider any partition of the domain of integration.
1. Evaluation in Cartesian Coordinates
The domain of integration can be considered as a volume V here. We prepare a decomposition of the domain by coordinate surfaces, in this case by planes, into in nitesimal parallelepipeds, i.e., their diameter is an in nitesimal quantity (Fig. 8.37). Then we perform the summation of all the products f (x y z) dV , starting the summation along the vertical columns, i.e., summation with respect to z, then in all columns of one slice, i.e., summation with respect to y, and nally in all such slices, i.e., summation with respect to x. Every single sum for any column is an approximation sum of an integral, and if the
8.4 Multiple Integrals 477
diameter of the parallelepipeds tends to zero, then the sums tend to the corresponding integrals, and if the integrand is continuous, then this repeated integral is equal to the triple integral. Analytically:
Z
V
8 '2(x)2 2 (xy) 3 9 > Zb > 4 f (x y z) dz75 dy> dx : " a '1 (x) 1 (xy) Zb 'Z2 (x) 2Z(xy) = f (x y z) dz dy dx:
(8.143a)
a '1 (x) 1 (xy)
Here z = 1 (x y) and z = 2 (x y) are the equations of the lower and upper part of the surface bounding the domain of integration V (see limiting curve ; in Fig. 8.37) dx dy dz is the elementary volume in the Cartesian coordinate system. y = '1 (x) and y = '2(x) are the functions describing the lower and upper part of the curve C which is the boundary line of the projection of the volume onto the x y plane, and x = a and x = b are the extreme values of the x coordinates of the points of the volume under consideration (and also the projection under consideration). We have the following postulates for the domain of integration: The functions '1(x) and '2 (x) are de ned and continuous in the interval a x b, and they satisfy the inequality '1(x) '2(x). The functions 1(x y) and 2 (x y) are de ned and continuous on the domain a x b, '1(x) y '2 (x), and also 1 (x y) 2 (x y) holds. In this way, every point (x y z) in V satis es the relations (8.143b) axb '1 (x) y '2(x) 1 (x y) z 2 (x y): Just as with double integrals, we can change the order of integration, then the limiting functions will change in the same sense. (Formally: the limits of the outermost integral must be constants, and any limit may contain variables only of exterior integrals.) Z Calculate the integral I = (y2 + z2 ) dV for a pyramid bounded by the coordinate planes and the V
planeZ x Z+ y +Zz = 1: Z 1 Z 1;x Z 1;x;y 1 1;x 1;x;y 2 1. I= (y + z2 ) dz dy dx = (y2 + z2 ) dz dy dx = 30 0
0
0
0
0
0
z
dz j 0dr dj r r dr dj x
z= a2-ρ2
y
Figure 8.38
2. Evaluation in Cylindrical Coordinates
x
ϕ
0 ρ
y
Figure 8.39
The domain of integration is decomposed into in nitesimal elementary cells by coordinate surfaces = const ' = const z = const (Fig. 8.38). The volume of an elementary domain in cylindrical coordinates is dV = dz d d': (8.144a)
478 8. Integral Calculus After de ning the integrand by cylindrical coordinates f ( ' z) the integral is:
Z V
f ( ' z) dV =
Z'2 Z2(') z2Z(') f ( ' z) dz d d':
(8.144b)
'1 1 (') z1 (')
Z
dV for a solid (Fig. 8.39) bounded by the x y plane, the x z plane, q = a2: z1 = 0 z2 = a2 ; x2 ; y2 = Z =2 Z a cos ' Z pa2 ;2 q2 2 dz d d' = a ; 1 = 0 2 = a cos ' '1 = 0 '2 = 2 . I = 0 0 0 9 2 3 p Z =2 8 Z Z 2 2 3 < a cos ' = 4 a ; dz 5 d " d' = a (3 ; 4) . Since f ( ' z) = 1, the integral is equal to : 18 Calculate the integral I =
V the cylindrical surface x2 + y2 = ax and the sphere x2 + y2 + z2
0
0
0
the volume of the solid.
z
z
dJ
j x
rdJ r sinJ dj r dr
r2=
h cos J 2a
y
dj
Figure 8.40
h
dr J
0 y
x
Figure 8.41
3. Evaluation in Spherical Coordinates
The domain of integration is decomposed into in nitesimal elementary cells by coordinate surfaces r = const ' = const = const (Fig. 8.40). The volume of an elementary domain in spherical coordinates is dV = r2 sin dr d d': (8.145a) For the integrand f (r ' ) in spherical coordinates, the integral is:
Z V
f (r ' ) dV =
Z'2 Z2 (') r2Z(') f (r ' ) r2 sin dr d d':
'1 1 (') r1 (')
(8.145b)
Z cos r2 dV for a cone whose vertex is at the origin, and its symmetry axis V is the z-axis. The angle at the vertex is 2, the altitude of the cone is h (Fig. 8.41). Consequently we have: r1 = 0 r2 = cosh 1 = 0 2 = '1 = 0 '2 = 2. "Z h= cos ) Z 2 Z Z h= cos cos Z 2 (Z 2 sin dr d d' = I= r cos sin dr d d' r2 0 0 0 0 0 0 Calculate the integral I =
8.5 Surface Integrals 479
= 2 h (1 ; cos ).
4. Evaluation in Arbitrary Curvilinear Coordinates u v w
The coordinates are de ned by the equations x = x(u v w) y = y(u v w) z = z(u v w) (8.146) (see 3.6.3.1, p. 244). The domain of integration is decomposed into in nitesimal elementary cells by the coordinate surfaces u = const v = const w = const. The volume of an elementary domain in arbitrary coordinates is: @x @x @x @u @v @w @y @y @y dV = jDj du dv dw with D = @u (8.147a) @w @z @v @z @z @u @v @w i.e., D is the Jacobian determinant. For the integrand f (u v w) in curvilinear coordinates u v w, the integral is:
Z
V
f (u v w) dV =
Zu2 vZ2 (u) w2Z(uv) f (u v w) jDj dw dv du:
u1 v1 (u) w1 (uv)
(8.147b)
Remark: The formulas (8.144b) and (8.145b) are special cases of (8.147b). 2
For cylindrical coordinates D = holds, for spherical coordinates D = r sin is valid. If the integrand is continuous, then we can change the order of integration in any coordinate system. We always try to choose a curvilinear coordinate system such that the determination of the limits of the integral (8.147b), and also the calculation of the integral, should be as easy as possible.
8.4.2.3 Applications of the Triple Integral
Some applications of the triple integral are collected in Table 8.11. The elementary areas corresponding to di erent coordinates are given in Table 8.8. The elementary volumes corresponding to di erent coordinates are given in Table 8.10. Table 8.10 Elementary volumes
Coordinates
Elementary volume
Cartesian coordinates x y z
dV = dx dy dz
Cylindrical coordinates ' z
dV = d d' dz
Spherical coordinates r '
dV = r2 sin dr d d'
Arbitrary curvilinear coordinates u v w dV = jDj du dv dw (D Jacobian determinant)
8.5 Surface Integrals
We distinguish surface integrals of the rst type, of the second type, and of general type, analogously to the three di erent line integrals (see 8.3, p. 462).
8.5.1 Surface Integral of the First Type
The surface integral or integral over a surface in space is the generalization of the double integral, similarly as the line integral of the rst type (see 8.3.1, p. 463) is a generalization of the ordinary integral.
480 8. Integral Calculus Table 8.11 Applications of the triple integral
General Cartesian formula coordinates 1. Volume of a solid V=
Z
ZZZ
dV =
V
dz dy dx
Cylindrical coordinates
ZZZ
dz d d'
Spherical coordinates
ZZZ
r2 sin dr d d'
2. Axial moment of inertia of a solid with respect to the z-axis ZZZ
Z Iz = 2 V = V
(x2 + y2) dz dy dx
ZZZ
3 dz d d'
3. Mass of a solid with the density function % Z M = % dV =
ZZZ
V
% dz dy dx
ZZZ
% dz d d'
4. Coordinates of the center of a homogeneous solid Z xC = V
Z yC =
V
Z zC = V
x dV V y dV V z dV V
ZZZ
=
ZZZ ZZZ
=
ZZZ ZZZ
=
ZZZ
x dz dy dx dz dy dx y dz dy dx dz dy dx z dz dy dx dz dy dx
ZZZ
2 cos ' d d' dz ZZZ d d' dz ZZZ 2 sin ' d d' dz ZZZ d d' dz ZZZ z d d' dz ZZZ d d' dz
ZZZ ZZZ
r4 sin3 dr d d'
%r2 sin dr d d'
ZZZ
r3 sin2 cos ' dr d d' ZZZ r2 sin dr d d' ZZZ r3 sin2 sin ' dr d d' ZZZ r2 sin dr d d' ZZZ r3 sin cos dr d d' ZZZ r2 sin dr d d'
8.5.1.1 Notion of the Surface Integral of the First Type
1. Denition
The surface integral of the rst type of a function u = f (x y z) of three variables de ned in a connected domain Z is the integral f (x y z) dS (8.148) S
over a region S of a surface. The numerical value of the surface integral of the rst kind is de ned in the following way (Fig. 8.42): 1. We decompose the region S in an arbitrary way into n elementary regions. 2. We choose an arbitrary point Pi(xi yi zi) inside or on the boundary of each elementary region.
8.5 Surface Integrals 481 z
3. We multiply the value f (xi yi zi) of the function at this point by the area %Si of the corresponding elementary region. 4. We add the products f (xi yi zi)%Si so obtained. 5. We determine the limit of the sum
y Pi
n X
S DS i
i=1
0
f (xi yi zi) %Si
(8.149a)
x as the diameter of each elementary region tends to zero, so %Si tends to zero, Figure 8.42 hence, their number n tends to 1 (see 8.4.1.1, 1., p. 471). If this limit exists and is independent of the particular decomposition of the region S into elementary regions and also of the choice of the points Pi(xi yi zi), then it is called the surface integral of the rst type of the function u = f (x y z ) over the region S , and we write:
Z S
f (x y z) dS = lim Si !0
n X
n!1 i=1
f (xi yi zi) %Si:
(8.149b)
2. Existence Theorem
If the function f (x y z) is continuous on the domain, and the functions de ning the surface have continuous derivatives here, the surface integral of the rst type exists.
8.5.1.2 Evaluation of the Surface Integral of the First Type
The evaluation of the surface integral of the rst type is reduced to the evaluation of a double integral over a planar domain (see 8.4.1, p. 471). 1. Explicit Representation of the Surface If the surface S is given by the equation z = z (x y ) (8.150) in explicit form, then Z ZZ q f (x y z) dS = f x y z(x y)] 1 + p2 + q2 dx dy (8.151a) S
S0
is valid, where S 0 is the projection of S onto the x y plane and p and q are the partial derivatives @z q = @z . Here we assume that to every point of the surface S there corresponds a unique point p = @x @y in S 0 in the x y plane, i.e., the points of the surface are de ned uniquely by their coordinates. If it does not hold, we decompose S into several parts each of which satis es the condition. Then the integral on the total surface can be calculated as the algebraic sum of the integrals over these parts of S . The equation (8.151a) can be written in the form Z ZZ f (x y z) dS = f x y z(x y)] dcosSxy (8.151b) S Sxy y Z ;z since the equation of the surface normal of (8.150) has the form X p; x = Y ; q = ;1 (see Table 3.29, p. 246), since for the angle between the direction of the normal and the z-axis, cos = p1 + 1p2 + q2 holds. In evaluating a surface integral of the rst type, this angle is always considered as an acute angle, so cos > 0 always holds.
482 8. Integral Calculus 2. Parametric Representation of the Surface
If the surface S is given in parametric form by the equations x = x(u v) y = y (u v ) z = z (u v ) (Fig. Z8.43), then f (x y z) dS
u1 m (u) v=v 1
A v= v
=
2 (u
)
u= n con st.
S
B
u2
ZZ p f x(u v) y(u v) z(u v) EG ; F 2 du dv
(8.152a)
(8.152b)
where the functions E F , and G are the quantities given in 3.6.3.3, 1., p. 245. The elementary region in parametric form is p EG ; F 2 du dv = dS (8.152c)
Figure 8.43 and % is the domain of the parameters u and v corresponding to the given surface region. The evaluation is performed by a repeated integration with respect to v and u:
Z S
(u v) dS =
Zu2 vZ2 (u) p (u v) EG ; F 2 dv du = f x(u v) y(u v) z(u v)]:
u1 v1 (u)
(8.152d)
Here u1 and u2 are coordinates of the extreme coordinate lines u = const enclosing the region S _ _ (Fig. 8.43), and v = v1(u) and v = v2 (u) are the equations of the curves AmB and AnB of the boundary of S . Remark: The formula (8.151a) is a special case of (8.152b) for u = x v = y E = 1 + p2 F = p q G = 1 + q 2 : (8.153)
3. Elementary Regions of Curved Surfaces
The elementary regions of curved surfaces are given in Table 8.12. Table 8.12 Elementary regions of curved surfaces
Coordinates
Elementary region
v ! ! u u @z 2 + @z 2 dx dy Cartesian coordinates x y z = z(x y) dS = t1 + @x @y Cylindrical lateral surface, dS = R d' dz R (const. radius), coordinates ' z Spherical surface R (const. radius), dS = R2 sin d d' coordinates ' p Arbitrary curvilinear coordinates u v dS = EG ; F 2 du dv (E F G see di erential of arc, p. 245)
8.5.1.3 Applications of the Surface Integral of the First Type
1. Surface Z Area of a Curved Surface S = dS: S
(8.154)
8.5 Surface Integrals 483
2. Mass of an Inhomogeneous Curved Surface S
With the coordinate-dependent density % = f (x y z) we have: Z MS = % dS:
(8.155)
S
8.5.2 Surface Integral of the Second Type
The surface integral of the second type , also called an integral over a projection , is a generalization of the notion of double integral similarly to the surface integral of the rst type.
8.5.2.1 Notion of the Surface Integral of the Second Type
1. Notion of an Oriented Surface
A surface usually has two sides, and one of them can be chosen arbitrarily as the exterior one. If the exterior side is xed, we call it an oriented surface. We do not discuss surfaces for which we cannot de ne two sides (see 8.7]).
2. Projection of an Oriented Surface onto a Coordinate Plane
If we project a bounded part S of an oriented surface onto a coordinate plane, e.g., onto the x,y plane, we can consider this projection Prxy S as positive or negative in the following way (Fig. 8.44):
0
0 x
a)
z
z
z
− PrxyS
y
+ PrxyS
b)
x
PrxzS
A' A
B
x
−
+
y
+
exterior side
C
interior side
−
C' D'
d)
c)
PrxyS
PryzS
z B'
y
0
D y
x
Figure 8.44
a) If the x y plane is looked at from the positive direction of the z-axis, and we see the positive side of the surface S , where the exterior part is considered to be positive, then the projection Prxy S has a positive sign, otherwise it has a negative sign (Fig. 8.44 a,b). b) If one part of the surface shows its positive side and the other part its negative side, then the projection Prxy S is regarded as the algebraic sum of the positive and negative projections (Fig. 8.44c). The Fig. 8.44d shows the projections Prxz S and Pryz S of a surface S one of them is positive the other one is negative. The projection of a closed oriented surface is equal to zero.
484 8. Integral Calculus
3. Denition of the Surface Integral of the Second Type over a Projection onto a Coordinate Plane
The surface integral of the second type of a function f (x y z) of three variables de ned in a connected domain is the integral Z f (x y z) dx dy (8.156) S
over the projection of an oriented surface S onto the x y plane, where S is in the same domain where the function is de ned, and if there is a one-to-one correspondence between the points of the surface and its projection. The numerical value of the integral is obtained in the same way as the surface integral of the rst type except that in the third step the function value f (xi yi zi ) is not multiplied by the elementary region %Si , but by its projection Prxy %Si , oriented according to 8.5.2.1, 2., p. 483 on the x y plane. Then we get:
Z S
f (x y z) dx dy = lim Si !0
n X
n!1 i=1
f (xi yi zi) Prxy %Si:
(8.157a)
We de ne analogously the surface integrals of the second type over the projections of the oriented surface S onto the y z plane and onto the z x plane:
Z S
Z S
f (x y z) dy dz = lim Si !0
n X
n!1 i=1
f (x y z) dz dx = lim Si !0
n X
n!1 i=1
f (xi yi zi ) Pryz %Si
(8.157b)
f (xi yi zi) Przx %Si:
(8.157c)
4. Existence Theorem for the Surface Integral of the Second Type
The surface integral of the second type (8.157a,b,c) exists if the function f (x y z) is continuous and the equations de ning the surface are continuous and have continuous derivatives.
8.5.2.2 Evaluation of Surface Integrals of the Second Type The principal method is to reduce it to the evaluation of double integrals.
1. Surface Given in Explicit Form
If the surface S is given by the equation z = '(x y) in explicit form, then the integral (8.157a) is calculated by the formula Z Z f (x y z) dx dy = f x y '(x y)] dSxy S
Prxy S
S
Pryz S
S
PrzxS
(8.158) (8.159a)
where Sxy = Prxy S . The surface integral of the function f (x y z) over the projections of the surface S onto the other coordinate planes is calculated similarly: Z Z f (x y z) dy dz = f (y z) y z] dSyz (8.159b) where we substitute x = (y z), the equation of the surface S solved for x, and Syz = Pryz S . Z Z f (x y z) dz dx = f x (z x) z] dSzx (8.159c) where we substitute y = (z x), the equation of the surface S solved for y, and Szx = PrzxS . If the orientation of the surface is changed, i.e., if we interchange the exterior and interior sides, then the integral over the projection changes its sign.
8.5 Surface Integrals 485
2. Surface Given in Parametric Form
If the surface is given by the equations x = x(u v) y = y (u v ) z = z(u v) (8.160) in parametric form,we calculate the integral (8.157a,b,c) with help of the following formulas: Z Z (x y) du dv (8.161a) f (x y z) dx dy = f x(u v) y(u v) z(u v)] D D (u v ) S
Z
Z D(y z) du dv f (x y z) dy dz = f x(u v) y(u v) z(u v)] D (u v ) S Z Z (z x) du dv: f (x y z) dz dx = f x(u v) y(u v) z(u v)] D D (u v ) S
(8.161b) (8.161c)
(x y) D(y z) D(z x) Here the expressions D D(u v) D(u v) D(u v) are the Jacobian determinants of pairs of functions x y z with respect to the variables u and v % is the domain of u and v corresponding to the surface S.
8.5.3 Surface Integral in General Form
8.5.3.1 Notion of the Surface Integral in General Form
If P (x y z) Q(x y z) R(x y z) are three functions of three variables de ned in a connected domain and S is an oriented surface contained in this domain, the sum of the integrals of the second type taken over the projections on the three coordinate planes is called the surface integral in general form: Z Z Z Z (P dy dz + Q dz dx + R dx dy) = P dy dz + Q dz dx + R dx dy: (8.162) S
S
S
S
The formula reducing the surface integral to a double integral is:
Z Z " D(y z) D(z x) + R D(x y) du dv (P dy dz + Q dz dx + R dx dy) = P D + Q (8.163) (u v ) D(u v) D(u v) S where the quantities D(x y) D(y z) D(z x) , and % have the same meaning, as above. D (u v ) D ( u v ) D (u v ) Remark: The surface integral of vector-valued functions is discussed in the chapter about the theory of vector elds (see 13.3.2. p. 663).
8.5.3.2 Properties of the Surface Integrals
1. If Zthe domain of integration, i.e., the surface S , is decomposed into two parts S1 and S2, then Z S
(P dy dz + Q dz dx + R dx dy) =
S1
(P dy dz + Q dz dx + R dx dy)
Z
+ (P dy dz + Q dz dx + R dx dy): S2
(8.164)
2. If the orientation of the surface is reversed, i.e., if we interchange the exterior and interior sides, the integral changes its sign: Z Z (P dy dz + Q dz dx + R dx dy) = ; (P dy dz + Q dz dx + R dx dy) S+
S;
(8.165)
486 8. Integral Calculus where S + and S ; denote the same surface with di erent orientation. 3. A surface integral depends, in general, on the line bounding the surface region S as well as on the surface itself. Thus the integrals taken over two di erent non-closed surface regions S1 and S2 spanned by the same closed curve C are, in general, not equal (Fig. 8.45): Z (P dy dz + Q dz dx + R dx dy)
ZS1
6= (P dy dz + Q dz dx + R dx dy): S2
8.5.3.3 An Application of the Surface Integral
(8.166)
S1
C S2
Figure 8.45
The volume V of a solid bounded by a closed surface S can be expressed and calculated by a surface integral Z V = 13 (x dy dz + y dz dx + z dx dy) (8.167) S where S is oriented so that its exterior side is positive.
487
9 DierentialEquations
1. A Di erential Equation is an equation, in which one or more variables, one or more functions of these variables, and also the derivatives of these functions with respect to these variables occur. The order of a di erential equation is equal to the order of the highest occurring derivative. 2. Ordinary and Partial Di erential Equations di er from each other in the number of their independent!variables in the rst case there is only one, in the second case there are several. dy 2 ; xy5 dy +sin y = 0. @ 2 z = xyz @z @z . A: dx B: xd2 ydx ; dy(dx)2 = ey (dy)3. C: @x@y dx @x @y
9.1 Ordinary Dierential Equations
1. General Ordinary Di erential Equation of Order n
in implicit form has the equation h i F x y(x) y0(x) : : : y(n)(x) = 0: (9.1) ( n ) If this equation is solved for y (x), then it is the explicit form of an ordinary di erential equation of order n.
2. Solution or Integral
of a di erential equation is every function satisfying the equation in an interval a x b which can be also in nite. A solution, which contains n arbitrary constants c1 c2 : : : cn, is called the general solution or general integral. If the values of these constants are xed, a particular integral or a particular solution is obtained. The value of these constants can be determined by n further conditions. If the substitution values of y and its derivatives up to order n ; 1 are prescribed at one of the endpoints of the interval, then the problem is called an initial value problem. If there are given substitution values at both endpoints of the interval, then the problem is called a boundary value problem. The di erential equation ;y0 sin x + y cos x = 1 has the general solution y = cos x + c sin x. For the condition c = 0 we get the particular solution y = cos x.
3. Initial Value Problem 0
If the n values y(x0) y (x0 ) : : : y(n;1)(x0) are given at x0 for the solution y = y(x) of an n-th order ordinary di erential equation, then an initial value problem is given. The numbers are called the initial values or initial conditions. They form a system of n equations for the unknown constants c1 c2 : : : cn of the general solution of the n-th order ordinary di erential equation. The harmonic motion of a special elastic spring-mass system can be modeled by the initial value problem y00 + y = 0 with y(0) = y0 y0(0) = 0. The solution is y = y0 cos x.
4. Boundery Value Problem
If the solution of an ordinary di erential equation and/or its derivatives are given at several points of its domain, then these values are called the boundary conditions. We call a di erential equation with boundary conditions a boundary value problem. The bending line of a bar with xed endpoints and uniform load is described by the di erential equation y00 = x ; x2 with the boundary conditions y(0) = 0 y(1) = 0 (0 x 1). The solution is 3 x4 y = x6 ; 12 ; 12x .
488 9. Dierential Equations
9.1.1 First-Order Dierential Equations 9.1.1.1 Existence Theorems, Direction Field 1. Existence of a Solution
In accordance with the Cauchy existence theorem the di erential equation y 0 = f (x y ) (9.2) has at least one solution in a neighborhood of x0 such that it takes the value y0 at x = x0 if the function f (x y) is continuous in a neighborhood G of the point (x0 y0). For example, we may select G as the region given by jx ; x0 j < a and jy ; y0j < b with some a and b.
2. Lipschitz Condition
The Lipschitz condition with respect to y is satis ed by f (x y) if jf (x y1) ; f (x y2)j N jy1 ; y2j (9.3) holds for all (x y1) and (x y2) from G, where N is independent of x, y1, and y2. If this condition is satis ed, then the di erential equation (9.2) has a unique solution through (x0 y0). The Lipschitz condition is obviously satis ed if f (x y) has a bounded partial derivative @f=@y in this neighborhood. In 9.1.1.4, p. 493 we will see examples in which the assumptions of the Cauchy existence theorem are not satis ed.
3. Direction Field
If the graph of a solution y = '(x) of the di erential equation y0 = f (x y) goes through the point P (x y), then the slope dy=dx of the tangent line of the graph at this point can be determined from the di erential equation. So, at every point (x y) the di erential equation de nes the slope of the tangent line of the solution passing through the considered point. The collection of these directions (Fig. 9.1) forms the direction eld. An element of the direction eld is a point together with the direction associated to it. Integration of a rst-order di erential equation geometrically means to connect the elements of a direction eld into an integral curve, whose tangents have the same slopes at all points as the corresponding elements of the direction eld. y
y
P(x0 ,y0) x
0
Figure 9.1
4. Vertical Directions
0
x
Figure 9.2
If in a direction eld we have vertical directions, i.e., if the function f (x y) has a pole, we exchange the role of the independent and dependent variables and we consider the di erential equation dx = 1 (9.4) dy f (x y) as an equivalent equation to (9.2). In the region where the conditions of the existence theorems are ful lled for the di erential equations (9.2) or (9.4), there exists a unique integral curve (Fig. 9.2) through every point P (x0 y0).
9.1 Ordinary Dierential Equations 489
5. General Solution
The set of all integral curves can be characterized by one parameter and it can be given by the equation F (x y C ) = 0 (9.5a) of the corresponding one-parameter family of curves. The parameter C , an arbitrary constant, can be chosen freely and it is a necessary part of the general solution of every rst-order di erential equation. A particular solution y = '(x), which satis es the condition y0 = '(x0 ), can be obtained from the general solution (9.5a) if C is expressed from the equation F (x0 y0 C ) = 0: (9.5b)
9.1.1.2 Important Solution Methods 1. Separation of Variables
If a di erential equation can be transformed into the form M (x)N (y)dx + P (x)Q(y)dy = 0 (9.6a) then it can be rewritten as R(x)dx + S (y)dy = 0 (9.6b) where the variables x and y are separated into two terms. To get this form, we divide equation (9.6a) by P (x)N (y). For the general solution we have Z M (x) Z Q( y ) dx + (9.7) P (x) N (y) dy = C: If for some values x = x or y = y, the functions P (x) or N (y) or both are equal to zero, then the constant functions x = x or/and y = y are also solutions of the di erential equation. They are called singular solutions. Z dy Z dx xdy + ydx = 0 y + x = C ln jyj + ln jxj = C = ln jcj yx = c. If we allow also c = 0 in this nal equation, then we have the singular solutions y 0 and x 0.
2. Homogeneous Equations
If M (x y) and N (x y) are homogeneous functions of the same order (see 2.18.2.6, 1., p. 121), then in the equation M (x y)dx + N (x y)dy = 0 (9.8) we separate the variables by substitution of u = y=x. x(x ; y)y0 + y2 = 0 with y = u(x)x we get (1 ; u)u0 + u=x = 0. By separation of the variables Z (1 ; u) Z du = ; 1 dx. After integration: ln jxj + ln ju ; uj = C = ln jcj ux = ceu y = cey=x . u x As we have seen in the preceding paragraph, Separation of Variables, the line x = 0 is also an integral curve.
3. Exact Di erential Equations
An exact dierential equation is an equation of the form M (x y)dx + N (x y)dy = 0 or N (x y)y0 + M (x y) = 0 (9.9a) if there exists a function (x y) of two variables such that (9.9b) M (x y)dx + N (x y)dy d (x y) i.e., if the left side of (9.9a) is the total di erential of a function (x y) (see 6.2.2.1, p. 394). If functions M (x y) and N (x y) and their rst-order partial derivatives are continuous on a connected domain G, then the equality @M = @N (9.9c) @y @x
490 9. Dierential Equations is a necessary and sucient condition for equation (9.9a) to be exact. In this case the general solution of (9.9a) is the function (x y) = C (C = const) (9.9d) which can be calculated according to (8.3.4), 8.3.4.4, p. 470 as the integral
Zx Zy (x y) = M ( y) d + N (x0 ) d x0
y0
(9.9e)
where x0 and y0 can be chosen arbitrarily from G. An example is given in 4. Integrating Factor.
4. Integrating Factor
A function (x y) is called an integrating factor or a multiplier if the equation Mdx + Ndy = 0 (9.10a) multiplied by (x y) becomes an exact di erential equation. The integrating factor satis es the di erential equation ln ; M @ ln = @M ; @N : N @ @x (9.10b) @y @y @x Every particular solution of this equation is an integrating factor. To give a general solution of this partial di erential equation is much more complicated than to solve the original equation, so usually we are looking for the solution (x y) in a special form, e.g., (x) (y) (xy) or (x2 + y2). We now solve the di erential equation (x2 + y) dx ; x dy = 0. The equation for the integrating factor is ;x @ ln ; (x2 + y) @ ln = 2. An integrating factor which is independent of y must satisfy x @ ln = @x @y @x ;2, so = x12 . Multiplication of the given di erential equation by yields 1 + xy2 dx ; x1 dy = 0. The general solution according to (9.9e) with the selection of x0 = 1 y0 = 0 is then: ! Zx Zy y (x y) 1 + 2 d ; d = C or x ; y = C1 . x 1 0
5. First-Order Linear Di erential Equations
A rst-order linear dierential equation has the form y0 + P (x)y = Q(x) (9.11a) where the unknown function and its derivative occur only in rst degree, and P (x) and Q(x) are given functions. If P (x) and Q(x) are continuous functions on a nite, closed interval, then the di erential equation here satis es the conditions of the Picard{Lindelof theorem (see p. 610). The integrating factor is here Z = exp P dx (9.11b) the general solution is
Z Z Z y = exp ; P dx Q exp P dx dx + C : (9.11c) If we replace the inde nite integrals by de nite ones with lower bound x0 and upper bound x in this formula, then for the solution y(x0) = C (see 8.2.1.2, 1., p. 442). If y1 is any particular solution of the di erential equation, then the general solution of the di erential equation is given by the formula Z y = y1 + C exp ; P dx : (9.11d)
9.1 Ordinary Dierential Equations 491
If y1(x) and y2(x) are two linearly independent particular solutions (see 9.1.2.3, 2., p. 500), then we can get the general solution without any integration as y = y1 + C (y2 ; y1): (9.11e) 0 Solve the di erential equation y ; y tan x = cos x with the initial condition x = 0 y = 0. We 0 0 Zx calculate exp ; tan x dx = cos x and get the solution according to (9.11c): 0
Zx x : y = cos1 x cos2 x dx = cos1 x sin x cos2 x + x = sin2 x + 2 cos x 0
6. Bernoulli Di erential Equations
The Bernoulli dierential equation is an equation of the form y0 + P (x)y = Q(x)yn (n 6= 0 n 6= 1) (9.12) which can be reduced to a linear di erential equation if it is divided by yn and the new variable z = y;n+1 is introduced. Solve the di erential equation y0 ; 4y = xpy . Since n = 1=2, dividing by py and introducing the x dz ; 2z = x . By using the formulas for the solution of a new variable z = py leads to the equation dx x 2
Z
R linear di erential equation we have exp( P dx) = x12 and z = x2 x2 x12 dx + C = x2 21 ln jxj + C . 2 So, nally, y = x4 12 ln jxj + C .
7. Riccati Di erential Equations
The Riccati dierential equation is the equation y0 = P (x)y2 + Q(x)y + R(x) (9.13a) which cannot usually be solved by elementary integration. However, if we know a particular solution y1 of the Riccati di erential equation, then we can reduce the equation to a linear di erential equation for z by substituting y = y1 + z1 : (9.13b) If we also know a second particular solution y2, then z1 = y ;1 y (9.13c) 2 1 is a particular solution of the linear di erential equation for the function z, so its solution can be simpli ed. If we know three particular solutions y1 y2, and y3, then the general solution of the Riccati di erential equation is y ; y2 : y3 ; y2 = C: (9.13d) y ; y1 y3 ; y1 By the substitution of y = P u(x) + (x) (9.13e) the Riccati di erential equation can be transformed into the normal form du = u2 + R(x): (9.13f) dx
492 9. Dierential Equations With the substitution 0 y = ; P (vx)v (9.13g) we get a second-order linear di erential equation (see 9.1.2.6, 1., p. 507) from (9.13a) P v00 ; (P 0 + PQ)v0 + P 2R v = 0: (9.13h) 1 4 0 2 Solve the di erential equation y + y + x y ; x2 = 0. We substitute y = z + (x) into the equation, and for the coecient of the rst power of z we get the expression 2 + 1=x, which disappears if we substitute (x) = ;1=2x. In this way we get z0 ; z2 ; 415 x2 = 0. We are looking for particular solutions a 3 in the form z1 = x and we get the solutions a1 = ; 2 , a2 = 52 by substitution, i.e., the two particular solutions are z1 = ; 23x , z2 = 25x . The substitution of z = u1 + z1 = u1 ; 23x results in the equation u0 + 3xu = 1. Substituting the particular solution u1 = z ;1 z = x4 we obtain the general solution 2 1 4+C 4 ; 2C x C x 1 3 1 2 x 1 u = 4 + x3 = 4x3 and therefore, y = u ; 2x ; 2x = x5 + C x1 .
9.1.1.3 Implicit Di erential Equations
1
1. Solution in Parametric Form
Suppose we have a di erential equation in implicit form F (x y y0) = 0: (9.14) There are n integral curves passing through a point P (x0 y0) if the following conditions hold: a) The equation F (x0 y0 p) = 0 (p = dy=dx) has n real roots p1 : : : pn at the point P (x0 y0). b) The function F (x y p) and its rst partial derivatives are continuous at x = x0 y = y0 p = pi furthermore @F=@p 6= 0. If the original equation can be solved with respect to y0, then it yields n equations of the explicit forms discussed above. Solving these equations we get n families of integral curves. If the equation can be written in the form x = '(y y0) or y = (x y0), then putting y0 = p and considering p as an auxiliary variable, after di erentiation with respect to y or x we obtain an equation for dp=dy or dp=dx which is solved with respect to the derivative. A solution of this equation together with the original equation (9.14) determines the desired solution in parametric form. Solve the equation x = yy0 + y02. We substitute y0 = p and get x = py + p2 . Di erentiation with 2 respect to y and substituting dx = 1 results in 1 = p + (y + 2p) dp or dy ; py 2 = 2p 2 . Solving dy p p dy dp 1 ; p 1 ; p p p1arcsin this equation for y we get y = ;p + c + ; p2 . Substituting into the initial equation we get the solution for x in parametric form.
2. Lagrange Di erential Equation
The Lagrange dierential equation is the equation a(y0)x + b(y0)y + c(y0) = 0: The solution can be determined by the method given above. If for p = p0 ,
a(p) + b(p)p = 0 (9.14b) then a(p0)x + b(p0)y + c(p0) = 0 is a singular solution of (9.14a).
(9.14a) (9.14c)
9.1 Ordinary Dierential Equations 493
3. Clairaut Di erential Equation
x-
xy
y
y= 27 4
y
=0
The Clairaut dierential equation is the special case of the Lagrange di erential equation if a(p) + b(p)p 0 (9.15a) and so it can be transformed into the form y = y0x + f (y0): (9.15b) The general solution is y = Cx + f (C ): (9.15c) Besides the general solution, the Clairaut di erential equation also has a singular solution, which can be obtained by eliminating the constant C from the equations y = Cx + f (C ) (9.15d) and 0 = x + f 0(C ) (9.15e) We can obtain the second equation by di erentiating the rst one with respect to C . Geometrically, the singular solution is the envelope (see 3.6.1.7, p. 237) of the solution family of lines (Fig. 9.3). Solve the di erential equation y = xy0 + y02. The general solution is y = Cx + C 2, and we get the singular solution with the help of the equation x + 2C = 0 to eliminate C , and hence x2 + 4y = 0. Fig. 9.3 shows this case.
0
x 0
Figure 9.3
x
Figure 9.4
9.1.1.4 Singular Integrals and Singular Points 1. Singular element 0
An element (x0 y0 y0) is called a singular element of the di erential equation, if in addition to the di erential equation F (x y y 0 ) = 0 (9.16a) it also satis es the equation @F = 0: (9.16b) @y0
2. Singular Integral
An integral curve from singular elements is called a singular integral curve the equation '(x y) = 0 (9.16c) of a singular integral curve is called a singular integral. The envelopes of the integral curves are singular integral curves (Fig. 9.3) they consist of the singular elements. The uniqueness of the solution (see 9.1.1.1, 1., p. 488) fails at the points of a singular integral curve.
3. Determination of Singular Integrals
Usually we cannot obtain singular integrals for any values of the arbitrary constants of the general solution. To determine the singular solution of a di erential equation (9.16a) with p = y0 we have to
494 9. Dierential Equations introduce the equation @F = 0 (9.16d) @p and to eliminate p. If the obtained relation is a solution of the given di erential equation, then it is a singular solution. The equation of this solution should be transformed into a form which does not contain multiple-valued functions, in particular no radicals where the complex values should also be considered. Radicals are expressions we get by nesting algebraic equations (see 2.2.1, p. 60). If the equation of the family of integral curves is known, i.e., the general solution of the given di erential equation is known, then we can determine the envelope of the family of curves, the singular integral, with the methods of di erential geometry (see 3.6.1.7, p. 237). A: Solve the di erential equation x;y ; 49 y02 + 278 y03 = 0. After we substitute y0 = p, the calculation of the additional equation with (9.16d) yields ; 89 p + 89 p2 = 0. Elimination of p results in equation a) 4 , where a) is not a solution, b) is a solution, a special case of the general x ; y = 0 and b) x ; y = 27 solution (y ; C )2 = (x ; C )3. The integral curves of a) and b) are shown in Fig. 9.4. B: Solve the di erential equation y0 ; ln jxj = 0. We transform the equation into the form ep ;jxj = 0. @F ep = 0. We get the singular solution x = 0 eliminating p. @p
4. Singular Points of a Di erential Equation
Singular points of a di erential equation are the points where the right side of the di erential equation y0 = f (x y) (9.17a) is not de ned. This is the case, e.g., in the di erential equations of the following forms:
1. Di erential Equation with a Fraction of Linear Functions dy ax + by 6 0) dx = cx + ey (ae ; bc =
(9.17b)
has an isolated singular point at (0 0), since the assumptions of the existence theorem are ful lled almost at every point arbitrarily close to (0 0) but not at this point itself. The conditions are not ful lled at the points where cx + ey = 0. We can force the ful llment of the conditions at these points if we exchange the role of the variables and we consider the equation dx = cx + ey : (9.17c) dy ax + by The behavior of the integral curve in the neighborhood of a singular point depends on the roots of the characteristic equation 2 ; (b + c) + bc ; ae = 0: (9.17d) We can distinguish between the following cases: Case 1: If the roots are real and they have the same sign, then the singular point is a branch point. The integral curves in a neighborhood of the singular point pass through it and if the roots of the characteristic equation do not coincide, except for one, they have a common tangent. If the roots coincide, then either all integral curves have the same tangent, or there is a unique integral curve passing through the singular point in each direction. dy = 2y the characteristic equation is 2 ; 3 + 2 = 0 = 2, A: For the di erential equation dx 1 x 2 2 = 1. The integral curves have the equation y = C x (Fig. 9.5). The general solution also contains
9.1 Ordinary Dierential Equations 495
the line x = 0 if we consider the form x2 = C1 y. dy = x + y is 2 ; 2 +1 = 0, = = 1. The integral curves B: The characteristic equation for dx 1 2 x are y = x ln jxj + Cx (Fig. 9.6). The singular point is a so-called node. dy = y is 2 ; 2 + 1 = 0, = = 1. The integral curves C: The characteristic equation for dx 1 2 x are y = C x (Fig. 9.7). The singular point is a so-called ray point. y y
y
x x
Figure 9.5
x
Figure 9.6
Figure 9.7
Case 2: If the roots are real and they have di erent signs, the singular point is a saddle point, and
two of the integral curves pass through it. dy = ; y is 2 ; 1 = 0, = +1, = ;1. The integral curves D: The characteristic equation for dx 1 2 x are x y = C (Fig. 9.8). For C = 0 we get the particular solutions x = 0, y = 0. Case 3: If the roots are conjugate complex numbers with a non-zero real part (Re() 6= 0), then the singular point is a spiral point which is also called a focal point, and the integral curves wind about this singular point. dy = x + y is 2 ; 2 + 2 = 0, = 1 + i, = 1 ; i. The E: The characteristic equation for dx 1 2 x;y integral curves in polar coordinates are r = C e' (Fig. 9.9). y y
y
x 0
Figure 9.8
0
x
Figure 9.9
Figure 9.10
x
Case 4: If the roots are pure imaginary numbers, then the singular point is a central point, or center, which is surrounded by the closed integral curves.
496 9. Dierential Equations dy = ; x is 2 + 1 = 0, = i, = ;i. The integral curves are F: The characteristic equation for dx 1 2 y x2 + y2 = C (Fig. 9.10). 2. Di erential Equation with the Ratio of Two Arbitrary Functions
dy P (x y) (9.18a) dx = Q(x y) has the singular points for the values of the variables where P (x y) = Q(x y) = 0: (9.18b) If P and Q are continuous functions and they have continuous partial derivatives, (9.18a) can be written in the form dy a(x ; x0 ) + b(y ; y0) + P1 (x y) (9.18c) dx = c(x ; x ) + e(y ; y ) + Q (x y) : 0
0
1
Here x0 and y0 are the coordinates of the singular point and P1(x y) and Q1 (x y) are in nitesimals of a higher order than the distance of the point (x y) from the singular point (x0 y0). With these assumptions the type of a singular point of the given di erential equation is the same as that of the approximate equation obtained by omitting the terms P1 and Q1 , with the following exceptions: a) If the singular point of the approximate equation is a center, the singular point of the original equation is either a center or a focal point. b) If a e ; b c = 0, i.e., ac = eb or a = c = 0 or a = b = 0, then the type of singular point should be determined by examining the terms of higher order.
9.1.1.5 Approximation Methods for Solution of First-Order Di erential Equations 1. Successive Approximation Method of Picard
The integration of the di erential equation y 0 = f (x y ) with the initial condition y = y0 for x = x0 results in the xed-point problem
Zx y = y0 + f (x y) dx:
(9.19a) (9.19b)
x0
If we substitute another function y1(x) instead of y into the right-hand side of (9.19b), then the result will be a new function y2(x), which is di erent from y1(x), if y1(x) is not already a solution of (9.19a). After substituting y2(x) instead of y into the right-hand side of (9.19b) we get a function y3(x). If the conditions of the existence theorem are ful lled (see 9.1.1.1, 1., p. 488), the sequence of functions y1 y2 y3 : : : converges to the desired solution in a certain interval containing the point x0. This Picard method of successive approximation is an iteration method (see 19.1.1, p. 884). Solve the di erential equation y0 = ex ; y2 with initial values x0 = 0 y0 = 0. Rewriting the equation in integral formZand using the successive approximation method with an initial approximation Z xh i x y0(x) 0, we get: y1 = ex dx = ex ; 1 y2 = ex ; (ex ; 1)2 dx = 3ex ; 21 e2x ; x ; 52 etc. 0
2. Solution by Series Expansion
0
The Taylor series expansion of the solution of a di erential equation (see 7.3.3.3, 1., p. 417) can be given in the form n 2 (9.20) y = y0 + (x ; x0 )y00 + (x ;2!x0 ) y000 + + (x ;n!x0 ) y0(n) +
9.1 Ordinary Dierential Equations 497
if the values y00 y000 : : : y0(n) : : : of all derivatives of the solution function are known at the initial value x0 of the independent variable. The values of the derivatives can be determined by successively di erentiating the original equation and substituting the initial conditions. If the di erential equation can be di erentiated in nitely many times, the obtained series will be convergent in a certain neighborhood of the initial value of the independent variable. We can use this method also for n-th order di erential equations. Remark: The above result is the Taylor series of the function, which may not represent the function itself (see 7.3.3.3, 1., p. 417). It is often useful to substitute the solution by an in nite series with undetermined coecients, and to determine them by comparing coecients. A: To solve the di erential equation y0 = ex ; y2, x0 = 0, y0 = 0 we consider the series y = a1 x + a2 x2 + a3 x3 + + an xn + . Substituting this into the equation considering the formula (7.88), p. 416 for the square of the series we get 2 3 a1 + 2a2x + 3a3x2 + + a12 x2 + 2a1a2 x3 + (a22 + 2a1a3 )x4 + ] = 1 + x + x2 + x6 + : Comparing coecients we get: a1 = 1 2a2 = 1 3a3 + a1 2 = 1 , 4a4 + 2a1a2 = 1 , etc. Solving 2 6 these equations successively and substituting the coecient values into the series representation we 2 3 get y = x + x ; x ; 5 x4 + . 2 6 24 B: The same di erential equation with the same initial conditions can also be solved in the following way: If we substitute x = 0 into the equation, we get y00 = 1. By successive di erentiation we get y00 = ex ; 2yy0, y000 = 1, y000 = ex ; 2y02 ; 2yy00, y0000 = ;1, y(4) = ex ; 6y0y00 ; 2yy000, y0(4) = ;5. From 2 3 4 the Taylor theorem (see 7.3.3.3, 1., p. 417) we get the solution y = x + x ; x ; 5x + . 2! 3! 4! y
3. Graphical Solution of Di erential Equations
y0
0
x0
x
Figure 9.11
The graphical integration of a di erential equation is a method, which is based on the direction eld (see 9.1.1.1, 3., p. 488). The integral curve in Fig. 9.11 is represented by a broken line which starts at the given initial point and is composed of short line segments. The directions of the line segments are always the same as the direction of the direction eld at the starting point of the line segment. This is also the endpoint of the previous line segment.
4. Numerical Solution of Di erential Equations
The numerical solutions of di erential equations will be discussed in detail in 19.4, p. 904. We use numerical methods to determine a solution of a di erential equation, if the equation y0 = f (x y) does not belong to the special cases dicussed above whose analytic solutions are known, or if the function f (x y) is too complicated. This can happen if f (x y) is non-linear in y.
9.1.2 Dierential Equations of Higher Order and Systems of Dierential Equations 9.1.2.1 Basic Results
1. Existence of a Solution
1. Reduction to a System of Di erential Equations Every explicit n-th order di erential equa-
tion
y(n) = f x y y0 : : : y(n;1)
(9.21a)
498 9. Dierential Equations by introducing the new variables y1 = y0 y2 = y00 : : : yn;1 = y(n;1) (9.21b) can be reduced to a system of n rst-order di erential equations dy dy1 dyn;1 (9.21c) dx = y1 dx = y2 : : : dx = f (x y y1 : : : yn;1): 2. Existence of a System of Solutions The system of n di erential equations dyi = f (x y y : : : y ) (i = 1 2 : : : n) (9.22a) n dx i 1 2 which is more general than system (9.21c), has a unique system of solutions yi = yi(x) (i = 1 2 : : : n) (9.22b) which is de ned in an interval x0 ; h x x0 + h and for x = x0 takes the previously given initial values yi(x0 ) = yi0 (i = 1 2 : : : n), if the functions fi(x y1 y2 : : : yn) are continuous with respect to all variables and satisfy the following Lipschitz condition. 3. Lipschitz condition For the values x yi and yi + %yi, which are in a certain neighborhood of the given initial values, the functions fi satisfy the following inequalities: jfi(x y1 + %y1 y2 + %y2 : : : yn + %yn) ; fi (x y1 y2 : : : yn)j K (j%y1j + j%y2j + + j%ynj) (9.23a) with a common constant K (see also 9.1.1.1, 2., p. 488). This fact implies that if the function f (x y y0 : : : y(n;1)) is continuous and satis es the Lipschitz condition (9.23a), then the equation y(n) = f x y y0 : : : y(n;1) (9.23b) 0 0 ( n ;1) ( n ;1) has a unique solution with the initial values y(x0) = y0 y (x0 ) = y0 : : : y (x0 ) = y0 , and it is (n ; 1) times continuously di erentiable.
2. General Solution
1. The general solution of the di erential equation (9.23b) contains n independent arbitrary con-
stants: y = y(x C1 C2 : : : Cn): (9.24a) 2. In the geometrical interpretation the equation (9.24a) de nes a family of curves depending on n parameters. Every single one of these integral curves, i.e., the graph of the corresponding particular solution, can be obtained by a suitable choice of the constants C1 C2 : : : Cn. If the solution has to satisfy the above initial conditions, then the values C1 C2 : : : Cn are determined from the following equations: y(x0 C1 : : : Cn) = y0 "
d y (x C : : : C ) = y00 (9.24b) 1 n dx x=x0 ": : :n:;:1: : : : : : : : : : : : : : : : : : : : : : d = y0(n;1) : dxn;1 y(x C1 : : : Cn) x=x0
If these equations are inconsistent for any initial values in a certain domain, then the solution is not general in this domain, i.e., the arbitrary constants cannot be chosen independently. 3. The general solution of system (9.22a) also contains n arbitrary constants. This general solution can be represented in two di erent ways: Either it is given in a form which is solved for the unknown functions y1 = F1 (x C1 : : : Cn) y2 = F2(x C1 : : : Cn) : : : yn = Fn(x C1 : : : Cn) (9.25a)
9.1 Ordinary Dierential Equations 499
or in the form which is solved for the constants '1(x y1 : : : yn) = C1 '2(x y1 : : : yn) = C2 : : : 'n(x y1 : : : yn) = Cn: (9.25b) In the case of (9.25b) each relation 'i(x y1 : : : yn) = Ci (9.25c) is a rst integral of the system (9.22a). The rst integral can be de ned independently of the general solution as a relation (9.25c). That is, (9.25c) will be an identity if we replace y1 y2 : : : yn by any particular solution of the given system and we replace the constant by the arbitrary constant Ci determined by this particular solution. If any rst integral is known in the form (9.25c), then the function 'i(x y1 : : : yn) satis es the partial di erent equation @'i + f (x y : : : y ) @'i + + f (x y : : : y ) @'i = 0: (9.25d) n n 1 n @x 1 1 @y1 @yn Conversely, each solution 'i(x y1 : : : yn) of the partial di erential equation (9.25d) de nes a rst integral of the system (9.22a) in the form (9.25c). The general solution of the system (9.22a) can be represented as a system of n rst integrals of system (9.22a), if the corresponding functions 'i(x y1 : : : yn) (i = 1 2 : : : n) are linearly independent (see 9.1.2.3, 2., p. 500).
9.1.2.2 Lowering the Order
One of the most important solution methods for n-th order di erential equations f x y y0 : : : y(n) = 0 (9.26) is the substitution of variables in order to obtain a simpler di erential equation, especially one of lower order. We can distinguish between di erent cases.
1. f = f (y y0 : : : y(n)) i.e. x does not appear explicitly:
f y y0 : : : y(n) = 0: (9.27a) By substitution dy d2y dp (9.27b) dx = p dx2 = p dy : : : we can reduce the order of the di erential equation from n to (n ; 1). We reduce the order of the di erential equation yy00 ; y02 = 0 to one. With the substitution y0 = p p dp=dy = y00 it becomes a rst-order di erential equation y p dp=dy ; p2 = 0, and y dp=dy ; p = 0 results in p = C y = dy=dx, y = C1eCx. Canceling p does not result in a loss of a solution, since p = 0 gives the solution y = C1, which is included in the general solution with C = 0. 2. f = f (x y0 : : : y(n)) i.e. y does not appear explicitly: f x y0 ::: y(n) = 0: (9.28a) The order of the di erential equation can be reduced from n to (n ; 1) by the substitution y0 = p: (9.28b) If the rst k derivatives are missing in the initial equation, then a suitable substitution is y(k+1) = p: (9.28c) The order of the di erential equation y00 ;xy000 +(y000)3 = 0 will be reduced by the substitution y00 = p, !3 so we get a Clairaut di erential equation p ; x dp + dp = 0 whose general solution is p = C1 x + C13 . dx dx 3 2 3 C 1 x C1 x Therefore, y = 6 ; 2 +C2x+C3 . From the singular solution of the Clairaut di erential equation
500 9. Dierential Equations
p
p
p = 2 3 3 x3=2 we get the singular solution of the original equation: y = 83153 x7=2 + C1x + C2. 3. f x y y0 : : : y(n) is a homogeneous function (see 2.18.2.4, 4., p. 120)
0 00 (n) in y y y :::y :
f x y y0 : : : y(n) = 0: We can reduce the order by the substitution R 0 z = yy i.e., y = e z dx :
(9.29a) (9.29b)
dz = We transform the di erential equation yy00 ; y02 = 0 by the substitution z = y0=y. Then dx yy00 ; y02 , so the order is reduced by one. We get z = C , therefore, ln jyj = C x + C , or y = CeC1 x 1 1 2 y2 with ln jC j = C2.
4. f = f (x y y0 : : : y(n)) is a function of only x:
y(n) = f (x): (9.30a) We get the general solution by n repeated integrations. It has the form y = C1 + C2x + C3 x2 + + Cnxn;1 + (x) (9.30b) with ZZ Z Zx (x) = f (x) (dx)n = (n ;1 1)! f (t)(x ; t)n;1 dt: (9.30c) x0 We mention here that x0 is not an additional arbitrary constant, since the change in x0 results in the change of Ck because of the relation Ck = (k ;1 1)! y(k;1)(x0 ): (9.30d)
9.1.2.3 Linear n-th Order Di erential Equations 1. Classication
A di erential equation of the form y(n) + an;1y(n;1) + an;2y(n;2) + + a1y0 + a0 y = F (9.31) is called an n-th order linear di erential equation. Here F and the coecients ai are functions of x, which are supposed to be continuous in a certain interval. If a0 a1 : : : an;1 are constants, we call it a dierential equation with constant coecients . If F 0, then the linear di erential equation is homogeneous, and if F 6 0, then it is inhomogeneous.
2. Fundamental System of Solutions
A system of n solutions y1 y2 : : : yn of a homogeneous linear di erential equation is called a fundamental system if these functions are linearly independent on the considered interval, i.e., their linear combination C1 y1 + C2 y2 + + Cn yn is not identically zero for any system of values C1 C2 : : : Cn, except for the values C1 = C2 = = Cn = 0. The solutions y1 y2 : : : yn of a linear homogeneous differential equation form a fundamental system on the considered interval if and only if their Wronskian determinant y1 y10 yy220 yynn0 W = . . . . . . . . . . . . . . . . . . . . . . . . (9.32) y (n;1) y (n;1) y (n;1) 1 2 n
9.1 Ordinary Dierential Equations 501
is non-zero. For every solution system of a homogeneous linear di erential equation the formula of Liouville is valid: 0 1
Zx W (x) = W (x0) exp @; a1 (x) dxA : x0
(9.33)
It follows from (9.33) that if the Wronskian determinant is zero somewhere in the solution interval, then it can be only identically zero. This means: The n solutions y1 y2 : : : yn of the homogeneous linear di erential equation are linearly dependent if even for a single point x0 of the considered interval W (x0) = 0. If the solutions y1 y2 : : : yn form a fundamental system of the di erential equation, then the general solution of the linear homogeneous di erential equation (9.31) is given as y = C1 y1 + C2 y2 + + Cn yn: (9.34) A linear n-th order homogeneous di erential equation has exactly n linearly independent solutions on an interval, where the coecient functions ai(x) are continuous.
3. Lowering the Order
If we know a particular solution y1 of a homogeneous di erential equation, by assuming y = y1(x)u(x) (9.35) we can determine further solutions from a homogeneous linear di erential equation of order n ; 1 for u0(x).
4. Superposition Principle
If y1 and y2 are two solutions of the di erential equation (9.31) for di erent right-hand sides F1 and F2 , then their sum y = y1 + y2 is a solution of the same di erential equation with the right-hand side F = F1 + F2. From this observation it follows that to get the general solution of an inhomogeneous di erential equation it is sucient to add any particular solution of the inhomogeneous di erential equation to the general solution of the corresponding homogeneous di erential equation.
5. Decomposition Theorem
If an inhomogeneous di erential equation (9.31) has real coecients and its right-hand side is complex in the form F = F1 + iF2 with some real functions F1 and F2, then the solution y = y1 + iy2 is also complex, where y1 and y2 are the two solutions of the two inhomogeneous di erential equations (9.31) with the corresponding right-hand sides F1 and F2.
6. Solution of Inhomogeneous Di erential Equations (9.31) by Means of Quadratures
If the fundamental system of the corresponding homogeneous di erential equation is already known, we have the following two solution methods to continue our calculations: 1. Method of Variation of Constants We are looking for the solution in the form y = C1y1 + C2y2 + + Cnyn (9.36a) where C1 C2 : : : Cn are functions of x. There are in nitely many such functions, but if we require that they satisfy the equations C10y1 + C20y2 + + Cn0 yn = 0 C10 y10 + C20 y20 + + Cn0yn0 = 0 (9.36b) ::::::::::::::: C10y1(n;2) + C20y2(n;2) + + Cn0 yn(n;2) = 0 and we substitute y into (9.31) with these equalities we get (9.36c) C10y1(n;1) + C20y2(n;1) + + Cn0 yn(n;1) = F: Because the Wronskian determinant of the coecients in the linear equation system (9.36b) and (9.36c) is di erent from zero, we get a unique solution for the unknown functions C1 0 C20 : : : Cn0 , and their integrals give the functions C1 C2 : : : Cn.
502 9. Dierential Equations y00 + 1 ;x x y0 ; 1 ;1 x y = x ; 1: (9.36d) In the interval x > 1 or x < 1 all assumptions on the coecients are ful lled. First we solve the homogeneous equation y00 + 1 ;x x y0 ; 1 ;1 x y = 0. A particular solution is '1 = ex. We are looking for a x 0 second one in the form '1 2 = e u(x), and with the notation u (x) = v(x) we get the rst-order di erential equation v0 + 1 + v = 0. A solution of this equation is v(x) = (1 ; x)e;x, and therefore, 1Z; x Z u(x) = v(x) dx = (1 ; x)e;x dx = xe;x. With this result we get '2 = x for the second element of the fundamental system. The general solution of the homogeneous equation is y(x) = C1ex + C2x. The variation of constants is now: y(x) = u1(x)ex + u2(x)x y0(x) = u1(x)ex + u2(x) + u10(x)ex + u20(x)x u10(x)ex + u20(x)x = 0 y00(x) = u1(x)ex + u10 (x)ex + u20 (x) u10 (x)ex + u20(x) = x ; 1 so u10(x) = xe;x u20 (x) = ;1 i.e., u1(x) = ;(1 + x)e;x + C1 u2(x) = ;x + C2: With this result the general solution of the inhomogeneous di erential equation is: y(x) = ;(1 + x2 ) + C1ex + (C2 ; 1)x = ;(1 + x2 ) + C1 ex + C2 x: 2. Method of Cauchy In the general solution y = C1y1 + C2y2 + + Cnyn (9.37a) of the homogeneous di erential equation associated to (9.31) we determine constants such that for an arbitrary parameter the equations y = 0 y0 = 0 : : : y(n;2) = 0 y(n;1) = F () are satis ed. In this way we get a particular solution of the homogeneous equation, denoted by '(x ), and then Zx y = '(x ) d (9.37b) x0
is a particular solution of the inhomogeneous di erential equation (9.31). This solution and their derivatives up to order (n ; 1) are equal to zero at the point x = x0. The general solution of the homogeneous equation associated to the di erential equation (9.36a) which we solved by the method variation of constants is y = C1ex + C2x. From this result we see that y() = C1e + C2 = 0, y0() = C1e + C2 = ; 1 and '(x ) = e; ex ; x, so the parsolution y(x) of the inhomogeneous di erential equation with y(x0) = y0(x0 ) = 0 is: y(x) = Zticular x ; x (e e ; x) d = (x0 + 1)ex;x0 + (x0 ; 1)x ; x2 ; 1. Therefore, we can already get the general x0 solution y(x) = C1 ex + C2 x ; (x2 + 1) of the inhomogeneous di erential equation.
9.1.2.4 Solution of Linear Di erential Equations with Constant Coecients 1. Operational Notation
The di erential equation (9.31) can be written symbolically in the form Pn(D)y Dn + an;1Dn;1 + an;2Dn;2 + + a1D + a0 y = F (9.38a) where D is a di erential operator: dy Dk y = dk y : (9.38b) Dy = dx dxk If the coecients ai are constants, then Pn(D) is a usual polynomial in the operator D of degree n.
9.1 Ordinary Dierential Equations 503
2. Solution of the Homogeneous Di erential Equation with Constant Coe cients
To determine the general solution of the homogeneous di erential equation (9.38a) with F = 0, i.e., Pn(D)y = 0 (9.39a) we have to nd the roots r1 r2 : : : rn of the characteristic equation Pn(r) = rn + an;1 rn;1 + an;2rn;2 + + a1 r + a0 = 0: (9.39b) Every root ri determines a solution erix of the equation Pn(D)y = 0. If a root ri has a higher multiplicity k, then xeri x x2 eri x : : : xk;1erix are also solutions. The linear combination of all these solutions is the general solution of the homogeneous di erential equation: y = C1er1 x + C2er2 x + + eri x Ci + Ci+1x + + Ci+k;1xk;1 + : (9.39c) If the coecients ai are all real, then the complex roots of the characteristic equation are pairwise conjugate with the same multiplicity. In this case, for r1 = + i and r2 = ; i we can replace the corresponding complex solution functions er1x and er2 x by the real functions e x cos x and e x sin x. The resulting expression C1 cos x + C2 sin x can be written in the form A cos(x + ') with some constants A and '. In the case of the di erential equation y(6) + y(4) ; y00 ; y = 0, the characteristic equation is r6 + r4 ; r2 ; 1 = 0 with roots r1 = 1, r2 = ;1, r34 = i, r56 = ;i. The general solution can be given in two forms: y = C1ex + C2e;x + (C3 + C4 x) cos x + (C5 + C6x) sin x or y = C1ex + C2e;x + A1 cos(x + '1 ) + xA2 cos(x + '2):
3. Hurwitz Theorem
In di erent applications, e.g., in vibration theory, it is important to know whether a solution of a given homogeneous di erential equation with constant coecients tend to zero for x ! +1 or not. It tends to zero, obviously, if the real parts of the roots of the characteristic equation (9.39b) are negative. According to the Hurwitz theorem an equation anxn + an;1xn;1 + + a1 x + a0 = 0 (9.40a) has only roots with negative real part if and only if all the determinants a a 0 : : : 0 a a 0 a a 1 0 1 0 1 0 D1 = a1 D2 = a3 a2 D3 = a3 a2 a1 : : : Dn = a. 3. .a.2. .a.1. .:.:.:.0. . a5 a4 a3 0 0 0 : : : an (with am = 0 for m > n) (9.40b) are positive. The determinants Dk have on their diagonal the coecients a1 a2 : : : ak (k = 1 2 : : : n), and the coecient-indices are decreasing from left to right. Coecients with negative indices and also with indices larger than n are all put to 0. For a cubic polynomial the determinants have in accordance to (9.40b) the following form: a a 0 1 0 D1 = a1 D2 = aa13 aa22 D3 = a3 a2 a1 : 0 0 a 3
4. Solution of Inhomogeneous Di erential Equations with Constant Coe cients
can be determined by the method variation of constants, or by the method of Cauchy, or with the operator method (see 9.2.2.3, 5., p. 534). If the right-hand side of the inhomogeneous di erential equation (9.31) has a some special form, a particular solution can be determined very easily. 1. Form: F (x) = Aex Pn() = 0 (9.41a)
6
504 9. Dierential Equations A particular solution is x y = PAe() : (9.41b) n If is a root of the characteristic equation of multiplicity m, i.e., if Pn() = Pn0 () = : : : = Pn(m;1) () = 0 (9.41c) me x Ax then y = (m) is a particular solution. These formulas can also be used by applying the decompoPn () sition theorem, if the right side is F (x) = Ae x cos !x or Ae x sin !x: (9.41d) The corresponding particular solutions are the real or the imaginary part of the solution of the same di erential equation for F (x) = Ae x (cos !x + i sin !x) = Ae( +i!)x (9.41e) on the right-hand side. A: For the di erential equation y00 ; 6y0 + 8y = e2x, the characteristic polynomial is P (D) = D2 ; 6D + 8 with P (2) = 0 and P 0(D) = 2D ; 6 with P 0(2) = 2 2 ; 6 = ;2, so the particular solution 2x is y = ; xe2 . B: The di erential equation y00 + y0 + y = ex sin x results in the equation (D2 + D + 1)y = e(1+i)x . (1+i)x x From its solution y = (1 + i)2e+ (1 + i) + 1 = e (cos2x++3ii sin x) we get a particular solution y1 = ex (2 sin x ; 3 cos x). Here y is the imaginary part of y. 1 13 2. Form: F (x) = Qn(x)ex Qn(x) is a polynomial of degree n (9.42) A particular solution can always be found in the same form, i.e., as an expression y = R(x)e x. R(x) is a polynomial of degree n multiplied by xm if is a root of the characteristic equation with a multiplicity m. We consider the coecients of the polynomial R(x) as unknowns and we substitute the expression into the inhomogeneous di erential equation. It must satisfy the equation, so we get a linear equation system for the coecients, and this equation system always has a unique solution. This method is very useful especially in the cases of F (x) = Qn(x) for = 0 and F (x) = Qn (x)erx cos !x or F (x) = Qn(x)erx sin !x for = r i!. There is a solution in the form y = xm erx Mn (x) cos !x + Nn(x) sin !x]. The roots of the characteristic equation associated to the di erential equation y(4) + 2y000 + y00 = 6x + 2x sin x are k1 = k2 = 0, k3 = k4 = ;1. Because of the superposition principle (see 9.1.2.3, 4., p. 501), we can calculate the particular solutions of the inhomogeneous di erential equation for the summands of the right-hand side separately. For the rst summand the substitution of the given form y1 = x2 (ax + b) results in a right-hand side 12a + 2b + 6ax = 6x, and so: a = 1 und b = ;6. For the second summand we substitute y2 = (cx + d) sin x +(fx + g) cos x. We get the coecients by coecient comparison from (2g +2f ; 6c +2fx) sin x ; (2c +2d +6f +2cx) cos x = 2x sin x, so c = 0 d = ;3 f = 1 g = ;1. Therefore, the general solution is y = c1 +c2 x;6x2 +x3 +(c3x+c4 )e;x ;3 sin x+(x;1) cos x.
3. Euler Di erential Equation The Euler di erential equation n X
k=0
ak (cx + d)k y(k) = F (x)
with the substitution cx + d = et
(9.43a) (9.43b)
9.1 Ordinary Dierential Equations 505
can be transformed into a linear di erential equation with constant coecients. The di erential equation x2 y00 ; 5xy0 + 8y = x2 is a special case of the Euler di erential equation for n = 2. With the substitution x = et it becomes the di erential equation discussed earlier in 2 example A, on p. 504: ddty2 ; 6 dy + 8y = e2t . The general solution is y = C1e2t + C2e4t ; 2t e2t = dt 2 C1x2 + C2x4 ; x2 ln jxj.
9.1.2.5 Systems of Linear Di erential Equations with Constant Coecients 1. Normal Form
The following simple case of a rst-order linear di erential equation system with constant coecients is called a normal system or a normal form: y10 = a11 y1 + a12 y2 + + a1nyn 9 > y20 = a21 y1 + a22 y2 + + a2nyn = (9.44a) ................................. > " 0 yn = an1y1 + an2y2 + + annyn: To nd the general solution of such a system, we have to nd rst the roots of the characteristic equation a ; r a a11 a12 ; r :: :: :: aa1n 22 2n 21 = 0: (9.44b) . . . . . . . . . . . . . . . . . . . . . . . . . an1 an2 : : : ann ; r To every single root ri of this equation there is a system of particular solutions y1 = A1 erix y2 = A2 erix : : : yn = An erix (9.44c) whose coecients Ak (k = 1 2 : : : n) are determined from the homogeneous linear equation system (a11 ; ri)A1 + a12 A2 + + a1n An = 0 ....................................... (9.44d) an1 A1 + an2A2 + + (ann ; ri)An = 0: This system gives the relations between the values of Ak (see Trivial solution and fundamental system in 4.4.2.1, 2., p. 273). For every ri, the particular solutions we get in this way will contain an arbitrary constant. If all the roots of the characteristic equation are di erent, the sum of these particular solutions contains n independent arbitrary constants, so in this way we get the general solution. If a root ri has a multiplicity m in the characteristic equation, the system of particular solutions corresponding to this root has the form y1 = A1 (x)erix y2 = A2(x)eri x : : : yn = An(x)eri x (9.44e) where A1 (x) : : : An(x) are polynomials of degree at most m ; 1. We substitute these expressions with unknown coecients of the polynomials Ak (x) into the di erential equation system. We can rst cancel the factor eri x, then we can compare the coecients of the di erent powers of x to have linear equations for the unknown coecients of the polynomials, and among them m can be chosen freely. In this way, we get a part of the solution with m arbitrary constants. The degree of the polynomials can be less than m ; 1. In the special case when the system (9.44a) is symmetric, i.e., when aik = aki, then it is sucient to substitute Ai (x) = const. For complex roots of the characteristic equation, the general solution can be transformed into a real form in the same way as shown earlier for the case of a di erential equation with constant coecients (see 9.1.2.4, p. 502). For the system y10 = 2y1 + 2y2 ; y3 y20 = ;2y1 + 4y2 + y3 y30 = ;3y1 + 8y2 + 2y3 the characteristic
506 9. Dierential Equations equation has the form 2 ; r 2 ;1 ;2 4 ; r 1 = ;(r ; 6)(r ; 1)2 = 0: ;3 8 2 ; r For the simple root r1 = 6 we get ;4A1 +2A2 ; A3 = 0, ;2A1 ; 2A2 + A3 = 0, ;3A1 +8A2 ; 4A3 = 0. From this system we have A1 = 0, A2 = 12 A3 = C1, y1 = 0, y2 = C1 e6x, y3 = 2C1e6x . For the multiple root r2 = 1 we get y1 = (P1 x + Q1)ex, y2 = (P2x + Q2 )ex, y3 = (P3x + Q3 )ex. Substitution into the equations yields P1x + (P1 + Q1 ) = (2P1 + 2P2 ; P3)x + (2Q1 + 2Q2 ; Q3) P2x + (P2 + Q2 ) = (;2P1 + 4P2 + P3)x + (;2Q1 + 4Q2 + Q3 ) P3x + (P3 + Q3 ) = (;3P1 + 8P2 + 2P3)x + (;3Q1 + 8Q2 + 2Q3 ) which implies that P1 = 5C2, P2 = C2, P3 = 7C2, Q1 = 5C3 ; 6C2, Q2 = C3, Q3 = 7C3 ; 11C2. The general solution is y1 = (5C2x + 5C3 ; 6C2)ex, y2 = C1e6x + (C2x + C3)ex, y3 = 2C1e6x + (7C2x + 7C3 ; 11C2)ex .
2. Homogeneous Systems of First-Order Linear Di erential Equations with Constant Coe cients
have the general form n X
k=1
aik yk 0 +
n X
k=1
bik yk = 0 (i = 1 2 : : : n):
(9.45a)
If the determinant det(aik ) does not disappear, i.e., det(aik ) 6= 0 (9.45b) then the system (9.45a) can be transformed into the normal form (9.44a). In the case of det(aik ) = 0 we need further investigations (see 9.15]). The solution can be determined from the general form in the same way as shown for the normal form. The characteristic equation has the form det(aik r + bik ) = 0: (9.45c) The coecients Ai in the solution (9.44c) corresponding to a single root rj are determined from the equation system n X
k=1
(aik rj + bik )Ak = 0 (i = 1 2 : : : n):
(9.45d)
Otherwise the solution method follows the same ideas as in the case of the normal form. The characteristic equation of the two di erential equations 5y10 +4y1 ;2y20 ;y2 = 0, y10 +8y1 ;3y2 = 0 is: 5r + 4 ;2r ; 1 r + 8 ;3 = 2r2 + 2r ; 4 = 0 r1 = 1 r2 = ;2: We get the coecients A1 and A2 for r1 = 1 from the equations 9A1 ; 3A2 = 0, 9A1 ; 3A2 = 0 so A2 = 3A1 = 3C1. For r2 = ;2 we get analogously A2 = 2A1 = 2C2. The general solution is y1 = C1ex + C2e;2x, y2 = 3C1ex + 2C2e;2x .
3. Inhomogeneous Systems of First-Order Linear Di ererential Equations have the general form n X
k=1
aik yk 0 +
n X
k=1
bik yk = Fi(x) (i = 1 2 : : : n):
(9.46)
9.1 Ordinary Dierential Equations 507
1. Superposition Principle: If yj (1) and yj (2) (j = 1 2 : : : n) are solutions of inhomogeneous
systems which di er from each other only in their right-hand sides Fi(1) and Fi(2) , then the sum yj = yj (1) + yj (2) (j = 1 2 : : : n) is a solution of this system with the right-hand side Fi (x) = Fi (1) (x) + Fi(2) (x). Because of this, to get the general solution of an inhomogeneous system it is enough to add a particular solution to the general solution of the corresponding homogeneous system. 2. The Variation of Constants can be used to get a particular solution of the inhomogeneous di erential equation system. To do this we use the general solution of the homogeneous system, and we consider the constants C1 C2 : : : Cn as unknown functions C1 (x) C2(x) : : : Cn(x). We substitute it into the inhomogeneous system. In the expressions of the derivatives of yk 0 we have the derivative of the new unknown functions Ck (x). Because y1 y2 : : : yn are solutions of the homogeneous system, the terms containing the new unknown functions will be canceled only their derivatives remain in the equations. We get for the functions Ck 0 (x) an inhomogeneous linear algebraic equation system which always has a unique solution. After n integrations we get the functions C1 (x) C2(x) : : : Cn(x). Substitution them into the solution of the homogeneous system instead of the constants results in the particular solution of the inhomogeneous system. For the system of two inhomogeneous di erential equations 5y10 + 4y1 ; 2y20 ; y2 = e;x, y10 + 8y1 ; 3y2 = 5e;x the general solution of the homogeneous system is (see p. 506) y1 = C1 ex + C2 e;2x, y2 = 3C1ex + 2C2e;x. Considering the constants C1 and C2 as functions of x and substituting into the original equations we get 50C10ex + 05C20e;2x ; 6C10ex ; 4C20 e;2x0 = e;x, C1 0ex + C20e;2x = 5e;x 0 ;2x 0 ;x ;x x or C2 e ; C1 e = e , C1 e + C2 e;2x = 5e;x. Therefore, 2C1 ex = 4e;x, C1 = ;e;2x + const, 2C20e;2x = 6e;x, C2 = 3ex + const. Since we are looking for a particular solution, we can replace every constant by zero and the result is y1 = 2e;x, y2 = 3e;x. The general solution is nally y1 = 2e;x + C1ex + C2 e;2x, y2 = 3e;x + 3C1ex + 2C2e;2x. 3. The Method of Unknown Coecients is especially useful if on the right-hand side there are special functions in the form Qn (x)e x. The application is similar to the one we used for di erential equations of n-th order (see 9.1.2.5, p. 505).
4. Second-Order Systems
The methods introduced above can also be used for di erential equations of higher order. For the system n X
k=1
aik yk 00 +
n X
k=1
bik yk 0 +
n X
k=1
cik yk = 0 (i = 1 2 : : : n)
(9.47)
we can determine particular solutions in the form yi = Aierix. To do this, we get ri from the characteristic equation det(aik r2 + bik r + cik ) = 0, and we get Ai from the corresponding linear homogeneous algebraic equations.
9.1.2.6 Linear Second-Order Di erential Equations
Many special di erential equations belong to this class, which often occur in practical applications. We discuss several of them in this paragraph. For more details of representation, properties and solution methods see 9.15].
1. General Methods
1. The Inhomogeneous Di erential Equation is
y00 + p(x)y0 + q(x)y = F (x): (9.48a) a) The general solution of the corresponding homogeneous di erential equation, i.e., with F (x) 0, is y = C1y1 + C2y2: (9.48b) Here y1 and y2 are two linearly independent particular solutions of this equation (see 9.1.2.3, 2., p. 500). If a particular solution y1 is already known, then the second one y2 can be determined by the equation R Z (9.48c) y2 = Ay1 exp (;y 2 p dx) dx 1
508 9. Dierential Equations which follows from the Liouville formula (9.33), where A can be chosen arbitrarily. b) A particular solution of the inhomogeneous equation can be determined by the formula Z Zx (9.48d) y = A1 F ( ) exp p( ) d y2(x)y1( ) ; y1(x)y2( )] d x0 where y1 and y2 are two particular solutions of the corresponding homogeneous di erential equation. c) A particular solution of the inhomogeneous di erential equation can be determined also by variation of constants (see 9.1.2.3, 6., p. 501).
2. If in the Inhomogeneous Di erential Equation
s(x)y00 + p(x)y0 + q(x)y = F (x) (9.49a) the functions s(x) p(x) q(x) and F (x) are polynomials or functions which can be expanded into a convergent power series around x0 in a certain domain, where s(x0 ) 6= 0, then the solutions of this di erential equation can also be expanded into a similar series, and these series are convergent in the same domain. We determine them by the method of undetermined coecients: The solution we are looking for as a series has the form y = a0 + a1 (x ; x0 ) + a2(x ; x0 )2 + (9.49b) and we substitute it into the di erential equation (9.49a). Equating like coecients (of the same powers of (x ; x0 )) results in equations to determine the coecients a0 a1 a2 : : : . To solve the di erential equation y00 + xy = 0 we substitute y = a0 + a1 x + a2 x2 + a3 x3 + , y0 = a1 + 2a2 x + 3a3x2 + , and y00 = 2a2 + 6a3x + . We get 2a2 = 0, 6a3 + a0 = 0 : : : . The solution of these equations is a2 = 0, a3 = ; 2a03 , a4 = ; 3a14 , a5 = 0 : : : , so the solution is ! ! 3 6 4 7 y = a0 1 ; 2x 3 + 2 3x 5 6 ; + a1 x ; 3x 4 + 3 4x 6 7 ; .
3. The Homogeneous Di erential Equation
x2 y00 + xp(x)y0 + q(x)y = 0 (9.50a) can be solved by the method of undetermined coecients if the functions p(x) and q(x) can be expanded as a convergent series of x. The solutions have the form y = xr (a0 + a1 x + a2x2 + ) (9.50b) whose exponent r can be determined from the dening equation r(r ; 1) + p(0)r + q(0) = 0: (9.50c) If the roots of this equation are di erent and their di erence is not an integer number, then we get two linearly independent solutions of (9.50a). Otherwise the method of undetermined coecients results only one solution. Then with the help of (9.48b) we can get a second solution or at least we can nd a form from which we can get a second solution with the method of undetermined coecients. For the Bessel di erential equation (9.51a) we get only one solution with the method of the undeter1 X mined coecients in the form y1 = ak xn+2k (a0 6= 0), which coincides with Jn(x) up to a constant k=0 Z factor. Since exp ; p dx = x1 we nd a second solution by using formula (9.48c) 1 Z Z P ck x2k 1 X dx y2 = Ay1 = Ay1 k=0x2n+1 dx = By1 ln x + x;n dk x2k : P 2 2 n 2 k xx ( a x ) k
k=0
The determination of the unknown coecients ck and dk is dicult from the ak 's. But we can use this last expression to get the solution with the method of undetermined coecients. Obviously this form is a series expansion of the function Yn(x) (9.52c).
9.1 Ordinary Dierential Equations 509
2. Bessel Di erential Equation 2 00 0 2 2
x y + xy + (x ; n )y = 0: 1. The De ning Equation is in this case r(r ; 1) + r ; n2 r2 ; n2 = 0 so, r = n. Substituting y = xn (a0 + a1 x + )
into (9.51a)and equating the coecients of xn+k to zero we have k(2n + k)ak + ak;2 = 0: For k = 1 we get (2n + 1)a1 = 0. For the values k = 2 3 : : : we obtain a2m+1 = 0 (m = 1 2 : : :) a2 = ; 2(2na0+ 2)
a4 = 2 4 (2n +a02)(2n + 4) : : :
a0 is arbitrary:
(9.51a) (9.51b) (9.51c) (9.51d)
(9.51e)
2. Bessel or Cylindrical Functions The series obtained above for a0 = 2n; (n1 + 1) , where ; is the gamma function (see 8.2.5, 6., p. 461), is a particular solution of the Bessel di erential equation (9.51a) for integer values of n. It de nes the Bessel or cylindrical function of the rst kind of index n ! n 2 4 Jn(x) = 2n; (xn + 1) 1 ; 2(2nx + 2) + 2 4 (2n +x 2)(2n + 4) ; n+2k k x 1 (;1) X = k!; (n +2k + 1) : (9.52a) k=0
The graphs of functions J0 and J1 are shown in Fig. 9.12. The general solution of the Bessel di erential equation for non-integer n has the form y = C1Jn(x) + C2J;n(x) (9.52b) where J;n(x) is de ned by the in nite series obtained from the series representation of Jn(x) by replacing n with ;n. For integer n, we have J;n(x) = (;1)nJn(x). In this case, the term J;n(x) in the general solution should be replaced with the Bessel function of the second kind Jm (x) cos m ; J;m(x) (9.52c) Yn(x) = mlim !n sin m which is also called the Weber function. For the series expansion of Yn(x) see, e.g., 9.15]. The graphs of the functions Y0 and Y1 are shown in Fig. 9.13. 3. Bessel Functions with Imaginary Variables In some applications we;nuse Bessel functions with pure imaginary variables. In this case we have to consider the product i Jn(ix) which will be denoted by In(x): x n x n+2 x n+4 2 2 ;n In(x) = i Jn(ix) = ; (n + 1) + 1!; (n + 2) + 2!;2(n + 3) + : (9.53a) The functions In(x) are solutions of the di erential equation x2 y00 + xy0 ; (x2 + n2 )y = 0: (9.53b) A second solution of this di erential equation is the MacDonald function (9.53c) Kn(x) = 2 I;n(xsin) ;nIn(x) :
510 9. Dierential Equations y y 1.0
0.5 y=J0(x)
0.5
0
y=J1(x)
0
y=Y0(x) y=Y1(x)
5
5
15 x
10
-0.5 15 x
10
-1.0
-0.5
y=I (x) 1
y 5 4 3 2
y=I (x) 0
Figure 9.12 Figure 9.13 If n converges to an integer number, this expression also converges. The functions In(x) and Kn(x) are called modied Bessel functions. The graphs of functions I0 and I1 are shown in Fig. 9.14 the graphs of functions K0 and K1 are illustrated in Fig. 9.15. The values of functions J0(x), J1 (x), Y0(x), Y1(x), I0 (x), I1(x), K0 (x), K1(x) are given in Table 21.11, p. 1064. y y
1.0
1 2 3 x
1.0
(x) y=K 0
-3 -2 -1 -2 -3 -4 -5
0.5 P4
(x) y=K 1
1.5
Figure 9.14
0.5
P1 -0.5 P2 1.0
1.5
2.0
x -1.0
Figure 9.15
4. Important Formulas for the Bessel Functions Jn(x) n(x) Jn;1(x) + Jn+1(x) = 2xn Jn(x) dJdx = ; nx Jn(x) + Jn;1(x):
The formulas (9.54a) are also valid for the Weber functions Yn(x). n(x) In;1(x) ; In+1(x) = 2nIxn(x) dIdx = In;1(x) ; nx In(x) n(x) Kn+1(x) ; Kn;1(x) = 2nKxn(x) dKdx = ;Kn;1(x) ; nx Kn(x): For integer numbers n the following formulas are valid:
Z=2 J2n(x) = 2 cos(x sin ') cos 2n' d' 0
P6
P7
P2
0
0.5 0
P1 P5
x P3
P4
0.5
1.0
Figure 9.16 (9.54a) (9.54b) (9.54c)
(9.54d)
9.1 Ordinary Dierential Equations 511
Z=2 J2n+1(x) = 2 sin(x sin ') sin(2n + 1)' d' 0
or, in complex form, n Z ix cos ' cos n' d': Jn(x) = ;(i) 0 e The functions Jn+1=2 (x) can be expressed by using elementary functions. In particular,
(9.54e)
(9.54f)
s s 2 sin x 2 cos x: J1=2 (x) = x (9.55a) J;1=2 (x) = x (9.55b) By applying the recursion formulas (9.54a){(9.54f) the expression for Jn+1=2 (x) for arbitrary integer n can be given. For large values of x we have the following asymptotic formulas: s 2 cos x ; n ; + O 1 Jn(x) = x (9.56a) 2 4 x
x In(x) = pe 1 + O x1 (9.56b) 2x s 2 sin x ; n ; + O 1 Yn(x) = x (9.56c) 2 4 x r
1 Kn(x) = 2x e;x 1 + O x : (9.56d) 1 The expression O x means an in nitesimal quantity of the same order as x1 (see the Landau symbol, p. 56). For further properties of the Bessel functions see 21.1].
3. Legendre Di erential Equation
Restricting our investigations to the case of real variables and integer parameters n = 0 1 2 : : : the Legendre di erential equation has the form (1 ; x2 )y00 ; 2xy0 + n(n + 1)y = 0 or ((1 ; x2 )y0)0 + n(n + 1)y = 0: (9.57a) 1. Legendre Polynomials or Spherical Harmonics of the First Kind are the particular solutions of the Legendre di erential equation for integer n, which can be expanded into the power series 1 X y = a x . By the method of undetermined coecients we get for jxj < 1 n = 0 1 2 : : : the =0 polynomials "
n)! xn ; n(n ; 1) xn;2 + n(n ; 1)(n ; 2)(n ; 3) xn;4 ; + Pn(x) = 2(2 (9.57b) n (n!)2 2(2n ; 1) 2 4(2n ; 1)(2n ; 3) n 2 n Pn(x) = F n + 1 ;n 1 1 ;2 x = 2n1n! d (xdx;n 1) (9.57c) where F denotes the hypergeometric series (see 4., p. 513). The rst eight polynomials have the following simple form (see 21.12, p. 1066): P0(x) = 1 (9.57d) P1(x) = x (9.57e)
512 9. Dierential Equations P2(x) = 12 (3x2 ; 1)
(9.57f)
P3(x) = 21 (5x3 ; 3x)
(9.57g)
P4(x) = 81 (35x4 ; 30x2 + 3)
(9.57h)
P5(x) = 81 (63x5 ; 70x3 + 15x)
(9.57i)
1 (231x6;315x4 +105x2;5) (9.57j) P (x) = 1 (429x7 ;693x5 +315x3 ;35x) :(9.57k) P6(x) = 16 7 16
The graphs of Pn(x) for the values from n = 1 to n = 7 are represented in Fig. 9.16. The numerical values can be calculated easily by pocket calculators or from function tables.
2. Properties of the Legendre Polynomials of the First Kind a) Integral Representation: Z Z p d' p 2 n+1 : Pn(x) = 1 (x cos ' x2 ; 1)n d' = 1 ( x cos ' x ; 1) 0 0
(9.58a)
The signs can be chosen arbitrarily in both equations.
b) Recursion Formulas:
(n + 1)Pn+1(x) = (2n + 1)xPn(x) ; nPn;1 (x) (n 1 P0(x) = 1 P1 (x) = x) n (x) (x2 ; 1) dPdx = n xPn (x) ; Pn;1(x)] (n 1):
c) Orthogonality Relation:
(9.58b) (9.58c)
80 for m 6= n, < Pn(x)Pm (x) dx = : 2 (9.58d) ;1 2n + 1 for m = n. d) Root Theorem: All the n roots of Pn(x) are real and single and are in the interval (;1 1). Z1
e) Generating Function: The Legendre polynomial of the rst kind can be represented as the power series expansion of the function 1 X p 1 = P (x)rn: 1 ; 2rx + r2 n=0 n For further properties of the Legendre polynomials of the rst kind see 21.1].
(9.58e)
3. Legendre Functions or Spherical Harmonics of the Second Kind We get a second particular solution Qn(x) of (9.57a), which is valid for jxj > 1 and linearly independent of Pn(x), see (9.58a), by the power series expansion
X
;(n+1)
=;1
b x :
n 2 Qn(x) = (22n(+n!)1)! x;(n+1) F n +2 1 n +2 2 2n 2+ 3 x12 n 2 " + 1)(n + 2) x;(n+3) = (22n(+n!)1)! x;(n+1) + (n 2(2 n + 3)
n + 2)(n + 3)(n + 4) x;(n+5) + : + (n +2 1)( 4 (2n + 3)(2n + 5) The representation of Qn(x) valid for jxj < 1 is: n 1 +x ;X Qn(x) = 12 Pn(x) ln 11 ; x k=1 k Pk;1(x)Pn;k (x):
(9.59a) (9.59b)
9.1 Ordinary Dierential Equations 513
We call the spherical harmonics of the rst and second kind also the associated Legendre functions (see also (9.118c), p. 543).
4. Hypergeometric Di erential Equation
The hypergeometric dierential equation is the equation x(1 ; x)y00 + ; ( + + 1)x]y0 ; y = 0 (9.60a) where are parameters. It contains several important special cases. a) For = n + 1 = ;n = 1, and x = 1 ;2 z it is the Legendre di erential equation. b) If 6= 0 or is not a negative integer, it has the hypergeometric series or hypergeometric function as a particular solution : 1) ( + 1) 2 F ( x) = 1 + 1 x + (1 + 2 ( + 1) x + ( + n) ( + 1) : : : ( + n) xn+1 + + (1 +2 :1): : :(:n: + (9.60b) 1) ( + 1) : : : ( + n) which is absolutely convergent for jxj < 1. The convergence for x = 1 depends on the value of = ; ; . For x = 1 it is convergent if > 0, it is divergent if 0. For x = ;1 it is absolutely convergent if < 0, it is conditionally convergent for ;1 < 0, and it is divergent for ;1. c) For 2 ; 6= 0 or not equal to a negative integer it has a particular solution y = x1; F ( + 1 ; + 1 ; 2 ; x): (9.60c) d) In some special cases the hypergeometric series can be reduced to elementary functions, e.g., (9.61b) F (1 x) = F ( 1 x) = 1 ;1 x (9.61a) F (;n ;x) = (1 + x)n 1 1 3 arcsin x (9.61d) F (1 1 2 ;x) = ln(1x+ x) (9.61c) F 2 2 2 x2 = x ! x = ex: (9.61e) lim F 1 1 !1
5. Laguerre Di erential Equation
If we restrict our investigation to integer parameters (n = 0 1 2 : : :) and real variables, the Laguerre dierential equation has the form xy00 + ( + 1 ; x)y0 + ny = 0: (9.62a) As a particular solution we have the Laguerre polynomial n n + ! (;x)k x ; dn X ;x n+ L(n ) (x) = e nx! dx ( e x ) = (9.62b) n k! : k=0 n ; k The recursion formula for n 1 is: (n + 1)L(n+1) (x) = (;x + 2n + + 1)L(n ) (x) ; (n + )L(n;)1(x) (9.62c) ( ) ( ) L0 (x) = 1 L1 = 1 + ; x: (9.62d) An orthogonality relation for > ;1 holds: 80 for m 6= n, > Z1 < ! ;x ( ) ( ) e x Lm (x)Ln (x) dx = > n + ; (1 + ) for m = n. (9.62e) : n 0 With ; we denote the gamma function (see 8.2.5, 6., p. 461).
514 9. Dierential Equations
6. Hermite Di erential Equation
Two de ning equations are often used in the literature:
a) De ning Equation of Type 1: y00 ; xy0 + ny = 0 (n = 0 1 2 : : :): b) De ning Equation of Type 2: y00 ; 2xy0 + ny = 0 (n = 0 1 2 : : :):
(9.63a)
(9.63b) Particular solutions are the Hermite polynomials, Hen(x) for the de ning equation of type 1, and Hn(x) for the de ning equation of type 2.
a) Hermite Polynomials for De ning Equation of Type 1: 2 dn x2 ; 2 Hen(x) = (;1)ne x2 dx n e !
!
!
= xn ; n2 xn;2 + 1 3 n4 xn;4 ; 1 3 5 n6 xn;6 + (n 2 IN): For n 1 the following recursion formulas are valid: Hen+1(x) = xHen(x) ; nHen;1 (x) (9.63d) He0(x) = 1 He1(x) = x: The orthogonality relation is: + Z1 2 e;x =2 Hem(x)Hen(x) dx = 0 p for m 6= n n! 2 for m = n. ;1
b) Hermite Polynomials for De ning Equation of Type 2: dn e;x2 (n 2 IN): Hn(x) = (;1)n ex2 dx n
The relation with the Hermite polynomials for de ning equation of type 1 is the following: ! Hen(x) = 2;n=2Hn px (n 2 IN): 2
(9.63c) (9.63e) (9.63f) (9.63g) (9.63h)
9.1.3 Boundary Value Problems 9.1.3.1 Problem Formulation
1. Notion of the Boundary Value Problem
In di erent applications, e.g., in mathematical physics, di erential equations must be solved as socalled boundary value problems (see 9.2.3, p. 534), where the solution we are looking for must satisfy previously given relations at the endpoints of an interval of the independent variable. A special case is the linear boundary value problem, where a solution of a linear di erential equation should satisfy linear boundary value conditions. In the followings we restrict our discussion to second-order linear di erential equations with linear boundary values.
2. Self-Adjoint Di erential Equation
Self-adjoint dierential equations are important special second-order di erential equations of the form py0]0 ; qy + %y = f: (9.64a) The linear boundary values are the homogeneous conditions A0y(a) + B0y0(a) = 0 A1 y(b) + B1 y0(b) = 0: (9.64b) The functions p(x) p0 (x) q(x) %(x), and f (x) are supposed to be continuous in the nite interval a x b. In the case of an in nite interval the results change considerably (see 9.5]). Furthermore, we suppose that p(x) > p0 > 0 %(x) > %0 > 0. The quantity , a parameter of the di erential equation, is a constant. For f = 0, it is called the homogeneous boundary value problem associated to
9.1 Ordinary Dierential Equations 515
the inhomogeneous boundary value problem. Every second-order di erential equation of the form Ay00 + By0 + Cy + Ry = F (9.64c) can be reduced to the self-adjoint equation (9.64a) by multiplying it by p=A if in a b] A 6= 0, and performing the following substitutions Z pR q = ; pC (9.64d) p = exp R A dx A %= A : To nd a solution satisfying the inhomogeneous conditions A0y(a) + B0y0(a) = C0 A1y(b) + B1y0(b) = C1 (9.64e) we return to the problem with homogeneous boundary conditions, but we change the right-hand side f (x). We substitute y = z + u where u is an arbitrary twice di erentiable function satisfying the inhomogeneous boundary conditions and z is a new unknown function satisfying the corresponding homogeneous conditions.
3. Sturm{Liouville Problem
For a given value of the parameter there are two cases: 1. Either the inhomogeneous boundary value problem has a unique solution for arbitrary f (x), while the corresponding homogeneous problem has only the trivial, identically zero solution, or, 2. The corresponding homogeneous problem also has non-trivial, i.e., not identically zero solutions, but in this case the inhomogeneous problem does not have a solution for arbitrary right-hand side and if a solution exists, it is not unique. The values of the parameter , for which the second case occurs, i.e., the homogeneous problem has a non-trivial solution, are called the eigenvalues of the boundary value problem, the corresponding nontrivial solutions are called the eigenfunctions. The problem of determining the eigenvalues and eigenfunctions of a di erential equation (9.64a) is called the Sturm{Liouville problem.
9.1.3.2 Fundamental Properties of Eigenfunctions and Eigenvalues
1. The eigenvalues of the boundary value problem (9.64a,b) form a monotone increasing sequence of
real numbers 0 < 1 < 2 < < n < (9.65a) tending to in nity. 2. The eigenfunction associated to the eigenvalue n has exactly n roots in the interval a < x < b. 3. If y(x) and z(x) are two eigenfunctions belonging to the same eigenvalue , they di er only in a constant multiplier c, i.e., z(x) = cy(x): (9.65b) 4. Two eigenfunctions y1(x) and y2(x), associated to di erent eigenvalues 1 and 2 , are orthogonal to each other with the weight function %(x)
Zb a
y1(x) y2(x) %(x) dx = 0:
(9.65c)
5. If in (9.64a) the coecients p(x) and q(x) are replaced by p~(x) p(x) and q~(x) q(x), then the eigenvalues will not decrease, i.e., ~n n, where ~n and n are the n-th eigenvalues of the modi ed and the original equations respectively. But if the coecient %(x) is replaced by %~(x) %(x), then the eigenvalues will not increase, i.e., ~n n. The n-th eigenvalue depends continuously on the coecients of the equation, i.e., small changes in the coecients will result in small variations of the n-th eigenvalue.
516 9. Dierential Equations 6. Reduction of the interval a b] into a smaller one does not result in smaller eigenvalues.
9.1.3.3 Expansion in Eigenfunctions 1. Normalization of the Eigenfunction
For every n an eigenfunction 'n(x) is chosen such that
Zb a
'n(x)]2 %(x) dx = 1:
(9.66a)
It is called a normalized eigenfunction.
2. Fourier Expansion
To every function g(x) de ned in the interval a b], we can assign its Fourier series
g(x) !
1 X
n=0
Zb cn'n(x) cn = g(x) 'n(x) %(x) dx a
(9.66b)
with the eigenfunctions of the corresponding boundary value problem, if the integrals in (9.66b) exist.
3. Expansion Theorem
If the function g(x) has a continuous derivative and satis es the boundary conditions of the given problem, then the Fourier series of g(x) (in the eigenfunctions of this boundary value problem) is absolutely and uniformly convergent to g(x).
4. Parseval Equation
If the integral on the left-hand side exists, then
Zb a
g(x)]2%(x) dx =
1 X
n=0
cn2
(Parseval equation)
(9.66c)
is always valid. The Fourier series of the function g(x) converges in this case to g(x) in mean, that is lim N !1
Zb " a
g(x) ;
N X
n=0
2 cn'n(x) %(x) dx = 0:
9.1.3.4 Singular Cases
(9.66d)
Boundary value problems of the above type very often occur in solving problems of theoretical physics by the Fourier method, however at the endpoints of the interval a b] some singularities of the di erential equation may occur, e.g., p(x) vanishes. At such singular points we impose some restrictions on the solutions, e.g., continuity or being nite or unlimited growth with a bounded order. These conditions play the role of homogeneous boundary conditions (see 9.1.3, p. 514). In addition, we often have the case where in certain boundary value problems homogeneous boundary conditions should be considered, such that they connect the values of the function or its derivative at di erent endpoints of the interval. We often have the relations y(a) = y(b) p(a)y0(a) = p(b)y0(b) (9.67) which represent periodicity in the case of p(a) = p(b). For such boundary value problems everything we introduced above remains valid, except statement (9.65b). For further discussion of this topic see 9.5].
9.2 Partial Dierential Equations 517
9.2 Partial Dierential Equations
9.2.1 First-Order Partial Dierential Equations
9.2.1.1 Linear First-Order Partial Di erential Equations
1. Linear and Quasilinear Partial Di erential Equations
The equation @z + X @z + + X @z = Y X1 @x (9.68a) 2 n @x2 @xn 1 is called a linear rst-order partial dierential equation. Here z is an unknown function of the independent variables x1 : : : xn, and X1 : : : Xn Y are given functions of these variables. If functions X1 : : : Xn Y depend also on z, the equation is called a quasilinear partial dierential equation. In the case of Y 0 (9.68b) the equation is called homogeneous.
2. Solution of a Homogeneous Partial Linear Di erential Equation
The solution of a homogeneous partial linear di erential equation and the solution of the so-called characteristic system dx1 = dx2 = = dxn (9.69a) X1 X2 Xn are equivalent. This system can be solved in two di erent ways: 1. Any xk , for which Xk 6= 0, can be chosen as an independent variable, so the system is transformed into the form dxj Xj (9.69b) dx = X (j = 1 : : : n): k
k
2. A more convenient way is to keep symmetry and introduce a new variable t, and then we get
dxj = X : (9.69c) j dt Every rst integral of the system (9.69a) is a solution of the homogeneous linear partial di erential equation (9.68b), and conversely, every solution of (9.68b) is a rst integral of (9.69a) (see 9.1.2.1, 2., p. 498). If the n ; 1 rst integrals 'i(x1 : : : xn) = 0 (i = 1 2 : : : n ; 1) (9.69d) are independent (see 9.1.2.3, 2., p. 500), then the general solution is z = ('1 : : : 'n;1): (9.69e) Here is an arbitrary function of the n ; 1 arguments 'i and a general solution of the homogeneous linear di erential equation.
3. Solution of Inhomogeneous Linear and Quasilinear Partial Di erential Equations
To solve an inhomogeneous linear and quasilinear partial di erential equation (9.68a) we try to nd the solution z in the implicit form V (x1 : : : xn z) = C . The function V is a solution of the homogeneous linear di erential equation with n + 1 independent variables @V + X @V + + X @V + Y @V = 0 (9.70a) X1 @x 2 n @x2 @xn @z 1 whose characteristic system dx1 = dx2 = = dxn = dz (9.70b) X1 X2 Xn Y
518 9. Dierential Equations is called the characteristic system of the original equation (9.68a).
4. Geometrical Representation and Characteristics of the System
In the case of the equation @z + Q(x y z) @z = R(x y z) P (x y z) @x (9.71a) @y with two independent variables x1 = x and x2 = y, a solution z = f (x y) is a surface in x y z space, and it is called the integral surface of the di erential equation. Equation!(9.71a) means that at every @z @z ;1 is orthogonal to the vector point of the integral surface z = f (x y) the normal vector @x @y (P Q R) given at that point. Here the system (9.70b) has the form dx dy dz (9.71b) P (x y z) = Q(x y z) = R(x y z) : It follows (see 13.1.3.5, p. 648) that the integral curves of this system, the so-called characteristics, are tangent to the vector (P Q R). Therefore, a characteristic having a common point with the integral surface z = f (x y) lies completely on this surface. Since the conditions for the existence theorem 13.1.3.5, 1., p. 498 hold, there is an integral curve of the characteristic system passing through every point of space, so the integral surface consists of characteristics.
5. Cauchy Problem
There are given n functions of n ; 1 independent variables t1 t2 : : : tn;1: x1 = x1 (t1 t2 : : : tn;1) x2 = x2 (t1 t2 : : : tn;1) : : : xn = xn (t1 t2 : : : tn;1): (9.72a) The Cauchy problem for the di erential equation (9.68a) is to nd a solution z = '(x1 x2 : : : xn) (9.72b) such that if we substitute (9.72a), the result is a previously given function (t1 t2 : : : tn;1): ' x1 (t1 t2 : : : tn;1 ) x2 (t1 t2 : : : tn;1 ) : : : xn(t1 t2 : : : tn;1)] = (t1 t2 : : : tn;1): (9.72c) In the case of two independent variables, the problem reduces to nd an integral surface passing through the given curve. If this curve has a tangent depending continuously on a point and it is not tangent to the characteristics at any point, then the Cauchy problem has a unique solution in a certain neighborhood of this curve. Here the integral surface consists of the set of all characteristics intersecting the given curve. For more mathematical discussion on theorems about the existence of the solution of the Cauchy problem see 9.15]. @z + (nx ; A: For the linear rst-order inhomogeneous partial di erential equation (mz ; ny) @x @z = ly ; mx (l m n are constants), the equations of the characteristics are dx = dy = lz) @y mz ; ny nx ; lz dz . The integrals of this system are lx + my + nz = C , x2 + y2 + z2 = C . We get circles as 1 2 ly ; mx characteristics, whose centers are on a line passing through the origin, and this line has direction cosines proportional to l m n. The integral surfaces are rotation surfaces with this line as an axis. @z + B: Determine the integral surface of the rst-order linear inhomogeneous di erential equation @x @z dx @y = z, which passes through the curve x = 0, z = '(y). The equations of characteristics are 1 = dy = dz . The characteristics passing through the point (x y z ) are y = x ; x + y , z = z ex;x0 . 0 0 0 0 0 0 1 z x A parametric representation of the required integral surface is y = x + y0 , z = e '(y0), if we substitute
9.2 Partial Dierential Equations 519
x0 = 0, z0 = '(y0). The elimination of y0 results in z = ex'(y ; x).
9.2.1.2 Non-Linear First-Order Partial Di erential Equations
1. General Form of First-Order Partial Di erential Equation is the implicit equation
! @z : : : @z = 0: F x1 : : : xn z @x (9.73a) @xn 1 1. Complete Integral is the solution z = '(x1 : : : xn a1 : : : an) (9.73b) depending on n parameters a1 : : : an if at the considered values of x1 : : : xn z the functional determinant (or Jacobian determinant, see 2.18.2.6, 3., p. 121) is non-zero: @ ('x1 : : : 'xn ) 6= 0: (9.73c) @ (a1 : : : an) 2. Characteristic Strip The solution of (9.73a) is reduced to the solution of the characteristic system dx1 = = dxn = dz ;dp1 ;dpn (9.73d) P1 Pn p1 P1 + + pnPn = X1 + p1Z = = Xn + pnZ with @F @z @F Z = @F (9.73e) @z Xi = @xi pi = @xi Pi = @pi (i = 1 : : : n): The solutions of the characteristic system satisfying the additional condition F (x1 : : : xn z p1 : : : pn) = 0 (9.73f) are called the characteristic strips .
2. Canonical Systems of Di erential Equations
Sometimes it is more convenient to consider an equation not involving explicitly the unknown function z. Such an equation can be obtained by introducing an additional independent variable xn+1 = z and an unknown function V (x1 , : : : xn, xn+1 ), which de nes the function z(x1 x2 : : : xn) with the equation (9.74a) V (x1 : : : xn z) = C: , @V @V (i = 1 : : : n) for @z in (9.73a). Then At the same time, we substitute the functions ; @x @xi i @xn+1 we solve the di erential equation (9.73a) for an arbitrary partial derivative of the function V . The corresponding independent variable will be denoted by x after a suitable renumbering of the other variables. Finally, we have the equation (9.73a) in the form @V p + H (x1 : : : xn x p1 : : : pn) = 0 p = @V (9.74b) @x pi = @xi (i = 1 : : : n): The system of characteristic di erential equations is transformed into the system dxi = @H dpi = ; @H (i = 1 : : : n) and (9.74c) dx @pi dx @xi dV = p @H + + p @H ; H dp = ; @H : (9.74d) n dx 1 @p1 @pn dx @x Equations (9.74c) represent a system of 2n ordinary di erential equations, which corresponds to an arbitrary function H (x1 : : : xn x p1 : : : pn) with 2n + 1 variables. It is called a canonical system or a normal system of di erential equations.
520 9. Dierential Equations Many problems of mechanics and theoretical physics lead to equations of this form. Knowing a complete integral V = '(x1 : : : xn x a1 : : : an) + a (9.74e) of the equation (9.74b) we can nd the general solution of the canonical system (9.74c), since the equa@' = b , @' = p (i = 1 2 : : : n) with 2n arbitrary parameters a and b determine a 2ntions @a i i i i @xi i parameter solution of the canonical system (9.74c).
3. Clairaut Di erential Equation
If the given di erential equation can be transformed into the form @z (i = 1 : : : n) z = x1 p1 + x2 p2 + + xnpn + f (p1 p2 : : : pn) pi = @x (9.75a) i it is called a Clairaut di erential equation. The determination of the complete integral is particularly simple, because a complete integral with the arbitrary parameters a1 a2 : : : an is z = a1x1 + a2 x2 + + anxn + f (a1 a2 : : : an): (9.75b) Two-Body Problem with Hamilton Function: Consider two particles moving in a plane under their mutual gravitational attraction according to the Newton eld (see also 13.4.3.2, p. 669). We choose the origin as the initial position of one of the particles, so the equations of motion have the form d2x = @V d2y = @V V = p k2 : (9.76a) dt2 @x dt2 @y x2 + y2 If we introduce the Hamiltonian function 2 H = 21 (p2 + q2) ; px2k+ y2 (9.76b) the system (9.76a) is transformed into the normal system (into the system of canonical di erential equations) dx = @H dy = @H dp = ; @H dq = ; @H (9.76c) dt @p dt @q dt @x dt @y with variables dy x y p = dx (9.76d) dt q = dt : Now, the partial di erential equation has the form 2 ! !3 @z + 1 4 @z 2 + @z 25 ; p k2 = 0: (9.76e) @t 2 @x @y x2 + y2 Introducing the polar coordinates , ' in (9.76e) we obtain a new di erential equation having the solution Z s 2 2 z = ;at ; b' + c ; 2a + 2k ; b2 dr (9.76f) r r 0 with the parameters a b c. We get the general solution of the system (9.76c) from the equations @z @z (9.76g) @a = ;t0 @b = ;'0 :
4. First-Order Di erential Equation in Two Independent Variables
For x1 = x x2 = y p1 = p p2 = q the characteristic strip (see 9.2.1.2, 1., p. 519) can be geometrically interpreted as a curve at every point (x y z) of which a plane p( ; x) + q( ; y) = ; z being tangent
9.2 Partial Dierential Equations 521
to the curve is prescribed. So, the problem of nding an integral surface of the equation ! @z @z = 0 (9.77) F x y z @x @y passing through a given curve, i.e., to solve the Cauchy problem (see 9.2.1.1, 5., p. 518), is transformed into another problem: To nd the characteristic strips passing through the points of the initial curve such that the corresponding tangent plane to each strip is tangent to that curve. We get the values p and q at the points of the initial curve from the equations F (x y z p q) = 0 and pdx + qdy = dz. We may have several solutions in the case of non-linear di erential equations. Therefore, under the formulation of the Cauchy problem, in order to obtain a unique solution we assume two continuous functions p and q satisfying the above relations along the initial curve. For the existence of solutions of the Cauchy problem see 9.15]. For the partial di erential equation p q = 1 and the initial curve y = x3 , z = 2x2 , we can choose p = x and q = 1=x along the curve. The characteristic system has the form dx = q dy = p dz = 2p q dp = 0 dq = 0: dt dt dt dt dt The characteristic strip with initial values x0 y0 z0 p0 and q0 for t = 0 satis es the equations x = x0 + q0 t y = y0 + p0t z = 2p0 q0t + z0 p = p0 q = q0 . For the case of p0 = x0 q0 = 1=x0 the equation of the curve belonging to the characteristic strip that passes through the point (x0 y0 z0 ) of the initial curve is x = x0 + xt y = x03 + tx0 z = 2t + 2x0 2 : 0 Eliminating the parameters x0 and t we get z2 = 4xy. For other chosen values of p and q along the initial curve we can get di erent solutions. Remark: The envelope of a one-parameter family of integral surfaces is also an integral surface. Considering this fact we can solve the Cauchy problem with a complete integral. We nd a one-parameter family of solutions tangent to the planes given at the points of the initial curve. Then we determine the envelope of this family. Determine the integral surface for the Clairaut di erential equation z ; px ; qy + pq = 0 passing through the curve y = x z = x2 . The complete integral of the di erential equation is z = ax + by ; ab. Since along the initial curve p = q = x, we determine the one-parameter family of integral surfaces by the condition a = b. Finding the envelope of this family we have z = 1 (x + y)2. 4
5. Linear First-Order Partial Di erential Equations in Total Di erentials
Equations of this kind have the form dz = f1 dx1 + f2dx2 + + fndxn (9.78a) where f1 f2 : : : fn are given functions of the variables x1 x2 : : : xn z. The equation is called a completely integrable or exact dierential equation when there exists a unique relation between x1 x2 : : : xn z with one arbitrary constant, which leads to equation (9.78a). Then there exists a unique solution z = z(x1 x2 : : : xn) of (9.78a), which has a given value z0 for the initial values x1 0 : : : xn0 of the independent variables. Therefore, for n = 2 x1 = x x2 = y a unique integral surface passes through every point of space. The di erential equation (9.78a) is completely integrable if and only if the n(n2; 1) equalities
@fi @fi @fk @fk @xk + fk @z = @xi + fi @z (i k = 1 : : : n)
(9.78b)
522 9. Dierential Equations in all variables x1 x2 : : : xn z are identically satis ed. If the di erential equation is given in symmetric form f1dx1 + + fndxn = 0 (9.78c) then the condition for complete integrability is ! ! ! @fk ; @fj + f @fi ; @fk + f @fj ; @fi = 0 fi @x (9.78d) j k @xk @xi @xi @xj j @xk for all possible combinations of the indices i j k. If the equation is completely integrable, then the solution of the di erential equation (9.78a) can be reduced to the solution of an ordinary di erential equation with n ; 1 parameters.
9.2.2 Linear Second-Order Partial Dierential Equations
9.2.2.1 Classication and Properties of Second-Order Di erential Equations with Two Independent Variables 1. General Form
of a linear second-order partial di erential equation with two independent variables x y and an unknown function u is an equation in the form @ 2 u + 2B @ 2 u + C @ 2 u + a @u + b @u + cu = f A @x (9.79a) 2 @x@y @y2 @x @y where the coecients A B C a b c and f on the right-hand side are known functions of x and y. The form of the solution of this di erential equation depends on the sign of the discriminant = AC ; B 2 (9.79b) in a considered domain. We distinguish between the following cases.
1. 2. 3. 4. 5.
< 0: Hyperbolic type. = 0: Parabolic type. > 0: Elliptic type.
changes its sign: Mixed type. An important property of the discriminant is that its sign is invariant with respect to arbitrary transformation of the independent variables, e.g., to introduction new coordinates in the x y plane. Therefore, the type of the di erential equation is invariant with respect to the choice of the independent variables.
2. Characteristics
of linear second-order partial di erential equations are the integral curves of the di erential equation p dy = B ; : Ady2 ; 2Bdxdy + Cdx2 = 0 or dx (9.80) A For the characteristics of the above three types of di erential equations the following statements are valid: 1. Hyperbolic type: There exist two families of real characteristics. 2. Parabolic type: There exists only one family of real characteristics. 3. Elliptic type: There exists no real characteristic. 4. A di erential equation obtained by coordinate transformation from (9.79a) has the same characteristics as (9.79a). 5. If a family of characteristics coincides with a family of coordinate lines, then the term with the second derivative of the unknown function with respect to the corresponding independent variable is missing in (9.79a). In the case of a parabolic di erential equation, the mixed derivative term is also missing.
9.2 Partial Dierential Equations 523
3. Normal Form or Canonical Form
We have the following possibilities to transform (9.79a) into the normal form of linear second-order partial di erential equations. 1. Transformation into Normal Form: The di erential equation (9.79a) can be transformed into normal form by introducing the new independent variables = '(x y) and = (x y) (9.81a) which according to the sign of the discriminant (9.79b) belongs to one of the three considered types: @ 2 u ; @ 2 u + = 0 < 0 hyperbolic type (9.81b) @ 2 @2 @ 2 u + = 0 = 0 parabolic type (9.81c) @2 2 2 @ u + @ u + = 0 > 0 elliptic type: (9.81d) @ 2 @2 The terms not containing second-order partial derivatives of the unknown function are denoted by dots. 2. Reduction of a Hyperbolic Type Equation to Canonical Form (9.81b): If, in the hyperbolic case, we chose two families of characteristics as the coordinate lines of the new coordinate system (9.81a), i.e., if we substitute 1 = '(x y), 1 = (x y), where '(x y) = constant, (x y) = constant are the equations of the characteristics, then (9.79a) becomes the form @2u (9.81e) @ 1 @1 + = 0: This form is also called the canonical form of a hyperbolic type dierential equation. From here we get the canonical form (9.81b) by the substitution = 1 + 1 = 1 ; 1 : (9.81f) 3. Reduction of a Parabolic Type Equation to Canonical Form (9.81c): The only family of characteristics given in this case is chosen for the family = const, where an arbitrary function of x and y can be chosen for , which must not be dependent on . 4. Reduction of an Elliptic Type Equation to Canonical Form (9.81d): If the coecients A(x y), B (x y), C (x y) are analytic functions (see 14.1.2.1, p. 672) in the elliptic case, then the characteristics de ne two complex conjugate families of curves '(x y) = constant, (x y) = constant. If we substitute = ' + , and = i(' ; ), the equation becomes the form (9.81d).
4. Generalized Form
Every statement for the classi cation and reduction to canonical form remains valid for equations given in a more general form ! 2 @ 2 u + C (x y) @ 2u + F x y u @u @u = 0 A(x y) @@xu2 + 2B (x y) @x@y (9.82) @y2 @x @y where F is a non-linear function of the unknown function u and its rst-order partial derivatives @u=@x and @u=@y, in contrast to (9.79a).
9.2.2.2 Classication and Properties of Linear Second-Order Di erential Equations with More than Two Independent Variables
1. General Form
A di erential equation of this kind has the form X @2u aik @x @x + = 0 i k ik
(9.83)
524 9. Dierential Equations where aik are given functions of the independent variables and the dots in (9.83) mean terms not containing second-order derivatives of the unknown function u. In general, the di erential equation (9.83) cannot be reduced to a simple canonical form by transforming the independent variables. However, there is an important classi cation, similar to the one introduced above in 9.2.2.1, p. 522 (see 9.5]).
2. Linear Second-Order Partial Di erential Equations with Constant Coe cients
If all coecients aik in (9.83) are constants, then the equation can be reduced by a linear homogeneous transformation of the independent variables into a simpler canonical form X @2u i @x 2 + = 0 (9.84) i i where the coecients i are 1 or 0. We can distinguish between several characteristic cases. 1. Elliptic Di erential Equation All the coecients i are di erent from zero, and they have the same sign. Then we have an elliptic dierential equation. 2. Hyperbolic and Ultrahyperbolic Di erential Equation All the coecients i are di erent from zero, but one has a sign di erent from the other's. Then we have a hyperbolic dierential equation. If both type of signs occur at least twice, then it is an ultrahyperbolic dierential equation. 3. Parabolic Di erential Equation One of the coecients i is equal to zero, the others are di erent from zero and they have the same sign. Then we have a parabolic dierential equation. 4. Simple Case for Elliptic and Hyperbolic Di erential Equations We have a relatively simple case if not only the coecients of the highest derivatives of the unknown function are constants, but also those of the rst derivatives. Then we can eliminate the terms of the rst derivatives, for which i 6= 0, by substitution. For this purpose, we substitute ! X u = v exp ; 12 bk xk (9.85) k where bk is the coecient of @u in (9.84) and the summation is performed for all i 6= 0. In this way, @xk every elliptic and hyperbolic di erential equation with constant coecients can be reduced to a simple form: 2 a) Elliptic Case: %v + kv = g: (9.86) b) Hyperbolic Case: @@tv2 ; %v + kv = g: (9.87) Here % denotes the Laplace operator (see 13.2.6.5, p. 657), t (time) a further independent variable.
9.2.2.3 Integration Methods for Linear Second-Order Partial Di erential Equations 1. Method of Separation of Variables
We can determine certain solutions of several di erential equations of physics by special substitutions, and although these are not general solutions, we get a family of solutions depending on arbitrary parameters. Linear di erential equations, especially those of second order, can often be solved if we are looking for a solution in the form of a product u(x1 : : : xn) = '1 (x1)'2 (x2) : : : 'n(xn): (9.88) Next, we try to separate the functions 'k (xk ), i.e., for each of them we want to determine an ordinary di erential equation containing only one variable xk . This separation of variables is successful in many cases when the trial solution in the form of a product (9.88) is substituted into the given di erential equation. In order to guarantee that the solution of the original equation satis es the required homogeneous boundary conditions, it may appear to be sucient that some of functions '1 (x1), '2(x2 ) : : : , 'n(xn) satisfy certain boundary conditions.
9.2 Partial Dierential Equations 525
By means of summation, di erentiation and integration, new solutions can be acquired from the obtained ones the parameters should be chosen so that the remaining boundary and initial conditions are satis ed (see examples). Finally, we must not forget that the solutions obtained in this way, often in nite series and improper integrals, are only formal solutions. That is, we have to check whether the solution makes a physical sense, e.g., whether it is convergent, satis es the original di erential equation and the boundary conditions, whether it is di erentiable termwise and whether the limit at the boundary exists. The in nite series and improper integrals in the examples of this paragraph are convergent if the functions de ning the boundary conditions satisfy the required conditions, e.g., the continuity assumption for the second derivativs in the rst and the second examples. A: Equation of the Vibrating String is a linear second-order partial di erential equation of hyperbolic type @ 2 u = a2 @ 2 u : (9.89a) @t2 @x2 It describes the vibration of a spanned string. The boundary and the initial conditions are: u = f (x) @u (9.89b) @t t=0 = '(x) ujx=0 = 0 ujx=l = 0: t=0 We seek a solution in the form u = X (x)T (t) (9.89c) and after substituting it into the given equation (9.89a) we have T 00 = X 00 : (9.89d) a2 T X The variables are separated, the right side depends on only x and the left side depends on only t, so each of them is a constant quantity. This constant must be negative, otherwise the bondary conditions cannot be satis ed. We get an ordinary linear second-order di erential equation with constant coecients for both variables. For the general solution see 9.1.2.4, p. 502. We denote this negative constant by ;2 and we get the linear di erential equations X 00 + 2 X = 0 (9.89e) T 00 + a2 2T = 0: (9.89f) We have X (0) = X (l) = 0 from the boundary conditions. Hence X (x) is an eigenfunction of the Sturm{Liouvilleboundary value problem and 2 is the corresponding eigenvalue (see 9.1.3.1, 3., p. 515). Solving the di erential equation (9.89e) for X with the corresponding boundary conditions we get X (x) = C sin x with sin l = 0 i.e., with = nl = n (n = 1 2 : : :): (9.89g) Solving equation (9.89f) for T yields a particular solution of the original di erential equation (9.89a) for every eigenvalue n: na t sin n x: un = an cos na t + b (9.89h) n sin l l l Requiring that for t = 0, 1 1 X X (9.89j) u = un is equal to f (x) (9.89i) and @ un is equal to '(x) @t n=1 n=1 we get with a Fourier series expansion in sines (see 7.4.1.1, 1., p. 420) Zl 2 Z l '(x) sin nx dx: an = 2l f (x) sin nx dx b (9.89k) n= l na 0 l 0
526 9. Dierential Equations B: Equation of Longitudinal Vibration of a Bar is a linear second-order partial di erential equation of hyperbolic type, which describes the longitudinal vibration of a bar with one end free and a constant force p a ecting the xed end. We have to solve the same di erential equation as in example A (p. 525), i.e., @ 2 u = a2 @ 2 u (9.90a) @t2 @x2 with the same initial but di erent boundary conditions: @u = 0 (free end) = '(x) (9.90b) (9.90c) u = f (x) @u @x x=0 @t t=0 t=0
@u = kp : (9.90d) @x x=l These conditions can be replaced by the homogeneous conditions @z @z (9.90e) @x x=0 = @x x=l = 0 where instead of u we introduce a new unknown function 2 z = u ; kpx (9.90f) 2l : The di erential equation becomes inhomogeneous: @ 2 z = a2 @ 2 z + a2kp : (9.90g) @t2 @x2 l We are looking for the solution in the form z = v + w, where v satis es the homogeneous di erential equation with the initial and boundary conditions for z, i.e., 2 @z = '(x) z = f (x) ; kpx (9.90h) 2 @t t=0 t=0 and w satis es the inhomogeneous di erential equation with zero initial and boundary conditions. So, 2 2 we get w = ka2lpt . Substituting the product form into the di erential equation v = X (x)T (t) (9.90i) we get the separated ordinary di erential equations as in the example A (p. 525) X 00 = T 00 = ;2 : (9.90j) X a2 T Integrating the di erential equation for X with the boundary conditions X 0(0) = X 0(l) = 0 we nd the eigenfunctions Xn = cos nx (9.90k) l and the corresponding eigenvalues 2 2 (9.90l) n2 = n l2 (n = 0 1 2 : : :): Proceeding as in example A (p. 525) we nally obtain ! 1 2 2 2 a b t + X ant + bn sin ant cos nx u = ka2lpt + kpx + a (9.90m) a 0+ 0 n cos 2l l l n l l n=1
9.2 Partial Dierential Equations 527
where an and bn (n = 0 1 2 : : :) are the coecients of the Fourier series expansion in cosines of the 2 l functions f (x) ; kpx 2 and a '(x) in the interval (0 l) (see 7.4.1.1, 1., p. 420).
C: Equation of a Vibrating Round Membrane xed along the boundary: The di erential equation is linear, partial and it is of hyperbolic type. It has the form in Cartesian and in polar coordinates (see 3.5.3.1, 6., p. 209) @2u @2u 1 @2u @ 2 u + 1 @u + 1 @ 2 u = 1 @ 2 u : (9.91b) (9.91a) @x2 + @y2 = a2 @t2 @ 2 @ 2 @'2 a2 @t2 The initial and boundary conditions are @u uj=R = 0: (9.91e) ujt=0 = f ( ') (9.91c) @t t=0 = F ( ') (9.91d) The substitution of the product form u = U ( )(')T (t) (9.91f) with three variables into the di erential equation in polar coordinates yields U 00 + U 0 + 00 = 1 T 00 = ;2 : (9.91g) U U 2 a2 T Three ordinary di erential equations are obtained for the separated variables analogously to examples A (p. 525) and B (p. 526): 2 U 00 + U 0 + 2 2 = ; 00 = 2 T 00 + a2 2T = 0 (9.91h) (9.91i) U 00 + 2 = 0: (9.91j) From the conditions (0) = (2), 0(0) = 0(2) it follows that: (') = an cos n' + bn sin n' 2 = n2 (n = 0 1 2 : : :): (9.91k) 2 n U and will be determined from the equations U 0 ]0 ; U = ;2 U and U (R) = 0. Considering the obvious condition of boundedness of U ( ) at = 0 and substituting = z we get z2 U 00 + zU 0 + (z2 ; n2 )U = 0 i.e., U ( ) = Jn(z) = Jn R (9.91l) where Jn are the Bessel functions (see 9.1.2.6, 2., p. 509) with = R and Jn( ) = 0. The function system Unk ( ) = Jn nk R (k = 1 2 : : :) (9.91m) with nk as the k-th positive root of the function Jn(z) is a complete system of eigenfunctions of the self-adjoint Sturm{Liouville problem which are orthogonal with the weight function . The solution of the problem can have the form of a double series: 1 X 1
X U= (ank cos n' + bnk sin n') cos a nk t R n=0 k=1 +(cnk cos n' + dnk sin n') sin a Rnk t Jn nk R : (9.91n)
528 9. Dierential Equations From the initial conditions at t = 0 we obtain 1 X 1 X f ( ') = (ank cos n' + bnk sin n')Jn nk R
(9.91o)
n=0 k=1
F ( ' ) = where ank =
1 X 1 X a nk (c cos n' + d sin n')J nk nk n nk R n=0 k=1 R
(9.91p)
f ( ') cos n'Jn nk R d (9.91q) 0 ZR bnk = d' f ( ') sin n'Jn nk R d : (9.91r) 0 In the case of n = 0, the numerator 2 should be changed to 1. To determine the coecients cnk and dnk we replace f ( ') by F ( ') in the formulas for ank and bnk and we multiply by a R . Z 2 2 2 2 R Jn;1( nk ) 0 Z 2 2 2 2 R Jn;1( nk ) 0
d'
ZR
nk
D: Dirichlet Problem (see 13.5.1, p. 670) for the rectangle 0 x a, 0 y b (Fig. 9.17):
y b
0
a x
Find a function u(x y) satisfying the elliptic type Laplace di erential equation %u = 0 (9.92a) and the boundary conditions u(0 y) = '1 (y) u(a y) = '2(y) u(x 0) = 1 (x) u(x b) = 2 (x): (9.92b)
Figure 9.17 First we determine a particular solution for the boundary conditions '1(y) = '2(y) = 0. Substituting the product form u = X (x)Y (y) (9.92c) into (9.92a) we get the separated di erential equations X 00 = ; Y 00 = ;2 (9.92d) X Y with the eigenvalue analogously to examples A (p. 525) through C (p. 527). Since X (0) = X (a) = 0, we get X = C sin x = n (9.92e) a = n (n = 1 2 : : :): In the second step we write the general solution of the di erential equation 2 2 n in the form Y = an sinh n (9.92g) Y 00 ; na2 Y = 0 (9.92f) a (b ; y) + bn sinh a y : From these equations we get a particular solution of (9.92a) satisfying the boundary conditions u(0 y) = u(a y) = 0, which has the form
n n un = an sinh n ( b ; y ) + b (9.92h) n sinh y sin x : a a a In the third step we consider the general solution as a series
u=
1 X
n=1
un
(9.92i)
9.2 Partial Dierential Equations 529
so from the boundary conditions for y = 0 and y = b we get 1 X n n u= an sinh n ( b ; y ) + b (9.92j) n sinh y sin x a a a n=1 with the coecients Za Za 2 n x dx b = 2 an = ( x ) sin 2 (x) sin n (9.92k) 1 n nb nb a a x dx: 0 0 a sinh a a sinh a The problem with the boundary conditions 1(x) = 2 (x) = 0 can be solved in a similar manner, and taking the series (9.92j) we get the general solution of (9.92a) and (9.92b). E: Heat Conduction Equation Heat conduction in a homogeneous bar with one end at in nity and the other end kept at a constant temperature is described by the linear second-order partial di erential equation of parabolic type @u = a2 @ 2 u (9.93a) @t @x2 which satis es the initial and boundary conditions ujt=0 = f (x) ujx=0 = 0 (9.93b) in the domain 0 x < +1, t 0. We also suppose that the temperature tends to zero at in nity. Substituting u = X (x)T (t) (9.93c) into (9.93a) we obtain the ordinary di erential equations T 0 = X 00 = ;2 (9.93d) a2 T X whose parameter is introduced analogously to the previous examples A (p. 525) through D (p. 528). We get T (t) = C e; 2 a2 t (9.93e) as a solution for T (t). Using the boundary condition X (0) = 0, we get X (x) = C sin x (9.93f) and so u = C e; 2 a2 t sin x (9.93g) where is an arbitrary real number. The solution can be obtained in the form
u(x t) =
Z1 0
C ()e; 2 a2 t sin x d:
(9.93h)
From the initial condition ujt=0 = f (x), so
Z1 f (x) = C () sin x d 0 which is satis ed if we substitute Z1 C () = 2 f (s) sin s ds 0 for the constant (see 7.4.1.1, 1., p. 420). Combining this equation and (9.93i) we get Z 1 2 2 Z1 u(x t) = 2 f (s) e; a t sin s sin x d ds 0 0
(9.93i) (9.93j) (9.93k)
530 9. Dierential Equations or after replacing the product of the two sines with one half of the di erence of two cosines ((2.122), p. 81) and using formula (21.27), p. 1058, we get 2 3 (x ; s)2 ; (x + s)2 Z1 ; 6 7 1 u(x t) = f (s) p 64e 4a2 t ; e 4a2 t 75 ds: (9.93l) 0 2a t
2. Riemann Method for Solving Cauchy's Problem for the Hyperbolic Di erential Equation
@ 2 u + a @u + b @u + cu = F (9.94a) @x@y @x @y 1. Riemann Function is a function v(x y ), where and are considered as parameters, satisfying the homogeneous equation @ 2 v @ (av) @ (bv) (9.94b) @x@y ; @x ; @y + cv = 0 which is the adjoint of (9.94a) and the conditions 0x 1 0Zy 1 Z B C @ v(x ) = exp @ b(s ) dsA v( y ) = exp a( s) dsA : (9.94c)
In general, linear second-order di erential equations and their adjoint di erential equations have the form X @ 2 u X @u X @ 2 (aik v) X @ (bi v) aik @x @x + bi @x + cu = f (9.94d) and ; @x + cv = 0: (9.94e) i k i i i i ik ik @xi @xk
2. Riemann Formula is the integral formula which is used to determine function u( ) satisfying the given di erential equation (9.94a) and taking the previously given values along the previously given curve ; (Fig. 9.18) together with its derivative in the direction of the curve normal (see 3.6.1.2, 2., p. 226): ! Z " @v dx u( ) = 21 (uv)P + 12 (uv)Q ; buv + 12 v @u ; u @x @x _ QP
y P Γ M(ξ,η)
0
Q x
" ! @v dy + Z Z Fv dx dy: ; auv + 12 v @u ; u @y @y PMQ
(9.94f)
Figure 9.18 The smooth curve ; (Fig. 9.18) must not have tangents parallel to the coordinate axes, i.e., the curve must not be tangent to the characteristics. The line integral in this formula can be calculated, since the values of both partial derivatives can be determined from the function values and from its derivatives in a non-tangential direction along the curve arc. In the Cauchy problem, the values of the partial derivatives of the unknown function, e.g., @u @y are often given instead of the normal derivative along the curve. Then we use another Riemann formula: ! ! Z @v dx ; auv + v @u dy + Z Z Fv dx dy: u( ) = (uv)P ; buv ; u @x (9.94g) @y _ PMQ QP
9.2 Partial Dierential Equations 531
Electric Circuit Equation (Telegraphic Equation) is a linear second-order partial di erential equation of hyperbolic type 2 2 + cu = @ u2 (9.95a) a @@tu2 + 2b @u @t @x where a > 0, b, and c are constants. The equation describes the current ow in wires. It is a generalization of the di erential equation of a vibrating string. We replace the unknown function u(x t) by u = ze;(b=a)t . Then (9.95a) is reduced to the form ! @ 2 z = m2 @ 2 z + n2z m2 = 1 n2 = b2 ; ac : (9.95b) @t2 @x2 a a2 Replacing the independent variables by n (mt + x) = n (mt ; x) =m (9.95c) m we nally get the canonical form @2z ; z = 0 (9.95d) @ @ 4 of a hyperbolic type linear partial di erential equation (see 9.2.2.1, 1., p. 523). The Riemann function v( 0 0) should satisfy this equation with unit value at = 0 and = 0 . If we choose the form w = ( ; 0)( ; 0 ) (9.95e) for w in v = f (w), then f (w) is a solution of the di erential equation d2f + df ; 1 f = 0 w dw (9.95f) 2 dw 4 with initial condition f (0) = 1. The substitution w = 2 reduces this di erential equation to Bessel's di erential equation of order zero (see 9.1.2.6, 2., p. 509) d2f + 1 df ; f = 0 (9.95g) d2 d hence the solution
q is v = I0 ( ; 0)( ; 0 ) : (9.95h) A solution of the original di erential equation (9.95a) satisfying the boundary conditions z = f (x) @z (9.95i) @t t=0 = g(x) t=0 can be obtained if we substitute the found value of v into the Riemann formula and then return to the original variables: z(x t) = 12 f (x ; mt) + f (x + mt)] nq nq 3 2 2 t2 ; (s ; x)2 2 t2 ; (s ; x)2 xZ+mt I m ntI m 0 1 66 77 + 21 ; f (s) qm 2 2 (9.95j) 4g(s) m 5 ds: 2 m m t ; ( s ; x ) x;mt
3. Green's Method of Solving the Boundary Value Problem for Elliptic Di erential Equations with Two Independent Variables
This method is very similar to the Riemann method of solving the Cauchy problem for hyperbolic di erential equations.
532 9. Dierential Equations If we want to nd a function u(x y) satisfying the elliptic type linear second-order partial di erential equation @ 2 u + @ 2 u + a @u + b @u + c u = f (9.96a) @x2 @y2 @x @y in a given domain and taking the prescribed values on its boundary, rst we determine the Green function G(x y ) for this domain, where and are regarded as parameters. The Green function must satisfy the following conditions: 1. The function G(x y ) satis es the homogeneous adjoint di erential equation @ 2 G + @ 2 G ; @ (a G) ; @ (b G) + c G = 0 (9.96b) @x2 @y2 @x @y everywhere in the given domain except at the point x = , y = . 2. The function G(x y ) has the form q U ln 1r + V (9.96c) with r = (x ; )2 + (y ; )2 (9.96d) where U has unit value at the point x = , y = and U and V are continuous functions in the entire domain together with their second derivatives. 3. The function G(x y ) is equal to zero on the boundary of the given domain. The second step is to give the solution of the boundary value problem with the Green function by the formula Z @ G(x y ) ds ; 1 ZZ f (x y)G(x y ) dx dy u( ) = 21 u(x y) @n (9.96e) 2 D S where D is the considered domain, S is its boundary on which the function is assumed to be known and @ @n denotes the normal derivative directed toward the interior of D. Condition 3 depends on the formulation of the problem. For instance, if instead of the function values the values of the derivative of the unknown function in the direction normal to the boundary of the domain are given, then in 3 we have the condition @G ; (a cos + b cos )G = 0 (9.96f) @n on the boundary. and denote here the angles between the interior normal to the boundary of the domain and the coordinate axes. In this case, the solution is given by the formula Z @u ZZ G ds ; 21 fG dx dy: (9.96g) u( ) = ; 21 @n S D
4. Green's Method for the Solution of Boundary Value Problems with Three Independent Variables
The solution of the di erential equation (9.97a) %u + a @u + b @u + c @u + e u = f @x @y @z should take the given values on the boundary of the considered domain. As the rst step, we construct again the Green function, but now it depends on three parameters , , and . The adjoint di erential equation satis ed by the Green function has the form a G) ; @ (b G) ; @ (c G) + e G = 0: %G ; @ (@x (9.97b) @y @z
9.2 Partial Dierential Equations 533
As in condition 2, the function G(x y z ) has the form q U 1r + V (9.97c) with r = (x ; )2 + (y ; )2 + (z ; )2: The solution of the problem is: ZZ 1 ZZZ fG dx dy dz: u( ) = 41 u @G ds ; @n 4 S
D
(9.97d) (9.97e)
r
Both methods, Riemann's and Green's, have the common idea rst to determine a special solution of the di erential equation, which can then be used to obtain a solution with arbitrary boundary conditions. An essential di erence between the Riemann and the Green function is that the rst one depends only on the form of the left-hand side of the di erential equation, while the second one depends also on the considered domain. Finding the Green function is, in practice, an extremely dicult problem, even if it is known to exist therefore, Green's method is used mostly in theoretical research. A: Construction of the Green function for the Dirichlet problem y of the Laplace di erential equation (see 13.5.1, p. 670) M1 %u = 0 (9.98a) for the case, when the considered domain is a circle (Fig. 9.19). The Green function is r1 M(x,h) G(x y ) = ln 1r + ln rR1 (9.98b) p where r = MP = OM , r1 = M1 P and R is the radius of the j r considered circle (Fig. 9.19). The points M and M1 are symmet0 R x ric with respect to the circle, i.e., both points are on the same ray starting from the center and OM OM1 = R2: (9.98c) The formula (9.96e) for a solution of Dirichlet's problem, after substituting the normal derivative of the Green function and after cerFigure 9.19 tain calculations, yields the so-called Poisson integral Z 2 2 ; 2 R 1 u(') d': (9.98d) u( ) = 2 0 R2 + 2 ; 2R cos( ; ') The notation is the same as above. The known values of u are given on the boundary of the circle by u('). For the coordinates of the point M ( ) we have: = cos , = sin . B: Construction of the Green function for the Dirichlet problem of the Laplace di erential equation (see 13.5.1, p. 670) %u = 0 (9.99a) for the case when the considered domain is a sphere with radius R. The Green function now has the form G(x y z ) = 1r ; rR (9.99b) 1 p with = 2 + 2 + 2 as the distance of the point ( ) from the center, r as the distance between the points (x y z) and ( ), and r1 as the distance of the point! (x y z) from the symmetric point of ( ) according to (9.98c), i.e., from the point R R R . In this case, the Poisson integral has (with the same notation as in example A (p. 533)) the form ZZ 2 2 u( ) = 41 R Rr;3 u ds: (9.99c) S
534 9. Dierential Equations
5. Operational Method
Operational methods can be used not only to solve ordinary di erential equations but also for partial di erential equations (see 15.1.6, p. 709). They are based on transition from the unknown function to its transform (see 15.1, p. 707). In this process, we regard the unknown function as a function of only one variable and we perform the transformation with respect to this variable. The remaining variables are considered as parameters. The di erential equation to determine the transform of the unknown function contains one less independent variable than the original equation. In particular, if the original equation is a partial di erential equation of two independent variables, then we obtain an ordinary di erential equation for the transform. If we can nd the transform of the unknown function from the obtained equation, then we determine the original function either from the formula for the inverse function or from the table of transforms.
6. Approximation Methods
In order to solve practical problems with partial di erential equations, we often use di erent approximation methods. We distinguish between analytical and numerical methods. 1. Analytical Methods make possible the determination of approximate analytical expressions for the unknown function. 2. Numerical Methods result in approximate values of the unknown function for certain values of the independent variables. We use the following methods (see 19.5, p. 911): a) Finite Di erence Method, or Lattice-Point Method: The derivatives are replaced by divided differences, so the di erential equation including the initial and boundary conditions becomes an algebraic equation system. A linear di erential equation with linear initial and boundary conditions becomes a linear equation system. b) Finite Element Method, or briey FEM, for boundary value problems: We assign a variational problem to the boundary value problem. We approximate the unknown function by a spline, whose coecients should be chosen to get the best possible solution. We decompose the domain of the boundary value problem into regular subdomains. The coecients are determined by solving an extreme value problem. c) Integral Equation Method (along a Closed Curve) for special boundary problems: We formulate the boundary value problem as an equivalent integral equation problem along the boundary of the domain of the boundary value problem. To do this, we apply the theorems of vector analysis, e.g., Green formulas. We determine the remaining integrals along the closed curve numerically by a suitable quadrature formula. 3. Physical Solutions of di erential equations can be given by experimental methods. This is based on the fact that various physical phenomena can be described by the same di erential equation. To solve a given equation, we rst construct a model by which we can simulate the given problem, and we obtain the values of the unknown function directly from this model. Since such models are often known and can be constructed by varying the parameters in a wide range, the di erential equation can also be investigated in a wide domain of the variables.
9.2.3 Some further Partial Dierential Equations from Natural Sciences and Engineering 9.2.3.1 Formulation of the Problem and the Boundary Conditions 1. Problem Formulation
The modeling and the mathematical treatment of di erent physical phenomena in classical theoretical physics, especially in modeling media considered structureless or continuously changing, such as gases, uids, solids, the elds of classical physics, leads to the introduction of partial di erential equations. Examples are the wave (see 9.2.3.2, p. 536) and the heat equations (see 9.2.3.3, p. 537). Many problems in non-classical theoretical physics are also governed by partial di erential equations. An important
9.2 Partial Dierential Equations 535
area is quantum mechanics, which is based on the recognition that media and elds are discontinuous. The most famous relation is the Schrodinger equation. Linear second-order partial di erential equations occur most frequently and they have special importance in today's natural sciences.
2. Initial and Boundary Conditions
The solution of the problems of physics, engineering, and the natural sciences must usually ful ll two basic requirements: 1. The solution must satisfy not only the di erential equation, but also certain initial and/or boundary conditions. There are problems with only initial condition or only with boundary conditions or with both. All the conditions together must determine the unique solution of the di erential equation. 2. The solution must be stable with respect to small changes in the initial and boundary conditions, i.e., its change should be arbitrarily small if the perturbations of these conditions are small enough. Then a correct problem formulation is given. We can assume that the mathematical model of the given problem to describe the real situation is adequate only in cases when these conditions are ful lled. For instance, the Cauchy problem (see 9.2.1.1, 5., p. 518) is correctly de ned with a di erential equation of hyperbolic type for investigating vibration processes in continuous media. This means that the values of the required function, and the values of its derivatives in a non-tangential (mostly in a normal) direction are given on an initial manifold, i.e., on a curve or on a surface. In the case of di erential equations of elliptic type, which occur in investigations of steady state and equilibrium problems in continuous media, the formulation of the boundary value problem is correct. If the considered domain is unbounded, then the unknown function must satisfy certain given properties with unlimited increase of the independent variables.
3. Inhomogeneous Conditions and Inhomogeneous Di erential Equations
The solution of homogeneous or inhomogeneous linear partial di erential equations with inhomogeneous initial or boundary conditions can be reduced to the solution of an equation which di ers from the original one only by a free term not containing the unknown function, and which has homogeneous conditions. It is sucient to replace the original function by its di erence from an arbitrary twice differentiable function satisfying the given inhomogeneous conditions. In general, we use the fact that the solution of a linear inhomogeneous partial di erential equation with given inhomogeneous initial or boundary conditions is the sum of the solutions of the same di erential equation with zero conditions and the solution of the corresponding homogeneous di erential equation with the given conditions. To reduce the solution of the linear inhomogeneous partial di erential equation @ 2 u ; L u] = g(x t) (9.100a) @t2 with homogeneous initial conditions (9.100b) u = 0 @u @t t=0 = 0 t=0 to the solution of the Cauchy problem for the corresponding homogeneous di erential equation, we substitute
Zt u = '(x t ) d : 0
Here '(x t ) is the solution of the di erential equation @ 2 u ; L u] = 0 @t2
(9.100c)
(9.100d)
536 9. Dierential Equations which satis es the boundary conditions u = 0 @u (9.100e) @t t= = g(x ): t= In this equation, x represents symbolically all the n variables x1 x2 : : : xn of the n-dimensional problem. L u] denotes a linear di erential expression, which may contain the derivative @u @t , but not higherorder derivatives with respect to t.
9.2.3.2 Wave Equation
The extension of oscillations in a homogeneous media is described by the wave equation @ 2 u ; a2 %u = Q(x t) (9.101a) @t2 whose right-hand side Q(x t) vanishes when there is no perturbation. The symbol x represents the n variables x1 : : : xn of the n-dimensional problem. The Laplace operator % (see also 13.2.6.5, 657,) is de ned in the following way: @2u + @2u + + @2u : %u = @x (9.101b) @xn 2 1 2 @x2 2 The solution of the wave equation is the wave function u. The di erential equation (9.101a) is of hyperbolic type.
1. Homogeneous Problem
The solution of the homogeneous problem with Q(x t) = 0 and with the initial conditions u = '(x) @u @t t=0 = (x) t=0 is given for the cases n = 1 2 3 by the following integrals.
Case n = 3 (Kirchho Formula): 2
(9.102)
3
1 66 ZZ (1 2 3) d + @ Z Z '(1 2 3) d77 u(x1 x2 x3 t) = 4a (9.103a) 5 24 t @t t (Sat) (Sat) where the integration is performed over the spherical surface Sat given by the equation (1 ; x1 )2 + (2 ; x2 )2 + (3 ; x3 )2 = a2 t2.
Case n = 2 (Poisson Formula):
1 ZZ q (1 2) d1d2 u(x1 x2 t) = 2a 2 t2 ; (1 ; x1 )2 ; (2 ; x2 )2 a (Cat) ZZ @ 1 d2 q 2 2 '(1 2) d (9.103b) + 2 2 @t (Cat) a t ; (1 ; x1 ) ; (2 ; x2) where the integration is performed along the circle Cat given by the equation (1 ; x1 )2 + (2 ; x2 )2 a2 t2.
Case n = 1 (d'Alembert formula): xZ1 +at u(x1 t) = '(x1 + at) +2 '(x1 ; at) + 21a () d: x1 ;at
(9.103c)
9.2 Partial Dierential Equations 537
2. Inhomogeneous Problem
In the case, when Q(x t) 6= 0, we have to add to the right-hand sides of (9.103a,b,c) the correcting terms: Caseqn = 3 (Retarded Potential): For a domain K given by r at with r = ( 1 ; x1)2 + ( 2 ; x2 )2 + ( 3 ; x3 )2, the correction term is r 1 ZZZ Q 1 2 3 t ; a d d d : (9.104a) 1 2 3 4a2 r (K ) 1 ZZZ q Q( 1 2 ) d 1d 2d Case n = 2: 2a (9.104b) 2 (t ; )2 ; ( 1 ; x1 )2 ; ( 2 ; x2 )2 a (K ) where K is a domain of 1 2 space de ned by the inequalities 0 t, ( 1 ; x1 )2 + ( 2 ; x2 )2 a2 (t ; )2.
ZZ Case n = 1: 21a Q( ) d d
(9.104c) (T ) where T is the triangle 0 t, j ; x1j ajt ; j. a denotes the wave velocity of the perturbation.
9.2.3.3 Heat Conduction and Di usion Equation for Homogeneous Media 1. Three-Dimensional Heat Conduction Equation
The propagation of heat in a homogeneous medium is described by a linear second-order partial di erential equation of parabolic type @u ; a2 %u = Q(x t) (9.105a) @t where % is the three-dimensional Laplace operator de ned in three directions of propagation x1 , x2 , x3 determined by the position vector ~r. If the heat ow has neither source nor sink, the right-hand side vanishes since Q(x t) = 0: The Cauchy problem can be posed in the following way: We want to determine a bounded solution u(x t) for t > 0, where ujt=0 = f (x). The requirement of boundedness guarantees the uniqueness of the solution. For the homogeneous di erential equation with Q(x t) = 0, we get the wave function + Z 1 +Z 1 +Z 1 u(x1 x2 x3 t) = p1 n f ( ) (2a t) ;1 ;1 ;1 1 2 3 2 2 2! exp ; (x1 ; 1) + (x24;a2t 2 ) + (x3 ; 3) d1d2d3: (9.105b) In the case of an inhomogeneous di erential equation with Q(x t) 6= 0, we have to add to the right-hand side of (9.105b) the following expression: Zt 2 +Z 1 +Z 1 +Z 1 Q(1 2 3) 4 q n 0 ;1 ;1 ;1 2a (t ; )]
2 2 2! exp ; (x1 ; 1 ) + (4xa22 (;t ;2 )) + (x3 ; 3) d1 d2d3 d : (9.105c)
538 9. Dierential Equations The problem of determining u(x t) for t < 0, if the values u(x 0) are given, cannot be solved in this way, since the Cauchy problem is not correctly formulated in this case. Since the temperature di erence is proportional to the heat, we often introduce u = T (~r t) (temperature eld) and a2 = DW (heat di usion constant or thermal conductivity) to get @T ; D %T = Q (~r t): (9.105d) W W @t
2. Three-Dimensional Di usion Equation
In analogy to the heat equation, the propagation of a concentration C in a homogeneous medium is described by the same linear partial di erential equation (9.105a) and (9.105d), where DW is replaced by the three-dimensional diusion coecient DC . The diusion equation is: @C ; D %C = Q (~r t): (9.106) C C @t We get the solutions by changing the symbols in the wave equations (9.105b) and (9.105c).
9.2.3.4 Potential Equation
The linear second-order partial di erential equation %u = ;4% (9.107a) is called the potential equation or Poisson dierential equation (see 13.5.2, p. 670), which makes the determination of the potential u(x) of a scalar eld determined by a scalar point function %(x) possible, where x has the coordinates x1 , x2 , x3 and % is the Laplace operator. The solution, the potential uM (x1 x2 x3) at the point M , is discussed in 13.5.2, p. 670. We get the Laplace dierential equation (see 13.5.1, p. 669) for the homogeneous di erential equation with % 0 %u = 0: (9.107b) The di erential equations (9.107a) and (9.107b) are of elliptic type.
9.2.3.5 Schrodinger's Equation
1. Notion of the Schrodinger Equation
1. Determination and Dependencies The solutions of the Schrodinger equation, the wave functions , describe the properties of a quantum mechanical system, i.e., the properties of the states of a particle. The Schrodinger equation is a second-order partial di erential equation with the second-order derivatives of the wave function with respect to the space coordinates and rst-order with respect to the time coordinate: h" 2 % + U (x x x t) = H^ = ; (9.108a) i h" @ 1 2 3 @t 2m 2 @ h" r: H^ 2p^m + U (~r t) p^ h"i @~ (9.108b) r i Here, % is the Laplace operator, h" = h is the reduced Planck's constant, i is the imaginary unit and 2 r is the nabla operator. The relation between the impulse p of a free particle with mass m and wave length is = h=p.
2. Remarks: a) In quantum mechanics, we assign an operator to every measurable quantity. The operator occurring in (9.108a) and (9.108b) is called the Hamilton operator H^ (\Hamiltonian") . It has the same role as the Hamilton function of classical mechanical systems (see, e.g., the example on Two-Body Problem on p. 520). It represents the total energy of the system which is divided into kinetic and potential energy. The rst term in H^ is the operator for the kinetic energy, the second one for the potential energy.
9.2 Partial Dierential Equations 539
b) The imaginary unit appears explicitly in the Schrodinger equation. Consequently, the wave func-
tions are complex functions. Both real functions occurring in (1) + i(2) are needed to calculate the observable quantities. The square j" j2 of the wave function, describing the probability dw of the particle being in an arbitrary volume element dV of the observed domain, must satisfy special further conditions. c) Besides the potential of the interaction, every special solution depends also on the initial and boundary conditions of the given problem. In general, we have a linear second-order boundary value problem, whose solutions have physical meaning only for the eigenvalues. The squares of the absolute value of meaningful solutions are everywhere unique and regular, and tend to zero at in nity. d) The microparticles also have wave and particle properties based on the wave{particle duality, so the Schrodinger equation is a wave equation (see 9.2.3.2, p. 536) for the De Broglie matter waves. e) The restriction to the non-relativistic case means that the velocity v of the particle is very small with respect to the velocity of light c (v * c). The application of the Schrodinger equations is discussed in detail in the literature of theoretical physics (see, e.g., 9.15], 9.7], 9.10], 22.14]). In this chapter we demonstrate only some most important examples.
2. Time-Dependent Schrodinger Equation
The time-dependent Schrodinger equation (9.108a) describes the general non-relativistic case of a spinless particle with mass m in a position-dependent and time-dependent potential eld U (x1 x2 x3 t). The special conditions, which must be satis ed by the wave function, are: a) The function must be bounded and continuous. b) The partial derivatives @=@x1 , @=@x2 , and @=@x3 must be continuous. c) The function jj2 must be integrable, i.e.,
ZZZ V
j(x1 x2 x3 t)j2 dV < 1:
(9.109a)
According to the normalization condition, the probability that the particle is in the considered domain must be equal to one. (9.109a) is sucient to guarantee the condition, since multiplying by a constant the value of the integral becomes one. A solution of the time-dependent Schrodinger equation has the form E (9.109b) (x1 x2 x3 t) = " (x1 x2 x3)e;i h t : The state of the particle is described by a periodic function of time with angular frequency ! = E=h" . If the energy of the particle has the xed value E = constant, then the probability dw of nding the particle in a space element dV is independent of time: d! = jj2 dV = dV: (9.109c) Then we talk about a stationary state of the particle.
3. Time-Independent Schrodinger Equation
If the potential U does not depend on time, i.e., U = U (x1 x2 x3 ), then it is the time-independent Schrodinger equation and the wave function " (x1 x2 x3 ) is sucient to describe the state. We can reduce it from the time-dependent Schrodinger equation (9.108a) with the solution (9.109b) and we get %" + 2m2 (E ; U )" = 0: (9.110a) h" In this non-relativistic case, the energy of the particle is 2 (9.110b) E = 2pm
540 9. Dierential Equations with impulse p = h . The wave functions " satisfying this di erential equation are the eigenfunctions they exist only for certain energy values E , which are given for the considered problem of the special boundary conditions. The union of the eigenvalues forms the energy spectrum of the particle. If U is a potential of nite depth and it tends to zero at in nity, then the negative eigenvalues form a discrete spectrum. If the considered domain is the entire space, then it can be required as a boundary condition that " is quadratically integrable in the entire space in the Lebesgue sense (see 12.9.3.2, p. 637 and 8.5]). If the domain is nite, e.g., a sphere or a cylinder, then we can require, e.g., " = 0 for the boundary as the rst boundary condition problem. We get the Helmholtz dierential equation in the special case of U (x) = 0: %" + " = 0 (9.111a) with the eigenvalue = 2mE : (9.111b) h" 2 " = 0 is often required here as a boundary condition. (9.111a) represents the initial mathematical equation for acoustic oscillation in a nite domain.
4. Force-Free Motion of a Particle in a Block
1. Formulation of the Problem A particle with a mass m is moving freely in a block with in-
penetrable walls of edge lengths a, b, c, therefore, it is in a potential box which is in nitely high in all three directions because of the inpenetracy of the walls. That is, the probability of the presence of the particle, and also the wave function " , vanishes outside the box. The Schrodinger equation and the boundary conditions for this problem are 8x = 0 x = a < @ 2 " + @ 2 " + @ 2 " + 2m E" = 0 (9.112a) " = 0 for : y = 0 y = b (9.112b) @x2 @y2 @z2 h" 2 z = 0 z = c.
2. Solution Separating the variables
" (x y z) = "x(x) "y (y) "z (z) (9.113a) and substituting into (9.112a) we get 2m 1 d2"x 1 d2"y 1 d2"z (9.113b) "x dx2 + "y dy2 + "z dz2 = ; h" 2 E = ;B: Every term on the left-hand side depends only on one independent variable. Their sum can be a constant ;B for arbitrary x, y, z only if every single term is a constant. In this case the partial di erential equation is reduced to three ordinary di erential equations: d2"x = ;k 2" d2"y = ;k 2" d2"z = ;k 2 " : (9.113c) x x y y z z dx2 dy2 dz2 The relation for the separation constants ;kx2 , ;ky 2 , ;kz 2 is kx2 + ky 2 + kz 2 = B (9.113d) consequently 2 E = 2h"m (kx2 + ky 2 + kz 2): (9.113e) 3. Solutions of the three equations (9.113c) are the functions "x = Ax sin kx x "y = Ay sin ky y "z = Az sin kz z (9.114a) with the constants Ax Ay Az . With these functions " satis es the boundary conditions " = 0 for x = 0, y = 0 and z = 0. sin kx a = sin ky b = sin kz c = 0 (9.114b)
9.2 Partial Dierential Equations 541
must be valid to satisfy also the relation " = 0 for x = a, y = b and z = c, i.e., the relations (9.114c) kx = na x ky = nb y kz = nc z must be satis ed, where nx, ny , and nz are integers. We get for the total energy 2 " 2 2 2 Enxny nz = 2 2h"m nax + nby + ncz (nx ny nz = 1 2 : : :): (9.114d) It follows from this formula that the changes of energy of a particle by interchange with the neighborhood is not continuous, which is possible only in quantum systems. The numbers nx, ny , and nz , belonging to the eigenvalues of the energy, are called the quantum numbers. After calculating the product of constants AxAy Az from the normalization condition ZaZbZc nxx (AxAy Az )2 sin2 a sin2 nby y sin2 ncz z dx dy dz = 1 (9.114e) 000 we get the complete eigenfunctions of the states characterized by the three quantum numbers s 8 sin nxx sin ny y sin nz z : "nxny nz = abc (9.114f) a b c The eigenfunctions vanish at the walls since one of the three sine functions is equal to zero. This is always the case outside the walls if the following relations are valid x = na n2a : : : (nx n; 1)a x x x b 2 b ( n y ; 1)b y = n n ::: (9.114g) ny y y z = nc n2c : : : (nz n; 1)c : z z z So, there are nx ; 1 and ny ; 1 and nz ; 1 planes perpendicular to the x- or y- or z-axis, in which " vanishes. These planes are called the nodal planes. 4. Special Case of a Cube, Degeneracy In the special case of a cube with a = b = c, a particle can be in di erent states which are described by di erent linearly independent eigenfunctions and they have the same energy. This is the case when the sum nx2 + ny 2 + nz 2 has the same value in di erent states. We call them degenerate states, and if there are i states with the same energy, we call it i-fold degeneracy. The quantum numbers nx, ny and nz can run through all real numbers, except zero. This last case would mean that the wave function is identically zero, i.e., the particle does not exist at any place in the box. The particle energy must remain nite, even if the temperature reaches absolute zero. This zero-point translational energy for a block is 2 E0 = 2 2h"m a12 + b12 + c12 : (9.114h) 5. Particle Movement in a Symmetric Central Field (see 13.1.2.2, p. 643) 1. Formulation of the Problem The considered particle moves in a central symmetric potential V (r). This model reproduces the movement of an electron in the electrostatic eld of a positively charged nucleus. Since this is a spherically symmetric problem, it is reasonable to use spherical coordinates (Fig. 9.20). We have the relations
542 9. Dierential Equations z J
0
r j
x
Figure 9.20
y
q r = x2 + y2 + z2 x = r sin cos ' = arccos zr y = r sin sin ' (9.115a) ' = arctan xy z = r cos where r is the absolute value of the radius vector, is the angle between the radius vector and the z-axis (polar angle) and ' is the angle between the projection of the radius vector onto the x y plane and the x-axis (azimuthal angle). For the Laplace operator we get
2 2 @" + 1 @ 2 " %" = @ "2 + 2 @" + 12 @ "2 + cos (9.115b) 2 @r r @r r @ r sin @ r2 sin2 @'2 so the time-independent Schrodinger equation is: ! ! 1 @ r2 @" + 1 @ sin @" + 1 @ 2 " + 2m E ; V (r)]" = 0: (9.115c) r2 @r @r r2 sin @ @ r2 sin2 @'2 h" 2 2. Solution We are looking for a solution in the form " (r ') = Rl (r)Ylm ( ') (9.116a) where Rl is the radial wave function depending only on r, and Ylm ( ') is the wave function depending on both angles. Substituting (9.116a) in (9.115c) we get ! 1 @ r2 @Rl Y m + 2m E ; V (r)]R Y m l l r2 @r @r l h" 2 ( ! ) 1 @ sin @Ylm R + 1 @ 2 YLm R : = ; r2 sin (9.116b) l l @ @ r2 sin2 @'2 Dividing by Rl Ylm and multiplying by r2 we get ! ( ! ) 1 d r2 dRl + 2mr2 E ; V (r)] = ; 1 1 @ sin @Ylm + 1 @ 2 Ylm : (9.116c) Rl dr dr Ylm sin @ @ sin2 @'2 h" 2 Equation (9.116c) can be satis ed if the expression on the left-hand side depending only on r and expression on the right side depending only on and ' are equal to a constant, i.e., both sides are independent of each other and they are equal to the same constant. From the partial di erential equation we get two ordinary di erential equations. If the constant is chosen equal to l(l + 1), then we get the so-called radial equation depending only on r and the potential V (r): ! "
1 d r2 dRl + 2m E ; V (r) ; l(l + 1)"h2 = 0: (9.116d) Rl r2 dr dr 2mr2 h" 2 We want to nd a solution for the angle-dependent part also in the separated form Ylm ( ') = !( )('): (9.116e) Substituting (9.116e) into (9.116c) we get ( ! ) 1 d sin d! + l(l + 1) = ; 1 d2 : sin2 (9.116f) ! sin d d d'2 If the separation constant is chosen as m2 in a reasonable way, then the so-called polar equation is ! 1 d sin d! + l(l + 1) ; m2 = 0 (9.116g) ! sin d d sin2
9.2 Partial Dierential Equations 543
and the azimuthal equation is d2 + m2 = 0: (9.116h) d'2 Both equations are potential-independent, so they are valid for every central symmetric potential. We have three requirements for (9.116a): It should tend to zero for r ! 1, it should be one-valued and quadratically integrable on the surface of the sphere. 3. Solution of the Radial Equation Beside the potential V (r) the radial equation (9.116d) also contains the separation constant l(l + 1). We substitute ul(r) = r Rl (r) (9.117a) since the square of the function ul(r) gives the last required probability jul (r)j2dr = jRl (r)j2r2dr of the presence of the particle in a spherical shell between r and r + dr. The substitution leads to the one-dimensional Schrodinger equation "
d2ul(r) + 2m E ; V (r) ; l(l + 1)"h2 u (r) = 0: (9.117b) l 2 2 2 dr 2mr h" This one contains the e ective potential Ve = V (r) + Vl (l) (9.117c) which has two parts. The rotation energy h2 Vl (l) = Vrot(l) = l(l2+mr1)" (9.117d) 2 is called the centrifugal potential. The physical meaning of l as the orbital angular momentum follows from analogy with the classical rotation energy !)2 = ~l 2 = ~l 2 Erot = 12 !~!2 = (!~ (9.117e) 2! 2! 2mr2 a rotating particle with moment of inertia ! = r2 and orbital angular momentum ~l = !~!: q ~l 2 = l(l + 1)"h2 ~l 2 = h" l(l + 1) : (9.117f) 4. Solution of the polar equation The polar equation (9.116g), containing both separation constants l(l + 1) and m2 , is a Legendre di erential equation (9.57a), p. 511. Its solution is denoted by !lm( ), and it can be determined by a power series expansion. Finite, single-valued and continuous solutions exist only for l(l + 1) = 0 2 6 12 : : : . We get for l and m: l = 0 1 2 : : : jmj l: (9.118a) So, m can take the (2l + 1) values ;l (;l + 1) (;l + 2) : : : (l ; 2) (l ; 1) l: (9.118b) We get the corresponding Legendre polynomial for m 6= 0, which is de ned in the following way: m l+m (cos2 ; 1)l 2 m=2 d Plm(cos ) = (;21) : (9.118c) l l! (1 ; cos ) (d cos )l+m We get the Legendre function of the rst kind (9.57c), p. 511, as a special case (l = n m = 0 cos = x). Its normalization results in the equation v u u m)! m m m !lm( ) = t 2l 2+ 1 ((ll ; (9.118d) + m)! Pl (cos ) = Nl Pl (cos ): 5. Solution of the Azimuthal Equation Since the motion of the particle in the potential eld V (r) is independent of the azimuthal angle even in the case of the physical assignment of a space direction,
544 9. Dierential Equations e.g., by a magnetic eld, we specify the general solution = ei m ' + e;i m ' by xing m(') = Aei m ' because in this case jm j2 is independent of '. The requirement for uniqueness is m(' + 2) = m (') so m can take on only the values 0 1 2 : : :. It follows from the normalization
Z2
that
0
jj2 d' = 1 = jAj2
Z 2 0
d' = 2jAj2
m(') = p1 ei m ' (m = 0 1 2 : : :): 2 The quantum number m is called the magnetic quantum number.
(9.119a) (9.119b) (9.119c) (9.119d)
6. Complete Solution for the Dependency of the Angles In accordance with (9.116e), the
solutions for the polar and the azimuthal equations should be multiplied by each other: Ylm ( ') = !( ) (') = p1 Nlm Plm(cos )eim' : (9.120a) 2 The functions Ylm( ') are the so-called surface spherical harmonics. When the radius vector ~r is reected with respect to the origin (~r ! ;~r), the angle becomes ; and ' becomes ' + , so the sign of Ylm may change: Ylm ( ; ' + ) = (;1)l Ylm ( '): (9.120b) We get the parity of the considered wave function P = (;1)l : (9.121a) 7. Parity The parity property serves the characterization of the behavior of the wave function under space inversion ~r ! ;~r (see 4.3.5.1, 1., p. 269). It is performed by the inversion or parity operator P: P" (~r t) = " (;~r t). If we denote the eigenvalue of the operator by P , then applying P twice it must yield P P " (~r t) = P P " (~r t) = " (~r t), the original wave function. So: P 2 = 1 P = 1: (9.121b) We call it an even wave function if its sign does not change under space inversion, and it is called an odd wave function if its sign changes.
6. Linear Harmonic Oscillator
1. Posing the Problem Harmonic oscillation occurs when the drag forces in the oscillator satisfy Hooke's law F = ;kx. For the frequency of the oscillation, for the frequency of the oscillation circuit and for the potential energy we get: s s 2 1 k Epot = 12 kx2 = m !2 x2 : (9.122c)
= 2 m (9.122a) ! = mk (9.122b) Substituting into (9.111a), the Schrodinger equation becomes
" d2" + 2m E ; !2 mx2 " = 0: (9.123a) dx2 h" 2 2 With the substitutions r y = x m! (9.123b) = 2h"E! (9.123c) h"
9.2 Partial Dierential Equations 545
where is a parameter and not the wavelength, (9.123a) can be transformed into the simpler form of the Weber di erential equation d 2 " + ( ; y 2 )" = 0 : (9.123d) dy2 2. Solution We get a solution for the Weber di erential equation in the form " (y) = e;y2=2 H (y): (9.124a) Di erentiation shows that "
d2" = e;y2 =2 d2H ; 2y dH + (y2 ; 1)H : (9.124b) dy2 dy2 dy Substitution into the Schrodinger equation (9.123d) yields d2H ; 2y dH + ( ; 1)H = 0: (9.124c) dy2 dy We determine a solution in the form of a series 1 1 1 2 X X X H = aiyi with dH = iai yi;1 d H2 = i(i ; 1)aiyi;2: (9.125a) dy i=1 dy i=2 i=0 Substitution of (9.125a) into (9.124c) results in 1 X
i=2
i(i ; 1)aiyi;2 ;
1 X
i=1
2iai yi +
1 X
i=0
i( ; 1)aiyi = 0:
(9.125b)
Comparing the coecients of yj we get the recursion formula (j + 2)(j + 1)aj+2 = 2j ; ( ; 1)]aj (j = 0 1 2 : : :): (9.125c) We get the coecients aj for even powers of y from a0 , the coecients for odd powers from a1. So, a0 and a1 can be chosen arbitrarily. 3. Physical Solutions We want to determine the probability of the presence of a particle in the di erent states. This will be described by a quadratically integrable wave function " (x) and by an eigenfunction which has physical meaning, i.e., normalizable and for large values of y it tends to zero. The exponential function exp(;y2=2) in (9.124a) guarantees that the solution " (y) tends to zero for y ! 1 if the function H (y) is a polynomial. To get a polynomial, the coecients aj in (9.125a), starting from a certain n, must vanish for every j > n: an 6= 0, an+1 = an+2 = an+3 = : : : = 0. The recursion formula (9.125c) with j = n is ( ; 1) a : an+2 = (2nn+;2)( (9.126a) n + 1) n an+2 = 0 can be satis ed for an 6= 0 only if 2n ; ( ; 1) = 0 = 2E = 2n + 1: (9.126b) h" ! The coecients an+2, an+4 : : : vanish for this choice of . Also an;1 = 0 must hold to make the coecients an+1 , an+3 : : : equal to zero. We get the Hermite polynomials from the second de ning equation (see 9.1.2.6, 6., p. 514) for the special choice of an = 2n, an;1 = 0. The rst six of them are: H0 (y) = 1 H3(y) = ;12y + 8y3 H1 (y) = 2y H4(y) = 12 ; 48y2 + 16y4 (9.126c) 2 H5(y) = 120y ; 160y3 + 32y5: H2 (y) = ;2 + 4y
546 9. Dierential Equations The solution " (y) for the vibration quantum number n is "n = Nne;y2 =2Hn(y) (9.127a) Z 2 where Nn is the normalizing factor. We get it from the normalization condition "n dy = 1 as r r p Nn2 = 2n1n! with = xy = m! (9.127b) h" (see (9.123b), p. 544): From the terminating condition of the series (9.123c) we get En = h" ! n + 21 (n = 0 1 2 : : :) (9.127c) for the eigenvalues of the vibration energy. The spectrum of the energy levels is equidistant. The summand +1=2 in the parentheses means that in contrast to the V(x) E(x) classical case the quantum mechanical oscillator has energy even in the deepest energetic level with n = 0, which is known Y(x) as the zero-point vibration energy. 5 Fig. 9.21 shows a graphical representation of the equidistant spectra of the energy states, the corresponding wave functions 4 from "0 to "5 and also the function of the potential energy 3 (9.122c). The points of the parabola of the potential energy represent the reversal points of the classical oscillator, which 2 are calculated from the energy E = 1 m!2 a2 as the amplitude 1 2 s hn 0 a = !1 2mE . The quantum mechanical probability of nding a x particle in the interval (x x + dx) is given by dwqu = j" (x)j2 dx. It is di erent from zero also outside of these points. Figure 9.21
s
So we get for, e.g., n = 1, hence for E = (3=2)"h!, according to dwqu = 2 e; x2 dx, the maximum of the probability of presence at s 1 h" : p xmaxqu = = m! (9.127d) For a corresponding classical oscillator, this is s s 2 E 3"h : xmaxkl = a = m!2 = m! (9.127e) The quantum mechanical probability density function approaches the classical one for large quantum number n in its mean value.
9.2.4 Non-LinearPartialDierentialEquations: Solitons,Periodic Patterns, and Chaos 9.2.4.1 Formulation of the Physical-Mathematical Problem 1. Notion of Solitons
Solitons, also called solitary waves, from the viewpoint of physics, are pulses, or also localized disturbances of a non-linear medium or eld the energy related to such propagating pulses or disturbances is concentrated in a narrow spatial region. They occur: in solids, e.g., in anharmonic lattices, in Josephson contacts, in glass bres and in quasi-one-dimensional
9.2 Partial Dierential Equations 547
conductors, in uids as surface waves or spin waves, in plasmas as Langmuir solitons, in linear molecules, in classical and quantum eld theory. Solitons have both particle and wave properties they are localized during their evolution, and the domain of the localization, or the point around which the wave is localized, travels as a free particle in particular it can also be at rest. A soliton has a permanent wave structure: based on a balance between nonlinearity and dispersion, the form of this structure does not change. Mathematically, solitons are special solutions of certain non-linear partial di erential equations occurring in physics, engineering and applied mathematics. Their special features are the absence of any dissipation and also that the non-linear terms cannot be handled by perturbation theory. Important examples of equations with soliton solutions are: a) Korteweg de Vries (KdV) Equation ut + 6uux + uxxx = 0 (9.128) 2 b) Non-Linear Schrodinger (NLS) Equation i ut + uxx 2juj u = 0 (9.129) c) Sine{Gordon (SG) Equation utt ; uxx + sin u = 0: (9.130) The subscripts x and t denote partial derivatives, e.g., uxx = @ 2 u=@x2 . We consider the one-dimensional case in these equations, i.e., u has the form u = u(x t), where x is the spatial coordinate and t is the time. The equations are given in a scaled form, i.e., the two independent variables x and t are here dimensionsless quantities. In practical applications, they must be multiplied by quantities having the corresponding dimensions and being characteristic of the given problem. The same holds for the velocity.
2. Interaction between Solitons
If two solitons, moving with di erent velocities, collide, they appear again after the interaction as if they had not collided. Every soliton asymptotically keeps its form and velocity there is only a phase shift. Two solitons can interact without disturbing each other asymptotically. This is called an elastic interaction which is equivalent to the existence of an N -soliton solution, where N (N = 1 2 3 : : :) is the number of solitons. Solving an initial value problem with a given initial pulse u(x 0) that disaggregatesRinto solitons, the number of solitons does not depend on the shape of the pulse but on its total +1 u(x 0) dx. amount ;1
3. Non-Linear Phenomena in Dissipative Systems
In dissipative systems (hence friction or damped systems), periodic patterns and non-linear waves can appear through the impact of external forces. One striking example is a uid (gas or liquid), where the combined action of gravitation and a tmpereature gradient causes (with increasing temperature di erence) a transition from a purely heat conducting state (without convection) to a very special, namely regular cell convection state nally up to turbulence. Depending on the magnitude of the temparature di erence bifurcation and chaos can appear (see 17.3, p. 829). Important examples for equations of such phenomena are: a) Ginsburg{Landau (GGL) Equation ut ; u ; (1 + ib)uxx + (1 + ic)juj2u = 0 (9.131) 2 b) Kuramoto{Sivashinsky (KS) Equation ut + uxx + uxxxx + ux = 0: (9.132) In contrast to the dissipationless KdV, NLS, SG, equations, the equations (9.131) and (9.132) are non-linear dissipative equations, which have, besides spatio-temporal periodic solutions, also spatiotemporal disordered (chaotic) solutions.
4. Non-Linear Evolution Equation
An evolution equation describes the evolution of a physical quantity in time. Examples for such evolution equations are the wave equation (see 9.2.3.2, p. 536), the heat equation (see 9.2.3.3, p. 537) and the Schrodinger equation (see 9.2.3.5, 1., p. 538). The solutions of the evolution equations are called evolution functions.
548 9. Dierential Equations In contrast to linear evolution equations, the non-linear evolution equations (9.128), (9.129), and (9.130) contain non-linear terms u@u=@x, juj2u and sin u. These equations are (with the exception of (9.131)) parameter-free. From the viewpoint of physics non-linear evolution equations describe structure formations like solitons (dispersive structures) as well as periodic patterns and non-linear waves (dissipative structures).
9.2.4.2 Korteweg de Vries Equation (KdV) 1. Occurrance
The KdV equation is used in the discussion of surface waves in shallow water, anharmonic vibrations in non-linear lattices, problems of plasma physics and non-linear electric networks.
u(x,t) 0,5 0,4 0,3
2. Equation and Solutions
0,2
The KdV equation for the evolution function u is 0,1 ut + 6uux + uxxx = 0: (9.133) It has the soliton solution 6 -6 -4 -2 0 4 2 x-vt-ϕ v h i u(x t) = : (9.134) p 2 cosh2 12 v(x ; vt ; ') Figure 9.22 This KdV soliton is uniquely de ned by the two dimensionless parameters v (v > 0) and '. In Fig. 9.22 v = 1 is chosen. A typical non-linear e ect is that the velocity of the soliton v determines the amplitude and the width of the soliton: KdV solitons with larger amplitude and smaller width move quicker than those with smaller amplitude and larger width (taller waves travel faster than shorter ones). The soliton phase ' describes the position of the maximum of the soliton at time t = 0. Equation (9.133) also has N -soliton solutions. Such an N -soliton solution can be represented asymptotically for t ! 1 by the linear superposition of one-soliton solutions:
u(x t) !
N X
n=1
un(x t):
(9.135)
Here every evolution function un(x t) is characterized by a velocity vn and a phase 'n . The initial phases ';n before the interaction or collision di er from the nal phases after the collision '+n , while the velocities v1 v2 : : : vN have no changes, i.e., it is an elastic interaction. For N = 2, (9.133) has a two-soliton solution. It cannot be represented for a nite time by a linear p superposition, and with kn = 1 vn and n = 1 pvn(x ; vn t ; 'n) (n = 1 2) it has the form: 2 2 " 2 2 2 ( + 1 2 1 k1 e + k2 e + (k1 ; k2) e 2 ) 2 + (k +1 k )2 k12e 1 + k22e 2 1 2 u(x t) = 8 : (9.136) 2 32 !2 41 + e 1 + e 2 + k1 ; k2 e( 1 + 2) 5 k +k 1
2
Equation (9.136) describes for t ! ;1 asymptotically two non-interacting solitons with velocities v1 = 4k12 and v2 = 4k22, which transform after their mutual interaction again into two non-interacting solitons with the same velocities for t ! +1 asymptotically. The non-linear evolution equation wt + 6(wx)2 + wxxx = 0 (9.137a) F where w = Fx has the following properties:
9.2 Partial Dierential Equations 549
a) For F (x t) = 1 + exp() = 12 pv(x ; vt ; ') it has a soliton solution and
!
2 b) for F (x t) = 1 + exp(1) + exp(2 ) + kk1 ;+ kk2 exp(1 + 2) 1 2
(9.137b) (9.137c)
it has a two-soliton solution. With 2wx = u the KdV equation (9.133) follows from (9.137a). Equation (9.136) and the expression w following from (9.137c) are examples of a non-linear superposition. If the term +6uux is replaced by ;6uux in (9.133), then the right-hand side of (9.134) has to be multiplied by (;1). In this case the notation antisoliton is used.
9.2.4.3 Non-Linear Schrodinger Equation (NLS) 1. Occurrence
The NLS equation occurs in non-linear optics, where the refractive index n depends on the electric eld strength E~ , as, e.g., for ~ ) = n0 + n2 jE~ j2 with n0 n2 = constant holds, and the Kerr e ect, where n(E in the hydrodynamics of self-gravitating discs which allow us to describe galactic spiral arms.
2. Equation and Solution
The NLS equation for the evolution function u and its solution are: (i 2 x + 4( 2 ; 2)t ; ]) : (9.139) u(x t) = 2 exp cosh i ut + uxx 2juj2u = 0 (9.138) 2(x + 4 t ; ')] Here u(x t) is complex. The NLS soliton is characRe u(x,t) terized by the four dimensionless parameters ', 1 and . The envelope of the wave packet moves with the velocity ;4 the phase velocity of the wave packet is 2(2 ; 2)= . In contrast to the KdV soliton (9.134), the amplitude and the velocity can be chosen independently of each other. -2 -4 6 2 4 In the case of N interacting solitons, we can char- -6 x-vt-ϕ acterize them by 4N arbitrary chosen parameters: n n 'n n (n = 1 2 : : : N ) . If the solitons have di erent velocities, the N -soliton solution splits asymptotically for t ! 1 into a sum of N -1 individual solitons of the form (9.139). Fig. 9.23 displays the real part of (9.139) with v = ;4 = 1=2 and = 2=5. Figure 9.23
9.2.4.4 Sine{Gordon Equation (SG) 1. Occurrence
The SG equation is obtained from the Bloch equation for spatially inhomogeneous quantum mechanical two-level systems. It describes the propagation of ultra-short pulses in resonant laser media (self-induced transparency), the magnetic ux in large surface Josephson contacts, i.e., in tunnel contacts between two superconductors and spin waves in superuid helium 3 (3 He). The soliton solution of the SG equation can be illustrated by a mechanical model of pendula and springs. The evolution function goes continuously from 0 to a constant value c. The SG solitons are often called
550 9. Dierential Equations
2. Equation and Solution
The SG equation for the evolution function u is utt ; uxx + sin u = 0: (9.140) It has the following soliton solutions:
1. Kink Soliton
6
u(x t) = 4 arctan exp (x ; x0 ; vt) (9.141) where = p 1 2 and ;1 < v < +1. 1;v The kink soliton (9.141) for v = 1=2 is given in Fig. 9.24. The kink soliton is determined by two dimensionless parameters v and x0 . The velocity is independent of the amplitude. The time and the position derivatives are ordinary localized solitons: ; uvt = ux = cosh (x2; x ; vt) : (9.142)
u(x,t)
2π
4 π 2
-10
-5
0 x-x0-vt
5
10
Figure 9.24 kink solitons. If the evolution function changes from the constant value c to 0, it describes a so-called antikink soliton. Walls of domain structures can be described with this type of solutions. 0
2. Antikink Soliton
u(x t) = 4 arctan exp ; (x ; x0 ; vt) : (9.143) 3. Kink-Antikink Soliton We get a static kink-antikink soliton from (9.141, 9.143) with v = 0: u(x t) = 4 arctan exp (x ; x0 ) : (9.144) Further solutions of (9.140) are: 4. Kink-Kink Collision "
sinh x : u(x t) = 4 arctan v cosh (9.145) vt
5. Kink-Antikink Collision "
u(x t) = 4 arctan 1 sinh vt : v cosh x
6. Double or Breather Soliton, also called Kink-Antikink Doublet "p
2 sin p !t 2 : u(x t) = 4 arctan 1 !; ! cosh 1 ; ! x
(9.146) (9.147)
Equation (9.147) represents a stationary wave, whose envelope is modulated by the frequency !.
7. Local Periodic Kink " Lattice ! u(x t) = 2 arcsin sn px ; vt 2 k + : k 1;v The relation between the wavelength and the lattice constant k is p = 4K (k)k 1 ; v2 : For k = 1, i.e., for ! 1, we get u(x t) = 4 arctan exp (x ; vt)
which is the kink soliton (9.141) and the antikink soliton (9.143) again, with x0 = 0.
(9.148a) (9.148b) (9.148c)
9.2 Partial Dierential Equations 551
Remark: sn x is a Jacobian elliptic function with parameter k and quarter-period K (see 14.6.2, p. 703): snx = sin '(x k) sinZ'(xk)
(9.149a)
Z=2 dq p d-2 2 : (9.149c) (9.149b) K ( k ) = 1 ; k sin (1 ; q2)(1 ; k2 q2) 0 0 Equation (9.149b) comes from (14.102a), p. 703, by the substitution of sin = q. The series expansion of the complete elliptic integral is given as equation (8.104), p. 462. x=
q
9.2.4.5 Further Non-linear Evolution Equations with Soliton Solutions 1. Modied KdV Equation 2
ut 6u ux + uxxx = 0: The even more general equation ut + upux + uxxx = 0 has the soliton 2 3 1p 1 jvj(p + 1)(p + 2) 66 2 q 775 u(x t) = 4 1 2 cosh 2 p jvj(x ; vt ; ') as its solution.
(9.150) (9.151) (9.152)
2. Sinh{Gordon Equation utt ; uxx + sinh u = 0:
(9.153)
3. Boussinesq Equation 2
uxx ; utt + (u )xx + uxxxx = 0: (9.154) This equation occurs in the description of non-linear electric networks as a continuous approximation of the charge{voltage relation.
4. Hirota Equation 2
ut + i3juj ux + uxx + iuxxx + juj2u = 0
5. Burgers Equation
= :
(9.155)
ut ; uxx + uux = 0: (9.156) This equation occurs when modeling turbulence. With the Hopf{Cole transformation it is transformed into the di usion equation, i.e., into a linear di erential equation.
6. Kadomzev{Pedviashwili Equation
The equation (ut + 6uux + uxxx)x = uyy (9.157a) has the soliton @ 2 ln 1 + x + iky ; 3k2 t2 u(x y t) = 2 @x (9.157b) 2 k2 as its solution. The equation (9.157a) is an example of a soliton equation with a higher number of independent variables, e.g., with two spatial variables.
552 10. Calculus of Variations
10 CalculusofVariations 10.1 Dening the Problem
1. Extremum of an Integral Expression
A very important problem of the di erential calculus is to determine for which x values the given function y(x) has extreme values. The calculus of variations discusses the following problem: For which functions has a certain integral, whose integrand depends also on the unknown function and its derivatives, an extremum value? The calculus of variations concerns itself with determining all the functions y(x) for which the integral expression
Zb I y] = F (x y(x) y0(x) : : : y(n)(x))dx a
(10.1)
has an extremum, if the functions y(x) are from a previously given class of functions. Here, we may de ne some boundary and side conditions for y(x) and for its derivatives.
2. Integral Expressions of Variational Calculus
There can also be several variables instead of x in (10.1). In this case, the occurring derivatives are partial derivatives and the integral in (10.1) is a multiple integral. In the calculus of variations, the following types of integral expressions are discussed:
Zb I y] = F (x y(x) y0(x)) dx
(10.2)
Zb I y1 y2 : : : yn] = F (x y1(x) : : : yn(x) y10 (x) : : : yn0 (x)) dx
(10.3)
a
a
Zb I y] = F (x y(x) y0(x) : : : y(n)(x)) dx a ZZ I u] = F (x y u ux uy ) dx dy:
Here the unknown function is u = u(x y), and ) represents a plane domain of integration. ZZZ I u] = F (x y z u ux uy uz ) dx dy dz: R
(10.4) (10.5) (10.6)
The unknown function is u = u(x y z), and R represents a space region of integration. Additionally, boundary values can be given for the solution of a variational problem, at the endpoints of the interval a and b in the one-dimensional case, and at the boundary of the domain of integration ) in the twodimensional case. Besides, various further side conditions can be de ned, e.g., in integral form or as a di erential equation. A variational problem is called rst-order or higher-order depending whether the integrand F contains only the rst derivative y0 or higher derivatives y(n) (n > 1) of the function y.
3. Parametric Representation of the Variational Problem
A variational problem can also be posed in parametric form. If we consider a curve in parametric form x = x(t) y = y(t) ( t ), then, e.g., the integral expression (10.2) has the form
Z I x y] = F (x(t) y(t) x_ (t) y_ (t)) dt:
(10.7)
10.2 Historical Problems 553
10.2 Historical Problems
10.2.1 Isoperimetric Problem
The general isoperimetric problem is to determine the plane region with the largest area among the plane regions with a given perimeter. The solution of this problem, a circle with a given perimeter, originates from queen Dido, who was allowed, as legend has it, to take such an area for the foundation of Carthago which she could be surround by one bull's leather. She cut the leather into ne stripes, and formed a circle with them. A special case of the isoperimetric problem is to nd the equation y of the curve y = y(x) in a Cartesian coordinate system connecty(x) ing the points A(a 0) and B (b 0) and having the given length l, for which the area determined by the line segment AB and the curve 0 A(a, 0) B(b, 0) x is the largest possible (Fig. 10.1). The mathematical formalization is: We have to determine a one-time continuously di erentiable Figure 10.1 function y(x) such that
Zb I y] = y(x) dx = max
(10.8a)
a
holds, where the side condition (10.8b) and the boundary conditions (10.8c) are satis ed:
G y] =
Zb q a
1 + y02(x) dx = l
(10.8b)
y(a) = y(b) = 0
(10.8c)
10.2.2 Brachistochrone Problem
The brachistochrone problem was formulated in 1696 by J. Bernoulli, and it is the following: A point mass descends from the point P0(x0 y0) to the origin in the vertical plane x y only under the inuence of gravity. We should determine the curve y = y(x) along which the point reaches the origin in the shortest possible time from P0 (Fig. 10.2). Considering the formula for the time of fall, T , we get the mathematical description: We have to determine a one-time continuously di erentiable function y = y(x), for which y Zx0 p 02 y0 P0(x0, y0) T y] = q 1 + y dx = min (10.9) 2g(y0 ; y) 0 (g is the acceleration due to gravity) and the boundary value condiy(x) tions are x 0 x0 y(0) = 0 y(x0) = y0: (10.10) We see that there is a singularity for x = x0 in (10.9). Figure 10.2
10.3 Variational Problems of One Variable
10.3.1 Simple Variational Problems and Extremal Curves
A simple variational problem is to determine the extreme value of the integral expression given in the form
Zb I y] = F (x y(x) y0(x)) dx a
(10.11)
where y(x) is a twice continuously di erentiable function satisfying the boundary conditions y(a) = A and y(b) = B . The values a b and A B , and the function F are given.
554 10. Calculus of Variations The integral expression (10.11) is an example of a so-called functional. A functional assigns a real number to every function y(x) from a certain class of functions. If the functional I y] in (10.11) takes, e.g., its relative maximum for a function y0(x), then I y0 ] I y ] (10.12) for every twice continuously di erentiable function y satisfying the boundary conditions. The curve y = y0(x) is called an extremal curve. Sometimes all the solutions of the Euler di erential equation of the variational calculus are called extremal curves.
10.3.2 Euler Dierential Equation of the Variational Calculus
We get a necessary condition for the solution of the variational problem in the following way: We construct an auxiliary curve or comparable curve for the extremal y0(x) characterized by (10.12) y(x) = y0(x) + (x) (10.13) with a twice continuously di erentiable function (x) satisfying the special boundary conditions (a) = (b) = 0. is a real parameter. Substituting (10.13) in (10.11) we get a function depending on instead of the functional I y]
Zb I ( ) = F (x y0 + y00 + 0) dx a
(10.14)
and the functional I y] has an extreme value for y0(x) if the function I ( ), as a function of , has an extreme value for = 0. Now, we deduce the variational problem to an extreme value problem with the necessary condition dI = 0 for = 0: (10.15) d Supposing that the function F , as a function of three independent variables, is di erentiable as many times as needed, by its Taylor expansion we get (see 7.3.3.3, p. 417)
Zb " @F 0 0 0 2 I ( ) = F (x y0 y00 ) + @F ( x y (10.16) 0 y0 ) + 0 (x y0 y0 ) + O ( ) dx: @y @y a The necessary condition (10.15) results in the equation Zb @F Zb @y dx + 0 @F dx = 0: (10.17) 0 a a @y By partial integration of this equation and considering the boundary conditions for (x), we get Zb @F d @F !! @y ; dx @y0 dx = 0: (10.18) a From the assumption of continuity and because the integral in (10.18) must disappear for any considerable (x), ! @F ; d @F = 0 (10.19) @y dx @y0 must hold. The equation (10.19) gives a necessary condition for the simple variational problem and it is called the Euler dierential equation of the calculus of variations. The di erential equation (10.19) can be written in the form @F ; @ 2 F ; @ 2 F y0 ; @ 2 F y00 = 0: (10.20) @y @x@y0 @y@y0 @y02 It is an ordinary second-order di erential equation if Fy0 y0 6= 0 holds.
10.3 Variational Problems of One Variable 555
The Euler di erential equation has a simpler form in the following special cases: Case 1: F (x y y0) = F (y0), i.e., x and y do not appear explicitly. Then instead of (10.19) we get ! @F = 0 d @F = 0: (10.21a) and (10.21b) @y dx @y0 Case 2: F (x y y0) = F (y y0), i.e., x does not appear explicitly. We consider ! ! !! d F ; y0 @F = @F y0 + @F y00 ; y00 @F ; y0 d @F = y0 @F ; d @F (10.22a) dx @y0 @y @y0 @y0 dx @y0 @y dx @y0 and because of (10.19), we get ! d F ; y0 @F = 0 (10.22b) i.e., F ; y0 @F (10.22c) dx @y0 @y0 = c (c const) as a necessary condition for the solution of the simple variational problem in the case F = F (y y0). A: The functional to determine the shortest curve connecting the points P1(a A) and P2 (b B ) in the x y plane is:
Z bq I y] = 1 + y02 dx = min: (10.23a) a p It follows from (10.21b) for F = F (y0) = 1 + y02 that ! d @F = y00 (10.23b) 3 = 0 p 0 dx @y 1 + y 02 so y00 = 0, i.e., the shortest curve is the straight line. B: We connect the points P1(a A) and P2(b B ) by a curve y(x), and we rotate it around the x-axis. Then the surface area is Zb q I y] = 2 y 1 + y02 dx: (10.24a) a For which curve y(x) will the surface area be the smallest? It follows from (10.22c) with F = F (y y0) = 2 q p 2y 1 + y02 that y = 2c 1 + y02 or y02 = cy ; 1 with c1 = 2c . This di erential equation is 1 separable (see 9.2.2.3, 1., p. 524), and its solution is y = c1 cosh cx + c2 (c1 c2 const) (10.24b) 1 the equation of the so-called catenary curve (see 2.15.1, p. 105). We determine the constants c1 and c2 from the boundary values y(a) = A and y(x) = B . We have to solve a non-linear equation system (see 19.2.2, p. 896), which cannot be solved for every boundary value.
10.3.3 Variational Problems with Side Conditions
These problems are usually isoperimetric problems (see 10.2.1, p. 553): The simple variational problem (see 10.2.1, p. 553), given by the functional (10.11), is completed by a further side condition in the form
Zb a
G(x y(x) y0(x)) dx = l (l const)
(10.25)
where the constant l and the function G are given. A method to solve this problem is given by Lagrange (extreme values with side conditions in equation form, see 6.2.5.6, p. 403). We consider the expression (10.26) H (x y(x) y0(x) ) = F (x y(x) y0(x)) + (G(x y(x) y0(x)) ; l)
556 10. Calculus of Variations where is a parameter, and we consider the problem
Zb a
H (x y(x) y0(x) ) = extreme!
(10.27)
i.e., an extreme value problem without side condition. The corresponding Euler di erential equation is: ! @H ; d @H = 0: (10.28) @y dx @y0 The solution y = y(x ) depends on the parameter , which can be determined by substituting y(x ) into the side condition (10.25). For the isoperimetric problem 10.2.1, p. 553, we get q H (x y(x) y0(x) ) = y + 1 + y02 : (10.29a) Because the variable x does not appear in H , we get instead of the Euler di erential equation (10.28), analogously to (10.22c), the di erential equation q 02 q 2 ; (c1 ; y)2 y 0 2 y + 1 + y02 ; q = c1 or y = (c1 const) (10.29b) c1 ; y 1 + y 02 whose solution is the family of circles (x ; c2)2 + (y ; c1)2 = 2 (c1 c2 const): (10.29c) The values c1 c2 and are determined from the conditions y(a) = 0 y(b) = 0 and from the requirement that the arclength between A and B should be l. We get a non-linear equation for , which should be solved by an appropriate iterative method.
10.3.4 Variational Problems with Higher-Order Derivatives We consider two types of problems.
1. F = F (x y y0 y00)
The variational problem is:
Zb I y] = F (x y y0 y00) dx = extreme! a
(10.30a)
with the boundary values y(a) = A y(b) = B y0(a) = A0 y0(b) = B 0 (10.30b) where the numbers a b A B A0, and B 0, and the function F are given. Similarly as in 10.3.2, p. 554, we introduce comparable curves y(x) = y0(x) + (x) with (a) = (b) = 0(a) = 0(b) = 0, and we get the Euler dierential equation ! ! @F ; d @F + d2 @F = 0 (10.31) 0 2 00 @y dx @y dx @y as a necessary condition for the solution of the variational problem (10.30a). The di erential equation (10.31) represents a fourth-order di erential equation. Its general solution contains four arbitrary constants which can be determined by the boundary values (10.30b). Consider the problem
I y] =
Z1 0
(y002 ; y02 ; y2) dx = extreme!
(10.32a)
10.3 Variational Problems of One Variable 557
with the given constants and for F = F (y y0 y00) = y002 ; y02 ; y2. Then: d (F 0 ) = ;2y00, d2 (F 00 ) = ;2y(4) , and the Euler di erential Fy = ;2y, Fy0 = ;2y0, Fy00 = 2y00, dx y dx2 y equation is y(4) + y00 ; y = 0: (10.32b) This is a fourth-order linear di erential equation with constant coecients (see 9.1.2.3, p. 500).
2. F = F (x y y0 : : : y(n))
In this general case, when the functional I y] of the variational problem depends on the derivatives of the unknown function y up to order n (n 1), the corresponding Euler di erential equation is ! ! ! @F ; d @F + d2 @F ; + (;1)n dn @F = 0 (10.33) @y dx @y0 dx2 @y00 dxn @y(n) whose solution should satisfy the boundary conditions analogously to (10.30b) up to order n ; 1.
10.3.5 Variational Problem with Several Unknown Functions Suppose the functional of the variational problem has the form
Zb I y1 y2 : : : yn] = F (x y1 y2 : : : yn y10 y20 : : : yn0 ) dx a
(10.34)
where the unknown functions y1(x) y2(x) : : : yn(x) should take given values at x = a and x = b. We consider n twice continuously di erentiable comparable functions yi(x) = yi0(x) + ii(x) (i = 1 2 : : : n) (10.35) where the functions i(x) should disappear at the endpoints. (10.34) becomes I ( 1 2 : : : n) with (10.35), and from the necessary conditions @I (10.36) @ i = 0 (i = 1 2 : : : n) for the extreme values of a function of several variables, we get the n Euler di erential equations ! ! ! @F ; d @F = 0 @F ; d @F = 0 : : : @F ; d @F = 0 (10.37) @y1 dx @y10 @y2 dx @y20 @yn dx @yn0 whose solutions y1(x) y2(x) : : : yn(x) must satisfy the given boundary conditions.
10.3.6 Variational Problems using Parametric Representation
For some variational problems it is useful to determine the extremal, not in the explicit form y = y(x), but in the parametric form x = x(t) y = y(t) (t1 t t2 ) (10.38) where t1 and t2 are the parameter values corresponding to the points (a A) and (b B ). Then the simple variational problem (see 10.3.1, p. 553) is
Zt2 I x y] = F (x(t) y(t) x_ (t) y_ (t))dt = extreme! t1
(10.39a)
with the boundary conditions x(t1 ) = a x(t2 ) = b y(t1) = A y(t2) = B: (10.39b) Here x_ and y_ denote the derivatives of x and y with respect to the parameter t, as usual in the parametric representation. The variational problem (10.39a) makes sense only if the value of the integral is independent of the
558 10. Calculus of Variations parametric representation of the extremal curve. To ensure the integral in (10.39a) is independent of the parametric representation of the curve connecting the points (a A) and (b B ), F must be a positive homogeneous function of the degree 1 of homogeneity (2.18.2.5, 4., p. 120), i.e., F (x y x_ y_ ) = F (x y x_ y_ ) ( > 0) (10.40) must hold. Because the variational problem (10.39a) can be considered as a special case of (10.34), the corresponding Euler di erential equations are ! ! @F ; d @F = 0 @F ; d @F = 0: (10.41) @x dt @ x_ @y dt @ y_ They are not independent of each other, but they are equivalent to the so-called Weierstrass form of the Euler di erential equation: @ 2 F @ 2 F + M (x_ y ; xy_ ) = 0 (10.42a) @x@ y_ ; @ x@y _ with 2 2F 1 @2F : withM = y_12 @@ x_F2 = ; x_1y_ @@x@ = (10.42b) _ y_ x_ 2 @ y_ 2 Starting with the calculation of the radius of curvature R of a curve given in parametric representation (see 3.6.1.1, 1., p. 225), we calculate the radius of curvature of the extremal curve considering (10.42a) with 2 2 3=2 (x_ + y_ ) M (x_ 2 + y_ 2)3=2 R = x_ y ; xy_ = F ; F : (10.42c) xy_ xy_ The isoperimetric problem (10.8a to 10.8c) (see 10.2.1, p. 553) has the form in parametric representation: Z t2 Z t2 q x_ 2 (t) + y_ 2(t) dt = l: (10.43b) I x y] = y(t)x_ (t)dt = max! (10.43a) with G x y] = t1
t1
This variational problem with the side condition becomes a variational problem without the side condition according to (10.26) with q H = H (x y x_ y_ ) = yx_ + x_ 2 + y_ 2: (10.43c) We see that H satis es the condition (10.40), so it is a positive homogeneous function of rst degree. Furthermore, we have M = y_12 Hx_ x_ = 2 2 3=2 Hxy_ = 1 Hxy_ = 0 (10.43d) (x_ + y_ ) so (10.42c) yields that the radius of curvature is R = jj. Since is a constant, the extremals are circles.
10.4 Variational Problems with Functions of Several Variables 10.4.1 Simple Variational Problem
One of the simplest problems with a function of several variables is the following variational problem for a double integral: ZZ I u] = F (x y u(x y) ux uy ) dx dy = extreme! (10.44) (G)
10.4 Variational Problems with Functions of Several Variables 559
Here, the unknown function u = u(x y) should take given values on the boundary ; of the domain G. Analogously to 10.3.2, p. 554, we introduce the comparable function in the form u(x y) = u0(x y) + (x y) (10.45) where u0(x y) is a solution of the variational problem (10.44) and it takes the given boundary values, while (x y) satis es the condition (x y) = 0 on the boundary ; (10.46) and together with u0(x y), they are di erentiable as many times as needed. The quantity is a parameter. We determine a surface by u = u(x y) which is close to the solution surface u0(x y). I u] becomes I ( ) with (10.45), i.e., the variational problem (10.44) becomes an extreme value problem which must satisfy the necessary conditions dI = 0 for = 0: (10.47) d We get from this the Euler dierential equation ! ! @F ; @ @F ; @ @F = 0 (10.48) @u @x @ux @y @uy as a necessary condition for the solution of the variational problem (10.44). A free membrane, xed at the perimeter ; of a domain G of the x y plane, covers a surface with area ZZ I1 = dx dy: (10.49a) (G)
If the membrane is deformed by a load so that every point has an elongation u = u(x y) in the zdirection, then its area is calculated by the formula
I2 =
ZZ q
(G)
1 + u2x + u2y dx dy:
(10.49b)
If we linearize the integrand in (10.49b) using Taylor series (see 6.2.2.3, p. 396), then we get the relation ZZ I2 I1 + 12 u2x + u2y dx dy: (10.49c) (G) We have ZZ U = (I2 ; I1) = 2 u2x + u2y dx dy (G)
(10.49d)
ZZ u2x + u2y dx dy
(10.49e)
for the potential energy U of the deformed membrane, where the constant denotes the tension of the membrane. We obtain the so-called Dirichlet variational problem in this way: We have to determine the function u = u(x y) so that the functional
I u] =
(G)
should have an extremum, and u vanishes on the boundary ; of the plane domain G. The corresponding Euler di erential equation is @ 2 u + @ 2 u = 0: (10.49f) @x2 @y2
560 10. Calculus of Variations It is the Laplace di erential equation for functions of two variables (see 13.5.1, p. 669).
10.4.2 More General Variational Problems
We should consider two generalizations of the simple variational problem.
1. F = F (x y u(x y) ux uy uxx uxy uyy )
The functional depends on higher-order partial derivatives of the unknown function u(x y). If the partial derivatives occur up to second order, then the Euler di erential equation is: ! ! ! ! ! @F ; @ @F ; @ @F + @ 2 @F + @ 2 @F + @ 2 @F = 0: (10.50) @u @x @ux @y @uy @x2 @uxx @x@y @uxy @y2 @uyy
2. F = F (x1 x2 : : : xn u(x1 : : : xn) ux1 : : : ux ) n
In the case of a variational problem with n independent variables x1 x2 : : : xn, the Euler di erential equation is: ! n @ @F ; X @F = 0: (10.51) @u k=1 @xk @uxk
10.5 Numerical Solution of Variational Problems Most often two ways are used to solve variational problems in practice.
1. Solution of the Euler Di erential Equation and Fitting the Found Solution to the Boundary Conditions
Usually, exact solution of the Euler di erential equation is possible only in the simplest cases, so we have to use a numerical method to solve the boundary value problem for ordinary or for partial di erential equations (see 19.5, p. 911 or 20.4.4, p. 992 ).
2. Direct Methods
The direct methods start directly from the variational problem and do not use the Euler di erential equation. The most popular and probably the oldest procedure is the Ritz method. It belongs to the socalled approximation methods which are also used for approximate solutions of di erential equations (see 19.4.2.2, p. 909 and 19.5.2, p. 912), and we demonstrate it with the following example. Solve numerically the isoperimetric problem
Z1 0
y02(x) dx = extreme! (10.52a)
for
Z1 0
y2(x) dx = 1 and y(0) = y(1) = 0: (10.52b)
The corresponding variational problem without side condition according to 10.3.3, p. 555, is:
Z 1h i I y] = y02(x) dx ; y2(x) = extreme! (10.52c) 0 We want to nd the best solution of the form y(x) = a1 x(x ; 1) + a2x2 (x ; 1): (10.52d) Both approximation functions x(x ; 1) and x2 (x ; 1) are linearly independent, and satisfy the boundary conditions. (10.52c) is reduced with (10.52d) to 2 a 2 + 1 a a ; 1 a2 + 1 a 2 + 1 a a (10.52e) I (a1 a1 ) = 13 a21 + 15 2 3 1 2 30 1 105 2 30 1 2 @I = @I = 0 result in he homogeneous linear equation system and the necessary conditions @a @a2 ! !1 ! ! 1 ; a + 4 ; 2 a = 0: 2; a + 1; a =0 (10.52f) 1 2 1 3 15 3 30 3 30 15 105 2
10.6 Supplementary Problems 561
This system has a non-trivial solution only if the determinant of the coecient matrix is equal to zero. So, we get: 2 ; 52 + 420 = 0 i.e., 1 = 10 2 = 42: (10.52g) For = 1 = 10 we get from (10.52f) a2 = 0 a1 arbitrary, so the normed solution belonging to 1 = 10 is: y = 5:48x(x ; 1): (10.52h) To make a comparison, consider the Euler di erential equation belonging to (10.52f). We get the boundary value problem y00 + y = 0 with y(0) = y(1) = 0 (10.52i) with the eigenvalues k = k22 (k = 1 2 : : :) and the solution yk = ck sin kx. The normed solution, e.g., for the case k = 1, i.e., 1 = 2 9:87 is p (10.52j) y = 2 sin x which is really very close to the approximate solution (10.52h). Remark: With today's level of computers and science we have to apply, rst of all, the nite element method (FEM) for numerical solutions of variational problems. The basic idea of this method is given in 19.5.3, p. 913, for numerical solutions of di erential equations. The correspondence between di erential and variational equations will be used there, e.g., by Euler di erential equations or bilinear forms according to (19.146a,b). Also the gradient method can be used for the numerical solution of variational problems as an ecient numerical method for non-linear optimization problems (see 18.2.7, p. 871).
10.6 Supplementary Problems
10.6.1 First and Second Variation
In the derivation of the Euler di erential equation with a comparable function (see 10.3.2, p. 554), we stopped after the linear term with respect to of the Taylor expansion of the integrand of
Zb I ( ) = F (x y0 + y00 + 0) dx: a
If we consider also quadratic terms, then we get
Zb " I ( ) ; I (0) = @F (x y0 y00 ) + @F0 (x y0 y00 )0 dx @y a @y
(10.53) (10.54)
2 Zb " @ 2 F 2 2 0 2 + 2 @ F (x y y 0 ) 0 + @ F (x y y 0 ) 02 + O ( ) dx: ( x y y ) 0 0 0 0 0 0 2 @y@y0 @y02 a @y
+ 2
If we denote as 1. Variation I of the functional I y] the expression
Zb " @F @F 0 0 0 ( x y 0 y0 ) + 0 (x y0 y0 ) dx and as @y a @y 2 2. Variation I of the functional I y] the expression
Zb " 2 @ 2 F (x y y0 )0 + @ 2 F (x y y0 )02 dx 2I = @ F2 (x y0 y00 )2 + 2 @y@y 0 0 0 0 0 02 @y a @y I =
(10.55)
(10.56)
562 10. Calculus of Variations then we can write: 2 I ( ) ; I (0) I + 2 2 I: (10.57) We can formalize the di erent optimality conditions with these variations for the functional I y] (see 10.7]).
10.6.2 Application in Physics
Variational calculus has a determining role in physics. We can derive the fundamental equations of Newtonian mechanics from a variational principle and arrive at the Jacobi{Hamilton theory. Variational calculus is also very important in both atomic theory and quantum physics. It is obvious that the extension and generalization of classical mathematical notions is undoubtedly necessary. So, the calculus of variations must be discussed today by modern mathematical disciplines, e.g., functional analysis and optimization. Unfortunately, we can only give a brief account of the classical part of the calculus of variations (see 10.4], 10.5], 10.7]).
563
11 LinearIntegralEquations 11.1 Introduction and Classication 1. Denitions
An integral equation is an equation in which the unknown function appears under the integral sign. There is no universal method for solving integral equations. Solution methods and even the existence of a solution depend on the particular form of the integral equation. An integral equation is called linear if linear operations are performed on the unknown function. The general form of a linear integral equation is:
g(x)'(x) = f (x) +
bZ(x)
a(x)
K (x y)'(y) dy
c x d:
(11.1)
The unknown function is '(x), the function K (x y) is called the kernel of the integral equation, and f (x) is the so-called perturbation function. These functions can take complex values as well. The integral equation is homogeneous if the function f (x) is identically zero over the considered domain, i.e., f (x) 0, otherwise it is inhomogeneous. is usually a complex parameter. Two types of equation (11.1) are of special importance. If the limits of the integral are independent of x, i.e., a(x) a and b(x) b, we call it a Fredholm integral equation (11.2a,11.2b). If a(x) a and b(x) = x, we call it a Volterra integral equation (11.2c, 11.2d). If the unknown function '(x) appears only under the integral sign, i.e., g(x) 0 holds, we have an integral equation of the rst kind as (11.2a), (11.2c). The equation is called an integral equation of the second kind if g (x) 1 as in (11.2b), (11.2d).
Zb
0 = f (x) + K (x y)'(y) dy a
Zx
0 = f (x) + K (x y)'(y) dy a
(11.2a)
Zb '(x) = f (x) + K (x y)'(y) dy
(11.2c)
Zx '(x) = f (x) + K (x y)'(y) dy: (11.2d)
2. Relations with Di erential Equations
a
(11.2b)
a
The problems of physics and mechanics relative rarely lead directly to an integral equation. These problems can be described mostly by di erential equations. The importance of integral equations is that many of these di erential equations, together with the initial and boundary values, can be transformed into integral equations. From the initial value problem y0(x) = f (x y) with x x0 and y(x0) = y0 by integration from x0 to x we get Zx y(x) = y0 + f ( y( )) d : (11.3) x0 The unknown function y(x) appears on the left-hand side of (11.3) and also under the integral sign. The integral equation (11.3) is linear if the function f ( y( )) has the form f ( ( )) = a( ) y( ) + b( ), i.e., the original di erential equation is also linear. Remark: In this chapter 11 we only deal with integral equations of the rst and second kind of Fredholm and Volterra types, and with some singular integral equations.
564 11. Linear Integral Equations
11.2 Fredholm Integral Equations of the Second Kind 11.2.1 Integral Equations with Degenerate Kernel
If the kernel K (x y) of an integral equation is the nite sum of products of two functions of one variable, i.e., one depends only on x and the other one only on y, it is called a degenerate kernel or a product kernel.
1. Solution in the Case of a Degenerate Kernel
The solution of a Fredholm integral equation of the second kind with a degenerate kernel leads to the solution of a nite-dimensional equation system. Consider the integral equation
Zb '(x) = f (x) + K (x y)'(y) dy with
(11.4a)
a
(11.4b) K (x y) = 1 (x)1 (y) + 2(x)2 (y) + : : : + n(x)n(y): The functions 1(x) : : : n(x) and 1 (x) : : : n(x) are given on the interval a b] and are supposed to be continuous. Furthermore, the functions 1(x) : : : , n(x) are supposed to be linearly independent of one another, i.e., the equality n X ck k (x) 0 (11.5) k=1
with constant coecients ck holds for every x in a b] only if c1 = c2 = : : : = cn = 0. Otherwise, K (x y) can be expressed as the sum of a smaller number of products. From (11.4a) and (11.4b) we get:
Zb Zb '(x) = f (x) + 1 (x) 1 (y)'(y) dy + : : : + n(x) n (y)'(y) dy: a
(11.6a)
a
The integrals are nolonger functions of the variable x, they are constant values. Let's denote them by Ak :
Zb Ak = k (y)'(y) dy a
k = 1 : : : n:
(11.6b)
The solution function '(x), if any exists, is the sum of the perturbation function f (x) and a linear combination of the functions 1(x) : : : n(x): '(x) = f (x) + A1 1(x) + A22 (x) + : : : + Ann(x): (11.6c)
2. Calculation of the Coe cients of the Solution
The coecients A1 : : : An are calculated as follows. Equation (11.6c) is multiplied by k (x) and its integral is calculated with respect to x with the limits a and b:
Zb a
Zb Zb Zb k (x)'(x) dx = k (x)f (x) dx + A1 k (x)1(x) dx + : : : + An k (x)n (x) dx: (11.7a) a
a
a
The left-hand side of this equation is equal to Ak according to (11.6b). Using the following notation
Zb Zb bk = k (x)f (x) dx and ckj = k (x)j (x) dx a
a
we obtain for k = 1 : : : n: Ak = bk + ck1A1 + ck2A2 + : : : + cknAn:
(11.7b) (11.7c)
11.2 Fredholm Integral Equations of the Second Kind 565
It is possible that the exact values of the integrals in (11.7b) cannot be calculated. When this is the case, their approximate values must be calculated by one of the formulas given in 19.3, p. 898. The linear equation system (11.7c) contains n equations for the unknown values A1 : : : An: (1 ; c11 )A1 ;c12 A2 ; : : : ;c1n An = b1 ;c21 A1 +(1 ; c22)A2 ; : : : ;c2n An = b2 (11.7d) ...................................................... ;cn1 A1 ;cn2 A2 ; : : : +(1 ; cnn)An = bn :
3. Analyzing the Solution, Eigenvalues and Eigenfunctions
It is known from the theory of linear equation systems that (11.7d) has one and only one solution for A1 : : : An if the determinant of the matrix of the coecients is not equal to zero, i.e., (1 ; c ) ;c12 : : : ;c1n 11 ;c21 (1 ; c22 ) : : : ;c2n D() = 6= 0: (11.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;cn1 ;cn2 : : : (1 ; cnn) Obviously D() is not identically zero, as D(0) = 1 holds. So there is a number R > 0 such that D() 6= 0 if jj < R. For further investigation we have to consider two di erent cases.
Case D() 6= 0:
The integral equation has exactly one solution in the form (11.6c), and the coecients A1 : : : An are given by the solution of the equation system (11.7d). If in (11.4a) we have a homogeneous integral equation, i.e., f (x) 0, then b1 = b2 = : : : = bn = 0. Then the homogeneous equation system (11.7d) has only the trivial solution A1 = A2 = : : : = An = 0. In this case only the function '(x) 0 satis es the integral equation.
Case D() = 0:
D() is a polynomial of no higher than n-th degree, so it can have at most n roots. For these values of the homogeneous equation system (11.7d) with b1 = b2 = : : : = bn = 0 also has non-trivial solutions, so besides the trivial solution '(x) 0 the homogeneous equation system has other solutions of the form '(x) = C (A1 1(x) + A22 (x) + : : : + Ann(x)) (C is an arbitrary constant.) Because 1 (x) : : : n(x) are linearly independent, '(x) is not identically zero. The roots of D() are called the eigenvalues of the integral equation. The corresponding non-vanishing solutions of the homogeneous integral equation are called the eigenfunctions belonging to the eigenvalue . Several linearly independent eigenfunctions can belong to the same eigenvalue. If we have an integral equation with a general kernel, we consider all values of eigenvalues, for which the homogeneous integral equation has non-trivial solutions. Some authors call the with D() = 0 the characteristic number, and = 1 is called the eigenvalue corresponding to an equation form '(x) =
4. Adjoint Integral Equation
Zb a
K (x y)'(y) dy.
Now we need to investigate the conditions under which the inhomogeneous integral equation will have solutions if D() = 0. For this purpose we introduce the adjoint or transposed integral equation of (11.4a):
Zb (x) = g(x) + K (y x)(y) dy: a
(11.9a)
566 11. Linear Integral Equations Let be an eigenvalue and '(x) a solution of the inhomogeneous integral equation (11.4a). It is easy to show that is also an eigenvalue of the adjoint equation. Now multiply both sides of (11.4a) by any solution (x) of the homogeneous adjoint integral equation and evaluate the integral with respect to x between the limits a and b:
1 Zb Zb 0 Zb '(x)(x) dx = f (x)(x) dx + @ K (x y)(x) dxA '(y) dy: (11.9b) a a a a Zb Zb Assuming that (y) = K (x y)(x) dx, we get f (x)(x) dx = 0. a a That is: The inhomogeneous integral equation (11.4a) has a solution for some eigenvalue if and only if the perturbation function f (x) is orthogonal to every non-vanishing solution of the homogeneous adjoint integral equation belonging to the same . This statement is valid not only for integral equations with degenerate kernels, but also for those with general kernels. Z +1 A: '(x) = x + ;1 (x2 y + xy2 ; xy)'(y) dy 1 (x) = x2 2 (x) = x 3 (x) = ;x 1(y) = y 2 (y) = y2 3(y) = y. The functions kZ(x) are linearly dependent. This is why we transform the +1 2 integral equation into the form '(x) = x + x y + x(y2 ; y)]'(y) dy. For this integral equation we ;1 have 1(x) = x2 2 (x) = x 1(y) = y 2(y) = y2 ; y. If any solution '(x) exists, it has the form '(x) =Z x + A1 x2 + A2 x. Z +1 Z +1 +1 3 c11 = x dx = 0 c12 = x2 dx = 32 b1 = x2 dx = 23 ;1 ;1 ;1 Z +1 Z +1 Z +1 c21 = (x4 ; x3 ) dx = 52 c22 = (x3 ; x2 ) dx = ; 23 b2 = (x3 ; x2) dx = ; 23 : ;1 ;1 ;1 2 2 With these values we have the equations for A1 and A2 : A1 ; 3 A2 = 3 ; 25 A1 + 1 + 23 A2 = ; 23 , 10 A = ; 2 and '(x) = x + 10 x2 ; 2 x = 10 x2 + 5 x. which in turn yield that A1 = 21 2 7 21 7 21 7 Z B: '(x) = x + 0 sin(x + y)'(y) dy, i.e.: K (x y) = sin(x + y) = sin x cos y + cos x sin y '(x) Z Z = x + sin x cos y '(y) dy + cos x sin y '(y) dy: Z 0 Z0 Z c11 = sin x cos x dx = 0 c12 = cos2 x dx = 2 b1 = x cos x dx = ;2 0 0 Z Z Z0 2 c21 = sin x dx = 2 c22 = cos x sin x dx = 0 b2 = x sin x dx = : 0 0 0 With these values the system (11.7d) is A1 ; 2 A2 = ;2 ; 2 A1 + A2 = . It has a unique solution 2 ; 2 2 1 ; for any with D() = 2 = 1 ; 2 6= 0. So A1 = 2 2 A2 = (1 ; )2 , and 4 ; 2 1 1; 2 1 ; 2 4
" 2 ! 4 the solution of the integral equation is '(x) = x + 2 ; 2 sin x + (1 ; ) cos x . The 2 1 ; 2 4 eigenvalues of the integral equation are 1 = 2 2 = ; 2 . Z The homogeneous integral equation '(x) = k sin(x + y)'(y) dy has non-trivial solutions of the Zb
0
11.2 Fredholm Integral Equations of the Second Kind 567
form 'k (x) = k (A1 sin x + A2 cos x) (k = 1 2). For 1 = 2 we get A1 = A2, and with an arbitrary constant A we have '1(x) = A(sin x + cos x). Similarly for 2 = ; 2 we get '2(x) = B (sin x ; cos x) with an arbitrary constant B . Remark: The previous solution method is fairly simple but it only works in the case of a degenerate kernel. By this method, however, we can get a good approximate solution in the case of a general kernel too if we can approximate the general kernel by a degenerate one closely enough (see 11.3]).
11.2.2 Successive Approximation Method, Neumann Series 1. Iteration Method
Similarly to the Picard iteration method (see 9.1.1.5, 1., p. 496) for the solution of ordinary di erential equations, an iterative method needs to be given to solve Fredholm integral equations of the second kind. Starting with the equation
Zb '(x) = f (x) + K (x y)'(y) dy
(11.10)
a
we de ne a sequence of functions '0 (x) '1(x) '2(x) : : : . Let the rst be '0(x) = f (x). We get the subsequent 'n(x) by the formula
Zb 'n(x) = f (x) + K (x y)'n;1(y) dy (n = 1 2 : : : '0(x) = f (x)): a
(11.11a)
Following the given method our rst step is
Zb '1(x) = f (x) + K (x y)f (y) dy:
(11.11b)
a
According to the iteration formula this expression of '(y) is substituted into the right-hand side of (11.10). To avoid the accidental confusion of the integral variables, let's denote y by in (11.11b).
2 3 Zb Zb '2(x) = f (x) + K (x y) 4f (y) + K (y )f () d5 dy a
Zb
a
= f (x) + K (x y)f (y) dy + 2
(11.11c)
ZbZb
K (x y)K (y )f () dy d: (11.11d) Zb Introducing the notation of K1 (x y) = K (x y) and K2(x y) = K (x )K ( y) d , and renaming a as y, we can write '2 (x) in the form Zb Zb '2(x) = f (x) + K1(x y)f (y) dy + 2 K2 (x y)f (y) dy: (11.11e) Denoting
a
aa
a
a
Zb Kn(x y) = K (x )Kn;1( y) d (n = 2 3 : : :) a
(11.11f)
568 11. Linear Integral Equations we get the representation of the n-th iterated 'n(x):
Zb Zb 'n(x) = f (x) + K1(x y)f (y) dy + : : : + n Kn(x y)f (y) dy: a
We call Kn(x y) the n-th iterated kernel of K (x y).
(11.11g)
a
2. Convergence of the Neumann Series
To get the solution '(x), we have to discuss the convergence of the power series of
f (x) +
Zb 1 X n Kn(x y)f (y) dy
n=1
(11.12)
a
which is called the Neumann series. If the functions K (x y) and f (x) are bounded, i.e., the inequalities jK (x y)j < M (a x b a y b) and jf (x)j < N (a x b) (11.13a) hold, then the series
N
1 X
n=0
jM (b ; a)jn
(11.13b)
is a majorant series for the power series (11.12). This geometric series is convergent for all jj < M (b1; a) : (11.13c) The Neumann series is absolute and uniformly convergent for all values of satisfying (11.13c). By a sharper estimation of the terms of the Neumann series we can give the convergence interval more precisely. According to this, the Neumann series is convergent for 1 jj < s : (11.13d)
Rb Rb jK (x y)j2 dx dy aa
This limit for the parameter does not mean that there are no solutions for any jj outside the bounds set by (11.13d), but only that we cannot get it by the Neumann series. Let's denote by
; (x y ) =
1 X
n=1
n;1Kn(x y)
(11.14a)
the resolvent or solving kernel of the integral equation. Using the resolvent we get the solution in the form
Zb '(x) = f (x) + ; (x y )f (y) dy: a
(11.14b)
Z1
For the inhomogeneous Fredholm integral equation of the second kind '(x) = x + xy '(y) dy 0 Z1 1 1 we have K1 (x y) = xy K2(x y) = x y dy = 3 xy K3(x y) = 9 xy : : : Kn(x y) = 3xy n;1 0 1 n! X and from this ; (x y ) = xy n . With the limit (11.13c) the series is de nitely convergent for n=0 3 jj < 1, because jK (x y)j M = 1 holds. The resolvent ; (x y ) = xy ! is a geometric series 1; 3
11.2 Fredholm Integral Equations of the Second Kind 569
Z1
xy2 ! dy = x . 1; 1; 3 3 Remark: If for a given the relation (11.13d) does not hold, 1then we can2 decompose any continuous kernel into the sum of two continuous kernels K (x y) = K (x y) + K (x y), where K 1 (x y) is a degenerate kernel, and K 2 (x y) is so small that for this kernel (11.13d) holds. This way we have an exact solution method for any which is not an eigenvalue (see 11.6]).
which is convergent even for jj < 3. Thus from (11.14b) we get '(x) = x+
0
11.2.3 Fredholm Solution Method, Fredholm Theorems 11.2.3.1 Fredholm Solution Method
1. Approximate Solution by Discretization A Fredholm integral equation of the second kind
Zb '(x) = f (x) + K (x y)'(y) dy a
(11.15)
can be approximately represented by a linear equation system. We need to assume that the functions K (x y) and f (x) are continuous for a x b a y b. We will approximate the integral in (11.15) with the so-called left-hand rectangular formula (see 19.3.2.1, p. 899). It is also possible to use any other quadrature formula (see 19.3.1, p. 898). With an equidistant partition yk = a + (k ; 1)h (k = 1 2 n h = b ;n a ) (11.16a) we get the approximation '(x) f (x) + h K (x y1)'(y1) + : : : + K (x yn)'(yn)] : (11.16b) Let's replace '(x) in this expression by a function '(x) exactly satisfying (11.16b): '(x) = f (x) + h K (x y1)'(y1) + : : : + K (x yn)'(yn)] : (11.16c) To determine this approximate solution, we need the substitution values of '(x) at the interpolation nodes xk = a + (k ; 1)h. If we substitute x = x1 x = x2 : : : x = xn into (11.16c), we get a linear equation system for the required n substitution values of '(xk ). Using the shorthand notation Kjk = K (xj yk ) 'k = '(xk ) fk = f (xk ) (11.17a) we get (1 ; hK11 )'1 ;hK12 '2 ; : : : ;hK1n 'n = f1 ;hK21 '1 +(1 ; hK22)'2 ; : : : ;hK2n 'n = f2 (11.17b) ............................................................ ;hKn1 '1 ;hKn2 '2 ; : : : +(1 ; hKnn)'n = fn: This system has the determinant of the coecients (1 ; hK ) hK12 : : : ;hK1n ;hK11 (1 ;;hK ) : : : 2n Dn() = . . . . . . . . . . 21. . . . . . . . . . . . .22. . . . . . . . . . . . ;. .hK (11.17c) . . . . . . : ;hKn1 ;hKn2 : : : (1 ; hKnn) This determinant has the same structure as the determinant of the coecients in the solution of an integral equation with a degenerate kernel. The equation system (11.17b) has a unique solution for every where Dn() 6= 0. The solution gives the approximate substitution values of the unknown function ' (x) at the interpolation nodes. The values of with Dn() = 0 are approximations of the
570 11. Linear Integral Equations eigenvalues of the integral equations. The solution of (11.17b) can be written in quotient form (see Cramer rule, 4.4.2.3, p. 275): Dkn() '(x ) k = 1 : : : n: 'k = D (11.18) k n ( ) Here we get Dkn() from Dn() by replacing the elements of the k-th column by f1 f2 : : : fn.
2. Calculation of the Resolvent
If n tends to in nity, so the number of rows and columns of the determinant Dkn() and Dn(), too. We use the determinant D() = nlim Dn() (11.19a) !1 to get the solution kernel (resolvent) ; (x y ) (see 11.2.2, p. 567) in the form x y ) : ; (x y ) = D(D( (11.19b) ) It is true that every root of D() is a pole of ; (x y ). Exactly these values of , for which D() = 0, are the eigenvalues of the integral equation (11.15), and in this case the homogeneous integral equation has non-vanishing solutions, the eigenfunctions belonging to the eigenvalue . In the case of D() 6= 0, knowing the resolvent ; (x y ), we have an explicit form of the solution: Zb Zb '(x) = f (x) + ; (x y )f (y) dy = f (x) + D() D(x y )f (y) dy: (11.19c) a a To get the resolvent, we need the power series of D(x y ) and D() with respect to : 1 P (;1)nKn(x y) n D( x y ) n ; (x y ) = D() = =0 P (11.20a) 1 (;1)ndn n n=0
where d0 = 1 K0 (x y) = K (x y), and we get further coecients from the recursive formula: Zb Zb dn = n1 Kn;1(x x) dx Kn(x y) = K (x y) dn ; K (x t)Kn;1 (t y) dt: (11.20b) a a
A: '(x) = sin x +
Z
2
sin x cos y '(y) dy. The exact solution of this integral equation is 2 '(x) = 2 ; sin x. For n = 3 with x1 = 0 x2 = 6 x3 = 3 h = 6 we have 1 0 0 p p !2 p 3 ; 24 = 1 ; 3 ; 2 2 = 1 ; 3 : = p12 2:205 is D3() = ; 12 1 ; 24 24 192 12 3 p p3 3 1 ; 3 ; ; 12 24 24 an approximation of the exact eigenvalue = 2. From the rst equation of the equation system (11.17b) for f1 = 0 we get the solution '1p= 0. !Substituting this result into the secondpand third p we ! equation 1 3 3 3 have the equation system: 1 ; 24 '2 ; 24 '3 = 2 ; 24 '2 + 1 ; 24 '3 = 23 . This 0
system has the solution '2 =
p1
2 ; 63
'3 =
p p3
2 ; 63
. If = 1, then '1 = 0
'2 = 0:915 '3 =
11.2 Fredholm Integral Equations of the Second Kind 571
1:585. The substitution values of the exact solution are: '(0) = 0 ' = 1 ' = 1:732. 6 3 In order to achieve better accuracy, the number of interpolation nodes needs to be increased. Z1 Z1 B: '(x) = x + 0 (4xy ; x2 )'(y) dy d0 = 1 K0 (x y) = 4xy ; x2 d1 = 0 3x2 dx = 1, Z1 Z1 1, K1(x y) = 4xy ; x2 ; (4xt ; x2 )(4ty ; t2 ) dt = x +2x2 y ; 34 x2 ; 43 xy d2 = 12 K1(x x) dx = 18 0 Z 0 1 (4xy ; x2 ) ; 1 K (x t)K (t y) dt = 0. With these the values d K (x y) and all the folK2(x y) = 18 1 3 3 0
4xy ; x2 ; x + 2x2y ; 4 x2 ; 4 xy 3 3 lowing values of dk and Kk (x y) are equal to zero. ; (x y ) = . 2 1 ; + 18 2 p From 1 ; + = 0, we get the two eigenvalues 12 = 9 3 7. If is not an eigenvalue, we have the 18 Z1 solution '(x) = x + ; (x y )f (y) dy = 3x(22 ;;183x+ +186) . 0
11.2.3.2 Fredholm Theorems
For the Fredholm integral equation of the second kind
Zb '(x) = f (x) + K (x y)'(y) dy
(11.21a)
a
the correspondent adjoint integral equation is given by
Zb (x) = g(x) + K (y x)(y) dy:
(11.21b)
a
For this pair of integral equations the following statements are valid (see also 11.2.1, p. 564). 1. A Fredholm integral equation of the second kind can only have nite or countably in nite eigenvalues. The eigenvalues cannot accumulate in any nite interval, i.e., for any positive R there are only a nite number of for which jj < R. 2. If is not an eigenvalue of (11.21a), then both of the inhomogeneous integral equations have a unique solution for any perturbation function f (x) or g(x), and the corresponding homogeneous integral equations have only trivial solutions. 3. If is a solution of (11.21a), then is also an eigenvalue of the adjoint equation (11.21b). Both homogeneous integral equations have non-vanishing solutions, and the number of linearly independent solutions are the same for both equations. 4. For an eigenvalue the homogeneous integral equation can be solved if and only if the perturbation function is orthogonal to every solution of the homogeneous adjoint integral equation, i.e., for every solution of the integral equation
Zb (x) = K (x y)(y) dy a
(11.22a)
Zb a
f (x)(x) dx = 0 holds:
(11.22b)
The Fredholm alternative theorem follows from these statements: Either the inhomogeneous integral equation can be solved for any perturbation function f (x) or the corresponding homogeneous equation
572 11. Linear Integral Equations has non-trivial solutions (see 11.6]).
11.2.4 NumericalMethodsforFredholmIntegralEquationsofthe Second Kind
Often it is either impossible or takes too much work to get the exact solution of a Fredholm integral equation of the second kind
Zb '(x) = f (x) + K (x y)'(y) dy
(11.23)
a
by the solution methods given in 11.2.1, p. 564, 11.2.2, p. 567 and 11.2.3, p. 569. In such cases certain numerical methods can be used for approximation. Three di erent methods are given below to get the numerical solution of an integral equation of the form (11.23). These solution approaches often lead to large linear equation systems with dense coecient matrices. Multigrid methods are especially suitable for solving linear equation systems arising from the numerical solution of integral equations of the second kind (see 11.3]).
11.2.4.1 Approximation of the Integral 1. Semi-Discrete Problem
Working on the integral equation (11.23) we replace the integral by an approximation formula. These approximation formulas are called quadrature formulas. They take the form
Zb a
f (x) dx Qab](f ) =
n X
k=1
!k f (xk )
(11.24)
i.e., instead of the integral we have a sum of the substitution values of the function at the interpolation nodes xk weighted by the values !k . The numbers !k should be suitably chosen (so as to be independent of f ). Equation (11.23) can be written in the approximate form:
'(x) f (x) + Qab](K (x )'()) = f (x) +
n X
k=1
!k K (x yk )'(yk ):
(11.25a)
The quadrature formula Qab](K (x )'()) also depends on the variable x. The dot in the argument of the function means that the quadrature formula will be used with respect to the variable y. De ning the relation n X '(x) = f (x) + !k K (x yk )'(yk ): (11.25b) k=1
'(x) is an approximation of the exact solution '(x). We can consider (11.25b) as a semi-discrete problem, because the variable y is turned into discrete values while the variable x can still be arbitrary. If the equation (11.25b) holds for a function '(x) for every x 2 a b], it must also be valid for the interpolation nodes x = xk : n X '(xk ) = f (xk ) + !j K (xk yj )'(yj ) k = 1 2 : : : n: (11.25c) j =1
This is a linear equation system containing n equations for the n unknown values '(xk ). Substituting these solutions into (11.25b) we have the solution of the semi-discrete problem. The accuracy and the amount of calculations of this method depend on the quadrature formula used. For example if we use the left-hand rectangular formula (see 19.3.2.1, p. 899) with an equidistant partition yk = xk = a + h(k ; 1) h = (b ; a)=n (k = 1 : : : n):
Zb a
K (x y)'(y)dy
n X
k=1
hK (x yk )'(yk ):
(11.26a)
11.2 Fredholm Integral Equations of the Second Kind 573
With the notation Kjk = K (xj yk ) fk = f (xk ) 'k = '(xk ) (11.26b) the system (11.25c) has the form: (1 ; hK11 )'1 ;hK12 '2 ; : : : ;hK1n 'n = f1 ;hK21 '1 +(1 ; hK22'2 ) ; : : : ;hK2n 'n = f2 (11.26c) ............................................................. ;hKn1 '1 ;hKn2 '2 ; : : : +(1 ; hKnn)'n = fn: We had the same system in the Fredholm solution method (see 11.2.3, p. 569). As the rectangular formula is not accurate enough, for a better approximation of the integral we have to increase the number of interpolation nodes, along with an increase in the dimension of the equation system. Hence we get the idea of looking for another quadrature formula.
2. Nystrom Method
In the so-called Nystrom method we use the Gauss quadrature formula for the approximation of the integral (see 19.3.3, p. 900). In order to derive this, we consider the integral
Zb I = f (x) dx:
(11.27a)
a
We replace the integrand by a polynomial p(x), namely the interpolation polynomial of f (x) at the interpolation nodes xk : n X p(x) = Lk (x)f (xk ) with k=1
xk;1)(x ; xk+1) : : : (x ; xn ) : (11.27b) Lk (x) = (x(x ;; xx1)) ::::::(x(x ; k 1 k ; xk;1 )(xk ; xk+1 ) : : : (xk ; xn ) For this polynomial, p(xk ) = f (xk ) k = 1 : : : n. The replacement of the integrand f (x) by p(x) results in the quadrature formula Zb Zb Zb Zb n n X X f (x) dx p(x) dx = f (xk ) Lk (x) dx = !k f (xk ) with !k = Lk (x) dx: (11.27c) a
a
k=1
a
k=1
a
For the Gauss quadrature formula the interpolation nodes cannot be chosen arbitrarily but we have to choose them by the formula: xk = a +2 b + b ;2 a tk k = 1 2 : : : n: (11.28a) The n values tk are the n roots of the Legendre polynomial of the rst kind (see 9.1.2.6, 3., p. 511) n 2 n Pn(t) = 2n 1 n! d (tdt;n 1) ] : (11.28b) These roots are in the interval ;1 +1]. We calculate the coecients !k by the substitution x ; xk = b ; a (t ; t ), so: k 2
Z1 Zb tk;1)(t ; tk+1) : : : (t ; tn ) dt !k = Lk (x) dx = (b ; a) 21 (t (t;;t t)1): :: :: :(t(t ; k 1 k ; tk;1 )(tk ; tk+1 ) : : : (tk ; tn ) a ;1 = (b ; a)Ak : (11.29) In Table 11.1 we give the roots of the Legendre polynomial of the rst kind and the weights Ak for n = 1 : : : 6.
574 11. Linear Integral Equations
n
t
Table 11.1 Roots of the Legendre polynomial of the rst kind
A
n
t
A
t1 = ;0:9062 A1 = 0:1185 t2 = ;0:5384 A2 = 0:2393 t3 = 0 A3 = 0:2844 3 t4 = 0:5384 A4 = 0:2393 t5 = 0:9062 A5 = 0:1185 6 t1 = ;0:9324 A1 = 0:0857 4 t2 = ;0:6612 A2 = 0:1804 t3 = ;0:2386 A3 = 0:2340 t4 = 0:2386 A4 = 0:2340 t5 = 0:6612 A5 = 0:1804 t6 = 0:9324 A6 = 0:0857 Z 1 Solve the integral equation '(x) = cos x + x2 +x 2 (ex +1)+ exy '(y) dy by the Nystrom method 0 for n = 3. n = 3 : x1 = 0:1127 x2 = 0:5 x3 = 0:8873 A1 = 0:2778 A2 = 0:4444 A3 = 0:2778 f1 = 0:96214 f2 = 0:13087 f3 = ;0:65251 K11 = 1:01278 K22 = 1:28403 K33 = 2:19746 K12 = K21 = 1 05797 K13 = K31 = 1:10517 K23 = K32 = 1:55838: The equation system (11.25c) for '1 '2, and '3 is 0:71864'1 ; 0:47016'2 ; 0:30702'3 = 0:96214 ;0:29390'1 + 0:42938'2 ; 0:43292'3 = 0:13087 ;0:30702'1 ; 0:69254'2 + 0:38955'3 = ;0:65251: The solution of the system is: '1 = 0:93651 '2 = ;0:00144 '3 = ;0:93950. The substitution values of the exact solution at the interpolation nodes are: '(x1) = 0:93797 '(x2) = 0 '(x3) = ;0:93797. 1 2
t1 = 0 t1 = ;0:5774 t2 = 0:5774 t1 = ;0:7746 t2 = 0 t3 = 0:7746 t1 = ;0:8612 t2 = ;0:3400 t3 = 0:3400 t4 = 0:8612
A1 = 1 A1 = 0:5 A2 = 0:5 A1 = 0:2778 A2 = 0:4444 A3 = 0:2778 A1 = 0:1739 A2 = 0:3261 A3 = 0:3261 A4 = 0:1739
5
11.2.4.2 Kernel Approximation
Replace the kernel K (x y) by a kernel K (x y) so that K (x y) K (x y) for a x b, a y b. Try to choose a kernel making the solution of the integral equation
Zb '(x) = f (x) + K (x y)'(y) dy
the easiest possible.
(11.30)
a
1. Tensor Product Approximation
A frequently-used approximation of the kernel is the tensor product approximation in the form
K (x y) K (x y) =
n X n X
j =0 k=0
djk j (x)k (y)
(11.31a)
with given linearly independent functions 0 (x) : : : n(x) and 0(y) : : : n(y) whose coecients djk must be chosen so that the double sum approximates the kernel closely enough in a certain sense.
11.2 Fredholm Integral Equations of the Second Kind 575
Rewrite (11.31a) in a degenerate kernel:
K (x y) =
n X
j =0
j (x)
"X n
k=0
n n X X djkk (y) j (y) = djk k (y) K (x y) = j (x) j (y): j =0
k=0
(11.31b)
Now, the solution method 11.2.1, p. 564 can be used for the integral equation
3 Zb 2X n 4 '(x) = f (x) + j (x) j (y)5 '(y) dy: a
(11.31c)
j =0
Functions 0 (x) : : : n(x) and 0(y) : : : n(y) should be chosen so that the coecients djk in (11.31a) can be calculated easily and also that the solution of (11.31c) isn't too dicult.
2. Special Spline Approach Let's choose
8 > < 1 ; n x ; k for k ; 1 x k + 1 n k (x) = k (x) = > n n :0 otherwise
(11.32)
for a special kernel approximation on the interval of integration a b!] = 0 1]. The function k (x) has non-zero values only in the so called carrier interval k ; 1 k + 1 , (Fig. 11.1). n n To calculate the coecients djk in (11.31a), consider K (x y) αk(x) at the points x = l=n, y = i=n (l i = 0 1 : : : n). We get 1 ! ( j=l k=i j nl k ni = 10 for (11.33) otherwise and consequently ! K (l=n i=n!) = dli. Hence, we substitute i l i l 0 x dli = K n n = K n n . Now (11.31a) has the form k-1 n
k n
Figure 11.1
k+1 n
K (x y) =
! K nj nk j (x)k (y): j =0 k=0 n X n X
(11.34)
As we know, the solution of (11.31c) has the form '(x) = f (x) + A0 0(x) + : : : + Ann(x): (11.35) The expression A00 (x) + : : : + Ann(x) is a piecewise linear function with substitution values Ak at the points xk = k=n. Solving (11.31c) by the method given for the degenerate kernel, we get a linear equation system for the numbers A0 : : : An: (1 ; c00 )A0 ;c01 A1 ; : : : ;c0n An = b0 ;c10 A0 + (1 ; c11 A1) ; : : : ;c1n An = b1 (11.36a) ....................................................... ;cn0 A0 ;cn1 A1 ; : : : + (1 ; cnn)An = bn where
Z1 Z1 "X n K nj ni j (x) k (x) dx cjk = j (x)k (x) dx = 0 0 i=0
576 11. Linear Integral Equations Z1 Z1 (11.36b) = K nj n0 0 (x)k (x) dx + : : : + K nj nn n(x)k (x) dx: 0 0 81 For the integrals we get > for j = 0 k = 0 and j = n k = n > > 3n > > > Z1 < 2 for j = k 1 j < n Ijk = j (x)k (x) dx = > 3n (11.36c) > 1 for j = k + 1 j = k ; 1 0 > > > > 6n : 0 otherwise. The numbers bk in (11.36a) are given by 2n 3 ! Z1 X k j bk = f (x) 4 K n n j (x)5 dx: (11.36d) j =0 0 Taking a matrix C with numbers cjk from (11.36a), a matrix B with the values K (j=n k=n) and a matrix A with the values Ijk respectively, a vector b from the numbers b0 : : : bn, and a vector a from the unknown values A0 : : : An, the equation system (11.36a) has the form (I ; C)a = (I ; BA)a = b: (11.36e) In the case when the matrix (I ; BA) is regular, this system has a unique solution a = (A0 : : : An).
11.2.4.3 Collocation Method
Suppose the n functions '1 (x) : : : 'n(x) are linearly independent in the interval a b]. They can be used to form an approximation function '(x) of the solution '(x): '(x) '(x) = a1 '1(x) + a2'2 (x) + : : : + an'n(x): (11.37a) The problem is now to determine the coecients a1 : : : an. Usually, there are no values a1 : : : an such that the function '(x) given in this form represents the exact solution '(x) = '(x) of the integral equation (11.23). Therefore, n interpolation points x1 : : : xn are de ned in the interval of integration, and it is required that the approximation function (11.37a) satis es the integral equation at least at these points: '(xk ) = a1 '1(xk ) + : : : + an'n(xk ) (11.37b)
Zb
= f (xk ) + K (xk y) a1'1 (y) + : : : + an 'n(y)] dy (k = 1 : : : n): a
(11.37c)
With some transformations this equation system takes the form:
2 3 2 3 Zb Zb 4'1 (xk ) ; K (xk y)'1(y) dy5 a1 + : : : + 4'n(xk ) ; K (xk y)'n(y) dy5 an a
a
= f (xk ) (k = 1 : : : n): Let's de ne the matrices 0 ' (x ) ' (x ) 1 0 1 1 1 n 1 11 1n Zb B C . . ... C .. A B = B A = @ .. A with jk = K (xj y)'k (y) dy @ ... a n1 nn '1(xn ) 'n(xn) and the vectors a = (a1 : : : an)> b = (f (x1 ) : : : f (xn))> :
(11.37d) (11.37e) (11.37f)
11.3 Fredholm Integral Equations of the First Kind 577
Then the equation system to determine the numbers a1 : : : an can be written in matrix form: (11.37g) (A ; B) a = b: px Z 1 p '(x) = 2 + xy '(y) dy. The approximation function is '(x) = a1 x2 + a2 x + a3 '1 (x) = 0 2 x '2 (x) = x '3(x) = 1. The 0 interpolation nodes 1 are x10= 0 x21= 0:5 x3 = 1. 00 0 11 0 0 0 0 B p p p CC B B B B C C 1 C B C 2 2 2 B B C 1 1 B C p A = BB@ 4 2 1 CCA B = BB 7 5 3 CC b = BBB 2 2 CCCC : B @ 1 A @ 2 2 2 C A 1 1 1 2 7 5 3 The system of equations is a3 = 0 p! p! p! 1 ; 2 a + 1 ; 2 a + 1 ; 2 a = p1 4 7 1 2 5 2 3 3 2 2 5a 3 + 5 a2 + 31 a3 = 12 7 1 whose solutions are a1 = ;0:8197 a2 = 1:8092 a3 = 0 and with these '(x) = ;0:8197 x2 + 1:8092 x and so '(0) = 0 '(0:5) = 0:6997 '(1) = 0:9895. p The exact solution of the integral equation is '(x) = x with the values '(0) = 0 '(0:5) = 0:7071 '(1) = 1. In order to improve the accuracy in this example, it is not a good idea to increase the degree of the polynomial, as polynomials of higher degree are numerically unstable. It is much better to use di erent spline approximations, e.g., a piecewise linear approximation '(x) = a1 '1(x)+ a2 '2(x)+ + an 'n(x) with the functions introduced in 11.2.4.2 8 k k;1 k+1 > < 'k (x) = > 1 ; n x ; n for n x n :0 otherwise. In this case, the solution '(x) is approximated by a polygon '(x). Remark: There is no theoretical restriction as to the choice of the interpolation nodes for the collocation method. In the case, however, when the solution function oscillates considerably in a subinterval, we have to increase the number of interpolation points in this interval.
11.3 Fredholm Integral Equations of the First Kind 11.3.1 Integral Equations with Degenerate Kernels 1. Formulation of the Problem
Consider the Fredholm integral equation of the rst kind with degenerate kernel
Zb f (x) = (1 (x)1(y) + : : : + n(x)n(y))'(y) dy (c x d) a
(11.38a)
and introduce the notation similar to that used in 11.2, p. 564,
Zb Aj = j (y)'(y) dy (j = 1 2 : : : n): a
(11.38b)
578 11. Linear Integral Equations Then (11.38a) has the form f (x) = A1 1(x) + : : : + Ann(x) (11.38c) i.e., the integral equation has a solution only if f (x) is a linear combination of the functions 1 (x) : : : n(x). If this assumption is ful lled, the constants A1 : : : An are known.
2. Initial Approach
We are looking for the solution in the form '(x) = c1 1(x) + : : : + cnn(x) where the coecients c1 : : : cn are unknown. Substituting in (11.38b)
Zb Zb Ai = c1 i(y)1(y) dy + : : : + cn i(y)n(y) dy (i = 1 2 : : : n) a
and introducing the notation
a
Zb Kij = i(y)j (y) dy a
we have the following equation system for the unknown coecients c1 : : : cn: K11 c1 + : : : + K1ncn = A1 ... ... ... Kn1 c1 + : : : + Knncn = An:
(11.39a) (11.39b) (11.39c)
(11.39d)
3. Solutions
The matrix of the coecients is non-singular if the functions 1(y) : : : n(y) are linearly independent (see 12.1.3, p. 598). However, the solution obtained in (11.39a) is not the only one. Unlike the integral equations of the second kind with a degenerate kernel, the homogeneous integral equation always has a solution. Suppose 'h(x) is a solution of the homogeneous equation and '(x) is a solution of (11.38a). Then '(x) + 'h(x) is also a solution of (11.38a). To determine all the solutions of the homogeneous equation, let us consider the equation (11.38c) with f (x) = 0. If the functions 1 (x) : : : n(x) are linearly independent, the equation holds if and only if
Zb Aj = j (y)'(y) dy = 0 (j = 1 2 : : : n) a
(11.40)
i.e., every function 'h(y) orthogonal to every function j (y) is a solution of the homogeneous integral equation.
11.3.2 Analytic Basis 1. Initial Approach
Several methods for the solution of Fredholm integral equations of the rst kind
Zb f (x) = K (x y)'(y) dy (c x d) a
(11.41)
determine the solution '(y) as a function series of a given system of functions (n(y)) = f1 (y) 2(y) : : :g, i.e., we are looking for the solution in the form
'(y) =
1 X
j =1
cj j (y)
(11.42)
where we have to determine the unknown constants cj . When choosing the system of functions we have to consider that the functions (n(y)) should generate the whole space of solutions, and also that the
11.3 Fredholm Integral Equations of the First Kind 579
calculation of the coecients cj should be easy. For an easier survey, we will discuss only real functions in this section. All of the statements can be extended to complex-valued functions, too. Because of the solution method we are going to establish, certain properties of the kernel function K (x y) are required. We assume that these requirements are always ful lled. Next, we discuss some relevant information.
2. Quadratically Integrable Functions
A function (y) is quadratically integrable over the interval a b] if
Zb a
j(y)j2 dy < 1
(11.43)
holds. For example, every continuous function on a b] is quadratically integrable. The space of quadratically integrable functions over a b] will be denoted by L2 a b].
3. Orthonormal System
Two quadratically integrable functions i(y) j (y) y 2 a b] are considered orthogonal to each other if the equality
Zb a
i (y)j (y) dy = 0
(11.44a)
holds. We call a system of functions (n(y)) in the space L2 a b] an orthonormal system if the following equalities are true: Zb i=j i (y)j (y) dy = 10 for (11.44b) for i 6= j: a An orthonormal system of functions is complete if there is no function ~(y) 6= 0 in L2 a b] orthogonal to every function of this system. A complete orthonormal system contains countably many functions. These functions form a basis of the space L2 a b]. To transform a system of functions (n(y)) into an orthonormal system (n(y)) we can use the Schmidt orthogonalization procedure. This determines the coecients bn1 bn2 : : : bnn for n = 1 2 : : : successively so that the function
n(y) =
n X
j =1
bnj j (y)
(11.44c)
is normalized and orthogonal to every function 1 (y) : : : n;1(y).
4. Fourier Series
If (n(y)) is an orthonormal system and (y) 2 L2 a b], we call the series 1 X
j =1
dj j (y) = (y)
(11.45a)
the Fourier series of (y) with respect to (n(y)), and the numbers dj are the corresponding Fourier coecients. Based on (11.44b) we have:
Zb a
k (y)(y) dy =
Zb 1 X dj j (y)k (y) dy = dk :
j =1
a
(11.45b)
If (n(y)) is complete, we have the Parseval equality
Zb a
j(y)j2 dy =
1 X
j =1
jdj j2:
(11.45c)
580 11. Linear Integral Equations
11.3.3 Reduction of an Integral Equation into a Linear System of Equations
A linear equation system is needed in order to determine the Fourier coecients of the solution function '(y) with respect to an orthonormal system. First, we have to choose a complete orthonormal system (n(y)) y 2 a b]. A corresponding complete orthonormal system (n(x)) can be chosen for the interval x 2 c d]. With respect to the system (n(x)) the function f (x) has the Fourier series
f (x) =
1 X
i=1
Zd fii(x) with fi = i(x)f (x) dx: c
(11.46a)
If the integral equation (11.41) is multiplied by i (x) and the integral is evaluated for x running from c to d alone, we get:
fi =
Zd Zb
K (x y)'(y)i(x) dy dx 9 = = : K (x y)i(x) dx" '(y) dy (i = 1 2 : : :): c a Zb 8 < for j (y ) 62 nh(y ) cj = > j (11.53) : arbitrary for j (y) 2 nh(y) :
11.3.6 Iteration Method To solve the integral equation
Zb f (x) = K (x y)'(y) dy (c x d)
(11.54a)
a
starting with 0 (x) = f (x) we determine the functions
Zd n(y) = K (x y)n;1(x) dx (11.54b) c
Zb
and n(x) = K (x y)n(y) dy a
(11.54c)
for n = 1 2 : : : . If there is a quadratically integrable solution '(y) of (11.54a) then the following equalities hold:
Zb a
'(y)n(y) dy =
Zb Zd a c Zd
'(y)K (x y)n;1(x) dx dy
= f (x)n;1(x) dx c
(n = 1 2 : : :):
(11.54d)
By the orthogonalizationand normalizationof the function systems that we have obtained from (11.54b), (11.54c) we get the orthonormal systems (n(x)) and (n(y)). Using the Schmidt orthogonalization method we have n(y) in the form
n (y ) =
n X
j =1
bnj j (y)
(n = 1 2 : : :):
(11.54e)
We need to show that the solution '(y) of (11.54a) has the representation by the series
'(y) =
1 X
j =1
cnn(y):
(11.54f)
In this case we have for the coecients cn regarding (11.54d):
Zb Zb Zd n n X X cn = '(y)n(y) dy = bnj '(y)j (y) dy = bnj f (x)j;1(x) dx: a
j =1
a
j =1
c
(11.54g)
11.4 Volterra Integral Equations 585
To have a solution in the form (11.54f) the following conditions are both necessary and sucient:
1:
Zd c
f (x)]2 dx =
1 Zd X f (x) (x) dx2 n
n=1 c
(11.55a)
2:
1 X
n=1
jcnj2 < 1:
(11.55b)
11.4 Volterra Integral Equations 11.4.1 Theoretical Foundations
A Volterra integral equation of the second kind has the form
Zx '(x) = f (x) + K (x y)'(y) dy: a
(11.56)
The solution function '(x) with the independent variable x from the closed interval I = a b] or from the semi-open interval I = a 1) is required. We have the following theorem about the solution of a Volterra integral equation of the second kind: If the functions f (x) for x 2 I and K (x y) on the triangular region x 2 I and y 2 a x] are continuous, then there exists a unique solution '(x) of the integral equation such that it is continuous for x 2 I . For this solution '(a) = f (a) (11.57) holds. In many cases, the Volterra integral equation of the rst kind can be transformed into an equation of the second kind. Hence, theorems about existence and uniqueness of the solution are valid with some modi cations.
1. Transformation by Di erentiation
Assuming that '(x), K (x y), and Kx(x y) are continuous functions, we can transform the integral equation of the rst kind
Zx f (x) = K (x y)'(y) dy
into the form
a
(11.58a)
Zx @ f 0(x) = K (x x)'(x) + @x K (x y)'(y) dy (11.58b) a by di erentiation with respect to x. If K (x x) 6= 0 for all x 2 I , then dividing the equation by K (x x) we get an integral equation of the second kind.
2. Transformation by Partial Integration
Assuming that '(x) , K (x y) and Ky (x y) are continuous, we can evaluate the integral in (11.58a) by partial integration. Substituting
Zx
gives
a
'(y) dy = (x)
! h iy=x Zx @ f (x) = K (x y)(y) y=a ; @y K (x y) (y) dy a ! Zx @ = K (x x)(x) ; @y K (x y) (y) dy: a
(11.59a)
(11.59b)
586 11. Linear Integral Equations If K (x x) 6= 0 for x 2 I , then dividing by K (x x) we have an integral equation of the second kind: ! Zx @ (x) = Kf((xx)x) + K (x1 x) @y K (x y) (y) dy: (11.59c) a Di erentiating the solution (x) we get the solution '(x) of (11.58a).
11.4.2 Solution by Dierentiation
In some Volterra integral equations the integral vanishes after di erentiation with respect to x, or it can be suitably substituted. Assuming that the functions K (x y), Kx(x y), and '(x) are continuous or, in the case of an integral equation of the second kind, '(x) is di erentiable, and di erentiating
Zx f (x) = K (x y)'(y) dy (11.60a) a
with respect to x we get
Zx
or '(x) = f (x) + K (x y)'(y) dy a
(11.60b)
Zx @ K (x y) '(y) dy or (11.60c) f 0(x) = K (x x)'(x) + @x a Zx @ '0(x) = f 0(x) + K (x x)'(x) + @x K (x y) '(y) dy: (11.60d) a
Zx Find the solution '(x) for x 2 0 of the equation cos(x ; 2y)'(y) dy = 1 x sin x (I). Di eren2 2 Z x0 1 tiating it twice with respect to x we have '(x) cos x ; sin(x ; 2y)'(y) dy = (sin x + x cos x) (II a), 2 0 Zx and '0(x) cos x ; cos(x ; 2y)'(y) dy = cos x ; 1 x sin x (IIb). The integral in the second equation 2 0 is the same as that in the original problem, so we can substitute it. We get '0(x) cos x = cos x and because cos x 6= 0 for x 2 0 2 , '0(x) = 1 so '(x) = x + C . To determine the constant C substitute x = 0 in (IIa) to obtain '(0) = 0. Consequently C = 0, and the solution of (I) is '(x) = x. Remark: If the kernel of a Volterra integral equation is a polynomial, then we can transform the integral equation by di erentiation into a linear di erential equation. Suppose the highest power of x in the kernel is n. After di erentiating the equation (n + 1) times with respect to x we have a di erential equation of n-th order in the case of an integral equation of the rst kind, and of the order n + 1 in the case of an integral equation of the second kind. Of course we have to assume that '(x) and f (x) are di erentiable as many times as necessary. Zx 2(x ; y)2 + 1]'(y) dy = x3 (I*). After di erentiating three times with respect to x we have 0 Z Zx x '(x)+4 (x;y)'(y) dy = 3x2 (II*a), '0(x)+4 '(y) dy = 6x (II*b), '00 (x)+4'(x) = 6 (II*c). 0 0 The general solution of this di erential equation is '(x) = A sin 2x + B cos 2x + 32 . Substituting x = 0 in (II*a) and (II*b) results in '(0) = 0 '0(0) = 0, so we have A = 0 B = ;1:5. The solution of the integral equation (I*) is '(x) = 3 (1 ; cos 2x). 2
11.4 Volterra Integral Equations 587
11.4.3 Solution of the Volterra Integral Equation of the Second Kind by Neumann Series
We can solve Volterra integral equations of the second kind by using Neumann series (see 11.2.2, p. 567). If we have the equation
Zx '(x) = f (x) + K (x y)'(y) dy
(11.61)
a
we substitute ( (x y) for y x K (x y) = K 0 for y > x: With this transformation (11.61) is equivalent to a Fredholm integral equation
Zb '(x) = f (x) + K (x y)'(y) dy
(11.62a) (11.62b)
a
allowing b = 1 as well. The solution has the representation
'(x) = f (x) +
Zb 1 X n Kn(x y)f (y) dy:
n=1
(11.62c)
a
The iterated kernels K1 K2 : : : are de ned by the following equalities:
Zb Zx K1(x y) = K (x y) K2 (x y) = K (x )K ( y) d = K (x )K ( y) d : : : a
and in general:
y
Zx Kn(x y) = K (x )Kn;1( y) d: y
(11.62d) (11.62e)
The equalities Kj (x y) 0 for y > x (j = 1 2 : : :) are also valid for iterated kernels. Contrary to Fredholm integral equations if (11.61) has any solution, the Neumann series converges to it regardless of the value of . Z Zx x '(x) = 1 + ex;y '(y) dy. K1(x y) = K (x y) = ex;y K2 (x y) = ex; e;y d = ex;y (x ; 0 y x;y e n ;1 y) : : : Kn(x y) = (n ; 1)! (x ; y) . 1 n X Consequently the resolvent is: ; (x y ) = ex;y n! (x ; y)n = e(x;y)( +1) . It is well-known that n=0 this series is convergent Z xfor any value of the parameterZ x. We get '(x) = 1 + e(x;y)( +1) dy = 1 + e( +1)x e;( +1)y dy in particular if = ;1: '(x) = 0 0 1 ; x 6= ;1: '(x) = +1 1 1 + e( +1)x .
11.4.4 Convolution Type Volterra Integral Equations If the kernel of a Volterra integral equation has the special form 0yx K (x y) = k0 (x ; y) for for 0 x < y
(11.63a)
588 11. Linear Integral Equations we can use the Laplace transformation to solve the equations
Zx 0
k(x ; y)'(y) dy = f (x) (11.63b)
Zx
or '(x) = f (x) + k(x ; y)'(y) dy: 0
(11.63c)
If the Laplace transforms Lf'(x)g = (p), Lff (x)g = F (p), and Lfk(x)g = K (p) exist, then the transformed equations have the form (see 15.2.1.2, 11., p. 713) K (p)(p) = F (p) (11.64a) or (p) = F (p) + K (p)(p) resp. (11.64b) From these we get F (p) (p) = K (11.64c) or (p) = 1 ;F (Kp)(p) resp. (11.64d) (p) The inverse transformation gives the solution '(x) of the original problem. Rewriting the formula for the Laplace transform of the solution of the integral equation of the second kind we have (p) F (p): (p) = 1 ;F (Kp)(p) = F (p) + 1 ;KK (11.64e) (p) The formula K (p) (11.64f) 1 ; K (p) = H (p) depends only on the kernel, and if we denote its inverse by h(x), the solution is
Zx '(x) = f (x) + h(x ; y)f (y) dy: 0
(11.64g)
The function h(x ;Z y) is the resolvent kernel of the integral equation. x 1 '(x) = f (x) + ex;y '(y) dy : (p) = F (p) + p ;1 1 (p), i.e., (p) = pp ; ; 2 F (p). The inverse 0 transformation gives '(x). From H (p) = 1 it follows that h(x) = e2x. By (11.64g) the solution is p;2 Zx '(x) = f (x) + e2(x;y) f (y) dy. 0
11.4.5 Numerical Methods for Volterra Integral Equation of the Second Kind
We are to nd the solution for the integral equation
Zx '(x) = f (x) + K (x y)'(y) dy a
(11.65)
for x from the interval I = a b]. The purpose of numerical methods is somehow to approximate the integral by a quadrature formula:
Zx a
K (x y)'(y) dy Qax](K (x :)'(:)):
(11.66a)
Both the interval of integration and the quadrature formula depend on x. This fact is emphasized by the index a x] of Qax](: : :). We get the following equation as an approximation of (11.65): '(x) = f (x) + Qax](K (x :)'(:)): (11.66b)
11.4 Volterra Integral Equations 589
The function '(x) is an approximation of the solution of (11.65). The number and the arrangement of the interpolation nodes of the quadrature formula depend on x, so as to allow little choice. If is an interpolation node of Qax](K (x :)'(:)) , then (K (x )'( )) and especially '( ) must be known. For this purpose, the right-hand side of (11.66b) should be evaluated rst for x = , which is equivalent to a quadrature over the interval a ]. As a consequence, the use of the popular Gauss quadrature formula is not possible. We solve the problem by choosing the interpolation nodes as a = x0 < x1 < : : : < xk < : : : and we use a quadrature formula Qaxn] with the interpolation nodes x0 x1 : : : xn. The substitution values of the function at the interpolation nodes are denoted by the brief notation 'k = '(xk ) (k = 0 1 2 : : :). For '0 we have (see 11.4.1, p. 585) '0 = f (x0 ) = f (a) (11.66c) and with this: '1 = f (x1) + Qax1 ](K (x1 :)'(:)): (11.66d) Qax1] has the interpolation points x0 and x1 and consequently it has the form Qax1](K (x1 :)'(:) = w0K (x1 x0)'0 + w1K (x1 x1)'1 (11.66e) with suitable coecients w0 and w1. Continuing this procedure, the values 'k are successively determined from the general relation: 'k = f (xk ) + Qaxk ](K (xk :)'(:)) k = 1 2 3 : : : : (11.66f) The quadrature formulas Qaxk] have the following form:
Qaxk ](K (xk :)'(:) =
k X
j =0
wjk K (xk xj )'j :
(11.66g)
Hence, (11.66f) takes the form:
'k = f (xk ) +
k X
wjkK (xk xj )'j :
j =0
(11.66h)
The simplest quadrature formula is the left-hand rectangular formula (see 19.3.2.1, p. 899). For this the coecients are wjk = xj+1 ; xj for j < k and wkk = 0: (11.66i) We have the system '0 = f (a) (11.67a) '1 = f (x1 ) + (x1 ; x0 )K (x1 x0 )'0 '2 = f (x2 ) + (x1 ; x0 )K (x2 x0 )'0 + (x2 ; x1 )K (x2 x1)'1 and generally
'k = f (xk ) +
kX ;1 j =0
(xj+1 ; xj )K (xk xj )'j :
(11.67b)
More accurate approximations of the integral can be obtained by using the trapezoidal formula (see 19.3.2.2, p. 899). To make it simple, we choose equidistant interpolation nodes xk = a + kh k = 0 1 2 ::: : 2 3 Zb kX ;1 (11.67c) g(x) dx h2 4g(x0) + 2 g(xj ) + g(xk )5 : j =1 a Using this approximation for (11.66f) we get: '0 = f (a) (11.67d)
2 3 kX ;1 h 4 'k = f (xk ) + 2 K (xk x0)'0 + K (xk xk )'k + 2 K (xk xj )'j 5 : j =1
(11.67e)
590 11. Linear Integral Equations Altough the unknown values also appear on the right-hand side of the equation, they are easy to express. Remark: With the previous method we can approximate the solution of non-linear integral equations as well. If we use the trapezoidal formula to determine the values 'k we have to solve a non-linear equation. We can avoid this by using the trapezoidal formula for the interval a xk;1], and we use the rectangular formula for the interval xk;1 xk ]. If h is small enough, this quadrature error does not have a signi cant e ect on the solution (see 11.3]). Zx Let's give the approximate values of the solution of the integral equation '(x) = 2+ (x;y)'(y) dy 0 by the formula (11.66f) using the rectangular formula. The interpolation nodes are the equidistant points xk = k 0:1, and hence h = 0:1. x exact rectangular trapezoidal formula formula '0 = 2 '1 = f (x1 ) + hK (x1 x0) '0 0.2 2.0401 2.0602 2.0401 = 2 + 0:1 0:1 2 = 2:02 2.2030 2.1620 '2 = f (x2 ) + h(K (x2 x0) '0 + K (x2 x1 ) '1) 0.4 2.1621 0.6 2.3709 2.4342 2.3706 = 2 + 0:1(0:2 2 + 0:1 2:02) = 2:0602 etc. 0.8 2.6749 2.7629 2.6743 1.0 3.0862 3.2025 3.0852 In the table the values of the exact solution are given, as well as the approximate values calculated by the rectangular and the trapezoidal formulas, respectively, so the accuracies of these methods can be compared. The step size used is h = 0:1.
11.5 Singular Integral Equations
An integral equation is called a singular integral equation if the range of the integral in the equation is not nite, or if the kernel has singularities inside of the range of integration. We suppose that the integrals exist as improper integrals, or as Cauchy principal values (see 8.2.3, p. 453 .). The properties and the conditions for the solutions of singular integral equations are very di erent from those in the case of \ordinary" integral equations. We will discuss only some special problems in the following paragraph. For further discussions see 11.2, 11.6, 11.8]. y0
y
P1 (0,0)
P0 ( x0 ,y0 )
x0
Figure 11.2
x
11.5.1 Abel Integral Equation
One of the rst applications of integral equations for a physical problem was considered by Abel. A particle is moving in a vertical plane along a curve under the inuence only of gravity from the point P0 (x0 y0) to the point P1 (0 0) (Fig. 11.2). The velocity of the particle at a point of the curve is q v = ds (11.68) dt = 2g(y0 ; y) : By integration we calculate the time of fall as a function of y0:
Zl T (y0) = q ds : 2g(y0 ; y) 0 If s is considered as a function of y, i.e., s = f (y), then Zy0 0 T (y0) = p12g pfy (y;) y dy: 0 0
(11.69a)
(11.69b)
11.5 Singular Integral Equations 591
The next problem is to determine the shape of the curve as a function of y0 if the time of the fall is given. By substitution q 2g T (y0) = F (y0) and f 0(y) = '(y) (11.69c) and changing the notation of the variable y0 into x, a Volterra integral equation of the rst kind is obtained: Zx y) F (x) = p'x(; (11.69d) y dy: 0 We will consider the slightly more general equation Zx f (x) = (x';(yy)) dy with 0 < < 1: (11.70) a The kernel of this equation is not bounded for y = x. In (11.70), the variable y is formally replaced by and the variable x by y. By these substitutions the solution is obtained in the form ' = '(x). If both sides of (11.70) are multiplied by the term (x ; 1y)1; and integrated with respect to y between the limits a and x, it yields the equation 0Zy 1 Zx 1 '( ) d A dy = Zx f (y) dy: @ (11.71a) 1; 1; a (x ; y ) a (y ; ) a (x ; y ) Changing the order of integration on the left-hand side we have
9 > Zx f (y) = d = 1; dy: > " a a (x ; y ) The inner integral can be evaluated by the substitution y = + (x ; )u: Zx Z1 dy du = 1; (y ; ) 1; = sin( ) : ( x ; y ) u (1 ; u ) 0 Zx
8x > 1 . The kernel K (x y)(y ; x);1 has a strong singularity for x = y. The integral exists as a Cauchy principal value. With K (x x) = b(x) and k(x y) = K (x y) ; K (x x) we have (11.72) in the form y;x Z '(y) Z b ( x ) (L')(x) := a(x)'(x) + i y ; x dy + 1i k(x y)'(y) dy = f (x) x 2 ;: (11.74a) ; ; The expression (L')(x) denotes the left-hand side of the integral equation in abbreviated form. L is a singular operator. The function k(x y) is a weakly singular kernel. It is assumed that the normality condition a(x)2 ; b(x)2 6= 0 x 2 ; holds. The equation Z (L0')(x) = a(x)'(x) + b(x) '(y) dy = f (x) x 2 ; (11.74b) i ; y;x is the characteristic equation pertaining to (11.74a). The operator L0 is the characteristic part of the operator L. From the adjoint integral equation of (11.74a) we get the equality ! Z Z (L>)(y) = a(y)(y) ; b(y) (x)dx + 1 k (x y) ; b(x) ; b(y) (x) dx i ; x;y i; x;y = g(y) y 2 ;: (11.74c)
11.5.2.2 Existence of a Solution
The equation (L')(x) = f (x) has a solution '(x) if and only if for every solution (y) of the homogeneous adjoint equation (L>)(y) = 0 the condition of orthogonality
Z
;
f (y)(y) dy = 0
(11.75a)
is satis ed. Similarly, the adjoint equation (L>)(y) = g(y) has a solution if for every solution '(x) of the homogeneous equation (L')(x) = 0 the following is valid:
Z
;
g(x)'(x) dx = 0:
11.5.2.3 Properties of Cauchy Type Integrals
(11.75b)
We call the function Z (z) = 21 i y';(y)z dy z 2 C (11.76a) ; a Cauchy type integral over ; . For z 2= ; the integral exists in the usual sense and the result is a holomorphic function (see 14.1.2, p. 672). We also have (1) = 0. For z = x 2 ; in (11.76a) we consider the Cauchy principal value Z '(y) (H')(x) = 1 dy x 2 ;: (11.76b) 2 i ; y ; x
11.5 Singular Integral Equations 593
The Cauchy type integral (z) can be extended continuously over ; from S + and from S ;. Approaching the point x 2 ; with z we denote the limit by +(x) and ;(x), respectively. The formulas of Plemelj and Sochozki are valid: +(x) = 12 '(x) + (H')(x) ; (x) = ; 12 '(x) + (H')(x): (11.76c)
11.5.2.4 The Hilbert Boundary Value Problem 1. Relations
The solution of the characteristic integral equation and the Hilbert boundary value problem strongly correlate. If '(x) is a solution of (11.74b), then (11.76a) is a holomorphic function on S + and S ; with (1) = 0. Because of the formulas of Plemelj and Sochozki (11.76c) we have: '(x) = +(x) ; ;(x) 2(H')(x) = +(x) + ;(x) x 2 ;: (11.77a) With the notation b(x) f (x) G(x) = aa((xx)) ; (11.77b) + b(x) and g(x) = a(x) + b(x) the characteristic integral equation has the form: +(x) = G(x); (x) + g(x) x 2 ;:
2. Hilbert Boundary Value Problem
(11.77c)
We are looking for a function (z) which is holomorphic on S + and S ;, and vanishes at in nity, and satis es the boundary conditions (11.77c) over ; . A solution (z) of the Hilbert problem can be given in the form (11.76a). So, as a consequence of the rst equation of (11.77a), a solution '(x) of the characteristic integral equation is determined.
11.5.2.5 Solution of the Hilbert Boundary Value Problem (in short: Hilbert Problem)
1. Homogeneous Boundary Conditions + ;
(x) = G(x) (x) x 2 ;: (11.78) During a single circulation of the point x along the curve ;l the value of log G(x) changes by 2 il , where l is an integer. The change of the value of the function log G(x) during a single traverse of the complete curve system ; is n X l=0
2 i l = 2 i :
(11.79a)
The number = P l is called the index of the Hilbert problem. We compose a function n
l=0
with
G0(x) = (x ; a0 );+(x)G(x)
(11.79b)
+(x) = (x ; a1) 1 (x ; a2 ) 2 (x ; an) n (11.79c) where a0 2 S + and al (l = 1 : : : n) are arbitrarily xed points inside ;l . If ; = ;0 is a simple closed curve (n = 0), then we de ne +(x) = 1. With Z I (z) := 21 i logy G;0z(y) dy (11.79d) ; the following particular solution of the homogeneous Hilbert problems is obtained, which is called the fundamental solution: ;1 (z) exp I (z) for z 2 S + X (z) = + (11.79e) (z ; a0); exp I (z) for z 2 S ;:
594 11. Linear Integral Equations This function doesn't vanish for any nite z. The most general solution of the homogeneous Hilbert problem, which vanishes at in nity, for > 0 is h(z) = X (z)P;1(z) z 2 C (11.80) with an arbitrary polynomial P;1(z) of degree at most ( ; 1). For 0 there exists only the trivial solution /h (z) = 0 which satis es the condition /h(1) = 0, so in this case P;1(z) 0. For > 0 the homogeneous Hilbert problem has linearly independent solutions vanishing at in nity.
2. Inhomogeneous Boundary Conditions
The solution of the inhomogeneous Hilbert problem is the following: 1 Z g(y)dy (z) = X (z)R(z) + h(z) (11.81) with R(z) = 2 i X +(y)(y ; z) : ;
(11.82)
If < 0 holds, for the existence of a solution vanishing at in nity the following necessary and sucient conditions must be ful lled: Z yk g(y)dy (11.83) X +(y) = 0 (k = 0 1 : : : ; ; 1): ;
11.5.2.6 Solution of the Characteristic Integral Equation 1. Homogeneous Characteristic Integral Equation
If h(z) is the solution of the corresponding homogeneous Hilbert problem, from (11.77a) we have the solution of the homogeneous integral equation 'h(x) = +h (x) ; ;h (x) x 2 ;: (11.84a) For 0 only the trivial solution 'h(x) = 0 exists. For > 0 the general solution is 'h(x) = X +(x) ; X ;(x)]P;1(x) (11.84b) with a polynomial P;1 of degree at most ; 1.
2. Inhomogeneous Characteristic Integral Equation
If (z) is a general solution of the inhomogeneous Hilbert problem, the solution of the inhomogeneous integral equation can be given by (11.77a): '(x) = +(x) ; ; (x) (11.85a) = X +(x)R+ (x) ; X ;(x)R; (x) + +h (x) ; ;h (x) x 2 ;: (11.85b) Using the formulas of Plemelj and Sochozki (11.76c) for R(z) we have R+(x) = 12 Xg+(x(x) ) + H Xg+ (x) R;(x) = ; 12 Xg+(x(x) ) + H Xg+ (x) : (11.85c) Substituting (11.85c) into (11.85a) and considering (11.76b) and g(x) = f (x)=(a(x) + b(x)) nally results in the the solution: + ; '(x) = 2(aX(x)(x+) b+(xX))X(x+)(x) f (x) Z +(X +(x) ; X ;(x)) 21 i (a(y) + b(yf))(Xy)+(y)(y ; x) dy + 'h(x) ;
x 2 ;:
(11.86)
According to (11.83) in the case < 0 the following relations must hold simultaneously in order to ensure the existence of a solution: Z yk f (y) dy = 0 (k = 0 1 : : : ; ; 1): (11.87) (a(y) + b(y))X +(y) ;
11.5 Singular Integral Equations 595
The characteristic integral equation is given with constant coecients a and b: Z a'(x) + bi y';(yx) dy = f (x). Here ; is a simple closed curve, i.e., ; = ;0 (n = 0). From (11.77b) ; b f (x) we get G = aa ; = 0. Therefore, +(x) = 1 and + b and g(x) = a + b . G is a constant, consequently 8 < log a ; b z 2 S + a;b 1 Z 1 b G0 = G = aa ; + b . I (z) = log a + b 2i ; y ; z dy = : 0 a + b z 2 S ;: 8a;b > < + b X ; = 1. X (z) = > a + b z 2 S i.e., X + = aa ; + b :1 ; z2S Since = 0 holds, the homogeneous Hilbert boundary value problem has only the function h(z) = 0 as the solution vanishing at in nity. From (11.86) we have Z Z + ; + ; '(x) = 2(Xa ++b)XX + f (x) + 2(Xa +;b)XX + 1i yf;(y)x dy = a2 ;a b2 f (x) ; a2 ;b b2 1i yf;(y)x dy. ; ;
596 12. Functional Analysis
12 FunctionalAnalysis 1. Functional Analysis
Functional analysis arose after the recognition of a common structure in di erent disciplines such as the sciences, engineering and economics. General principles were discovered that resulted in a common and uni ed approach in calculus, linear algebra, geometry, and other mathematical elds, showing their interrelations.
2. Innite Dimensional Spaces
There are many problems, the mathematical modeling of which requires the introduction of in nite systems of equations or inequalities. Di erential or integral equations, approximation, variational or optimization problems could not be treated by using only nite dimensional spaces.
3. Linear and Non-Linear Operators
In the rst phase of applying functional analysis { mainly in the rst half of the twentieth century { linear or linearized problems were thoroughly examined, which resulted in the development of the theory of linear operators. More recently the application of functional analysis in practical problems required the development of the theory of non-linear operators, since more and more problems had to be solved that could be described only by non-linear methods. Functional analysis is increasingly used in solving di erential equations, in numerical analysis and in optimization, and its principles and methods became a necessary tool in engineering and other applied sciences.
4. Basic Structures
In this chapter only the basic structures will be introduced, and only the most important types of abstract spaces and some special classes of operators in these spaces will be discussed. The abstract notion will be demonstrated by some examples, which are discussed in detail in other chapters of this book, and the existence and uniqueness theorems of the solutions of such problems are stated and proved there. Because of its abstract and general nature it is clear that functional analysis o ers a large range of general relations in the form of mathematical theorems that can be directly used in solving a wide variety of practical problems.
12.1 Vector Spaces
12.1.1 Notion of a Vector Space
A non-empty set V is called a vector space or linear space over the eld IF of scalars if there exist two operations on V { addition of the elements and multiplication by scalars from IF { such that they have the following properties: 1. for any two elements x y 2 V, there exists an element z = x + y 2 V, which is called their sum. 2. For every x 2 V and every scalar (number) 2 IF there exists an element x 2 V, the product of x and the scalar so that the following properties, the axioms of vector spaces (see also 5.3.7.1, p. 316), are satis ed for arbitrary elements x y z 2 V and scalars 2 IF: (V1) x + (y + z) = (x + y) + z: (12.1) (V2) There exists an element 0 2 V the zero element, such that x + 0 = x: (12.2) (V3) To every vector x there is a vector ; x such that x + (;x) = 0: (12.3) (V4) x + y = y + x: (12.4) (V5) 1 x = x 0 x = 0: (12.5) (V6) (x) = ( )x: (12.6) (V7) ( + )x = x + x: (12.7) (V8) (x + y) = x + y: (12.8)
12.1 Vector Spaces 597
V is called a real or complex vector space, depending on whether IF is the eld IR of real numbers or the eld C of complex numbers. The elements of V are called either points or, according to linear algebra, vectors. In functional analysis, we do not use the vector notation ~x or x. We can also de ne in V the di erence x ; y of two arbitrary vectors x y 2 V as x ; y = x + (;y). From the previous de nition, it follows that the equation x + y = z can be solved uniquely for arbitrary elements y and z. The solution is x = z ; y. Further properties follow from axioms (V1){(V8): the zero element is uniquely de ned, x = x and x 6= 0, imply = , x = y and 6= 0, imply x = y, ;(x) = (;x).
12.1.2 Linear and Ane Linear Subsets 1. Linear Subsets
A non-empty subset V0 of a vector space V is called a linear subspace or a linear manifold of V if together with two arbitrary elements x y 2 V0 and two arbitrary scalars 2 IF, their linear combination x + y is also in V0. V0 is a vector space in its own right, and therefore satis es the axioms (V1){ (V8). The subspace V0 can be V itself or only the zero point. In these cases the subspace is called trivial.
2. A ne Subspaces
A subset of a vector space V is called an ane linear subspace or an ane manifold if it has the form fx0 + y : y 2 V0g (12.9) where x0 2 V is a given element and V0 is a linear subspace. It can be considered (in the case x0 6= 0) as the generalization of the lines or planes not passing through the origin in IR3.
3. The Linear Hull
The intersection of an arbitrary number of subspaces in V is also a subspace. Consequently, for every non-empty subset E V, there exists a smallest linear subset lin(E ) or E ] in V containing E , namely the intersection of all the linear subspaces, which contain E . The set lin(E ) is called the linear hull of the set E , or the linear subspace generated by the set E . It coincides with the set of all ( nite) linear combinations 1x1 + 2 x2 + : : : + nxn (12.10) comprised of elements x1 x2 : : : xn 2 E and scalars 1 2 : : : n 2 IF.
4. Examples for Vector Spaces of Sequences
A Vector Space IFn: Let n be a given natural number and V the set of all n-tuples, i.e., all nite sequences consisting of n scalar terms f( 1 : : : n) : i 2 IF i = 1 : : : ng. The operations will be de ned componentwise or termwise, i.e., if x = ( 1 : : : n) and y = (1 : : : n) are two arbitrary elements from V and is an arbitrary scalar, 2 IF, then x + y = ( 1 + 1 : : : n + n) (12.11a) x = ( 1 : : : n): (12.11b) n In this way, we get the vector space IF . In the special case of n = 1 we get the linear spaces IR or C. This example can be generalized in two di erent ways (see examples B and C). B VectorSpace s of allSequences: If we consider the in nite sequences as elements x = f ng1n=1, n 2 IF and de ne the operations componentwise, similar to (12.11a) and (12.11b), then we get the vector space s of all sequences. C Vector Space '(also c00) of all Finite Sequences: Let V be the subset of all elements of s containing only a nite number of non-zero components, where the number of non-zero components depends on the element. This vector space { the operations are again introduced termwise { is denoted by ' or also by c00, and it is called the space of all nite sequences of numbers.
598 12. Functional Analysis D Vector Space m (also l1) of all Bounded Sequences: A sequence x = f ng1n=1 belongs to m if and only if there exists Cx > 0 with j nj Cx 8 n = 1 2 : : : . This vector space is also denoted by l1. E Vector Space c of all Convergent Sequences: A sequence x = f ng1n=1 belongs to c if and only if there exists a number 0 2 IF such that for 8 " > 0 there exists an index n0 = n0(") such that for all n > n0 one has j n ; 0j < " (see 7.1.2, p. 405). F Vector Space c0 of all Null Sequences: The vector space c0 of all null sequences, i.e., the subspace of c consisting of all sequences converging to zero ( 0 = 0). G Vector Space lp: The vector space of all sequences x = f ng1n=1 such that P1n=1 j njp is convergent, is denoted by lp (1 p < 1). It can be shown by the Minkowski inequality that the sum of two sequences from lp also belongs to lp, (see 1.4.2.13, p. 32). Remark: For the vector spaces introduced in examples A{G, the following inclusions hold: ' c0 c m s and ' lp lq c0 where 1 p < q < 1: (12.12)
5. Examples of Vector Spaces of Functions
A Vector Space F (T ): Let V be the set of all real or complex valued functions de ned on a given set T , where the operations are de ned pointwise, i.e., if x = x(t) and y = y(t) are two arbitrary elements of V and 2 IF is an arbitrary scalar, then we de ne the elements (functions) x + y and x by the rules (x + y)(t) = x(t) + y(t) 8 t 2 T (12.13a) (x)(t) = x(t) 8 t 2 T: (12.13b) We denote this vector space by F (T ). We introduce some of its subspaces in the following examples. B Vector Space B(T ) or M(T ): The space B(T ) is the space of all functions bounded on T . This vector space is often denoted by M(T ). In the case of T = IN, we get the space M(IN) = m from example D of the previous paragraph. C Vector Space ( a b]): The set C ( a b]) of all functions continuous on the interval a b] (see 2.1.5.1, p. 57). D Vector Space (k)( a b]): Let k 2 IN k 1. The set C (k) ( a b]) of all functions k-times continuously di erentiable on a b] (see 6.1, p. 379{384) is a vector space. At the endpoints a and b of the interval a b], the derivatives have to be considered as right-hand and left-hand derivatives, respectively. Remark: For the vector spaces of examples A{D of this paragraph, and T = a b] the following subspace relations hold: C (k) ( a b]) C ( a b]) B( a b]) F ( a b]): (12.14) E Vector Subspace of C ( a b]): For any given point t0 2 a b], the set fx 2 C ( a b]): x(t0 ) = 0g forms a linear subspace of C ( a b]).
C C
12.1.3 Linearly Independent Elements
1. Linear Independence
A nite subset fx1 : : : xng of a vector space V is called linearly independent if 1x1 + + nxn = 0 implies 1 = = n = 0: (12.15) Otherwise, it is called linearly dependent. If 1 = = n = 0, then for arbitrary vectors x1 : : : xn from V, the vector 1x1 + + nxn is trivially the zero element of V. Linear independence of the vectors x1 : : : xn means that the only way to produce the zero element 0 = 1 x1 + + nxn is when all coecients are zero 1 = = n = 0. This important notion is well known from linear algebra
12.1 Vector Spaces 599
(see 5.3.7.2, p. 317) and was used for the de nition of a fundamental system of homogeneous di erential equations (see 9.1.2.3, 2., p. 500). An in nite subset E V is called linearly independent if every nite subset of E is linearly independent. Otherwise, E is called linearly dependent. If we denote by ek the sequence whose k-th term is equal to 1 and all the others are 0, then ek is in the space ' and consequently in any space of sequences. The set fe1 e2 : : :g is linearly independent in every one of these spaces. In the space C ( 0 ]), e.g., the system of functions 1 sin nt cos nt (n = 1 2 3 : : :) is linearly independent, but the functions 1 cos 2t cos2 t are linearly dependent (see (2.97), p. 79).
2. Basis and Dimension of a Vector Space
A linearly independent subset B from V, which generates the whole space V, i.e., lin(B ) = V holds, is called an algebraic basis or a Hamel basis of the vector space V (see 5.3.7.2, p. 317).PB = fx : 2 0g is a basis of V if and only if every vector x 2 V can be written in the form x = x , where the 2 coecients are uniquely determined by x and only a nite number of them (depending on x) can be di erent from zero. Every non-trivial vector space V, i.e., V 6= f0g, has at least one algebraic basis, and for every linearly independent subset E of V, there exists at least one algebraic basis of V, which contains E . A vector space V is m-dimensional if it possesses a basis consisting of m vectors. That is, there exist m linearly independent vectors in V, and every system of m + 1 vectors is linearly dependent. A vector space is innite dimensional if it has no nite basis, i.e., if for every natural number m there are m linearly independent vectors in V. The space IFn is n-dimensional, and all the other spaces in examples B{E are in nite dimensional. The subspace lin(f1 t t2g) C ( a b]) is three-dimensional. In the nite dimensional case, every two bases of the same vector space have the same number of elements. Also in an in nite dimensional vector space any two bases have the same cardinality, which is denoted by dim(V). The dimension is an invariant quantity of the vector space, it does not depend on the particular choice of an algebraic basis.
12.1.4 Convex Subsets and the Convex Hull 12.1.4.1 Convex Sets
A subset C of a real vector space V is called convex if for every pair of vectors x y 2 C all vectors of the form x + (1 ; )y 0 1, also belong to C . In other words, the set C is convex, if for any two elements x and y, the whole line segment fx + (1 ; )y : 0 1g (12.16) (which is also called an interval), belongs to C . (For examples of convex sets in IR2 see the sets denoted by A and B in Fig. 12.5, p. 626.) The intersection of an arbitrary number of convex sets is also a convex set, where the empty set is agreed to be convex. Consequently, for every subset E V there exists a smallest convex set which contains E , namely, the intersection of all convex subsets of V containing E . It is called the convex hull of the set E and it is denoted by co (E ). co (E ) is identical to the set of all nite convex linear combinations of elements from E , i.e., co (E ) consists of all elements of the form 1x1 + + nxn, where x1 : : : xn are arbitrary elements from E and i 2 0 1] satisfy the equality 1 + + n = 1. Linear and ane subspaces are always convex.
12.1.4.2 Cones
A non-empty subset C of a (real) vector space V is called a convex cone if it satis es the following properties: 1. C is a convex set.
600 12. Functional Analysis 2. From x 2 C and 0, it follows that x 2 C . 3. From x 2 C and ;x 2 C , it follows that x = 0. A cone can be characterized also by 3. together with x y 2 C and 0 imply x + y 2 C: (12.17) A: The set IRn+ of all vectors x = ( 1 : : : n) with non-negative components is a cone in IRn. B: The set C+ of all real continuous functions on a b] with only non-negative values is a cone in the space C ( a b]). C: The set of all sequences of real numbers f ng1n=1 with only non-negative terms, i.e., n 0 8 n, is a cone in s. Analogously we obtain cones in the spaces of examples C{G on p. 597, if we consider the sets of non-negative sequences in these spaces. D: The set C lp (1 p < 1), consisting of all sequences f ng1n=1, such that for some a > 0 1 X
j njp a (12.18) is a convex set in lp, but obviously, not a cone. E: Examples from IR2 see Fig. 12.1: a) convex set, not a cone, b) not convex, c) convex hull. n=1
y
y
y -x
(0,1)
a)
x
_ 0} E={(x,y): y=e ,x >
x
b)
co (E)
c)
x
Figure 12.1
12.1.5 Linear Operators and Functionals
12.1.5.1 Mappings
A mapping T : D ;! Y from the set D X into the set Y is called injective, if T (x) = T (y) =) x = y (12.19) surjective, if for (12.20) 8 y 2 Y there exists an element x 2 D such that T (x) = y bijective, if T is both injective and surjective. D is called the domain of the mapping T and is denoted by DT or D(T ), while the subset fy 2 Y : 9 x 2 DT with T (x) = yg of Y is called the range of the mapping T and is denoted by R(T ) or Im(T ).
12.1.5.2 Homomorphism and Endomorphism
Let X and Y be two vector spaces over the same eld IF and D a linear subset of X. A mapping T : D ;! Y is called linear (or a linear transformation, linear operator or homomorphism), if for arbitrary x y 2 D and 2 IF, T (x + y) = Tx + Ty: (12.21) For a linear operator T we prefer the notation Tx, which is similarly used for linear functions, while the notation T (x) is used for general operators. N (T ) = fx 2 X : Tx = 0g is the null space or kernel
12.1 Vector Spaces 601
of the operator T and is also denoted by ker(T ). A mapping of the vector space X into itself is called an endomorphism. If T is an injective linear mapping, then the mapping de ned on R(T ) by y 7;! x such that Tx = y y 2 R(T ) (12.22) is linear. It is denoted by T ;1 : R(T ) ;! X and is called the inverse of T . If Y is the vector space IF, then a linear mapping f: X ;! IF is called a linear functional or a linear form.
12.1.5.3 Isomorphic Vector Spaces
A bijective linear mapping T : X ;! Y is called an isomorphism of the vector spaces X and Y. Two vector spaces are called isomorphic provided an isomorphism exists.
12.1.6 Complexi cation of Real Vector Spaces
Every real vector space V can be extended to a complex vector space V~ . The set V~ consists of all pairs (x y) with x y 2 V. The operations (addition and multiplication by a complex number a + ib 2 C) are de ned as follows: (x1 y1) + (x2 y2) = (x1 + x2 y1 + y2) (12.23a) (a + ib)(x y) = (ax ; by bx + ay): (12.23b) Since the special relations (x y) = (x 0) + (0 y) and i(y 0) = (0 + i1)(y 0) = (0 y ; 1 0 1y + 0 0) = (0 y) (12.24) hold, the pair (x y) can also be written as x + iy. The set V~ is a complex vector space, where the set V is identi ed with the linear subspace V~ 0 = f(x 0): x 2 Vg, i.e., x 2 V is considered as (x 0) or as x + i0. This procedure is called the complexication of the vector space V. A linearly independent subset in V is also linearly independent in V~ . The same statement is valid for a basis in V, so dim(V) = dim(V~ ).
12.1.7 Ordered Vector Spaces
12.1.7.1 Cone and Partial Ordering
If a cone C is xed in a vector space V, then an order can be introduced for certain pairs of vectors in V. Namely, if x ; y 2 C for some x y 2 V then we write x y or y x and say x is greater than or equal to y or y is smaller than or equal to x. The pair (V C ) is called an ordered vector space or a vector space partially ordered by the cone C . An element x is called positive, if x 0 or, which means the same, if x 2 C holds. Moreover C = fx 2 V: x 0g: (12.25) If we consider the vector space IR2 ordered by its rst quadrant as the cone C (= IR2+), then a typical phenomenon of ordered vector spaces will be seen. This is referred to as \partially ordered" or sometimes as \semi-ordered". Namely, only several pairs of two vectors are comparable. Considering the vectors x = (1 ;1) and y = (0 2), neither the vector x ; y = (1 ;3) nor y ; x = (;1 3) is in C , so neither x y nor x y holds. An ordering in a vector space, generated by a cone, is always only a partial ordering. It can be shown that the binary relation has the following properties: (O1) x x 8 x 2 V (reexivity): (12.26) (O2) x y and y z imply x z (transitivity):
(12.27)
(O3) x y and 0 2 IR imply x y:
(12.28)
(O4) x1 y1 and x2 y2 imply x1 + x2 y1 + y2:
(12.29)
602 12. Functional Analysis Conversely, if in a vector space V there exists an ordering relation, i.e., a binary relation is de ned for certain pairs of elements and satis es axioms (O1){(O4), and if one puts V+ = fx 2 V: x 0g (12.30) then it can be shown that V+ is a cone. The order V+ in V induced by V+ is identical to the original order consequently, the two possibilities of introducing an order in a vector space are equivalent. A cone C X is called generating or reproducing if every element x 2 X can be represented as x = u ; v with u v 2 C . We also write X = C {C . A: An obvious order in the space s (see example B, p. 597) is induced by means of the cone C = fx = f ng1 (12.31) n=1 : n 0 8 ng (see example C, p. 600). We usually consider the natural coordinatewise order in the spaces of sequences (see (12.12), p 598). This is de ned by the cone obtained as the intersection of the considered space with C (see (12.31), p. 602). The positive elements in these ordered vector spaces are then the sequences with non-negative terms. It is clear that we can de ne other orders by other cones, as well. Then we obtain orderings di erent from the natural ordering (see 12.17], 12.19]). B: In the real spaces of functions F (T ) B(T ) C ( a b]) and C (k) ( a b]) (see 12.1.2, 5., p. 598), we de ne the natural order x y for two functions x and y by x(t) y(t) 8 t 2 T or 8 t 2 a b]. Then x 0 if and only if x is a non-negative function in T . The corresponding cones are denoted by F+(T ) B+(T ), etc. We can also obtain C+ = C+(T ) = F+(T ) \ C (T ) if T = a b].
12.1.7.2 Order Bounded Sets
Let E be an arbitrary non-empty subset of an ordered vector space V. An element z 2 V is called an upper bound of the set E if for every x 2 E , x z. An element u 2 V is a lower bound of E if u x 8 x 2 E . For any two elements x y 2 V with x y, the set x y] = fv 2 V: x v yg (12.32) is called an order interval or (o)-interval. Obviously, the elements x and y are a lower bound and an upper bound of the set x y], respectively, where they even belong to the set. A set E V is called order bounded or simply (o)-bounded, if E is a subset of an order interval, i.e., if there exist two elements u z 2 V such that u x z 8 x 2 E or, equivalently, E u z]. A set is called bounded above or bounded below if it has an upper bound, or a lower bound, respectively.
12.1.7.3 Positive Operators
A linear operator (see 12.2], 12.17]) T : X ;! Y from an ordered vector space X = (X X+) into an ordered vector space Y = (Y Y+) is called positive, if T (X+) Y+ i.e., Tx 0 for all x 0: (12.33)
12.1.7.4 Vector Lattices 1. Vector Lattices1
In the vector space IR of the real numbers the notions of (o)-boundedness and boundedness (in the usual sense) are identical. It is known that every set of real numbers which is bounded from above has a supremum: the smallest of its upper bounds (or the least upper bound, sometimes denoted by lub). Analogously, if a set of reals is bounded from below, then it has an inmum, the greatest lower bound, sometimes denoted by glb. In a general ordered vector space, the existence of the supremum and in mum cannot be guaranteed even for nite sets. They must be given by axioms. An ordered vector space V is called a vector lattice or a linear lattice or a Riesz space, if for two arbitrary elements x y 2 V there exists an element z 2 V with the following properties: 1. x z and y z, 2. if u 2 V with x u and y u, then z u.
12.1 Vector Spaces 603
Such an element z is uniquely determined, is denoted by x _ y, and is called the supremum of x and y (more precisely: supremum of the set consisting of the elements x and y). In a vector lattice, there also exists the in mum for any x and y, which is denoted by x ^ y. For applications of positive operators in vector lattices see, e.g., 12.2], 12.3] 12.15]. A vector lattice in which every non-empty subset E that is order bounded from above has a supremum lub(E ) (equivalently, if every non-empty subset that is order bounded from below has an in mum glb(E )) is called Dedekind complete or a K-space (Kantorovich space). A: In the vector lattice F ( a b]) (see 12.1.2, 5., p. 598), the supremum of two functions x y is calculated pointwise by the formula 1 (x _ y)(t) = maxfx(t) y(t)g 8 t 2 a b]: (12.34) In the case of a b] = 0 1], x(t) = 1 ; 23 t and y(t) = t2 (Fig. 12.2), we get ( 3 t 21 y(t) (x _ y)(t) = 1 ; 2t2t ifif 01 (12.35) 2 t 1: B: The spaces C ( a b]) and B( a b]) (see 12.1.2, 5., p. 598) are 1 _ x(t) 1 t 0 also vector lattices, while the ordered vector space C (1) ( a b]) is not 2 a vector lattice, since the minimum or maximum of two di erentiable functions may not be di erentiable on a b], in general. - 1_2 A linear operator T : X ;! Y from a vector lattice X into a vector lattice Y is called a vector lattice homomorphism or homomorphism Figure 12.2 of the vector lattice, if for all x y 2 X
T (x _ y) = Tx _ Ty and T (x ^ y) = Tx ^ Ty:
(12.36)
2. Positive and Negative Parts, Modulus of an Element
For an arbitrary element x of a vector lattice V, the elements x+ = x _ 0 x; = (;x) _ 0 and jxj = x+ + x; (12.37) are called the positive part, negative part, and modulus of the element x, respectively. For every element x 2 V, the three elements x+ x; jxj are positive, where for x y 2 V the following relations are valid: x x+ jxj x = x+ ; x; x+ ^ x; = 0 jxj = x _ (;x) (12.38a) (x + y)+ x+ + y+ (x + y); x; + y; jx + yj jxj + jyj (12.38b) x y implies x+ y+ and x; y; (12.38c) and for arbitrary 0 ( x)+ = x+ ( x); = x; j xj = jxj: (12.38d) a
a 0
x(t)
b
0
x+(t) b
0 a
x-(t) b
0 a
|x|(t) b
Figure 12.3 In the vector spaces F ( a b]) and C ( a b]), we get the positive part, the negative part, and the modulus of a function x(t) by means of the following formulas (Fig. 12.3):
604 12. Functional Analysis 0 x+(t) = x(t0) ifif xx((tt)) 0
(12.39a) (12.39b)
jxj(t) = jx(t)j 8 t 2 a b]:
(12.39c)
12.2 Metric Spaces
12.2.1 Notion of a Metric Space
Let X be a set, and suppose a real, non-negative function (x y) (x y 2 X) is de ned on X X. If this function : X X ;! IR1+ satis es the following properties (M1){(M3) for arbitrary elements x y z 2 X, then it is called a metric or distance in the set X, and the pair X = (X ) is called a metric space. The axioms of metric spaces are: (M1) (x y) 0 and (x y) = 0 if and only if x = y (non-negativity) (12.40) (M2) (x y) = (y x) (symmetry) (12.41) (M3) (x y) (x z) + (z y) (triangle inequality): (12.42) A metric can be de ned on every subset Y of a metric space X = (X ) in a natural way if we restrict the metric of the space X to the set Y, i.e., if we consider only on the subset Y Y. The space (Y ) of X X is called a subspace of the metric space X. A: The sets IRn and Cn are metric spaces with the Euclidean metric de ned for points x = ( 1 : : : n) and y = (1 : : : n) as
v u n uX (x y) = t j k ; k j2: k=1
(12.43)
B: The function
(x y) = 1max j k ; k j (12.44)
k n n n for vectors x = ( 1 : : : n) and y = (1 : : : n) also de nes a metric in IR and C , the so-called maximum metric. If x~ = ( ~1 : : : ~n) is an approximation of the vector x, then it is of interest to know how much is the maximal deviation between the coordinates: 1max j k ; ~k j.
k n The function n X (x y) = j k ; k j (12.45) k=1
for vectors x y 2 IRn (or Cn) de nes a metric in IRn and C, the so-called absolute value metric. The metrics (12.43), (12.44) and (12.45) are reduced in the case of n = 1 to the absolute value jx ; yj in the spaces IR = IR1 and C (the sets of real and complex numbers). C: Finite 0-1 sequences, e.g., 1110 and 010110, are called words in coding theory. If we count the number of positions where two words of the same length n have di erent digits, i.e., for x = ( 1 : : : n) y = (1 : : : n) k k 2 f0 1g, we de ne (x y) as the number of the k 2 f1 : : : ng values such that k 6= k , then the set of words with a given length n is a metric space, and the metric is the so-called Hamming distance, e.g., ((1110) (0100)) = 2. D: In the set m and in its subsets c and c0 (see (12.12), p. 598) a metric is de ned by (x y) = sup j k ; k j (x = ( 1 2 : : :) y = 1 2 : : :)): (12.46) k
12.2 Metric Spaces 605
E: In the set lp (1 p < 1) of sequences x = ( 1 2 : : :) with absolutely convergent series nP=1 j njp 1
a metric is de ned by
v u 1 uX p (x y) = t j n ; njp
(x y 2 lp):
(12.47)
F: In the set C ( a b]) a metric is de ned by (x y) = tmax jx(t) ; y(t)j: 2ab]
(12.48)
n=1
G: In the set C (k) ( a b]) a metric is de ned by k X
max jx(l) (t) ; y(l)(t)j (12.49) l=0 t2ab] where (see (12.14) C (0) ( a b]) is understood as C ( a b])). H: Consider the set Lp()) (1 p < 1) of the equivalence classes of Lebesgue measurable functions
(x y) =
Z
which are de ned almost everywhere on a bounded domain ) IRn and jx(t)jp d < 1 (see also
12.9, p. 635). A metric in this set is de ned by
v uZ t jx(t) ; y(t)jp d : (x y) = pu
(12.50)
12.2.1.1 Balls, Neighborhoods and Open Sets
In a metric space X = (X ), whose elements are also called points, the following sets B (x0 r) = fx 2 X : (x x0 ) < rg (12.51) B (x0 r) = fx 2 X : (x x0 ) rg (12.52) de ned by means of a real number r > 0 and a xed point x0 , are called an open and closed ball with radius r and center at x0 , respectively. The balls (circles) de ned by the metrics (12.43) and (12.44) and (12.45) in the vector space IR2 are represented in Fig. 12.4a,b with x0 = 0 and r = 1. y
y
(0,1)
y
(0,1) 2
(0,1)
2
{(x, y): x +y £1} (1,0) x a)
(1,0) x
(1,0) x c)
b)
Figure 12.4 A subset U of a metric space X = (X ) is called a neighborhood of the point x0 if U contains x0 together with an open ball centered at x0 , in other words, if there exists an r > 0 such that B (x0 r) U . A neighborhood U of the point x is also denoted by U (x). Obviously, every ball is a neighborhood of its center an open ball is a neighborhood of all of its points. A point x0 is called an interior point of a set A X if x0 belongs to A together with some of its neighborhood, i.e., there is a neighborhood N of x0 such that x0 2 U A. A subset of a metric space is called open if all of its points are interior points. Obviously, X is an open set. The open balls in every metric space, especially the open intervals in IR, are the prototypes of open sets.
606 12. Functional Analysis The set of all open sets satis es the following axioms of open sets: If G is open for 8 2 I , then the set S2I G is also open. n If G1 G2 : : : Gn are nitely many arbitrary open sets, then the set kT=1 Gk is also open. The empty set is open by de nition. A subset A of a metric space is bounded if for a certain element x0 (which does not necessarily belong to A) and a real number R > 0 the set A is in the ball B (x0 R), i.e., (x x0 ) < R for all x 2 A.
12.2.1.2 Convergence of Sequences in Metric Spaces
Let X = (X ) be a metric space, x0 2 X and fxng1 n=1 xn 2 X a sequence of elements of X. The sequence fxng1 n=1 is said to be convergent to the point x0 if for every neighborhood U (x0 ) there is an index n0 = n0(U (x0 )) such that for all n > n0, xn 2 U (x0 ). We use the usual notation xn ! x0 (n ! 1) or nlim x = x0 (12.53) !1 n and call the point x0 the limit of the sequence fxng1 . The limit of a sequence is uniquely determined. n=1 Instead of an arbitrary neighborhood of the point x0 , it is sucient to consider only open balls with arbitrary radii, so (12.53) is equivalent to the following: 8 " > 0 (we are at once thinking about the open ball B (x0 ")), there is an index n0 = n0("), such that if n > n0 , then (xn x0 ) < ". Notice that (12.53) means (xn x0 ) ! 0. With these notions introduced in special metric spaces we can calculate the distance between points and investigate the convergence of point sequences. This has a great importance in numerical methods and in approximating functions by certain classes of functions (see, e.g., 19.6, p. 917). In the space IRn, equipped with one of the metrics given above, convergence always means coordinatewise convergence. In the spaces B( a b]) and C ( a b]), the convergence introduced by (12.48) means uniform convergence of the function sequence on the set a b] (see 7.3.2, p. 414). In the space L2 ()) convergence with respect to the metric (12.50) means convergence in the (quadratic) mean, i.e., xn ! x0 if
Z
jxn ; x0 j2 d ! 0 for n ! 1 :
(12.54)
12.2.1.3 Closed Sets and Closure 1. Closed Sets
A subset F of a metric space X is called closed if X n F is an open set. Every closed ball in a metric space, especially every interval of the form a b] a 1) (;1 a] in IR, is a closed set. Corresponding to the axioms of open sets, the collection of all closed sets of a metric space has the following properties: If F are closed for 8 2 I , then the set T2I F is closed. n If F1 : : : Fn are nitely many closed sets, then the set kS=1 Fk is closed. The empty set is a closed set by de nition. The sets and X are open and closed at the same time. A point x0 of a metric space X is called a limit point of the subset A X if for every neighborhood U (x0 ), U (x0 ) \ A 6= : (12.55) If this intersection always contains at least one point di erent from x0 , then x0 is called an accumulation point of the set A. A limit point, which is not an accumulation point, is called an isolated point. An accumulation point of A does not need to belong to the set A, e.g., the point a with respect to the set A = (a b], while an isolated point of A must belong to the set A.
12.2 Metric Spaces 607
A point x0 is a limit point of the set A if there exists a sequence fxng1 n=1 with elements xn from A, which converges to x0 . If x0 is an isolated point, then xn = x0 8 n n0 for some index n0 .
2. The Closure of a Set
Every subset A of a metric space X obviously lies in the closed set X. Therefore, there always exists a smallest closed set containing A, namely the intersection of all closed sets of X, which contain A. This set is called the closure of the set A and it is usually denoted by A. A is identical to the set of all limit points of A we get A from the set A by adding all of its accumulation points to it. A is a closed set if and only if A = A. Consequently, closed sets can be characterized by sequences in the following way: A is closed if and only if for every sequence fxng1 n=1 of elements of A, which converges in X to an element x0 (2 X), the limit x0 also belongs to A. Boundary points of A are de ned as follows: x0 is a boundary point of A if for every neighborhood U (x0 ), U (x0 ) \ A 6= and also U (x0 ) \ (X n A) 6= . x0 itself does not need to belong to A. Another characterization of a closed set is the following: A is closed if it contains all of its boundary points. (The set of boundary points of the metric space X is the empty set.)
12.2.1.4 Dense Subsets and Separable Metric Spaces
A subset A of a metric space X is called everywhere dense if A = X, i.e., each point x 2 X is a limit point of the set A. That is, for each x 2 X, there is a sequence fxng xn 2 A such that xn ! x. A: According to the Weierstrass approximation theorem, every continuous function on a bounded closed interval a b] can be approximated arbitrarily well by polynomials in the metric space of the space C ( a b]), i.e., uniformly. This theorem can now be formulated as follows: The set of polynomials on the interval a b] is everywhere dense in C ( a b]). B: Further examples for everywhere dense subsets are the set of rational numbers Q and the set of irrational numbers in the space of the real numbers IR . A metric space X is called separable if there exists a countable everywhere dense subset in X. A countable everywhere dense subset in IRn is, e.g., the set of all vectors with rational components. The space l = l1 is also separable, since a countable everywhere dense subset is formed, for example, by the set of its elements of the form x = (r1 r2 : : : rN 0 0 : : :) , where ri are rational numbers and N = N (x) is an arbitrary natural number. The space m is not separable.
12.2.2 Complete Metric Spaces 12.2.2.1 Cauchy Sequences
Let X = (X ) be a metric space. A sequence fxng1 n=1 with xn 2 X is called a Cauchy sequence if for 8 " > 0 there is an index n0 = n0 (") such that for 8 n m > n0 there holds the inequality (xn xm ) < ": (12.56) Every Cauchy sequence is a bounded set. Furthermore, every convergent sequence is a Cauchy sequence. In general, the converse statement is not true, as is shown in the following example. the space l1 with the metric (12.46) of the space m. Obviously, the elements x(n) = Consider 1 1 1 : : : n1 0 0 : : : belong to l1 for every n = 1 2 : : : and the sequence fx(n) g1 n=1 is a Cauchy 2 3 ( n ) 1 sequence in this space. If the sequence (of sequences) 1 1 fx 1 gn=11 converges, then it has to be convergent (0) also coordinate-wise to the element x = 1 2 3 : : : n n + 1 : : : . However, x(0) does not belong 1 X to l1, since 1 = +1 (see 7.2.1.1, 2., p. 406, harmonic series). n=1 n
608 12. Functional Analysis
12.2.2.2 Complete Metric Spaces
A metric space X is called complete if every Cauchy sequence converges in X. Hence, complete metric spaces are the spaces for which the Cauchy principle, known from real calculus, is valid: A sequence is convergent if and only if it is a Cauchy sequence. Every closed subspace of a complete metric space (considered as a metric space on its own) is complete. The converse statement is valid in a certain way: If a subspace Y of a (not necessary complete) metric space X is complete, then the set Y is closed in X. Complete metric spaces are, e.g., the spaces: m lp (1 p < 1), c B(T ), C ( a b]), C (k) ( a b]), Lp(a b) (1 p < 1).
12.2.2.3 Some Fundamental Theorems in Complete Metric Spaces
The importance of complete metric spaces can be illustrated by a series of theorems and principles, which are known and used in real calculus, and which we want to apply even in the case of in nite dimensional spaces.
1. Theorem on Nested Balls
Let X be a complete metric space. If B (x1 r1) % B (x2 r2) % % B (xn rn) % (12.57) is a sequence of nested closed balls with rn ;! 0, then the intersection of all of those balls is nonempty and consists of only a single point. If this property is valid in some metric space for any sequence satisfying the assumptions, then the metric space is complete.
2. Baire Category Theorem
S F = X. Then Let X be a complete metric space and fFk g1 k k=1 a sequence of closed sets in X with k=1 there exists at least one index k0 such that the set Fk0 has an interior point. 1
3. Banach Fixed-Point Theorem
Let F be a non-empty closed subset of a complete metric space (X ). Let T: X ;! X be a contracting operator on F , i.e., there exists a constant q 2 0 1) such that (Tx Ty) q (x y) for all x y 2 F: (12.58) Suppose, if x 2 F , then Tx 2 F . Then the following statements are valid: a) For an arbitrary initial point x0 2 F the iteration xn+1 := Txn (n = 0 1 2 : : :) (12.59) is well de ned, i.e., xn 2 F for every n. b) The iteration sequence fxng1n=0 converges to an element x 2 F . c) Tx = x i.e., x is a xed point of the operator T: (12.60) d) The only xed point of T in F is x . e) The following error estimation is valid: n (x xn) 1 q; q (x1 x0 ): (12.61) The Banach xed-point theorem is sometimes called the contraction mapping principle .
12.2.2.4 Some Applications of the Contraction Mapping Principle 1. Iteration Method for Solving a System of Linear Equations The given linear (n n) system of equations a11 x1 +a12x2 + : : : + a1n x1 = b1 a21 x1 +a22x2 + : : : + a2nxn = b2 ................................. an1 x1 +an2x2 + : : : + annxn = bn
(12.62a)
12.2 Metric Spaces 609
can be transformed according to 19.2.1, p. 890, into the equivalent system x1 ;(1 ; a11 )x1 +a12 x2 + +a1n xn = b1 x2 +a21 x1 ;(1 ; a22 )x2 + +a2n xn = b2 ...................................................... xn +an1x1 +an2 x2 + ;(1 ; ann)xn = bn: If the operator T : IFn ;! IFn is de ned by
Tx = x1 ;
n X
k=1
a1k xk + b1 : : : xn ;
n X
k=1
ank xk + bn
!T
(12.62b)
(12.63)
then the last system is transformed into the xed-point problem x = Tx (12.64) in the metric space IFn, where an appropriate metric is considered: The Euclidean (12.43), the maxin mum (12.44) or the absolute value metric (x y) = P jxk ; yk j (compare with (12.45)). If one of the k=1 numbers v
u n u t X jajk j2
n X max ja j 1 j n k=1 jk
jk=1
n X max ja j 1 k n j =1 jk
(12.65)
is smaller than one, then T turns out to be a contracting operator. It has exactly one xed point according to the Banach xed-point theorem, which is the componentwise limitof the iteration sequence started from an arbitrary point of IFn.
2. Fredholm Integral Equations
The Fredholm integral equation of the second kind (see also 11.2, p. 564)
Zb '(x) ; K (x y)'(y) dy = f (x) a
x 2 a b]
(12.66)
with a continuous kernel K (x y) and continuous right-hand side f (x) can be solved by iteration. By means of the operator T : C ( a b]) ;! C ( a b]) de ned as
Zb T'(x) = K (x y)'(y) dy + f (x) 8 ' 2 C ( a b]) a
(12.67)
it is transformed into a xed-point problem T' = ' in the metric space C ( a b]) (see example A in Zb 12.1.2, 4., p. 597). If amax jK (x y)j dy < 1, then T is a contracting operator and the xed-point
x b a theorem can be applied. The unique solution is now obtained as the uniform limit of the iteration sequence f'ng1 n=1 , where 'n = T'n;1 , starting with an arbitrary function '0 (x) 2 C ( a b]). It is clear that 'n = T n'0 and the iteration sequence is fT n'0g1 n=1 .
3. Volterra Integral Equations
The Volterra integral equation of the second kind (see 11.4, p. 585)
Zx '(x) ; K (x y)'(y) dy = f (x) a
x 2 a b]
(12.68)
with a continuous kernel and a continuous right-hand side can be solved by means of the Volterra integral operator
Zx
(V ')(x) := K (x y)'(y) dy 8 ' 2 C ( a b]) a
(12.69)
610 12. Functional Analysis and T' = f + V ' as the xed-point problem T' = ' in the space C ( a b]).
4. Picard{Lindelof Theorem
Consider the di erential equation x_ = f (t x) (12.70) withna continuous mapping f : I G ;! IRn, where I is an open interval of IR and G is an open domain of IR . Suppose the function f satis es a Lipschitz condition with respect to x (see 9.1.1.1, 2. p. 488), i.e., there is a positive constant L such that %(f (t x1 ) f (t x2)) L%(x1 x2) 8 (t x1 ) (t x2) 2 I G (12.71) where % is the Euclidean metric in IRn. (Using the norm (see 12.3.1, p. 611) and the formula (12.81) %(x y) = kx ; yk (12.71) can be written as kf (t x1) ; f (t x2 )k kx1 ; x2 kn.) Let (t0 x0 ) 2 I G. Then there are numbers > 0 and r > 0 such that the set ) = f(t x) 2 IR IR : jt ; t0 j %(x x0 ) rg lies in I G. Let M = max %(f (t x) 0) and = minf r g. Then there is a number b > 0 such M that for each x~ 2 B = fx 2 IRn : %(x x0 ) bg, the initial value problem x_ = f (t x) x(t0 ) = x~ (12.72) has exactly one solution '(t x~), i.e., '_ (t x~) = f (t '(t x~)) for 8 t satisfying jt ; t0j and '(t0 x~) = x~. The solution of this initial value problem is equivalent to the solution of the integral equation
Zt '(t x~) = x~ + f (s '(s x~)) ds t 2 t0 ; t0 + ]: t0
(12.73)
If X denotes the closed ball f'(t x): d('(t x) x0) rg in the complete metric space C ( t0 ; t0 + ] B IRn) with metric d(' ) = (tx)2fjmax %('(t x) (t x)) (12.74) t;t j g B 0
then X is a complete metric space with the induced metric. If the operator T : X ;! X is de ned by
Zt T'(t x) = x~ + f (s '(s x~)) ds t0
(12.75)
then T is a contracting operator and the solution of the integral equation (12.73) is the unique xed point of T which can be calculated by iteration.
12.2.2.5 Completion of a Metric Space
Every (non-complete) metric space X can be completed more precisely, there exists a metric space X~ with the following properties: a) X~ contains a subspace Y isometric to X (see 12.2.3, 2., p. 611). b) Y is everywhere dense in X~ . c) X~ is a complete metric space. d) If Z is any metric space with the properties a){c), then Z and X~ are isometric. The complete metric space, de ned uniquely in this way up to isometry, is called the completion of the space X.
12.2.3 Continuous Operators 1. Continuous Operators
Let T : X ;! Y be a mapping of the metric space X = (X ) into the metric space Y = (Y %). T is said to be continuous at the point x0 2 X if for every neighborhood V = V (y0) of the point y0 = T (x0 )
12.3 Normed Spaces 611
there is a neighborhood U = U (x0 ) such that: T (x) 2 V for all x 2 U: (12.76) T is called continuous on the set A X if T is continuous at every point of A. Equivalent properties for T to be continuous on X are: a) For any point x 2 X and any arbitrary sequence fxng1n=1 xn 2 X with xn ! x there always holds T (xn) ;! T (x). Hence (xn x0 ) ! 0 implies %(T (xn) T (x0)) ! 0. b) For any open subset G Y the inverse image T ;1(G) is an open subset in X. c) For any closed subset F Y the inverse image T ;1(F ) is a closed subset in X. d) For any subset A X one has T (A) T (A).
2. Isometric Spaces
If there is a bijective mapping T : X ;! Y for two metric spaces X = (X ) and Y = (Y %) such that (12.77) (x y) = %(T (x) T (y)) 8 x y 2 X then the spaces X and Y are called isometric, and T is called an isometry.
12.3 Normed Spaces
12.3.1 Notion of a Normed Space 12.3.1.1 Axioms of a Normed Space
Let X be a vector space over the eld IF. A function k k : X ;! IR1+ is called a norm on the vector space X and the pair X = (X k k) is called a normed space over the eld IF, if for arbitrary elements x y 2 X and for any scalar 2 IF the following properties, the so-called axioms of a normed space, are ful lled: (N1) kxk 0 and kxk = 0 if and only if x = 0 (12.78) (N2) kxk = jj kxk (homogenity), (12.79) (N3) kx + yk kxk + kyk (triangle inequality): (12.80) A metric can be introduced by means of (x y) = kx ; yk x y 2 X (12.81) in any normed space. The metric (12.81) has the following additional properties which are compatible with the structure of the vector space: (x + z y + z) = (x y) z2X (12.82a) (x y) = jj (x y) 2 IF: (12.82b) So, in a normed space there are available both the properties of a vector space and the properties of a metric space. These properties are compatible in the sense of (12.82a) and (12.82b). The advantage is that most of the local investigations can be restricted to the unit ball B (0 1) = fx 2 X : kxk < 1g or B (0 1) = fx 2 X : kxk 1g (12.83) since B (x r) = fy 2 X : ky ; xk < rg = x + rB (0 1) 8 x 2 X and 8 r > 0: (12.84) Moreover, the algebraic operations in a vector space are continuous, i.e., xn ! x yn ! y n ! imply xn + yn ! x + y nxn ! x kxn k ! kxk: (12.85) In normed spaces instead of (12.53) we may write for convergent sequences kxn ; x0 k ;! 0 (n ! 1): (12.86)
612 12. Functional Analysis
12.3.1.2 Some Properties of Normed Spaces
Among the linear metric spaces, those spaces are normable (i.e., a norm can be introduced by means of the metric, if one de nes kxk = (x 0)) whose metric satis es the conditions (12.82a) and (12.82b). Two normed spaces X and Y are called norm isomorphic if there is a bijective linear mapping T : X ;! Y with kTxk = kxk for all x 2 X. Let k k1 and k k2 be two norms on the vector space X, and denote the corresponding normed spaces by X1 and X2, i.e., X1 = (X k k1) and X2 = (X k k2). The norm k k1 is stronger than the norm k k2, if there is a number > 0 such that kxk2 kxk1 , for all x 2 X. In this case, the convergence of a sequence fxng1 n=1 to x with respect to the stronger norm k k1, i.e., kxn ; xk1 ! 0, implies the convergence to x with respect to the norm k k2, i.e., kxn ; xk2 ! 0. Two norms kk and kjkj are called equivalent if there are two numbers 1 > 0 2 > 0 such that 8 x 2 X there holds 1kxk kjxkj 2kxk. In a nite dimensional vector space all norms are equivalent to each other. A subspace of a normed space is a closed linear subspace of the space.
12.3.2 Banach Spaces
A complete normed space is called a Banach space. Every normed space X can be completed into a Banach space X~ by the completion procedure given in 12.2.2.5, p. 610, and by the natural extension of its algebraic operations and the norm to X~ .
12.3.2.1 Series in Normed Spaces
In a normed space X we can consider innite series. That means for a given sequence fxng1 n=1 of elements xn 2 X a new sequence fsk g1 k=1 is constructed by s1 = x1 s2 = x1 + x2 : : : sk = x1 + + xk = sk;1 + xk : (12.87) If the sequence fsk g1 is convergent, i.e., k s ; s k ! 0 ( k ! 1 ) for some s 2 X , then a convergent k k=1 series is de ned. The elements s1 s2 : : : sk : : : are called the partial sums of the series. The limit
s = klim !1
k X
n=1
xn
(12.88)
1 1 is the sum of the series, and we write s = P xn. A series P xn is called absolutely convergent if the
n=1
n=1
1 number series P kxnk is convergent. In a Banach space every absolutely convergent series is conver-
n=1
1 gent, and ksk P kxnk holds for its sum s.
n=1
12.3.2.2 Examples of Banach ! 1 Spaces A : IFn with kxk =
n X
k=1
j kjp
p
if 1 p < 1 kxk = 1max j k j
k n
if p = 1:
(12.89a)
These normed spaces over the same vector space IFn are often denoted by lp(n) (1 p 1). For 1 p < 1, we call them Euclidean spaces in the case of IF = IR, and unitary spaces in the case of IF = C. B : m with kxk = sup j k j: (12.89b) k C : c and c0 with the norm from m: (12.89c)
D : lp with kxk = kxkp =
1 X
n=1
j njp
! 1p
(1 p < 1):
(12.89d)
12.3 Normed Spaces 613
E : C ( a b]) with kxk = tmax jx(t)j: 2ab]
0Zb 1 p1 p p @ F : L ((a b)) (1 p < 1) with kxk = kxkp = jx(t)j dtA :
G : C (k) ( a b]) with kxk =
k X
a
max jx(l) (t)j
l=0 t2ab]
where x(0) (t) stands for x(t):
(12.89e) (12.89f) (12.89g)
12.3.2.3 Sobolev Spaces
Let ) IRn be a bounded domain, i.e., an open connected set, with a suciently smooth boundary @ ). For n = 1 or n = 2 3 we can imagine ) being something similar to an interval (a b) or a convex set. A function f : ) ;! IR is k-times continuously di erentiable on the closed domain ) if f is k-times continuously di erentiable on ) and each of its partial derivatives has a nite limit on the boundary, i.e., if x approaches an arbitrary point of @ ). In other words, all partial derivatives can be continuously extended on the boundary of ), i.e., each partial derivative is na continuous function on ). In this vector space (for p 2 1 1)) and with the Lebesgue measure in IR (see example C in 12.9.1, 2., p. 636) the following norm is de ned: 0 1 p1 Z X Z B p p kf kkp = kf k = @ jf (x)j d + jD f j dCA : (12.90)
1 j
j k
The resulting normed space is denoted by W~ kp()) or also by W~ pk ()) (in contrast to the space C (k)( a b]) which has a quite di erent norm). Here means a multi-index, i.e., an ordered n-tuple (1 : : : n) of non-negative integers, where the sum of the components of is denoted by jj = 1 + 2 + + n. For a function f (x) = f ( 1 : : : n) with x = ( 1 : : : n) 2 ) we use the brief notation as in (12.90): j j D f = @ 1@ f@ n : (12.91) 1 n The normed space W~ kp()) is not complete. Its completion is denoted by W kp()) or in the case of p = 2 by IHk ()) and it is called a Sobolev space.
12.3.3 Ordered Normed Spaces 1. Cones in a Normed Space
Let X be a real normed space with the norm kk. A cone X+ X (see 12.1.4.2, p. 599) is called a solid, if X+ contains a ball (with positive radius), or equivalently, X+ contains at least one interior point. The usual cones are solid in the spaces IR C ( a b]) c, but in the spaces Lp((a b)) and lp (1 p < 1) they are not solid. A cone X+ is called normal if the norm in X is semimonotonic, i.e., there exists a constant M > 0 such that 0 x y =) kxk M kyk: (12.92) If X is a Banach space ordered by a cone X+, then every (o)-interval is bounded with respect to the norm if and only if the cone X+ is normal. The cones of the vectors with non-negative components and of the non-negative functions in the spaces IRn m c c0 C lp and Lp, respectively, are normal. A cone is called regular if every monotonically increasing sequence which is bounded above, x1 x2 xn z (12.93)
614 12. Functional Analysis is a Cauchy sequence in X. In a Banach space every closed regular cone is normal. The cones in IRn lp and Lp for 1 p < 1 are regular, but in C and m they are not.
2. Normed Vector Lattices and Banach Lattices
Let X be a vector lattice, which is a normed space at the same time. X is called a normed lattice or normed vector lattice (see 12.15], 12.19], 12.22], 12.23]), if the norm satis es the condition jxj jyj implies kxk kyk 8 x y 2 X (monotonicity of the norm): (12.94) A complete (with respect to the norm) normed lattice is called a Banach lattice. The spaces C ( a b]) Lp lp B( a b]) are Banach lattices.
12.3.4 Normed Algebras
A vector space X over IF is called an algebra, if in addition to the operations de ned in the vector space X and satisfying the axioms (V1){(V8) (see 12.1.1, p. 596), a product x y 2 X is also de ned for every two elements x y 2 X, or with a simpli ed notation the product xy is de ned so that for arbitrary x y z 2 X and 2 IF the following conditions are satis ed: (A1) x(yz) = (xy)z (12.95) (A2) x(y + z) = xy + xz (12.96) (A3) (x + y)z = xz + yz (12.97) (A4) (xy) = (x)y = x(y): (12.98) An algebra is commutative if xy = yx holds for two arbitrary elements x y. A linear operator (see (12.21), p. 600) T : X ;! Y of the algebra X into the algebra Y is called an algebra homomorphism if for any x1 x2 2 X: T (x1 x2 ) = Tx1 Tx2: (12.99) An algebra X is called a normed algebra or a Banach algebra if it is a normed vector space or a Banach space and the norm has the additional property kx yk kxk kyk: (12.100) In a normed algebra all the operations are continuous, i.e., additionally to (12.85), if xn ;! x and yn ;! y, then also xnyn ;! xy (see 12.20]). Every normed algebra can be completed to a Banach algebra, where the product is extended to the norm completion with respect to (12.100). A: C ( a b]) with the norm (12.89e) and the usual (pointwise) product of continuous functions. B: The vector space W ( 0 2]) of all complex-valued functions x(t) continuous on 0 2] and having an absolutely convergent Fourier series expansion, i.e.,
x(t) =
1 X
n=;1
cneint
(12.101)
1 with the norm kxk = P jcnj and the usual multiplication.
n=;1
C: The space L(X) of all bounded linear operators on the normed space X with the operator norm and the usual algebraic operations (see 12.5.1.2, p. 619), where the product T S of two operators is de ned as the sequential application, i.e., TS (x) = T (S (x)) x 2 X. D: The space L1 (;1 1) of all measurable and absolutely integrable functions on the real axis (see 12.9, p. 635) with the norm Z1 kxk = jx(t)j dt ;1
is a Banach algebra if the multiplication is de ned as the convolution (x # y)(t) =
Z1 ;1
(12.102)
x(t ; s)y(s) ds.
12.4 Hilbert Spaces 615
12.4 Hilbert Spaces
12.4.1 Notion of a Hilbert Space 12.4.1.1 Scalar Product
A vector space V over a eld IF (mostly IF = C) is called a space with scalar product or an inner product space or pre-Hilbert space if to every pair of elements x y 2 V there is assigned a number (x y ) 2 IF (the scalar product of x and y), such that the axioms of the scalar product are satis ed, i.e., for arbitrary x y z 2 V and 2 IF: (H1) (x x) 0 (i.e., (x x) is real), and (x x) = 0 if and only if x = 0 (12.103) (H2) (x y) = (x y) (12.104) (H3) (x + y z) = (x z) + (y z) (12.105) (H4) (x y) = (y x): (12.106) (Here ! denotes the conjugate of the complex number !, which is denoted by ! in (1.133c). Sometimes the notation of a scalar product is hx yi.) In the case of IF = IR, i.e., in a real vector space, (H4) means the commutativity of the scalar product. Some further properties follow from the axioms: (x y) = "(x y) and (x y + z) = (x y) + (x z): (12.107)
12.4.1.2 Unitary Spaces and Some of their Properties
In a pre-Hilbert space IH a norm can be introduced by means of the scalar product as follows: q kxk = (x x) (x 2 IH): (12.108) A normed space IH = (IH kk) is called unitary if there is a scalar product satisfying (12.108). Based on the previous properties of scalar products and (12.108) in unitary spaces the following facts are valid:
a) Triangle Inequality: kx + yk2 (kxk + kyk)2 : (12.109) b) Cauchy{Schwarz Inequality or Schwarz{Buniakowski Inequality (see also 1.4.2.9, p. 31): q q j(x y)j (x x) (y y) : (12.110) c) Parallelogram Identity: This characterizes the unitary spaces among the normed spaces: kx + yk2 + kx ; yk2 = 2 kxk2 + kyk2 : (12.111) d) Continuity of the Scalar Product: xn ! x yn ! y imply (xn yn) ! (x y): (12.112)
12.4.1.3 Hilbert Space
A complete unitary space is called a Hilbert space. Since Hilbert spaces are also Banach spaces, they possess in particular, the properties of the last (see 12.3.1, p. 611 12.3.1.2, p. 612 12.3.2, p. 612). In addition they have the properties of unitary spaces 12.4.1.2, p. 615. A subspace of a Hilbert space is a closed linear subspace. A: l2(n) l2 and L2 ((a b)) with the scalar products (x y) =
n X
k=1
k k (x y) =
1 X
k=1
k k and (x y) =
Zb a
x(t)y(t) dt:
(12.113)
616 12. Functional Analysis B: The space IH2()) with the scalar product X Z
Z
(f g) = f (x)g(x) dx +
1 j
j k
D f (x)D g(x) dx:
(12.114)
C: Let '(t) be a measurable positive function on a b]. The complex space L2 ((a b) ') of all measurable functions, which are quadratically integrable with the weight function ' on (a b), is a Hilbert space if the scalar product is de ned as (x y) =
Zb a
x(t)y(t)'(t) dt:
(12.115)
12.4.2 Orthogonality
Two elements x y of a Hilbert space IH are called orthogonal (we write x ? y) if (x y) = 0 (the notions of this paragraph also make sense in pre-Hilbert spaces and in unitary spaces). For an arbitrary subset A IH, the set A? = fx 2 IH: (x y) = 0 8 y 2 Ag (12.116) of all vectors which are orthogonal to each vector in A is a (closed linear) subspace of IH and it is called the orthogonal space to A or the orthogonal complement of A. We write A ? B if (x y) = 0 for all x 2 A and y 2 B . If A consists of a single element x, then we write x ? B .
12.4.2.1 Properties of Orthogonality
The zero vector is orthogonal to every vector of IH. The following statements hold: a) x ? y and x ? z imply x ? (y + z) for any 2 C. b) From x ? yn and yn ! y it follows that x ? y. c) x ? A if and only if x ? lin(A), where lin(A) denotes the closed linear hull of the set A. d) If x ? A and A is a fundamental set, i.e., lin(A) is everywhere dense in IH, then x = 0. e) Pythagoras Theorem: If the elements x1 : : : xn are pairwise orthogonal, that is xk ? xl for all k 6= l, then
k
n X
k=1
xk k2 =
n X
k=1
kxk k2 :
(12.117)
f) Projection Theorem: If IH0 is a subspace of IH, then each vector x 2 IH can be written uniquely as
x0 2 IH0 x00 ? IH0 : (12.118) 0 g) Approximation Problem: Furthermore, the equation kx k = (x IH0 ) = inf y2IH0 fkx ; ykg holds, and so the problem kx ; yk ! inf y 2 IH0 (12.119) has the unique solution x0 in IH0 . In this statement IH0 can be replaced by a convex closed non-empty subset of IH. The element x0 is called the projection of the element x on IH0 . It has the smallest distance from x (to IH0), and the space IH can be decomposed: IH = IH0 $ IH?0 . x = x0 + x00
12.4.2.2 Orthogonal Systems
A set fx : 2 0g of vectors from IH is called an orthogonal system if it does not contain the zero vector and x ? x 6= , hence (x x ) = holds, where = = 10 for (12.120) for 6=
12.4 Hilbert Spaces 617
denotes the Kronecker symbol (see 4.1.2, 10., p. 253). An orthogonal system is called orthonormal if in addition kx k = 1 8 . In a separable Hilbert space an orthogonal system may contain at most countably many elements. In what follows we assume, therefore, 0 = IN. A: The system p1 p1 cos t p1 sin t p1 cos 2t p1 sin 2t : : : (12.121) 2 in the real space L2 ((; )) and the system p1 eint (n = 0 1 2 : : :) (12.122) 2 2 in the complex space L ((; )) are orthonormal systems. Both of these systems are called trigonometric . B: The Legendre polynomials of the rst kind (see 9.1.2.6, 2., p. 512) n Pn(t) = ddtn (t2 ; 1)n] (n = 0 1 : : :) (12.123) 2 form an orthogonal system of elements in the space L ((;1 1)). The corresponding orthonormal system is s P~n(t) = n + 12 (2n1)!! Pn(t): (12.124) C: The Hermite polynomials (see 9.1.2.6, 6., p. 514 and 9.2.3.5, 3., 545) according to the second de nition of the Hermite di erential equation (9.63b) n Hn(t) = et2 dtd n e;t2 (n = 0 1 : : :) (12.125) 2 form an orthogonal system in the space L ((;1 1)). D: The Laguerre polynomials form an orthogonal system (see 9.1.2.6, 5., p. 513) in the space L2 ((0 1)). Every orthogonal system is linearly independent, since the zero vector was excluded. Conversely, if we have a system x1 x2 : : : xn : : : of linearly independent elements in a Hilbert space IH, then there exist vectors e1 e2 : : : en : : :, obtained by the Gram{Schmidt orthogonalization method (see 4.5.2.2, 1., p. 280) which form an orthonormal system. They span the same subspace, and by the method they are determined up to a scalar factor with modulus 1.
12.4.3 Fourier Series in Hilbert Spaces 12.4.3.1 Best Approximation
Let IH be a separable Hilbert space and fen : n = 1 2 : : :g (12.126) a xed orthonormal system in IH. For an element x 2 IH the numbers cn = (x en) are called the Fourier coecients of x with respect to the system (12.126). The (formal) series 1 X
n=1
cnen
(12.127)
is called the Fourier series of the element x with respect to the system (12.126) (see 7.4.1.1, 1., p. 420). The n-th partial sum of the Fourier series of an element x has the property of the best approximation,
618 12. Functional Analysis i.e., for xed n, the n-th partial sum of the Fourier series
n =
n X
(x ek )ek
k=1
(12.128)
gives the smallest value of kx ; P k ek k among all vectors of IHn = lin(fe1 : : : eng). Furthermore, k=1 x ; n is orthogonal to IHn, and there holds the Bessel inequality: 1 X
n=1
n
jcnj2 kxk2 cn = (x en) (n = 1 2 : : :):
(12.129)
12.4.3.2 Parseval Equation, Riesz{Fischer Theorem
The Fourier series of an arbitrary element x 2 IH is always convergent. Its sum is the projection of the element x onto the subspace IH0 = lin(feng1 n=1 ). If an element x 2 IH has the representation 1 x = P nen, then n are the Fourier coecients of x (n = 1 2 : : :). If fng1 n=1 is an arbitrary n=1
sequence of numbers with the property P jnj2 < 1, then there is a unique element x in IH, whose n=1 Fourier coecients are equal to n and for which the Parseval equation holds: 1 X
1
j(x en)j2 =
1 X
jnj2 = kxk2 (Riesz{Fischer theorem): (12.130) An orthonormal system feng in IH is called complete if there is no non-zero vector y orthogonal to every 1 en it is called a basis if every vector x 2 IH has the representation x = P nen , i.e., n = (x en) and n=1 n=1
n=1
x is equal to the sum of its Fourier series. In this case, we also say that x has a Fourier expansion. The following statements are equivalent: a) feng is a fundamental set in IH. b) feng is complete in IH. c) feng is a basis in IH. d) For 8 x y 2 IH with the corresponding Fourier coecients cn and dn (n = 1 2 : : :) there holds 1 X (x y) = cndn: (12.131) n=1
e) For every vector x 2 IH, the Parseval equation (12.130) holds. A: The trigonometric system (12.121) is a basis in the space L2 ((; )). B: The system of the normalized Legendre polynomials (12.124) P~n(t) (n = 0 1 : : :) is complete and consequently a basis in the space L2 ((;1 1)).
12.4.4 Existence of a Basis, Isomorphic Hilbert Spaces
In every separable Hilbert space there exits a basis. From this fact it follows that every orthonormal system can be completed to a basis. Two Hilbert spaces IH1 and IH2 are called isometric or isomorphic as Hilbert spaces if there is a linear bijective mapping T : IH1 ;! IH2 with the property (Tx Ty)IH2 = (x y)IH1 (that is, it preserves the scalar product and because of (12.108) also the norm). Any two arbitrary in nite dimensional separable Hilbert spaces are isometric, in particular every such space is isometric to the separable space l2.
12.5 Continuous Linear Operators and Functionals 619
12.5 Continuous Linear Operators and Functionals
12.5.1 Boundedness, Norm and Continuity of Linear Operators
12.5.1.1 Boundedness and the Norm of Linear Operators
Let X = (X k k) and Y = (Y k k) be normed spaces. In the following discussion, we omit the index X in the notation k kX emphasizing that we are in the space X, because from the text it will be always clear which norm and which space we are talking about. An arbitrary operator T : X ;! Y is called bounded if there is a real number > 0 such that kT (x)k kxk 8 x 2 X: (12.132) A bounded operator with a constant \stretches" every vector at most times and it transforms every bounded set of X into a bounded set of Y, in particular the image of the unit ball of X is bounded in Y. This last property is characteristic of bounded linear operators. A linear operator is continuous (see 12.2.3, p. 610) if and only if it is bounded. The smallest constant , for which (12.132) still holds, is called the norm of the operator T and it is denoted by kT k, i.e., kT k := inf f > 0 : kTxk kxk x 2 Xg: (12.133) For a continuous linear operator the following equalities hold: kT k = sup kTxk = sup kTxk = sup kTxk (12.134) kxk 1
kxk r(T ), the operator (I ; T );1 has the representation in the form of a Neumann series (I ; T );1 = ;1 I + ;2T + : : : + ;nT n;1 + : : : (12.144) which is convergent for jj > r(T ) in the operator norm on L(X).
12.5.1.3 Convergence of Operator Sequences 1. Pointwise Convergence
of a sequence of linear continuous operators Tn : X ;! Y to an operator T : X ;! Y means that: Tnx ;! Tx in Y for each x 2 X: (12.145)
2. Uniform Convergence
The usual norm-convergence of a sequence of operators fTng1 n=1 in a space L(X Y) to T , i.e., kTn ; T k = sup kTnx ; Txk ! 0 (n ! 1) (12.146) kxk 1
is the uniform convergence on the unit ball of X. It implies pointwise convergence, while the converse statement is not true in general.
3. Applications
The convergence of quadrature formulas when the number n of interpolation nodes tends to 1, the performence principle of summation, limiting methods, etc.
12.5.2 Linear Continuous Operators in Banach Spaces
Suppose now X and Y to be Banach spaces.
1. Banach{Steinhaus Theorem (Uniform Boundedness Principle)
The theorem characterizes the pointwise convergence of a sequence fTng of linear continuous operators Tn to some linear continuous operator by the conditions: a) For every element from an everywhere dense subset D X, the sequence fTnxg has a limit in Y, b) there is a constant C such that kTnk C 8 n.
2. Open Mappings Theorem
The theorem tells us that a linear continuous operator mapping from X onto Y is open, i.e., the image T (G) of every open set G from X is an open set in Y.
3. Closed Graph Theorem
An operator T : DT ;! Y with DT X is called closed if xn 2 DT xn ! x0 in X and Txn ! y0 in Y imply x0 2 DT and y0 = Tx0. A necessary and sucient condition is that the graph of the operator T in the space X Y, i.e., the set ;T = f(x Tx): x 2 DT g (12.147) is closed, where here (x y) denotes an element of the set X Y. If T is a closed operator with a closed domain DT , then T is continuous.
4. Hellinger{Toeplitz Theorem
Let T be a linear operator in a Hilbert space IH. If (x Ty) = (Tx y) for every x y 2 IH, then T is continuous (here (x Ty) denotes the scalar product in IH).
12.5 Continuous Linear Operators and Functionals 621
5. Krein{Losanovskij Theorem on the Continuity of Positive Linear Operators
If X = (X X+ k k) and Y = (Y Y+ k k) are ordered normed spaces, where X+ is a generating cone, then the set L+(X Y) of all positive linear and continuous operators T , i.e., T (X+) Y+, is a cone in L(X Y). The theorem of Krein and Losanovskij asserts (see 12.17]): If X and Y are ordered Banach spaces with closed cones X+ and Y+, and X+ is a generating cone, then the positivity of a linear operator implies its continuity.
6. Inverse Operator
Let X and Y be arbitrary normed spaces and let T : X ;! Y be a linear, not necessarily continuous operator. T has a continuous inverse T ;1 : Y ;! X, if T (X) = Y and there exists a constant m > 0 such that kTxk mkxk for each x 2 X . Then kT ;1k m1 . In the case of Banach spaces X Y we have the
7. Banach Theorem on the Continuity of the Inverse Operator
If T is a linear continuous bijective operator from X onto Y, then the inverse operator T ;1 is also continuous. An important application is, e.g., the continuity of (I ; T );1 given the injectivity and surjectivity of I ; T . This fact has importance in investigating the spectrum of an operator (see 12.5.3.2, p. 622). It also applies to the
8. Continuous Dependence of the Solution
on the right-hand side and also on the initial data of initial value problems for linear di erential equations. We will demonstrate this fact by the following example. The initial value problem x(t) + p1(t)x_ (t) + p2(t)x(t) = q(t) t 2 a b] x(t0 ) = x_ (t0 ) = _ t0 2 a b] (12.148a) with coecients p1(t) p2(t) 2 C ( a b]) has exactly one solution x from C 2( a b]) for every right-hand side q(t) 2 C ( a b]) and for every pair of numbers _. The solution x depends continuously on q(t) and _ in the following sense. If qn(t) 2 C ( a b]) n _n 2 IR1 are given and xn 2 C ( a b]) denotes the solution of xn(t) + p1 (t)x_ n(t) + p2(t)xn (t) = qn(t) xn (a) = n x_ n(a) = _n (12.148b) for n = 1 2 : : :, then: 9 qn (t) ! q(t) in C ( a b]) > = 2 n ! (12.148c) > implies that xn ! x in the space C ( a b]): " _n ! _
9. Method of Successive Approximation
to solve an equation of the form x ; Tx = y (12.149) with a continuous linear operator T in a Banach space X for a given y. This method starts with an arbitrary initial element x0 , and constructs a sequence fxn g of approximating solutions by the formula xn+1 = y + Txn (n = 0 1 : : :) : (12.150) This sequence converges to the solution x in X of (12.149). The convergence of the method, i.e., xn ! x , is based on the convergence of the series (12.144) with = 1. Let kT k q < 1. Then the following statements are valid: a) The operator I ; T has a continuous inverse with k(I ; T );1k 1 ;1 q , and (12.149) has exactly one solution for each y.
622 12. Functional Analysis b) The series (12.144) converges and its sum is the operator (I ; T );1. c) The method (12.150) converges to the unique solution x of (12.149) for any initial element x0 , if the series (12.144) converges. Then the following estimation holds: n kxn ; x k 1 q; q kTx0 ; x0 k (n = 1 2 : : :): Equations of the type x ; Tx = y x ; Tx = y 2 IF can be handled in an analogous way (see 11.2.2, p. 567, and 12.8]).
(12.151) (12.152)
12.5.3 Elements of the Spectral Theory of Linear Operators 12.5.3.1 Resolvent Set and the Resolvent of an Operator
For an investigation of the solvability of equations one tries to rewrite the problem in the form (I ; T )x = y (12.153) with some operator T having a possible small norm. This is especially convenient for using a functional analytic method because of (12.143) and (12.144). In order to handle large values of kT k as well, we investigate the whole family of equations (I ; T )x = y x 2 X with 2 C (12.154) in a complex Banach space X. Let T be a linear, but in general not a bounded operator in a Banach space X. The set %(T ) of all complex numbers such that (I ; T );1 2 B (X) = L(X) is called the resolvent set and the operator R = R (T ) = (I ; T );1 is called the resolvent. Let T now be a bounded linear operator in a complex Banach space X. Then the following statements are valid: a) The set %(T ) is open. More precisely, if 0 2 %(T ) and 2 C satisfy the inequality j ; 0 j < kR1 k (12.155)
0 then R exists and 1 X R = R 0 + ( ; 0)R 2 0 + ( ; 0)2R 3 0 + : : : = ( ; 0)k;1R k0 : (12.156) k=1
b) f 2 C : jj > kT kg %(T ). More precisely, 8 2 C with jj > kT k, the operator R exists and 2 R = ; I ; T2 ; T3 ; : : : : (12.157) c) kR ; R 0 k ! 0, if ! 0 ( 0 2 %(T )), and kR k ! 0, if ! 1 ( 2 %(T )). % % d) %%% R ;; R 0 ; R 2 0 %%% ;! 0, if ! 0. 0 e) For an arbitrary functional f 2 X (see 12.5.4.1, p. 623) and arbitrary x 2 X the function F () = f (R (x)) is holomorphic on %(T ). f) For arbitrary 2 %(T ) and 6= one has: ; R : R R = R R = R ;
(12.158)
12.5.3.2 Spectrum of an Operator 1. Denition of the Spectrum
The set (T ) = C n %(T ) is called the spectrum of the operator T . Since I ; T has a continuous inverse (and consequently (12.153) has a solution, which continuously depends on the right-hand side) if and only if 1 2 %(T ), we must know the spectrum (T ) as well as possible. From the properties of the
12.5 Continuous Linear Operators and Functionals 623
resolvent set it follows immediately that the spectrum (T ) is a closed set of C which lies in the disk
f : jj kT kg, however, in many cases (T ) is much smaller than this disk. The spectrum of any linear continuous operator on a complex Banach space is never empty and r(T ) = sup jj:
(12.159)
2(T )
It is possible to say more about the spectrum in the cases of di erent special classes of operators. If T is an operator in a nite dimensional space X and if the equation (I ; T )x = 0 has only the trivial solution (i.e., I ; T is injective), then 2 %(T ) (i.e., I ; T is surjective). If this equation has a non-trivial solution in some Banach space, then the operator I ; T is not injective and (I ; T );1 is in general not de ned. The number 2 C is called an eigenvalue of the linear operator T , if the equation x = Tx has a nontrivial solution. All those solutions are called eigenvectors, or in the case when X is a function space (which occurs very often in applications), they are called eigenfunctions of the operator T associated to . The subspace spanned by them is called the eigenspace (or characteristic space) associated to . The set p(T ) of all eigenvalues of T is called the point spectrum of the operator T .
2. Comparison to Linear Algebra, Residual Spectrum
An essential di erence between the nite dimensional case which is considered in linear algebra and the in nite dimensional case discussed in functional analysis is that in the rst case (T ) = p(T ) always holds, while in the second case the spectrum usually also contains points which are not eigenvalues of T . If I ; T is injective and surjective as well, then 2 %(T ) due to the theorem on the continuity of the inverse (see 12.5.2, 7., p. 621). In contrast to the nite dimensional case where the surjectivity follows automatically from the injectivity, the in nite dimensional case has to be dealt with in a very di erent way. The set c(T ) of all 2 (T ), for which I ; T is injective and Im(I ; T ) is dense in X, is called the continuous spectrum and the set r (T ) of all with an injective I ; T and a non-dense image, is called the residual spectrum of operator T . For a bounded linear operator T in a complex Banach space X (T ) = p(T ) c(T ) r (T ) (12.160) where the terms of the right-hand side are mutually exclusive.
12.5.4 Continuous Linear Functionals 12.5.4.1 Denition
For Y = IF we call a linear mapping a linear functional or a linear form. In the following discussions, for a Hilbert space we consider the complex case in other situations almost every times the real case is considered. The Banach space L(X IF) of all continuous linear functionals is called the adjoint space or the dual space of X and it is denoted by X (sometimes also by X0). The value (in IF) of a linear continuous functional f 2 X on an element x 2 X is denoted by f (x), often also by (x f ) { emphasizing the bilinear relation of X and X { (compare also with the Riesz theorem (see 12.5.4.2, p. 624). A: Let t1 t2 : : : tn be xed points of the interval a b] and c1 c2 : : : cn real numbers. By the formula
f (x) =
n X
k=1
ck x(tk )
(12.161)
a linear continuous functional is de ned on the space C ( a b]) the norm of f is kf k = P jck j. A special k=1 case of (12.161) for a xed t 2 a b] is the functional t (x) = x(t) (x 2 C ( a b])): (12.162) n
624 12. Functional Analysis B: With an integrable function '(t) (see 12.9.3.1, p. 637) on a b]
Zb f (x) = '(t)x(t) dt (12.163) a is a linear continuous functional on C ( a b]) and also on B( a b]) in each case with the norm kf k = Zb j'(t)j dt. a
12.5.4.2 Continuous Linear Functionals in Hilbert Spaces. Riesz Representation Theorem
In a Hilbert space IH every element y 2 IH de nes a linear continuous functional by the formula f (x) = (x y), where its norm is kf k = kyk. Conversely, if f is a linear continuous functional on IH, then there exists a unique element y 2 IH such that f (x) = (x y) 8 x 2 IH (12.164) where kf k = kyk. According to this theorem the spaces IH and IH are isomorphic and might be identi ed. The Riesz representation theorem contains a hint on how to introduce the notion of orthogonality in an arbitrary normed space. Let A X and A X . We call the sets A? = ff 2 X: f (x) = 0 8 x 2 Ag and A ? = fx 2 X: f (x) = 0 8 f 2 A g (12.165) the orthogonal complement or the annulator of A and A , respectively.
12.5.4.3 Continuous Linear Functionals in L p
Let p 1. The number q is called the conjugate exponent to p if 1 + 1 = 1, where it is assumed that p q q = 1 in the case of p = 1. Based on the Holder integral inequality (see 1.4.2.12, p. 32) the functional (12.163) can be considered also in the spaces Lp( a b]) (1 p 1) (see 12.9.4, p. 639) if ' 2 Lq ( a b]) and p1 + 1q = 1. Its norm is then 8 Z !1 > > < b j'(t)jq dt q if 1 < p 1 kf k = k'k = > a (12.166) > : ess. sup j'(t)j if p = 1 t2ab]
(with respect to the de nition of ess. sup j'j see (12.217), p. 639). To every linear continuous functional f in the space Lp( a b]) there is a uniquely (up to its equivalence class) de ned element y 2 Lq ( a b]) such that 0 11 q Zb Zb f (x) = (x y) = x(t)y(t)dt x 2 Lp and kf k = kykq = @ jy(t)jq dtA :
a
For the case of p = 1 see 12.15].
a
(12.167)
12.5.5 Extension of a Linear Functional
1. Seminorm
A mapping p: X ;! IR of a vector space X is called a seminorm or pseudonorm, if it has the following properties: (HN1) p(x) 0 (12.168) (HN2) p(x) = jjp(x) (12.169) (HN3) p(x + y) p(x) + p(y): (12.170)
12.5 Continuous Linear Operators and Functionals 625
Comparison with 12.3.1, p. 611, shows that a seminorm is a norm if and only if p(x) = 0 holds only for x = 0. Both for theoretical mathematical questions and for practical reasons in applications of mathematics, the problem of the extension of a linear functional given on a linear subspace X0 X to the entire space (and, in order to avoid trivial and uninteresting cases) with preserving certain \ good" properties became a fundamental question. The solution of this problem is guaranteed by
2. Analytic Form of the Hahn{Banach Extension Theorem
Let X be a vector space over IF and p a pseudonorm on X. Let X0 be a linear (complex in the case of IF = C and real in the case of IF = IR) subspace of X, and let f0 be a (complex-valued in the case of IF = C and real-valued in the case of IF = IR) linear functional on X0 satisfying the relation jf0(x)j p(x) 8 x 2 X0 : (12.171) Then there exists a linear functional f on X with the following properties: f (x) = f0 (x) 8 x 2 X0 jf (x)j p(x) 8 x 2 X: (12.172) So, f is an extension of the functional f0 onto the whole space X preserving the relation (12.171). If X0 is a linear subspace of a normed space X and f0 is a continuous linear functional on X0 , then p(x) = kf0k kxk is a pseudonorm on X satisfying (12.171), so we get the Hahn{Banach theorem on the extension of continuous linear functionals. Two important consequences are: 1. For every element x 6= 0 there is a functional f 2 X with f (x) = kxk and kf k = 1. 2. For every linear subspace X0 X and x0 2= X0 with the positive distance d = inf x2X0 kx ; x0 k > 0 there is an f 2 X such that f (x) = 0 8 x 2 X0 f (x0 ) = 1 and kf k = d1 : (12.173)
12.5.6 Separation of Convex Sets
1. Hyperplanes
A linear subset L of the real vector space X , L 6= X, is called a hypersubspace or hyperplane through 0 if there exists an x0 2 X such that X = lin(x0 L). Sets of the form x + L (L a linear subset) are anelinear manifolds (see 12.1.2, p. 597). If L is a hypersubspace, these manifolds are called hyperplanes. There exist the following close relations between hyperplanes and linear functionals: in X, and a) The kernel f ;1(0) = fx 2 X: f (x) = 0g of a linear functional f on X is a ;hypersubspace for each number 2 IR there exists an element x 2 X with f (x ) = and f 1() = x + f ;1(0). b) For any given hypersubspace L X and each x;01 2= L and 6= 0 ( 2 IR) there always exists a uniquely determined linear functional f on X with f (0) = L and f (x0 ) = . The closedness of f ;1(0) in the case of a normed space X is equivalent to the continuity of the functional f.
2. Geometric Form of the Hahn{Banach Extension Theorem
Let X be a normed space, x0 2 X and L a linear subspace of X. Then for every non-empty convex open set K which does not intersect the ane-linear manifold x0 + L, there exists a closed hyperplane H such that x0 + L H and H \ K = .
3. Separation of Convex Sets
Two subsets A B of a real normed space X are said to be separated by a hyperplane if there is a functional f 2 X such that: sup f (x) yinf f (y): (12.174) 2B x2A
The separating hyperplane is then given by f ;1() with = supx2A f (x), which means that the two sets are contained in the di erent half-spaces
626 12. Functional Analysis A fx 2 X: f (x) g and B fx 2 X: f (x) g: (12.175) In Fig. 12.5b,c two cases of the separation by a hyperplane are shown. Their disjointness is less decisive for the separation of two sets. In fact, Fig. 12.5a shows two sets E and B , which are not separated although E and B are disjoint and B is convex. The convexity of both sets is the intrinsic property for separating them. In this case it is possible that the sets have common points which are contained in the hyperplane. E
A
A
¦-1(a)
-1
¦ (a)
B -1
a)
B
b)
c)
B
¦ (a)
Figure 12.5 If A is a convex set of a normed space X with a non-empty interior Int(A) and B X is a non-empty convex set with Int(A) \ B = , then A and B can be separated. The hypothesis Int(A) 6= in that statement cannot be dropped (see 12.3], example 4.47). A (real linear) functional f 2 X is then called a supporting functional to A of the set A at the point x0 2 A, if there is a real number 2 IR such that f (x0) = , and A fx 2 X : f (x) g. f ;1() is called the supporting hyperplane at the point x0. For a convex set K with a non-empty interior, there exists a supporting functional at each of its boundary points. Remark: The famous Kuhn{Tucker theorem (see 18.2, p. 862) which yields practical methods to determine the minimum of convex optimization problems (see 12.5]), is also based on the separation of convex sets.
12.5.7 Second Adjoint Space and Reexive Spaces
The adjoint space X of a normed space X is also a normed space if it is equipped with the norm kf k = sup jf (x)j, so (X ) = X . The second adjoint space to X can also be considered. The canonical kxk 1 embedding (12.176) J : X ;! X with Jx = Fx where Fx(f ) = f (x) 8 f 2 X is a norm isomorphism (see 12.3.1, p. 611), hence X is identi ed with the subset J (X) X . A Banach space X is called reexive if J (X) = X . Hence the canonical embedding is then a surjective norm isomorphism. Every nite dimensional Banach space and every Hilbert space is reexive, as well as the spaces Lp (1 p < 1), however C ( a b]) L1 ( 0 1]) c0 are examples of non-reexive spaces.
12.6 Adjoint Operators in Normed Spaces 12.6.1 Adjoint of a Bounded Operator
For a given linear continuous operator T : X ;! Y (X Y are normed spaces) to every g 2 Y there is assigned a functional f 2 X by f (x) = g(Tx) 8 x 2 X. In this way, we get a linear continuous operator T : Y ;! X (T g)(x) = g(Tx) 8 g 2 Y and 8 x 2 X (12.177) which is called the adjoint operator of T and has the following properties: (T + S ) = T + S (ST ) = S T kT k = kT k, where for the linear continuous operators T : X ! Y
12.6 Adjoint Operators in Normed Spaces 627
and S : Y ! Z (X Y Z normed spaces), the operator ST : X ! Z is de ned in the natural way as ST (x) = S (T (x)). With the notation introduced in 12.1.5, p. 600, and 12.5.4.2, p. 624, the following identities are valid for an operator T 2 B (X Y): Im(T ) = ker(T )? Im(T ) = ker(T )? (12.178) where the closedness of Im(T ) implies the closedness of Im(T ). The operator T : X ! Y , obtained as (T ) from T , is called the second adjoint of T . Due to (T (Fx))g = Fx(T g) = (T g)(x) = g(Tx) = FTx(g) the operator T has the following property: If Fx 2 X , then T Fx = FTx 2 Y . Hence, the operator T : X ! Y is an extension of T . In a Hilbert space IH the adjoint operator can also be introduced by means of the scalar product (Tx y) = (x T y) x y 2 IH. This is based on the Riesz representation theorem, where the identi cation of IH and IH implies (T ) = T , I = I and even T = T . If T is bijective, then the same holds for T , and also (T );1 = (T ;1) . For the resolvents of T and T there holds R (T )] = R (T ) (12.179) from which (T ) = f: 2 (T )g follows for the spectrum of the adjoint operator. A: Let T be an integral operator in the space Lp( a b]) (1 < p < 1)
Zb
(Tx)(s) = K (s t)x(t) dt (12.180) a with a continuous kernel K (s t). The adjoint operator of T is also an integral operator, namely
Zb
(T g)(t) = K (t s)yg (s) ds (12.181) a q p with the kernel K (s t) = K (t s), where yg is the element from L associated to g 2 (L ) according to (12.167). B: In a nite dimensional complex vector space the adjoint of an operator represented by the matrix A = (aij ) is de ned by the matrix A with aij = aji.
12.6.2 Adjoint Operator of an Unbounded Operator
Let X and Y be real normed spaces and T a (not necessarily bounded) linear operator with a (linear) domain D(T ) X and values in Y. For a given g 2 Y , the expression g(Tx), depending obviously linearly on x, is meaningful. Now the question is: Does there exist a well-de ned functional f 2 X such that f (x) = g(Tx) 8 x 2 D(T ): (12.182) Let D Y be the set of all those g 2 Y for which the representation (12.182) holds for a certain f 2 X . If D(T ) = X, then for the given g the functional f is uniquely de ned. So a linear operator T is de ned by f = T g with D(T ) = D . Then for arbitrary x 2 D(T ) and g 2 D(T ) one has g(Tx) = (T g)(x): (12.183) The operator T turns out to be closed and is called the adjoint of T . The naturalness of this general procedure stems from the fact that D(T ) = Y holds if and only if T is bounded on D(T ). In this case T 2 B (Y X ) and kT k = kT k hold.
12.6.3 Self-Adjoint Operators
An operator T 2 B (IH) (IH is a Hilbert space) is called self-adjoint if T = T . In this case the number (Tx x) is real for each x 2 IH. One has the equality kT k = sup j(Tx x)j (12.184) kxk=1
628 12. Functional Analysis and with m = m(T ) = kxinf (Tx x) and M = M (T ) = sup (Tx x) also the relations k=1 kxk=1
m(T )kxk2 (Tx x) M (T )kxk2 and kT k = r(T ) = maxfjmj M g: (12.185) The spectrum of a self-adjoint (bounded) operator lies in the interval m M ] and m M 2 (T ) holds.
12.6.3.1 Positive Denite Operators
A partial ordering can be introduced in the set of all self-adjoint operators of B (IH) if we de ne T 0 if and only if (Tx x) 0 8 x 2 IH: (12.186) An operator T with T 0 is called positive (or, more exactly positive denite). For any self-adjoint operator T (with (H1) from 12.4.1.1, p. 615), (T 2x x) = (Tx Tx) 0, so T 2 is positive de nite. Every positive de nite operator T possesses a square root, i.e., there exists a unique positive de nite operator W such that W 2 = T . Moreover, the vector space of all self-adjoint operators is a K-space (Kantorovich space, see 12.1.7.4, p. 602), where the operators p jT j = T 2 T + = 21 (jT j + T ) T ; = 12 (jT j ; T ) (12.187) are the corresponding elements with respect to (12.37). They are of particular importance for the spectral decomposition and spectral and integral representations of self-adjoint operators by means of some Stieltjes integral (see 8.2.3.1, 2., p. 453, and 12.1], 12.11], 12.12], 12.15], 12.18]).
12.6.3.2 Projectors in a Hilbert Space
Let IH0 be a subspace of a Hilbert space IH. Then every element x 2 IH has its projection x0 onto IH0 according to the projection theorem (see 12.4.2, p. 616), and therefore, an operator P with Px = x0 is de ned on IH with values in IH0 . P is called a projector onto IH0 . Obviously, P is linear, continuous, and kP k = 1. A continuous linear operator P in IH is a projector (onto a certain subspace) if and only if: a) P = P , i.e., P is self-adjoint, and b) P 2 = P , i.e., P is idempotent.
12.7 Compact Sets and Compact Operators 12.7.1 Compact Subsets of a Normed Space
A subset A of a normed spacey X is called compact, if every sequence of elements from A contains a convergent subsequence whose limit lies in A, relatively compact or precompact if its closure (see 12.2.1.3, p. 606) is compact, i.e., every sequence of elements from A contains a convergent subsequence (whose limit does not necessarily belong to A). This is the Bolzano{Weierstrass theorem in real calculus, and we say that such a set has the Bolzano{ Weierstrass property. Every compact set is closed and bounded. Conversely, if the space X is nite dimensional, then every such set is compact. The closed unit ball in a normed space X is compact if and only if X is nite dimensional. For some characterizations of relatively compact subsets in metric spaces (the Hausdor theorem on the existence of a nite "-net) and in the spaces s C (Arzela{Ascoli theorem) and in the spaces Lp(1 < p < 1) see 12.15].
12.7.2 Compact Operators
12.7.2.1 Denition of Compact Operator
An arbitrary operator T : X ;! Y of a normed space X into a normed space Y is called compact if the y
It is enough that X is a metric (or an even more general) space. We do not use this generality in what follows.
12.7 Compact Sets and Compact Operators 629
image T (A) of every bounded set A X is a relatively compact set in Y. If, in addition the operator T is also continuous, then it is called completely continuous. Every compact linear operator is bounded and consequently completely continuous. For a linear operator to be compact it is sucient to require that it transforms the unit ball of X into a relatively compact set in Y.
12.7.2.2 Properties of Linear Compact Operators
A characterization by sequences of the compactness of an operator from B (X Y) is the following: For 1 every bounded sequence fxng1 n=1 from X the sequence fTxn gn=1 contains a convergent subsequence. A linear combination of compact operators is also compact. If one of the operators U 2 B (W X) T 2 B (X Y) S 2 B (Y Z) in each of the following products is compact, then the operators TU and ST are also compact. If Y is a Banach space, then one has the following important statements. a) Convergence: If a sequence of compact operators fTng1n=1 is convergent in the space B (X Y), then its limit is a compact operator, too. b) Schauder Theorem: If T is a linear continuous operator, then either both T and T are compact or both are not.
c) Spectral Properties of a Compact Operator T in an (In nite Dimensional) Banach Space X: The zero belongs to the spectrum. Every non-zero point of the spectrum (T ) is an eigenvalue with a nite dimensional eigenspace X = fx 2 X : (I ; T )x = 0g, and 8 " > 0 there is always only a nite number of eigenvalues of T outside the circle fjj "g, where only the zero ;1
can be an accumulation point of the set of eigenvalues. If = 0 is not an eigenvalue of T , then T is unbounded if it exists.
12.7.2.3 Weak Convergence of Elements
A sequence fxn g1 n=1 of elements of a normed space X is called weakly convergent to an element x0 if for each f 2 X the relation f (xn) ! f (x0) holds (written as: xn * x0 ). Obviously: xn ! x0 implies xn * x0 . If Y is another normed space and T : X ;! Y is a continuous linear operator, then: a) xn * x0 implies Txn * Tx0 , b) if T is compact, then xn * x0 implies Txn ! Tx0 . A: Every nite dimensional operator is compact. From this it follows that the identity operator in an in nite dimensional space cannot be compact (see 12.7.1, p. 628). B: 0Suppose X = l21, and let T be the operator in l2 given by the in nite matrix t11 t12 t13 C B ! t21 t22 t23 C 1 1 B X X B C t B C with Tx = t x : : : t x : : : : (12.188) 31 1 k k nk k B C
@ A
k=1
k=1
1 P If jtnk j2 = M < 1, then T is a compact operator from l2 into l2 with kT k M . kn=1 C: The integral operator (12.136) is a compact operator in the spaces C ( a b]) and Lp((a b)) (1 < p < 1).
12.7.3 Fredholm Alternative
Let T be a compact linear operator in a Banach space X. We consider the following equations (of the second kind) with a parameter 6= 0: x ; Tx = y x ; Tx = 0 (12.189) f ; T f = g f ; T f = 0:
630 12. Functional Analysis The following statements are valid: a) dim(ker(I ; T )) = dim(ker(I ; T )) < +1, i.e., both homogeneous equations always have the same number of linearly independent solutions. b) Im(I ; T ) = ker(I ; T )? and z Im(I ; T ) = ker(I ; T )?. c) Im(I ; T ) = X if and only if ker(I ; T ) = 0. d) The Fredholm alternative (also called the Riesz{Schauder theorem): ) Either the homogeneous equation has only the trivial solution. In this case 2 %(T ), the operator (I ; T );1 is bounded, and the inhomogeneous equation has exactly one solution x = (I ; T );1y for arbitrary y 2 X. ) Or the homogeneous equation has at least one non-trivial solution. In this case is an eigenvalue of T , i.e., 2 (T ), and the inhomogeneous equation has a (non-unique) solution if and only if the righthand side y satis es the condition f (y) = 0 for every solution f of the adjoint equation T f = f . In this last case every solution x of the inhomogeneous equation has the form x = x0 + h, where x0 is a xed solution of the inhomogeneous equation and h 2 ker(I ; T ). Linear equations of the form Tx = y with a compact operator T are called equations of the rst kind. Their mathematical investigation is in general more dicult (see 12.11], 12.18]).
12.7.4 Compact Operators in Hilbert Space
Let T : IH ;! IH be a compact operator. Then T is the limit (in B (IH)) of a sequence of nite dimensional operators. The similarity to the nite dimensional case can be seen from the following: If C is a nite dimensional operator and T = I ; C , then the injectivity of T implies the existence of T ;1 and T ;1 2 B (IH). If C is a compact operator, then the following statements are equivalent: a) 9 T ;1 and it is continuous, b) x 6= 0 ) Tx 6= 0, i.e., T is injective, c) T (IH) = IH, i.e., T is surjective.
12.7.5 Compact Self-Adjoint Operators
1. Eigenvalues
A compact self-adjoint operator T 6= 0 in a Hilbert space IH possesses at least one non-zero eigenvalue. More precisely, T always has an eigenvalue with jj = kT k. The set of eigenvalues of T is at most countable. Any compact self-adjoint operator T has the representation T = P k P k (in B (IH)), where k are the k di erent eigenvalues of T and P denotes the projector onto the eigenspace IHP
. We say in this case that the operator T can be diagonalized. From this fact it follows that Tx = k (x ek )ek for every k x 2 IH, where fek g is the orthonormal system of the eigenvectors of T . If 2= (T ) and y 2 IH, then X the solution of the equation (I ; T )x = y can be represented as x = R (T )y = ;1 (y ek )ek .
2. Hilbert{Schmidt Theorem
k
k
If T is a compact self-adjoint operator in a separable Hilbert space IH, then there is a basis in IH consisting of the eigenvectors of T . The so-called spectral (mapping) theorems (see 12.8], 12.10], 12.12], 12.13], 12.18]) can be considered as the generalization of the Hilbert{Schmidt theorem for the non-compact case of self-adjoint (bounded or unbounded) operators. z
Here the orthogonality is considered in Banach spaces (see 12.5.4.2, p. 624).
12.8 Non-Linear Operators 631
12.8 Non-Linear Operators
In the theory of non-linear operator equations the most important methods are based on the following principles: 1. Principle of the Contracting Mapping, Banach Fixed-Point Theorem (see 12.2.2.3, p. 608, and 12.2.2.4, p. 608). For further modi cations of this principle see 12.8], 12.11], 12.12], 12.18]. 2. Generalization of the Newton Method (see 18.2.5.2, p. 869 and 19.1.1.2, p. 885) for the in nite dimensional case.
3. Schauder Fixed-Point Principle 4. Leray{Schauder Theory Methods based on principles 1 and 2 yield information on the existence, uniqueness, constructivity etc. of the solution, while methods based on principles 3 and 4, in general, allow \only" the qualitative statement of the existence of a solution. If further properties of operators are known then see also 12.8.6, p. 633, and 12.8.7, p. 634.
12.8.1 Examples of Non-Linear Operators
For non-linear operators the relation between continuity and boundedness discussed for linear operators in 12.5.1, p. 619 is no longer valid in general. In studying non-linear operator equations, e.g., nonlinear boundary value problems or integral equations, the following non-linear operators occur most often. Iteration methods described in 12.2.2.4, p. 608, can be succesfully applied for solving non-linear integral equations.
1. Nemytskij Operator
Let ) be an open measurable subset from IRn (12.9.1, p. 635) and f : ) IR ;! IR a function of two variables f (x s), which is continuous with respect to x for almost every s and measurable with respect to s for every x (Caratheodory conditions). The non-linear operator N to F ()) de ned as (N u)(x) = f x u(x)] (x 2 )) (12.190) is called the Nemytskij operator. It is continuous and bounded if it maps Lp()) into Lq ()), where 1 + 1 = 1. This is the case, e.g., if p q p
jf (x s)j a(x) + bjsj q with a(x) 2 Lq ()) (b > 0) or if f : ) IR ;! IR is continuous. The operator N is compact only in special cases.
(12.191)
2. Hammerstein Operator
Let ) be a relatively compact subset of IRn, f a function satisfying the Caratheodory conditions and K (x y) a continuous function on ) ). The non-linear operator H on F ()) Z (Hu)(x) = K (x y)f y u(y)] dy (x 2 )) (12.192)
is called the Hammerstein operator. H can be written in the form H = K N with the integral operator K determined by Zthe kernel K (Ku)(x) = K (x y)u(y) dy (x 2 )): (12.193)
If the kernel K (x y) satis es the additional condition Z jK (x y)jq dx dy < 1
(12.194)
and the function f satis es the condition (12.191), then H is a continuous and compact operator on Lp()).
632 12. Functional Analysis
3. Urysohn Operator n
Let ) IR be an open measurable subset and K (x y s) : ) ) IR ;! IR a function of three variables. Then the non-linear operator U on F ())
Z
(U u)(x) = K x y u(y)] dy (x 2 ))
(12.195)
is called the Urysohn operator. If the kernel K satis es the corresponding conditions, then U is a continuous and compact operator in C ()) or in Lp()), respectively.
12.8.2 Dierentiability of Non-Linear Operators
Let X Y be Banach spaces, D X be an open set and T : D ;! Y. The operator T is called Fr echet dierentiable (or, briey, di erentiable) at the point x 2 D if there exists a linear operator L 2 B (X Y) (in general depending on the point x) such that T (x + h) ; T (x) = Lh + !(h) with k!(h)k = o(khk) (12.196) or in an equivalent form lim kT (x + h) k;hTk (x) ; Lhk = 0 (12.197) khk!0 i.e., 8 " > 0 9 > 0, such that khk < implies kT (x + h) ; T (x) ; Lhk "khk. The operator L, which is usually denoted by T 0(x), T 0(x ) or T 0 (x)(), is called the Fr echet derivative of the operator T at the point x. The value dT (x h) = T 0(x)h is called the Fr echet dierential of the operator T at the point x (for the increment h). The di erentiability of an operator at a point implies its continuity at that point. If T 2 B (X Y), i,e., T itself is linear and continuous, then T is di erentiable at every point, and its derivative is equal to T.
12.8.3 Newton's Method
Let X D be as in the previous paragraph and T : D ;! X. Under the assumption of the di erentiability of T at every point of the set D an operator T 0 : D ;! B (X) is de ned by assigning the element T 0(x) 2 B (X) to every point x 2 D. Suppose the operator T 0 is continuous on D (in the operator norm) in this case T is called continuously dierentiable on D. Suppose Y = X and also that the set D contains a solution x of the equation T (x) = 0: (12.198) Furthermore, we suppose that the operator T 0(x) is continuously invertible for each x 2 D, hence T 0(x)];1 is in B (X). Because of (12.196) for an arbitrary x0 2 D one conjectures that the elements T (x0) = T (x0) ; T (x ) and T 0(x0 )(x0 ; x ) are \not far" from each other and therefore the element x1 de ned as x1 = x0 ; T 0(x0)];1 T (x0) (12.199) is an approximation of x (under the assumption we made). Starting with an arbitrary x0 the so-called Newton approximation sequence xn+1 = xn ; T 0(xn)];1 T (xn) (n = 0 1 : : :) (12.200) can be constructed. There are many theorems known from the literature discussing the behavior and the convergence properties of this method. We mention here only the following most important result which demonstrates the main properties and advantages of Newton's method: 8 " 2 (0 1) there exists a ball B = B (x0 ) = (") in X, such that all points xnn lie in B and the Newton sequence converges to the solution x of (12.198). Moreover, kxn ; x0 k " kx0 ; x k which yields a practical error estimation. The modied Newton's method is obtained if the operator T 0(x0 )];1 is used instead of T 0(xn)];1 in
12.8 Non-Linear Operators 633
formula (12.200). For further estimations of the speed of convergence and for the (in general sensitive) dependence of the method on the choice of the starting point x0 see 12.7], 12.12], 12.18].
12.8.4 Schauder's Fixed-Point Theorem
Let T : D ;! X be a non-linear operator de ned on a subset D of a Banach space X. The non-trivial question of whether the equation x = T (x) has at least one solution, can be answered as follows: If X = IR and D = ;1 1], then every continuous function mapping D into D has a xed point in D. If X is an arbitrary nite dimensional normed space (dimX 2), then Brouwer's xed-point theorem holds. 1. Brouwer's Fixed-Point Theorem Let D be a non-empty closed bounded and convex subset of a nite dimensional normed space. If T is a continuous operator, which maps D into itself, then T has at least one xed point in D. The answer in the case of an arbitrary in nite dimensional Banach space X is given by Schauder's xed-point theorem. 2. Schauder's Fixed-Point Theorem Let D be a non-empty closed bounded and convex subset of a Banach space. If the operator T : D ;! X is continuous and compact (hence completely continuous) and it maps D into itself, then T has at least one xed point in D. By using this theorem, it is proved, e.g., that the initial value problem (12.70), p. 610, always has a local solution for t 0, if the right-hand side is assumed only to be continuous.
12.8.5 Leray{Schauder Theory
For the existence of solutions of the equations x = T (x) and (I +T )(x) = y with a completely continuous operator T , a further principle is found which is based on deep properties of the mapping degree. It can be successfully applied to prove the existence of a solution of non-linear boundary value problems. We mention here only those results of this theory which are the most useful ones in practical problems, and for simplicity we have chosen a formulation which avoids the notion of the mapping degree. Leray{Schauder Theorem: Let D be an open bounded set in a real Banach space X and let T : D :;! X be a completely continuous operator. Let y 2 D be a point such that x + T (x) 6= y for each x 2 @D and 2 0 1], where @D denotes the boundary of the set D. Then the equation (I + T )(x) = y has at least one solution. The following version of this theorem is very useful in applications: Let T be a completely continuous operator in the Banach space X. If all solutions of the family of equations x = T (x) ( 2 0 1]) (12.201) are uniformly bounded, i.e., 9 c > 0 such that 8 and 8 x satisfying (12.201) the a priori estimation kxk c holds, then the equation x = T (x) has a solution.
12.8.6 Positive Non-Linear Operators
The successful application of Schauder's xed-point theorem requires the choice of a set with appropriate properties, which is mapped into itself by the considered operator. In applications, especially in the theory of non-linear boundary value problems, ordered normed function spaces and positive operators are often considered, i.e., which leave the corresponding cone invariant, or isotone increasing operators, i.e., if x y ) T (x) T (y). If confusions (see, e.g., 12.8.7, p. 634) are excluded, we also call these operators monotone. Let X = (X X+ k k) be an ordered Banach space, X+ a closed cone and a b] an order interval of X. If X+ is normal and T is a completely continuous (not necessarily isotone) operator that satis es T ( a b]) a b], then T has at least one xed point in a b] (Fig. 12.6b). Notice that the condition T ( a b]) a b] automatically holds for any isotone increasing operator T , which is de ned on an (o)-interval (order interval) a b] of the space X if it maps only the endpoints a b
634 12. Functional Analysis y b
b
y=x
y
y=f(x)
y=f(x)
a
a a)
y=x
a
x0
b x
b)
a x0
b x
Figure 12.6 into a b], i.e., when the two conditions T (a) a and T (b) b are satis ed. Then both sequences x0 = a and xn+1 = T (xn) (n 0) and y0 = b and yn+1 = T (yn) (n 0) (12.202) are well de ned, i.e., xn yn 2 a b] n 0. They are monotone increasing and decreasing, respectively, i.e., a = x0 x1 : : : xn : : : and b = y0 y1 : : : yn : : :. A xed point x of the operator T is called minimal , maximal, respectively, if for every xed point z of T the inequalities x z, z x hold, respectively. Now, we have the following statement (Fig. 12.6a)): Let X be an ordered Banach space with a closed cone X+ and T : D ;! X D X a continuous isotone increasing operator. Let a b] D be such that T (a) a and T (b) b. Then T ( a b]) a b], and the operator T has a xed point in a b] if one of the following conditions is ful lled: a) X+ is normal and T is compact b) X+ is regular. 1 Then the sequences fxng1 n=0 and fyngn=0 , de ned in (12.202), converge to the minimal and to the maximal xed points of T in a b], respectively. The notion of the super- and subsolutions is based on these results (see 12.14]).
12.8.7 Monotone Operators in Banach Spaces
1. Special Properties
An arbitrary operator T : D X ;! Y (X Y normed spaces) is called demi-continuous at the point 1 x0 2 D if for each sequence fxng1 n=1 D converging to x0 (in the norm of X) the sequence fT (xn )gn=1 converges weakly to T (x0 ) in Y. T is called demi-continuous on the set D if T is demi-continuous at every point of D. In this paragraph we introduce another generalization of the notion of monotonity known from real analysis. Let X now be a real Banach space, X its dual, D X and T : D ;! X a non-linear operator. T is called monotone if 8 x y 2 D the inequality (T (x) ; T (y) x ; y) 0 holds. If X = IH is a Hilbert space, then ( ) means the scalar product, while in the case of an arbitrary Banach space we refer to the notation introduced in 12.5.4.1, p. 623. The operator T is called strongly monotone if there is a constant c > 0 such that (T (x) ; T (y) x ; y) > ckx ; yk2 for 8 x y 2 D. An operator T : X ;! X (T (x) x) = 1. is called coercive if kxlim k!1 kxk
2. Existence Theorems
for solutions of operator equations with monotone operators are given here only exemplarily: If the operator T , mapping the real separable Banach space X into X (DT = X), is monotone demi-continuous and coercive, then the equation T (x) = f has a solution for arbitrary f 2 X . If in addition the operator T is strongly monotone, then the solution is unique. In this case the inverse operator T ;1 also exists. For a monotone, demi-continuous operator T : IH ;! IH in a Hilbert space IH with DT = IH, there holds
12.9 Measure and Lebesgue Integral 635
Im(I + T ) = IH, where (I + T );1 is continuous. If we suppose that T is strongly monotone, then T ;1 is bijective with a continuous T ;1 . Constructive approximation methods for the solution of the equation T (x) = 0 with a monotone operator T in a Hilbert space are based on the idea of Galerkin's method (see 19.4.2.2, p. 909, or 12.10], 12.18]). By means of this theory set-valued operators T : X ;! 2X can also be handled. The notion of monotonity is then generalized by (f ; g x ; y) 0 8 x y 2 DT and f 2 T (x) g 2 T (y).
12.9 Measure and Lebesgue Integral 12.9.1 Sigma Algebra and Measures
The initial point for introducing measures is a generalization of the notion of the length of an interval in IR, of the area, and of the volume of subsets of IR2 and IR3, respectively. This generalization is necessary in order to \measure" as many sets as possible and to \make integrable" as many functions as possible. For instance, the volume of an n-dimensional rectangular parallelepiped Y n Q = fx 2 IRn : ak xk bk (k = 1 2 : : : n)g has the value (bk ; ak ): (12.203) k=1
1. Sigma Algebra
Let X be an arbitrary set. A non-empty system A of subsets from X is called a -algebra if: a) A 2 A implies X n A 2 A and 1 (12.204a) # b) A1 A2 : : : An : : : 2 A implies An 2 A: (12.204b) n=1
Every -algebra contains the sets and X, the intersection of countably many of its sets and also the di erence sets of any two of its sets. In the following IR denotes the set IR of real numbers extended by the elements f;1g and f+1g (extended real line), where the algebraic operations and the order properties from IR are extended to IR in the natural way. The expressions (1) + (1) and 1 are meaningless, while 0 (+1) and 1 0 (;1) are assigned the value 0.
2. Measure
A function : A ;! IR+ = IR+ +1, de ned on a -algebra A, is called a measure if a) (A) 0 8 A 2 A b) ( ) = 0 ! X 1 1 # c) A1 A2 : : : An : : : 2 A Ak \ Al = (k 6= l) implies An = (An): n=1
n=1
(12.205a) (12.205b) (12.205c)
The property c) is called -additivity of the measure. If is a measure on A, and for the sets A B 2 A A B holds, then (A) (B ) (monotonicity). If An 2 A (n = 1 2 : : :) and A1 A2 , 1 S then An = nlim (An) (continuity from below). !1 n=1
Let A be a -algebra of subsets of X and a measure on A. The triplet X = (X A ) is called a measure space, and the sets belonging to A are called measurable or A-measurable. A: Counting Measure: Let X be a nite set fx1 x2 : : : xN g, A the -algebra of all subsets of X, and let assign a non-negative number pk to each xk (k = 1 : : : N ). Then the function de ned on A for every set A 2 A, A = fxn1 xn2 : : : xnk g by (A) = pn1 + pn2 + + pnk is a measure which takes on only nite values since (X) = p1 + + pN < 1. This measure is called the counting measure. B: Dirac Measure: Let A be a -algebra of subsets of a set X and a an arbitrary given point from X. Then a measure is de ned on A by a (A) = 10 ifif aa 22= A (12.206) A:
636 12. Functional Analysis It is called the function (concentrated on a). Obviously a (A) = a (A) = A (a) (see 12.5.4, p. 623), where A denotes the characteristic function of the set A. C: Lebesgue Measure: Let X be a metric space and B(X) the smallest -algebra of subsets of X which contains all the open sets from X. B(X) exists as the intersection of all the -algebras containing all the open sets, and is called the Borel -algebra of X. Every element from B(X) is called a Borel set (see 12.6]). Suppose now, X = IRn (n 1). Using an extension procedure we can construct a -algebra and a measure on it, which coincides with the volume on the set of all rectangularn parallelepipeds in IRn. More precisely: There exists a uniquely de ned -algebra A of subsets of IR and a uniquely de ned measure on A with the following properties: a) Each open set from IRn belongs to A, in other words: B(IRn) A. b) If A 2 A, (A) = 0 and B A then B 2 A and (B ) = 0. n c) If Q is a rectangular parallelepiped, then Q 2 A, and (Q) = Q (bk ; ak ). k=1
d) is translation invariant, i.e., for every vector x 2 IRn and every set A 2 A one has x + A = fx + y : y 2 Ag 2 A and (x + A) = (A). The elementsnof A are called Lebesgue measurable subsets of IRn. is the (n-dimensional) Lebesgue
measure in IR . Remark: In measure theory and integration theory one says that a certain statement (property, or condition) with respect to the measure is valid almost everywhere or -almost everywhere on a set X, if the set, where the statement is not valid, has measure zero. We write a.e. or -a.e.x For instance, if is the Lebesgue measure on IR and A B are two disjoint sets with IR = A B and f is a function on IR with f (x) = 1 8x 2 A and f (x) = 0 8 x 2 B , then f = 1 -a.e. on IR if and only if (B ) = 0.
12.9.2 Measurable Functions 12.9.2.1 Measurable Function
Let A be a -algebra of subsets of a set X. A function f : X ;! IR is called measurable if for an arbitrary 2 IR the set f ;1 (( +1]) = fx : x 2 X f (x) > g is in A. A complex-valued function g + ih is called measurable if both functions g and h are measurable. If A is the -algebra of the Lebesgue measurable sets of IRn and f : IRn ;! IR is a continuous function, then the set f ;1(( +1]) = f ;1(( +1)), according to 12.2.3, p. 610, is open for every 2 IR, hence f is measurable.
12.9.2.2 Properties of the Class of Measurable Functions
The notion of measurable functions requires no measure but a -algebra. Let A be a -algebra of subsets of the set X and let f g fn : X ;! IR be measurable functions. Then the following functions (see 12.1.7.4, p. 602) are also measurable: a) f for every 2 IR f g b) f+ f; jf j f _ g and f ^ g c) f + g, if there is no point from X where the expression (1) + (1) occurs d) sup fn inf fn lim sup fn (= nlim sup fk ) lim inf fn !1 k n e) the pointwise limit lim fn, in case it exists f) if f 0 and p 2 IR p > 0, the f p is measurable. A function f : X ;! IR is called elementary or simple if there is a nite number of pairwise disjoint sets x
Here and in the following \a.e." is an abbrevation for \almost everywhere".
12.9 Measure and Lebesgue Integral 637 n A1 : : : An 2 A and real numbers 1 : : : n such that f = P k k , where k denotes the characterk=1 istic function of the set Ak . Obviously, each characteristic function of a measurable set is measurable, so every elementary function is measurable. It is interesting that each measurable function can be approximated arbitrarily well by elementary functions: For each measurable function f 0 there exists a monotone increasing sequence of non-negative elementary functions, which converges pointwise to f .
12.9.3 Integration
12.9.3.1 Denition of the Integral
Z
Z
Let (X A ) be a measurable space. The integral f d (also denoted by f d in this section, X except point 5. we prefer the latter notation) for a measurable function f is de ned by means of the following steps: n 1. If f is an elementary function f = P k k , then
Z
f d =
n X
k=1
k (Ak ):
k=1
(12.207)
2. If f : X ;! IR (f 0), then Z
f d = sup
Z
g d : g is an elementary function with 0 g(x) f (x) 8 x 2 X : (12.208)
3. If f : X ;! IR and f+ f; are the positive and the negative parts of f , then Z Z Z f d = f+ d ; f; d
(12.209) under the condition that at least one of the integrals on the right side is nite (in order to avoid the meaningless expression 1 ; 1). 4. For a complex-valued function f = g + ih, if the integrals (12.209) of the functions g h are nite, put Z Z Z f d = g d + i h d : (12.210) 5. If for any measurable set A and a function f there exists the integral of the function fA then put Z Z f d = fA d : (12.211) A
The integral of a measurable function is in general a number from IR. AZfunction f : X ;! IR is called integrable or summable over X with respect to if it is measurable and jf j d < 1.
12.9.3.2 Some Properties of the Integral
Let (X A ) be a measure space, f g : X ;! IR be measurable functions and 2 IR. 1. If f is integrable, then f is nite a.e., i.e., fx 2 X: jf (x)j = +1g = 0.
Z
Z
2. If f is integrable, then f d jf j d . Z
3. If f is integrable and f 0, then f d 0.
Z
Z
4. If 0 g(x) f (x) on X and f is integrable, then g is also integrable, and g d f d .
638 12. Functional Analysis Z
Z
Z
5. If f g are integrable, then f + g is integrable, and (f + g) d = f d + g d . Z
Z
6. If f g are integrable on A 2 A, i.e., there exist the integrals A f d and A g d according to Z
Z
(12.211) and f = g -a.e. on A, then f d = g d . A A If X = IRn and is Lebesgue measure, then we have the notion of the (n-dimensional) Lebesgue integral (see also 8.2.3.1, 3., p. 454). In the case n = 1 and A = a b], for every continuous function f on Zb Z a b] both the Riemann integral f (x) dx (see 8.2.1.1, 2., p. 441) and the Lebesgue integral f d a ab] are de ned. Both values are nite and equal to each other. Furthermore, if f is a bounded Riemann integrable function on a b], then it is also Lebesgue integrable and the values of the two integrals coincide. The set of Lebesgue integrable functions is larger than the set of the Riemann integrable functions and it has several advantages, e.g., when passing to the limit under the integral sign and f , jf j is Lebesgue integrable simultaneously.
12.9.3.3 Convergence Theorems
Now Lebesgue measurable functions will be considered throughout.
1. B. Levi's Theorem on Monotone Convergence
Let ffng1 n=1 be an a.e. monotone increasing sequence of non-negative integrable functions with values in IR. ThenZ Z fn d = nlim f d : (12.212) nlim !1 !1 n
2. Fatou's Theorem
Let ffng1 be a sequence of non-negative IR-valued measurable functions. Then Z n=1 Z lim inf fn d lim inf fn d :
3. Lebesgue's Dominated Convergence Theorem
(12.213)
Let ffng be a sequence of measurable functions convergent on X a.e. to some function f . If there exists an integrable function g such that jfnj g a.e., then f = lim fn is integrable and Z Z lim fn d = lim fn d : (12.214)
4. Radon{Nikodym Theorem
a) Assumptions: Let (X A ) be a - nite measure space, i.e., there exists a sequence fAng, An 2 A 1 such that X = S An and (An) < 1 for 8 n. In this case the measure is called -nite. It is called n=1 nite if (X) < 1, and it is called a probability measure if (X) = 1. A real function ' de ned on A is called absolutely continuous with respect to if (A) = 0 implies '(A) = 0. We denote this property by ' , . R For an integrable function f , the function ' de ned on A by '(A) = A f d is -additive and absolutely continuous with respect to the measure . The converse of this property plays a fundamental role in many theoretical investigations and practical applications: b) Radon{Nikodym Theorem: Suppose a -additive function ' and a measure are given on a -algebra A, and let ' , . Then there exists a -integrable function f such that for each set A 2 A, Z '(A) = f d : (12.215) A
12.9 Measure and Lebesgue Integral 639
The function f is uniquely determined up to its equivalence class, and ' is non-negative if and only if f 0 -a.e.
12.9.4 p Spaces L
Let (X A ) be a measure space and p a real number 1 p < 1. For a measurable function f , according to 12.9.2.2, p. 636, the function jf jp is measurable as well, so the expression
Z p1 Np(f ) = jf jp d (12.216) is de ned (and may be equal to +1). A measurable function f : X ;! IR is called p-th power integrable, or an Lp-function if Np(f ) < +1 holds or, equivalent to this, if jf jp is integrable. For every p with 1 p < +1, we denote the set of all Lp-functions, i.e., all functions p-th power integrable with respect to on X, by Lp( ) or by Lp(X) or in full detail Lp(X A ). For p = 1 we use the simple notation L(X). For p = 2 the functions are called quadratically integrable. We denote the set of all measurable -a.e. bounded functions on X by L1( ) and de ne the essential supremum of a function f as N1(f ) = ess. sup f = inf fa 2 IR : jf (x)j a {a.e.g: (12.217) Lp( ) (1 p 1) equipped with the usual operations for measurable functions and taking into consideration Minkowski inequality for integrals (see 1.4.2.13, p. 32), is a vector space and Np() is a semi-norm on Lp( ). If f g means that f (x) g(x) holds -a.e., then Lp( ) is also a vector lattice and even a K -space (see 12.1.7.4, p. 602). Two functions f g 2 Lp( ) are called equivalent (or we declare them as equal) if f = g -a.e. on X. In this way, functions are identi ed if they are equal a.e. The factorization of the set Lp(X) modulo the linear subspace Np;1(0) leads to a set of equivalence classes on which the algebraic operations and the order can be transferred naturally. So we get a vector lattice (K -space) again, which is denoted now by Lp(X ) or Lp( ). Its elements are called functions, as before, but actually they are classes of equivalent functions. It is very important that kf^kp = Np(f ) is now a norm on Lp( ) (f^ stands here for the equivalence class of f , which will later be denoted simply by f ), and (Lp( ) kf kp) for every p with 1 p +1 is a Banach lattice with several good compatibility conditions between norm and order. For p = 2 with Z (f g) = fg d as a scalar product, L2 ( ) is also a Hilbert space (see 12.12]). Very often for a measurable subset ) IRn the space Lp()) is considered. Its de nition is not a problem because of step 5 in (12.9.3.1, p. 637). The spaces Lp() ), where is the n-dimensional Lebesgue measure, can also be introduced as the completions (see 12.2.2.5, p. 610 and 12.3.2, p. 612) of the non-complete normed spaces C ()) of all Z p1 continuous functions on the set ) IRn equipped with the integral norm kxkp = jxjp d (1 p < 1) (see 12.18]). Let X be a set with a nite measure, i.e., (X) < +1, and suppose for the real numbers p1 p2, 1 p1 < p2 +1. Then Lp2 (X ) Lp1 (X ), and with a constant C = C (p1 p2 (X)) > 0 independent of x, there holds the estimation kxk1 C kxk2 for x 2 Lp2 (here kxkk denotes the norm of the space Lpk (X ) (k = 1 2)).
12.9.5 Distributions
12.9.5.1 Formula of Partial Integration
For an arbitrary (open) domain ) IRn, C01()) denotes the set of all arbitrary many times in ) di erentiable functions ' with compact support, i.e., the set supp(') = fx 2 ) : '(x) 6= 0g is compact in IRn and lies in ). By L1loc()) we denote the set of all locally summable functions with respect to
640 12. Functional Analysis the measure in IRn, i.e., all the measurable functions f (equivalent classes) on ) such that R jfLebesgue ! j d < +1 for every bounded domain ! ). Both sets are vector spaces (with the natural algebraic operations). There holds Lp()) L1loc()) for 1 p 1, and L1loc()) = L1 ()) for a bounded ). If we consider the elements of C k ()) as the classes generated by them in Lp()), then the inclusion C k ()) Lp()) holds for bounded ), where C k ()) is at once dense. If ) is unbounded, then the set C01()) is dense (in this sense) in Lp()). For a given function f 2 C k ()) and an arbitrary function ' 2 C01()) the formula of partial integration has theZ form Z f (x)D '(x) d = (;1)j j '(x)D f (x) d (12.218)
8 with jj k (we have used the fact that D 'j@ = 0), which can be taken as the starting point for the de nition of the generalized derivative of a function f 2 L1loc()).
12.9.5.2 Generalized Derivative
Suppose f 2 L1loc()). If there exists a function g 2 L1loc()) such that 8 ' 2 C01()) with respect to some multi-index the equation Z Z f (x)D '(x) d = (;1)j j g(x)'(x) d (12.219)
holds, then g is called the generalized derivative (derivative in the Sobolev sense or distributional derivative) of order of f . We write g = D f for this as in the classical case. 1 1 We de ne the convergence of a sequence f'k g1 k=1 in the vector space C0 ()) to ' 2 C0 ()) as follows: a) 9 a compact K ) with supp('k ) K 8n 'k ;! ' if and only if b) D 'k ! D 'setuniformly on K for each multi-index : (12.220) 1 The set C0 ()), equipped with this convergence of sequences, is called the fundamental space, and is denoted by D()). Its elements are often called test functions.
12.9.5.3 Distributions
A linear functional ` on D()) continuous in the following sense (see 12.2.3, p. 610): 'k ' 2 D()) and 'k ;! ' implies `('k ) ;! `(') (12.221) is called a generalized function or a distribution. A: If f 2 L1loc()), then Z `f (') = (f ') = f (x)'(x) d ' 2 D()) (12.222) is a distribution. A distribution, de ned by a locally summable function as in (12.222), is called regular. Two regular distributions are equal, i.e., `f (') = `g (') 8' 2 D()), if and only if f = g a.e. with respect to . B: Let a 2 ) be an arbitrary xed point. Then `a (') = '(a) ' 2 D()) is a linear continuous functional on D()), hence a distribution, which is called the Dirac distribution, distribution or function. Since `a cannot be generated by any locally summable function (see 12.11], 12.24]), it is an example for a non-regular distribution. The set of all distributions is denoted by D0()). From a more general duality theory than that discussed in 12.5.4, p. 623, we get D0()) as the dual space of D()). Consequently, we should write D ()) instead. In the space D0()), it is possible to de ne several operations with its elements and with functions from C 1()), e.g., the derivative of a distribution or the convolution of two distributions, which make
12.9 Measure and Lebesgue Integral 641
D0()) important not only in theoretical investigations but also in practical applications in electrical engineering, mechanics, etc. For a review and for simple examples in applications of generalized functions see, e.g., 12.11], 12.24]. Here, we discuss only the notion of the derivative of a generalized function.
12.9.5.4 Derivative of a Distribution
If ` is a given distribution, then the distribution D ` de ned by (D `)(') = (;1)j j`(D ') ' 2 D()) (12.223) is called the distributional derivative of order of `. Let f be a continuously di erentiable function, say on IR (so f is locally summable on IR, and f can be considered as a distribution), let f 0 be its classical derivative and D1f its distributional derivative of order 1. Then: Z (D1f ') = f 0(x)'(x) dx (12.224a) IR
from which by partial Z integration there follows (D1f ') = ; f (x)'0(x) dx = ;(f '0 ): IR
In the case of a regular distribution `f with f 2 L1loc()) by using (12.222) we obtain Z (D `f )(') = (;1)j j`f (D ') = (;1)j j f (x)D ' d
(12.224b) (12.225)
and get the generalized derivative of the function f in the Sobolev sense (see (12.219)). A: For the regular distribution generated by the obviously locally summable Heaviside function x 0 !(x) = 10 for (12.226) for x < 0 we get the non-regular distribution as the derivative. B: In mathematical modeling of technical and physical problems we are faced with (in a certain sense idealized) inuences concentrated at one point, such as a \point-like" force, needle-deection, collision, etc., which can be expressed mathematically by using the or Heaviside function. For example, m a is the mass density of a point-like mass m concentrated at one point a (0 a l) of a beam of length l. The motion of a spring-mass system on which at time t0 there acts a momentary external force F is described by the equation x + !2x = F t0 . With the initial conditions x(0) = x_ (0) = 0 its solution is x(t) = F! sin(!(t ; t0 ))!(t ; t0).
642 13. Vector Analysis and Vector Fields
13 VectorAnalysisandVectorFields 13.1 Basic Notions of the Theory of Vector Fields 13.1.1 Vector Functions of a Scalar Variable 13.1.1.1 Denitions
1. Vector Function of a Scalar Variable t
A vector function of a scalar variable is a vector ~a whose components are real functions of t: ~a = ~a(t) = ax(t)~ex + ay (t)~ey + az (t)~ez : (13.1) The notions of limit, continuity, di erentiability are de ned componentwise for the vector ~a(t).
2. Hodograph of a Vector Function
If we consider the vector function ~a(t) as a position or radius vector ~r = ~r(t) of a point P , then this function describes a space curve while t varies (Fig. 13.1). This space curve is called the hodograph of the vector function ~a(t). P1 r1 0
P
P2
r2
r
r3
P3
Figure 13.1
0
dr dt
Dr r+Dr
Figure 13.2
13.1.1.2 Derivative of a Vector Function
30 25 20 15
Figure 13.3
The derivative of (13.1) with respect to t is also a vector function of t: d~a = lim ~a(t + %t) ; ~a(t) = dax(t)~e + day (t)~e + daz (t)~e : (13.2) dt t!0 %t dt x dt y dt z r The geometric representation of the derivative d~ dt of the radius vector is a vector pointing in the direction of the tangent of the hodograph at the point P (Fig. 13.2). Its length depends on the choice of the parameter t. If t is the time, then the vector ~r(t) describes the motion of a point P in space (the r space curve is its path), and d~ motion. If t = s dt has the direction and magnitude of the velocity of this d~ r is the arclength of this space curve, measured from a certain point, then obviously = 1. ds
13.1.1.3 Rules of Di erentiation for Vectors d dt d dt d dt
~ (~a ~b ~c) = d~a db d~c dt dt dt d~a ('~a) = d' dt ~a + ' dt a ~ d~b (~a~b) = d~ dt b + ~a dt
(13.3a) (' is a scalar function of t)
(13.3b) (13.3c)
13.1 Basic Notions of the Theory of Vector Fields 643
d d~b ~ d~a ~ dt (~a b) = dt b + ~a dt d d~a d' dt ~a '(t)] = d' dt
(the factors must not be interchanged)
(13.3d)
(chain rule):
(13.3e)
a d~a If j~a(t)j = const, i.e., ~a2 (t) = ~a(t) ~a(t) = const, then it follows from (13.3c) that ~a d~ dt = 0, i.e., dt and ~a are perpendicular to each other. Examples of this fact: A: Radius and tangent vectors of a circle in the plane and B: position and tangent vectors of a curve on the sphere. Then the hodograph is a spherical curve .
13.1.1.4 Taylor Expansion for Vector Functions
hn dn~a a h2 d2~a ~a(t + h) = ~a(t) + h d~ (13.4) dt + 2! dt2 + + n! dtn + : The expansion of a vector function in a Taylor series makes sense only if it is convergent. Because the limit is de ned componentwise, the convergence can be checked componentwise, so the convergence of this series with vector terms can be determined exactly by the same methods as the convergence of a series with complex terms (see 14.3.2, p. 691). So the convergence of a series with vector terms is reduced to the convergence of a series with scalar terms. The di erential of a vector function ~a(t) is de ned by: a d~a = d~ (13.5) dt %t:
13.1.2 Scalar Fields
13.1.2.1 Scalar Field or Scalar Point Function
If we assign a number (scalar value) U to every point P of a subset of space, then we write U = U (P ) and we call (13.6a) a scalar eld (or scalar function). Examples of scalar elds are temperature, density, potential, etc., of solids. A scalar eld U = U (P ) can also be considered as U = U (~r) where ~r is the position vector of the point P with a given pole 0 (see 3.5.1.1, 6., p. 181).
(13.6a)
(13.6b)
13.1.2.2 Important Special Cases of Scalar Fields 1. Plane Field
We have a plane eld, if the function is de ned only for the points of a plane in space.
2. Central Field
If a function has the same value at all points P lying at the same distance from a xed point C (~r1), called the center, then we call it a central symmetric eld or also a central or spherical eld. The function U depends only on the distance CP = j~rj: U = f (j ~r j): (13.7a) The eld of the intensity of a point-like source, e.g., the eld of brightness of a point-like source of light at the pole, can be described with j~rj = r as the distance from the light source: U = rc2 (c const) : (13.7b)
644 13. Vector Analysis and Vector Fields
3. Axial Field
If the function U has the same value at all points lying at an equal distance from a certain straight line (axis of the eld) then the eld is called cylindrically symmetric or an axially symmetric eld , or briey an axial eld .
13.1.2.3 Coordinate Denition of a Field
If the points of a subset of space are given by their coordinates, e.g., by Cartesian, cylindrical, or spherical coordinates, then the corresponding scalar eld (13.6a) is represented, in general, by a function of three variables: U = (x y z) U = " ( ' z) or U = (r '): (13.8a) In the case of a plane eld, a function with two variables is sucient. It has the form in Cartesian and polar coordinates: U = (x y) or U = " ( '): (13.8b) The functions in (13.8a) and (13.8b), in general, are assumed to be continuous, except, maybe, at some points, curves or surfaces of discontinuity. The functions have the form q q a) for a central eld: U = U ( x2 + y2 + z2 ) = U ( 2 + z2 ) = U (r) (13.9a) q 2 2 b) for an axial eld: U = U ( x + y ) = U ( ) = U (r sin ): (13.9b) Dealing with central elds is easiest using spherical coordinates, with axial elds using cylindrical coordinates.
13.1.2.4 Level Surfaces and Level Lines of a Field 1. Level Surface
A level surface is the union of all points in space where the function (13.6a) has a constant value U = const: (13.10a) Di erent constants U0 U1 U2 : : : de ne di erent level surfaces. There is a level surface passing through every point except the points where the function is not de ned. The level surface equations in the three coordinate systems used so far are: U = (x y z) = const U = " ( ' z) = const U = (r ') = const: (13.10b) Examples of level surfaces of di erent elds: A: U = ~c ~r = cxx + cy y + cz z: Parallel planes. B: U = x2 + 2y2 + 4z2 : Similar ellipsoids in similar positions. C: Central eld: Concentric spheres. D: Axial eld: Coaxial cylinders.
2. Level Lines
Level lines replace level surfaces in plane elds. They satisfy the equation U = const: (13.11) Level lines are usually drawn for equal intervals of U and each of them is marked by the corresponding value of U (Fig. 13.3). Well-known examples are the isobaric lines on a synoptic map or the contour lines on topographic maps. In particular cases, level surfaces degenerate into points or lines, and level lines degenerate into separate points. The level lines of the elds a) U = xy, b) U = xy2 , c) U = r2, d) U = 1r are represented in Fig. 13.4.
13.1 Basic Notions of the Theory of Vector Fields 645 1 2 34 y
y 4
-4 -3 -2 1 -1
3
-1 0 1 2 -1
1
2
43 2 1
1
3
1 -1 -2 -3
4
1
-1
x
-1 -4
-4-3 -2 -1
-1 -2 -3 -4
a)
b) y
1
1
2
y 1
5 4 3 4
-1 0
1
x
0
c)
13.1.3 Vector Fields
x
3
2
1
x
d) Figure 13.4
13.1.3.1 Vector Field or Vector Point Function
~ to every point P of a subset of space, then we denote it by If we assign a vector V V~ = V~ (P ) (13.12a) and we call (13.12a) a vector eld . Examples of vector elds are the velocity eld of a uid in motion, a eld of force, and a magnetic or electric intensity eld. ~ =V ~ (P ) can be regarded as a vector function A vector eld V ~V = V ~ (~r) (13.12b) ~ lie in a where ~r is the position vector of the point P with a given pole 0. If all values of ~r as well as V plane, then the eld is called a plane vector eld (see 3.5.2, p. 189).
13.1.3.2 Important Cases of Vector Fields 1. Central Vector Field
~ lie on straight lines passing through a xed point called the center In a central vector eld all vectors V
(Fig. 13.5a).
If we locate the pole at the center, then the eld is de ned by the formula V~ = f (~r) ~r
(13.13a)
646 13. Vector Analysis and Vector Fields where all the vectors have the same direction as the radius vector ~r. It often has some advantage to de ne the eld by the formula V~ = '(~r)~rr (13.13b) ~ and ~r is a unit vector. where '(~r) is the length of the vector V r c
a)
b)
c)
Figure 13.5
2. Spherical Vector Field
~ depends A spherical vector eld is a special case of a central vector eld, where the length of the vector V only on the distance j~rj (Fig. 13.5b). Examples are the Newton and the Coulomb force eld of a point-like mass or of a point-like electric charge: V~ = rc3 ~r = rc2 ~rr (c const) : (13.14) The special case of a plane spherical vector eld is called a circular eld .
3. Cylindrical Vector Field
a) All vectors V~ lie on straight lines intersecting a certain line (called the axis) and perpendicular to
it, and b) all vectors V~ at the points lying at the same distance from the axis have equal length, and they are directed either toward the axis or away from it (Fig. 13.5c). If we locate the pole on the axis parallel to the unit vector ~c, then the eld has the form V~ = '( )~r (13.15a) where ~r is the projection of ~r on a plane perpendicular to the axis: ~r = ~c (~r ~c): (13.15b) By intersecting this eld with planes perpendicular to the axis, we always get equal circular elds.
13.1.3.3 Coordinate Representation of Vector Fields 1. Vector Field in Cartesian Coordinates
The vector eld (13.12a) can be de ned by scalar elds V1(~r), V2(~r), and V3(~r) which are the coordinate ~ , i.e., the coecients of its decomposition into any three non-coplanar base vectors ~e1 , functions of V ~e2 , and ~e3 :
13.1 Basic Notions of the Theory of Vector Fields 647
V~ = V1~e1 + V2~e2 + V3~e3:
(13.16a) ~ ~ ~ If we take the coordinate unit vectors i, j, and k as the base vectors and express the coecients V1 , V2, V3 in Cartesian coordinates, then we get: V~ = Vx(x y z)~i + Vy (x y z)~j + Vz (x y z)~k: (13.16b) So, the vector eld can be de ned with the help of three scalar functions of three scalar variables.
2. Vector Field in Cylindrical and Spherical Coordinates
In cylindrical and spherical coordinates, the coordinate unit vectors ~e ~e' ~ez (= ~k) and ~er (= ~rr ) ~e ~e' (13.17a) are tangents to the coordinate lines at each point (Fig. 13.6, 13.7). In this order they always form a right-handed system. The coecients are expressed as functions of the corresponding coordinates: V~ = V( ' z)~e + V'( ' z)~e' + Vz ( ' z)~ez (13.17b) V~ = Vr (r ')~er + V(r ')~e + V'(r ')~e': (13.17c) At transition from one point to the other, the coordinate unit vectors change their directions, but remain mutually perpendicular. z z V
y ez=k
V
er ej
ej 0
er
eJ
ej V y
0
er
0
x
y x
x
Figure 13.6
Figure 13.7
13.1.3.4 Transformation of Coordinate Systems
Figure 13.8
See also Table 13.1.
1. Cartesian Coordinates in Terms of Cylindrical Coordinates Vx = V cos ' ; V' sin ' Vy = V sin ' + V' cos ' Vz = Vz : 2. Cylindrical Coordinates in Terms of Cartesian Coordinates V = Vx cos ' + Vy sin ' V' = ;Vx sin ' + Vy cos ' Vz = Vz : 3. Cartesian Coordinates in Terms of Spherical Coordinates Vx = Vr sin cos ' ; V' sin ' + V cos ' cos
(13.18) (13.19)
Vy = Vr sin sin ' + V' cos ' + V sin ' cos Vz = Vr cos ; V sin :
(13.20)
Vr = Vx sin cos ' + Vy sin sin ' + Vz cos V = Vx cos cos ' + Vy cos sin ' ; Vz sin
(13.21)
4. Spherical Coordinates in Terms of Cartesian Coordinates
648 13. Vector Analysis and Vector Fields V' = ; Vx sin ' + Vy cos ':
5. Expression of a Spherical Vector Field in Cartesian Coordinates q V~ = '( x2 + y2 + z2 )(x~i + y~j + z~k): 6. Expression of a Cylindrical Vector Field in Cartesian Coordinates q V~ = '( x2 + y2)(x~i + y~j):
(13.22)
(13.23) In the case of a spherical vector eld, spherical coordinates are most convenient for investigations, ~ = V (r)~er and for investigations in cylindrical elds, cylindrical coordinates are most i.e., the form V ~ = V (')~e'. In the case of a plane eld (Fig. 13.8), we have convenient, i.e., the form V V~ = Vx(x y)~i + Vy (x y)~j = V( ')~e + V'( ')~e' (13.24) and for a circular eld q V~ = '( x2 + y2)(x~i + y~j) = '( )~e: (13.25) Table 13.1 Relations between the components of a vector in Cartesian, cylindrical, and spherical coordinates
Cartesian coordinates V~ = Vx~ex + Vy~ey + Vz~ez Vx Vy
Cylindrical coord. Spherical coordinates V~e + V'~e' + Vz~ez Vr~er + V~e + V'~e' = V cos ' ; V' sin ' = Vr sin cos ' + V cos cos ' ; V' sin '
= V sin ' + V' cos ' = Vr sin sin ' + V cos sin ' + V' cos ' Vz = Vz = Vr cos ; V sin Vx cos ' + Vy sin ' = V = Vr sin + V cos ;Vx sin ' + Vy cos ' = V' = V' Vz = Vz = Vr cos ; V sin Vx sin cos ' + Vy sin sin ' + Vz cos = V sin + Vz cos = Vr Vx cos cos ' + Vy cos sin ' ; Vz sin = V cos ; Vz sin = V ;Vx sin ' + Vy cos ' = V' = V'
13.1.3.5 Vector Lines
A curve C is called a line of a vector or a vector line of the vector eld
V~ (~r) (Fig. 13.9) if the vector V~ (~r) is a tangent vector of the curve
at every point P . There is a vector line passing through every point of the eld. Vector lines do not intersect each other, except, maybe, ~ is not de ned, or where it is the zero at points where the function V vector. The di erential equations of the vector lines of a vector eld V~ given in Cartesian coordinates are
Figure 13.9
13.2 Dierential Operators of Space 649
dy dz a) in general: dx Vx = Vy = Vz (13.26a)
dy b) for a plane eld: dx Vx = Vy :
(13.26b)
To solve these di erential equations see 9.1.1.2, p. 489 or 9.2.1.1, p. 517. A: The vector lines of a central eld are rays starting at the center of the vector eld. B: The vector lines of the vector eld V~ = ~c ~r are circles lying in planes perpendicular to the vector ~c. Their centers are on the axis parallel to ~c.
13.2 Dierential Operators of Space
13.2.1 Directional and Space Derivatives
13.2.1.1 Directional Derivative of a Scalar Field
The directional derivative of a scalar eld U = U (~r) at a point P with position vector ~r in the direction ~c (Fig. 13.10) is de ned as the limit of the quotient @U = lim U (~r + "~c) ; U (~r) : (13.27) @~c "!0 " 0 If the derivative of the eld U = U (~r) at a point ~r in the direction of the unit vector ~c of ~c is denoted @U , then the relation between the derivative of the function with respect to the vector ~c and with by @~ c0 respect to its unit vector ~c 0 at the same point is @U = j~cj @U : (13.28) @~c @~c 0 The derivative @U0 with respect to the unit vector represents the speed of increase of the function U in @~c the direction of the vector ~c 0 at the point ~r. If ~n is the normal unit vector to the level surface passing through the point ~r, and ~n is pointing in the direction of increasing U , then @U has the greatest value @~n among all the derivatives at the point with respect to the unit vectors in di erent directions. Between the directional derivatives with respect to n~ and with respect to any direction ~c 0, we have the relation @U @U @U 0 0 (13.29) @~c 0 = @~n cos(~c ~n) = @~n cos ' = ~c grad U (see (13.35), p. 651) : In the following, directional derivatives always mean the directional derivative with respect to a unit vector.
0
n ϕ
r+εc
V( r ) c
r
V(
P
r
r+ε
a)
dV( r )
P
0
Figure 13.10
Figure 13.11
a
650 13. Vector Analysis and Vector Fields
13.2.1.2 Directional Derivative of a Vector Field
The directional derivative of a vector eld is de ned analogously to the directional derivative of a scalar ~ =V ~ (~r) at a point P with position vector ~r eld. The directional derivative of the vector eld V (Fig. 13.11) with respect to the vector ~a is de ned as the limit of the quotient ~ @V V~ (~r + "~a) ; V~ (~r) : = lim (13.30) " ! 0 @~a " ~ =V ~ (~r) at a point ~r in the direction of the unit vector ~a 0 of ~a is If the derivative of the vector eld V ~ @ V , then denoted by @~ a0 ~ ~ @V @V (13.31) @~a = j~aj @~a 0 : ~ = Vx~ex + Vy~ey + Vz~ez , ~a = ax~ex + ay~ey + az~ez , we have: In Cartesian coordinates, i.e., for V ~ @ V = (~a grad )V ~ = (~a grad Vx)e~x + (~a grad Vy )e~y + (~a grad Vz )e~z : (13.32a) @~a In general coordinates we have: ~ @V ~ @~a = (~a grad )V ~ ~a) + grad (~a V ~ ) + ~adiv V ~ ;V ~ div ~a ; ~a rot V ~ ;V ~ rot ~a: = 1 (rot (V (13.32b) 2
13.2.1.3 Volume Derivative
~ at a point ~r are quantities of three Volume derivatives of a scalar eld U = U (~r) or a vector eld V forms, which are obtained as follows: 1. We surround the point ~r of the scalar eld or of the vector eld by a closed surface 1. This surface can be represented in parametric form ~r = ~r(u v) = x(u v)~ex + y(u v)~ey + z(u v)~ez , so the corresponding vectorial surface element is @~r @~r du dv : d~S = @u (13.33a) @v 2. We evaluate the surface integral over the closed surface 1. Here, the following three types of integrals can beZZconsidered: ZZ ZZ - U d~S - V~ d~S - V~ d~S: (13.33b) ()
()
()
3. We determine the limits (if they exist) ZZ 1 ZZ- U d~S lim 1 ZZ- V ~ d~S lim 1 - V ~ d~S: lim V !0 V V !0 V V !0 V () () ()
(13.33c)
Here V denotes the volume of the region of space that contains the point with the position vector ~r inside, and which is bounded by the considered closed surface 1. The limits (13.33c) are called volume derivatives. The gradient of a scalar eld and the divergence and the rotation of a vector eld can be derived from them in the given order. In the following paragraphs, we discuss these notions in details (we will even de ne them again.)
13.2.2 Gradient of a Scalar Field
The gradient of a scalar eld can be de ned in di erent ways.
13.2 Dierential Operators of Space 651
13.2.2.1 Denition of the Gradient
The gradient of a function U is a vector grad U , which can be assigned to every point of a scalar eld U = U (~r), having the following properties: 1. The direction of grad U is always perpendicular to the direction of the level surface U = const, passing through the considered point, 2. grad U is always in the direction in which the function U is increasing, 3. jgrad U j = @U @~n , i.e., the magnitude of grad U is equal to the directional derivative of U in the normal direction. If the gradient is de ned in another way, e.g., as a volume derivative or by the di erential operator, then the previous de ning properties became consequences of the de nition.
13.2.2.2 Gradient and Volume Derivative
The gradient U of the scalar eld U = U (~r) at a point ~r can be de ned as its volume derivative. If the following limit exists, then we call it the gradient of U at ~r: ZZ - U d~S () grad U = Vlim (13.34) !0 V : Here V is the volume of the region of space containing the point belonging to ~r inside and bounded by the closed surface 1. (If the independent variable is not a three-dimensional vector, then the gradient is de ned by the di erential operator.)
13.2.2.3 Gradient and Directional Derivative
The directional derivative of the scalar eld U with respect to the unit vector ~c 0 is equal to the projection of grad U onto the direction of the unit vector ~c 0: @U 0 (13.35) @~c 0 = ~c grad U i.e., the directional derivative can be calculated as the dot product of the gradient and the unit vector pointing into the required direction. Remark: The directional derivative at certain points in certain directions may also exist if the gradient does not exist there.
13.2.2.4 Further Properties of the Gradient
1. The absolute value of the gradient is greater if the level lines or level surfaces drawn as mentioned in 13.1.2.4, 2., p. 644, are more dense. 2. The gradient is the zero vector (grad U = ~0) if U has a maximum or minimum at the considered point. The level lines or surfaces degenerate to a point there.
13.2.2.5 Gradient of the Scalar Field in Di erent Coordinates 1. Gradient in Cartesian Coordinates
grad U = @U (x@xy z)~i + @U (x@yy z)~j + @U (x@zy z) ~k:
2. Gradient in Cylindrical Coordinates (x = cos ' y = sin ' z = z) grad U = grad U~e + grad' U~e' + gradz U~ez with 1 @U @U gradU = @U @ grad'U = @' gradz U = @z :
(13.36) (13.37a) (13.37b)
652 13. Vector Analysis and Vector Fields
3. Gradient in Spherical Coordinates (x = r sin cos ' y = r sin sin ' z = r cos ) grad U = gradr U~er + grad U~e + grad' U~e' with gradr U = @U gradU = 1 @U grad'U = 1 @U : @r r @ r sin @'
(13.38a)
(13.38b)
4. Gradient in General Orthogonal Coordinates ( )
If ~r( ) = x( )~i + y( )~j + z( )~k, then we get grad U = grad U~e + grad U~e + grad U~e where grad U = 1 @U grad U = 1 @U grad U = 1 @U : @ @ @~ r @~ r @~r @ @ @ @
(13.39a) (13.39b)
13.2.2.6 Rules of Calculations
We assume in the followings that ~c and c are constant. grad c = ~0 grad (U1 + U2) = grad U1 + grad U2 grad (c U ) = c grad U: d' grad U: grad (U1 U2) = U1grad U2 + U2 grad U1 grad '(U ) = dU ~1V ~ 2 ) = (V ~ 1 grad )V ~ 2 + (V ~ 2 grad )V ~1+V ~ 1 rot V ~2+V ~ 2 rot V ~ 1: grad (V grad (~r ~c) = ~c:
1. Di erential of a Scalar Field as the Total Di erential of the Function U
(13.40) (13.41) (13.42) (13.43)
@U @U dU = grad U d~r = @U @x dx + @y dy + @z dz:
(13.44)
dU = @U dx + @U dy + @U dz : dt @x dt @y dt @z dt
(13.45)
2. Derivative of a Function U along a Space Curve~r(t) 3. Gradient of a Central Field
~ ~ grad U (r) = U 0 (r) rr (spherical eld) (13.46a) grad r = rr ( eld of unit vectors): (13.46b)
13.2.3 Vector Gradient
The relation (13.32a) inspires the notation ~ @V ~ (13.47a) @~a = ~a grad (Vx~ex + Vy~ey + Vz~ez ) = ~a grad V ~ where grad V is called the vector gradient. It follows from the matrix notation of (13.47a) that the vector gradient, as a tensor, can be represented by a matrix: 0 @V @V @V 1 0 @V @V @V 1 x x x x x x B C B B C B @x @y @z C @x @y @z C 0 1 B C C B ax B C C B @V @V @V @V @V @V y y y y y y B C C B ~ =B ~ =B @ A a (~a grad )V C C (13.47b) grad V : (13.47c) y B C C B @x @y @z @x @y @z B C C B a z B C C B @ @Vz @Vz @Vz A @ @Vz @Vz @Vz A @x @y @z @x @y @z
13.2 Dierential Operators of Space 653
These types of tensors have a very important role in engineering sciences, e.g., for the description of tension and elasticity (see 4.3.2, 4., p. 263, and p. 263).
13.2.4 Divergence of Vector Fields 13.2.4.1 Denition of Divergence
~ (~r) which is called its divergence . The divergence is We can assign a scalar eld to a vector eld V de ned as a space derivative of the vector eld at a point ~r: ZZ ~ ~ - V dS ~ = lim () div V (13.48) V !0 V : ~ is considered as a stream eld, then the divergence can be considered as the uid If the vector eld V output or source, because it gives the amount of uid given in a unit of volume during a unit of time ~ . In the case div V ~ > 0 the point is called a source, owing by the considered point of the vector eld V ~ in the case div V < 0 it is called a sink.
13.2.4.2 Divergence in Di erent Coordinates 1. Divergence in Cartesian Coordinates ~ = @Vx + @Vy + @Vz div V @x @y @z
(13.49a)
~ (x y z) = Vx~i + Vy~j + Vz~k: with V
(13.49b)
~ can be represented as the dot product of the nabla operator and the vector V ~ as The scalar eld div V ~ ~ div V = r V (13.49c) and it is translation and rotation invariant, i.e., scalar invariant (see 4.3.3.2, p. 265).
2. Divergence in Cylindrical Coordinates
~ = 1 @ ( V ) + 1 @V' + @Vz (13.50a) with V ~ ( ' z) = V~e + V'~e' + Vz~ez : (13.50b) div V @ @' @z
3. Divergence in Spherical Coordinates
2 ~ = 12 @ (r Vr ) + 1 @ (sin V ) + 1 @V' div V r @r r sin @ r sin @' ~ (r ') = Vr~er + V~e + V'~e': with V
4. Divergence( in General Orthogonal Coordinates ! !) ! 1 @ @~ r @~ r @ @~r @~r V + @ @~r @~r V ~ div V = D @ @ @ V + @ @ @ @ @ @ with ~r( ) = x( )~i + y( )~j + z( )~k ! @~r @~r @~r = @~r @~r @~r D = @ @ @ @ @ @ ~ ( ) = V~e + V~e + V ~e : and V
13.2.4.3 Rules for Evaluation of the Divergence div ~c = 0
~1+V ~ 2) = div V ~ 1 + div V ~2 div (V
~ ) = c div V ~: div (cV
(13.51a) (13.51b) (13.52a) (13.52b) (13.52c) (13.52d) (13.53)
654 13. Vector Analysis and Vector Fields !
especially div( r~c) = ~r ~c : r ~1V ~ 2) = V ~ 2 rot V ~1;V ~ 1 rot V ~2: div (V
~ ) = U div V ~ +V ~ grad U div (U V
(13.54) (13.55)
13.2.4.4 Divergence of a Central Field div ~r = 3
div '(r)~r = 3'(r) + r'0 (r):
(13.56)
13.2.5 Rotation of Vector Fields 13.2.5.1 Denitions of the Rotation 1. Denition
~ at the point ~r is a vector denoted by rot V ~ , curl V ~ or with the The rotation or curl of a vector eld V ~ , and de ned as the negative space derivative of the vector eld: nabla operator r V ZZ ZZ - V~ d~S - d~S V~ () ~ = ; lim () rot V = Vlim : (13.57) V !0 !0 V V
2. Denition
~ (~r) can be de ned in the following way: The vector eld of the rotation of the vector eld V a) We put a small surface sheet S (Fig. 13.12) through the point ~r. We describe this surface sheet by a vector ~S whose direction is the dio rection of the surface normal ~n and its absolute Vdr 90 value is equal to the area of this surface patch. Projn rot V=lim S S 0 The boundary of this surfaceIis denoted by C . n rot V b) We evaluate the integral V~ d~r along the C
S
(C )
P C r
Smax
Cmax
0
closed boundary curve C of the surface (the sense of the curve is positive looking to the surface from the direction of the surface normal (see Fig. 13.12). I 1 c) We nd the limit (if it exists) Slim V~ d~r, !0 S (C )
while the position of the surface sheet remains unchanged. d) We change the position of the surface sheet in order to get a maximum value of the limit. The surface area in this position is Smax and the corresponding boundary curve is Cmax. e) We determine the vector rot ~r at the point ~r, whose absolute value is equal to the maximum value found above and its direction coincides with the direction of the surface normal of the corresponding surface. We then get: I V~ d~r ( C ) ~ = lim max rot V (13.58a) Smax !0 Smax : ~ onto the surface normal ~n of a surface with area S , i.e., the component of the The projection of rot V ~ in an arbitrary direction ~n = ~l is vector rot V Figure 13.12
13.2 Dierential Operators of Space 655
I ~l rot V ~ = rot l V ~ = lim (C )
V~ d~r
S : ~ are called the curl lines of the vector eld V ~. The vector lines of the eld rot V S !0
(13.58b)
13.2.5.2 Rotation in Di erent Coordinates 1. Rotation in Cartesian Coordinates
! ~i ~j ~k ~ = ~i @Vz ; @Vy + ~j @Vx ; @Vz + ~k @Vy ; @Vx = @ @ @ : rot V (13.59a) @y @z @z @x @x @y @x @y @z Vx Vy Vz ~ can be represented as the cross product of the nabla operator and the vector V ~: The vector eld rot V ~ ~ rot V = r V: (13.59b) !
!
2. Rotation in Cylindrical Coordinates
~ = rot V ~ ~e + rot 'V ~ ~e' + rot z V ~ ~ez with rot V (13.60a) ( ) ~ = 1 @Vz ; @V' rot ' V ~ = @V ; @Vz rot z V ~ = 1 @ ( V') ; @V : (13.60b) rot V @' @z @z @ @ @'
3. Rotation in Spherical Coordinates ~ = rot r V ~ ~er + rot V ~ ~e + rot 'V ~ ~e' rot V ( ) @ (sin V ) ; @V ~ = 1 rot r V ' r sin @ @' 1 @V 1 @ r ~ rot V = r sin @' ; r @r (rV')
(
)
~ = 1 @ (rV) ; Vr : rot ' V r @r @
9 > > > > > = > > > > > "
with
(13.61a)
(13.61b)
4. Rotation in General Orthogonal Coordinates ~ = rot V ~ ~e + rot V ~ ~e + rot V ~ ~e with rot V ! " ! ~ = 1 @~r @ @~r V ; @ @~r V rot V D @ " @ @ ! @ @ ! ~ = 1 @~r @ @~r V ; @ @~r V rot V D @ " @ @ ! @ @ ! ~ = 1 @~r @ @~r V ; @ @~r V rot V D @ @ @ @ @
~r( ) = x( )~i + y( )~j + z( )~k
9 > > > > > = > > > > > "
(13.62b)
@~r @~r @~r D = @ @ @ :
13.2.5.3 Rules for Evaluating the Rotation rot (V~1 + V~2) = rot V~1 + rot V~2
(13.62a)
~ ) = c rot V ~: rot (cV
(13.62c) (13.63)
656 13. Vector Analysis and Vector Fields ~ ) = U rot V ~ + grad U V ~: rot (U V rot (V~1 V~2) = (V~2 grad )V~1 ; (V~1 grad )V~2 + V~1 div V~2 ; V~2 div V~1:
13.2.5.4 Rotation of a Potential Field
(13.64) (13.65)
This also follows from the Stokes theorem (see 13.3.3.2, p. 666) that the rotation of a potential eld is identically zero: ~ = rot (grad U) = ~0: rot V (13.66) ~ This also follows from (13.59a) for V = grad U , if the assumptions of the Schwarz interchanging theorem are ful lled (see 6.2.2.2, 1., p. 395). p For ~r = x~i + y~j + z~k with r = j~rj = x2 + y2 + z2 we have: rot ~r = 0 and rot ('(r)~r) = ~0, where '(r) is a di erentiable function of r.
13.2.6 Nabla Operator, Laplace Operator 13.2.6.1 Nabla Operator
The symbolic vector r is called the nabla operator. Its use simpli es the representation of and calculations with space di erential operators. In Cartesian coordinates we have: @ ~i + @ ~j + @ ~k: r = @x (13.67) @y @z The components of the nabla operator are considered as partial di erential operators, i.e., the symbol @ @x means partial di erentiation with respect to x, where the other variables are considered as constants. The formulas for spatial dierential operators in Cartesian coordinates can be obtained by formal mul~ . For instance, in the case of the tiplication of this vector operator by the scalar U or by the vector V operators gradient, vector gradient, divergence, and rotation: grad U = r U (gradient of U (see 13.2.2, p. 650)) (13.68a) ~ = rV ~ ~ (see 13.2.3, p. 652)) grad V (vector gradient of V (13.68b) ~ = rV ~ (divergence of V ~ div V (see 13.2.4, p. 653)) (13.68c) ~ = rV ~ (rotation or curl of V ~ (see 13.2.5, p. 654)): rot V (13.68d)
13.2.6.2 Rules for Calculations with the Nabla Operator P
1. If r stands in front of a linear combination ai Xi with constants ai and with point functions Xi,
then, independently of whether they are scalar or vector functions, we have the formula: X X r( ai Xi) = airXi: (13.69) 2. If r is applied to a product of scalar or vector functions, then we apply it to each of these functions after each other and add the result. There is a above the symbol of the function submitted to the operation r(XY Z ) = r( X Y Z ) + r(X Y Z ) + r(XY Z ) i.e. (13.70) r(XY Z ) = (rX )Y Z + X (rY )Z ) + XY (rZ ): We transform the products according to vector algebra so as the operator r is applied to only one factor with the sign . Having performed the computation we omit that sign.
13.2 Dierential Operators of Space 657
A: div (U V~ ) = r(U V~ ) = r( U V~ ) + r(U V~ ) = V~ rU + U r V~ = V~ grad U + U div V~ . B: grad (V~ 1V~ 2 ) = r(V~ 1V~ 2) = r( V~ 1 V~ 2) + r(V~ 1 V~ 2 ). Because ~b(~a~c) = (~a~b)~c + ~a (~b ~c) we ~ 1V ~ 2 ) = (V ~ 2r)V ~1+V ~ 2 (r V ~ 1) + (V ~ 1r)V ~2+V ~ 1 (r V ~ 2) get: grad (V ~ 2 grad )V ~1+V ~ 2 rot V ~ 1 + (V ~ 1grad )V ~2+V ~ 1 rot V ~ 2. = (V
13.2.6.3 Vector Gradient
~ is represented by the nabla operator as The vector gradient grad V ~ = rV ~: grad V ~: We get for the expression occurring in the vector gradient (~a r)V ~ = rot (V ~ ~a) + grad (~aV ~ ) + ~adiv V ~ ;V ~ div ~a ; ~a rot V ~ ;V ~ rot ~a: 2(~a r)V ~ ~ ~ In particular we get for ~r = xi + yj + zk: (~a r)~r = ~a:
13.2.6.4 Nabla Operator Applied Twice ~: For every eld V ~ r(r V) = div rot V~ = 0
r(rU ) = div grad U = %U:
(13.72a) r (rU ) = rot grad U = ~0
(13.71a) (13.71b) (13.71c)
(13.72b)
(13.72c)
13.2.6.5 Laplace Operator 1. Denition
The dot product of the nabla operator with itself is called the Laplace operator: % = r r = r2: (13.73) The Laplace operator is not a vector. It prescribes the summation of the second partial derivatives. It can be applied to scalar functions as well as to vector functions. The application to a vector function, componentwise, results in a vector. The Laplace operator is an invariant, i.e., it does not change during translation and/or rotation of the coordinate system.
2. Formulas for the Laplace Operator in Di erent Coordinates
In the following, we apply the Laplace operator to the scalar point function U (~r). Then the result is a ~ (~r) results in a vector %V ~ with components %Vx, scalar. The application of it for vector functions V %Vy , %Vz .
1. Laplace Operator in Cartesian Coordinates 2 2 2 %U (x y z) = @@xU2 + @@yU2 + @@zU2 :
(13.74)
2. Laplace Operator in Cylindrical Coordinates !
@ @U + 1 @ 2 U + @ 2 U : %U ( ' z) = 1 @ @ 2 @'2 @z2
(13.75)
658 13. Vector Analysis and Vector Fields 3. Laplace Operator in Spherical ! Coordinates
!
@ r2 @U + 1 @ sin @U + 1 @ 2 U : %U (r ') = r12 @r @r r2 sin @ @ r2 sin2 @'2
(13.76)
4. Laplace Operator2in General Orthogonal Coordinates 0 1 0 1 0
13 66 B C B C B C77 @B D @U C @ B D @U C @ B D @U C B C B C B C %U ( ) = D1 666 @ B C + B C + B C777 with (13.77a) 2 2 2 4 B @ @~r @ C A @ B @ @~r @ C A @ B @ @~r @ C A5 @ @ @ @~r @~r @~r ~r( ) = x( )~i + y( )~j + z( )~k (13.77b) D = @ @ @ : (13.77c)
3. Special Relations between the Nabla Operator and Laplace Operator r(r V~ ) = grad div V~ r (r V~ ) = rot rot V~ ~ r(r V) ; r (r V~ ) = %V~ where
(13.78) (13.79) (13.80)
!
2 2 2 ~ = (r r)V ~ = %Vx~i + %Vy~j + %Vz ~k = @ V2x + @ V2x + @ V2x ~i %V @x @y @z ! ! 2V 2V @ @ y @ 2 Vy @ 2 Vy ~ z @ 2 Vz @ 2 V z ~ + @x2 + @y2 + @z2 j + @x2 + @y2 + @z2 k:
(13.81)
13.2.7 Review of Spatial Dierential Operations
13.2.7.1 Fundamental Relations and Results (see Table 13.2) Table 13.2 Fundamental relations for spatial di erential operators
Operator
Symbol Relation Gradient grad U rU ~ rV ~ Vector gradient grad V ~ Divergence div V r V~ ~ Rotation rot V r V~ Laplace operator %U (r r)U ~ ~ Laplace operator %V (r r)V
Argument Result scalar vector vector vector scalar vector
vector tensor second order scalar vector scalar vector
Meaning
maximal increase source, sink curl potential eld source
13.2.7.2 Rules of Calculation for Spatial Di erential Operators ~ V ~1 V ~ 2 vector functions: U U1 U2 scalar functions c constant V grad (U1 + U2 ) = grad U1 + grad U2 : grad (cU ) = c grad U: grad (U1 U2) = U1 grad U2 + U2 grad U1: grad F (U ) = F 0(U ) grad U:
(13.82) (13.83) (13.84) (13.85)
13.2 Dierential Operators of Space 659
~1+V ~ 2) = div V ~ 1 + div V ~ 2: div (V ~ ) = c div V ~: div (cV ~)=V ~ grad U + U div V ~: div (U V ~1+V ~ 2) = rot V ~ 1 + rot V ~ 2: rot (V ~ ) = c rot V ~: rot (cV ~ ) = U rot V ~ ;V ~ grad U: rot (U V ~ div rot V 0: rot grad U ~0 (zero vector): div grad U = %U: ~ = grad div V ~ ; %V ~: rot rot V ~1V ~ 2) = V ~ 2 rot V ~1;V ~ 1 rot V ~ 2: div (V
(13.86) (13.87) (13.88) (13.89) (13.90) (13.91) (13.92) (13.93) (13.94) (13.95) (13.96)
13.2.7.3 Expressions of Vector Analysis in Cartesian, Cylindrical, and Spherical Coordinates (see Table 13.3)
Table 13.3 Expressions of vector analysis in Cartesian, cylindrical, and spherical coordinates
Cartesian coordinates Cylindrical coordinates Spherical coordinates ~ed + ~e' d' + ~ez dz ~er dr + ~erd + ~e'r sin d'
d~s = d~r ~exdx + ~ey dy + ~ez dz gradU ~ex @U + ~ey @U + ~ez @U @x @y @z @V x @Vy @Vz ~ divV @x + @y + @z ~ rotV
%U
! z @Vy ~ex @V ; @y @z ! @V z +~ey @zx ; @V @x ! +~ez @Vy ; @Vx @x @y
@2U + @2U + @2U @x2 @y2 @z2
1 @U @U ~e @U @ + ~e' @' + ~ez @z 1 @ ( V ) + 1 @V' + @Vz @ @' @z
1 @U 1 @U ~er @U @r + ~e r @ + ~e' r sin @' 1 @ 2 1 @ r2 @r (r Vr ) + r sin @ (V sin ) 1 @V' + r sin "@' !
@V @V @ (V sin ) ; @V 1 1 z ' ~e @' ; @z ~er r sin @ ' @' ! "
@V 1 1 @V @ @Vz r +~e' @z ; @ +~e r sin @' ; @r (rV') ! "
+~ez 1 @ ( V') ; 1 @V +~e' 1 @ (rV) ; @Vr @ @' r @r @ ! ! 1 @ @U + 1 @ 2 U 1 @ r2 @U @ @ 2 @'2 r2 @r @r ! 2U @ 1 @ sin @U + @z2 + r2 sin @ @ 2U @ 1 + 2 2 r sin @'2
660 13. Vector Analysis and Vector Fields
13.3 Integration in Vector Fields
Integration in vector elds is usually performed in Cartesian, cylindrical or in spherical coordinate systems. Usually we integrate along curves, surfaces, or volumes. The line, surface, and volume elements needed for these calculations are collected in Table 13.4. Table 13.4 Line, surface, and volume elements in Cartesian, cylindrical, and spherical coordinates
Cartesian coordinates
Cylindrical coordinates
Spherical coordinates
~ed + ~e' d' + ~ez dz ~er dr + ~e rd + ~e'r sin d' d~r ~exdx + ~ey dy + ~ez dz d~S ~exdydz + ~ey dxdz + ~ez dxdy ~e d'dz + ~e'd dz + ~ez d d' ~er r2 sin d d' +~er sin drd' +~e'rdrd dv dxdydz d d'dz r2 sin drd d' ~ex = ~ey ~ez ~e = ~e' ~ez ~er = ~e ~e' ~ey = ~ez ~ex ~e' = ~ez ~e ~e = ~e' ~er ~ez = ~ex ~ey ~ez = ~e ~e' ~e' = ~er ~e 0 i 6= j 0 i 6= j j ~ei ~ej = 1 i = j ~ei ~ej = 1 i = j ~ei ~ej = 01 ii 6= =j The indices i and j take the place of x y z or ' z or r ': * The volume is denoted here by v to avoid confusion with the absolute value of the vector ~ j = V: function jV
13.3.1 Line Integral and Potential in Vector Fields 13.3.1.1 Line Integral in Vector Fields
1. De nition The scalar-valued curvilinear integral or line integral of a vector function V~ (~r) along _ a recti cable curve AB (Fig. 13.13) is the scalar value Z ~ (~r) d~r: P= V (13.97a) _
AB
2. Evaluation of this Integral in Five Steps _ a) We divide the path AB (Fig. 13.13) by division points A1(~r1 ), A2 (~r2) : : : An;1(~rn;1) (A A0 , B An) into n small arcs which are approximated by the vectors ~ri ; ~ri;1 = %~ri;1. b) We choose arbitrarily the points Pi with position vectors ~ri lying inside or at the boundary of each
small arc. c) We calculate the dot product of the value of the function V~ (~ri) at these chosen points with the corresponding %~ri;1. d) We add all the n products. n X e) We calculate the limit of the sums got this way V~~ (~ri) %~ri;1 for %~ri;1 ! 0, while n ! 1 obviously.
i=1
13.3 Integration in Vector Fields 661
If this limit exists independently of the choice of the points Ai and Pi, then it is called the line integral Z n X V~ d~r = lim V~~ (~ri) %~ri;1 : (13.97b) ~r!0 n!1 i=1 _ AB ~ (~r) A sucient condition for the existence of the line integral (13.97a,b) is that the vector function V _ and the curve AB are continuous and the curve has a tangent varying continuously. A vector function V~ (~r) is continuous if its components, the three scalar functions, are continuous. An-1
A3 A2
A1 ∆r1 ∆r0 A=A0
C B
rn
r3
r2
B=An
Pn
rn-1
P3 ∆r 2
P2
∆rn-1
A
r1
P1
0
r0
Figure 13.13
Figure 13.14
13.3.1.2 Interpretation of the Line Integral in Mechanics
~ (~r) is a eld of force, i.e., V ~ (~r) = F~ (~r), then the line integral (13.97a) represents the work done by If V _ F~ while a particle m moves along the path AB (Fig. 13.13,13.14).
13.3.1.3 Properties of the Line Integral Z
Z
Z
V~ (~r) d~r = V~ (~r) d~r + V~ (~r) d~r:
_ _ _ ABC AB BC Z Z V~ (~r) d~r = ; V~ (~r) d~r (Fig. 13.14): _ _ BA AB Z h Z Z i ~ (~r) d~r = V ~ (~r) d~r + W ~ (~r) d~r: V~ (~r) + W _ _ _ AB AB AB Z Z ~ (~r) d~r = c V ~ (~r) d~r: cV _ _ AB AB
(13.98) (13.99) (13.100) (13.101)
13.3.1.4 Line Integral in Cartesian Coordinates In Cartesian coordinates, we have: Z Z V~ (~r) d~r = (Vx dx + Vy dy + Vz dz) : _ _ AB AB
(13.102)
662 13. Vector Analysis and Vector Fields
13.3.1.5 Integral Along a Closed Curve in a Vector Field
A line integral is called a contour integral if the path of integration is a closed curve. If the scalar value of the integral is denoted by P and the closed curve is denoted by C , then we use the notation: I ~ (~r) d~r: P= V (13.103) (C )
13.3.1.6 Conservative or Potential Field 1. Denition
If the value P of the line integral (13.97a) in a vector eld depends only on the initial point A and the endpoint B , and is independent of the path between them, then we call this eld a conservative eld or a potential eld. The value of the contour integral in a conservative eld is always equal to zero: I V~ (~r) d~r = 0: (13.104) A conservative eld is always irrotational: ~ = ~0 rot V (13.105) and conversely, this equality is a sucient condition for a vector eld to be conservative. Of course, we have to suppose that the partial derivatives of the eld function are continuous with respect to the ~ is simply connected. This condition, also called the corresponding coordinates, and the domain of V integrability condition (see 8.3.4.3, p. 469), has the form in Cartesian coordinates @Vx = @Vy @Vy = @Vz @Vz = @Vx : (13.106) @y @x @z @y @x @z 2. Potential of a Conservative Field, or its potential function or briey its potential is the scalar function
Z~r ~ (~r) d~r: '(~r) = V ~r0
(13.107a)
We can calculate it in a conservative eld with a xed initial point A(~r0) and a variable endpoint B (~r) from the integral Z ~ (~r) d~r: '(~r) = V (13.107b) _ AB Remark: In physics, the potential ' (~r) of a function V~ (~r) at the point ~r is considered with the opposite sign:
Z~r ~ (~r) d~r = ;'(~r): ' (~r) = ; V ~r0
(13.108)
3. Relations between Gradient, Line Integral, and Potential
~ (~r) = grad U (~r) holds, then U (~r) is the potential of the eld V ~ (~r), and conversely, V ~ (~r) If the relation V is a conservative or potential eld. In physics we consider the negative sign corresponding to (13.108).
4. Calculation of the Potential in a Conservative Field
~ (~r) is given in Cartesian coordinates V ~ = Vx~i + Vy~j + Vz~k, then for the total di erential If the function V of its potential function holds:
13.3 Integration in Vector Fields 663
dU = Vx dx + Vy dy + Vz dz: (13.109a) Here, the coecients Vx, Vy , Vz must ful ll the integrability condition (13.106). The determination of U follows from the equation system @U = V @U = V @U = V : (13.109b) y z @x x @y @z In practice, the calculation of the potential can be done by performing the integration along three straight line segments parallel to the coordinate axes and connected to each other (Fig. 13.15):
z B( r ) A( r0)
0 y x
Figure 13.15
Z~r Z ~ d~r = U (x0 y0 z0 ) + x Vx(x y0 z0 ) dx U= V x0 ~r0 Zy Zz + Vy (x y z0) dy + Vz (x y z) dz: y0
(13.110)
z0
13.3.2 Surface Integrals
13.3.2.1 Vector of a Plane Sheet
The vector representation of the surface integral of general type (see 8.3.4.2, p. 485) requires to assign a vector ~S to a plane surface region S , which is perpendicular to this region and its absolute value is equal to the area of S . Fig. 13.16a shows the case of a plane sheet. The positive direction in S is given by de ning the positive sense along a closed curve C according to the right-hand law (also called right screw rule): If we look from the initial point of the vector into the direction of its nal point, then the positive sense is the clockwise direction. By this choice of orientation of the boundary curve we x the exterior side of this surface region, i.e., the side on which the vector lies. This de nition works in the case of any surface region bounded by a closed curve (Fig. 13.16b,c). S
interior side
exterior side interior side
a)
b)
K
exterior side
c)
K
Figure 13.16
13.3.2.2 Evaluation of the Surface Integral
The de nition of a surface integral in scalar or vector elds is independent of whether the surface S is bounded by a closed curve or is itself a closed surface. The evaluation is performed in ve steps: a) We divide the surface region S on the exterior side de ned by the orientation of the boundary curve (Fig. 13.17) into n arbitrary elementary surfaces %Si so that each of these surface elements can be approximated by a plane surface element. We assign the vector %S~i to every surface element %Si as given in (13.33a). In the case of a closed surface, the positive direction is de ned so that the exterior side is where %S~i should start.
664 13. Vector Analysis and Vector Fields b) We choose an arbitrary point Pi with the position vector r~i inside or on the boundary of each surface
element. c) We produce the products U (r~i) %S~i in the case of a scalar eld and the product V~ (r~i) %S~i or V~ (r~i) %S~i in the case of a vector eld. d) We add all these products. e) We evaluate the limit while the diameters of %Si tend to zero, i.e., %S~i ! 0 for n ! 1. So, the surface elements tend to zero in the sense given in 8.4.1, 1., p. 471, for double integrals. If this limit exists independently of the partition and of the choice of the points r~i, then we call it the ~ on the given surface. surface integral of V z Syz Szx
DSi
S
V( ri )
Pi DSi
0 y
0
ri
Sxy
x
Figure 13.17
Figure 13.18
13.3.2.3 Surface Integrals and Flow of Fields 1. Vector Flown of a Scalar Field Z P~ = lim
X
Si !0 i=1 n!1
U (~ri) %~Si =
(S )
U (~r) d~S:
(13.111)
2. Scalar Flown of a Vector Field Z Q = lim
X~
Si !0 i=1 n!1
V(~ri) %~Si = V~ (~r) d~S:
(13.112)
(S )
3. Vector Flown of a Vector Field Z R~ = lim
X~
Si !0 i=1 n!1
V(~ri) %~Si = V~ (~r) d~S:
(13.113)
(S )
13.3.2.4 Surface Integral in Cartesian Coordinates as Surface IntegralsZofZ Second Type Z ZZ ZZ (S )
U d~S =
(Syz )
U dy dz ~i +
(Szx )
U dz dx~j +
(Sxy )
U dx dy ~k:
(13.114)
13.3 Integration in Vector Fields 665
Z (S )
Z
(S )
V~ d~S =
ZZ (Syz )
V~ d~S =
ZZ
(Syz )
ZZ
Vx dy dz +
(Szx )
Vy dz dx +
ZZ
(Vz~j ; Vy~k) dy dz +
(Szx )
ZZ (Sxy )
Vz dx dy:
(13.115)
(Vx~k ; Vz~i) dz dx +
ZZ (Sxy )
(Vy~i ; Vx~j) dx dy:(13.116)
The existence theorems for these integrals can be given similarly to those in 8.5.2, 4., p. 484. In the formulas above, each of the integrals is taken over the projection S on the corresponding coordinate plane (Fig. 13.18), where one of the variables x, y or z should be expressed by the others from the equation of S . Remark: Integrals over a closed surface are denoted by I ZZ I ZZ I ZZ U d~S = -U d~S V~ d~S = -V~ d~S V~ d~S = -V~ d~S: (13.117) (S )
(S )
(S )
A: Calculate the integral P~ =
(S )
Z (S )
(S )
(S )
xyz d~S, where the surface is the plane region x + y + z = 1 bounded
by the Zcoordinate planes. The upward Z Z Z side is the positive side: Z Z P~ = (1 ; y ; z)yz dy dz ~i + (1 ; x ; z)xz dz dx~j + (1 ; x ; y)xy dx dy ~k
Z Z (Syz )
(Syz )
(1 ; y ; z)yz dy dz =
Z 1 Z (S1;zxz) 0
(Sxy )
1 . We get the two further integrals (1 ; y ; z)yz dy dz = 120
0
~ = 1 (~i + ~j + ~k). analogously. The result is: P 120
Z
ZZ
(S )
(Syz ) 1 Z 1;x
B: Calculate the integral Q = ~r d~S = ZZ
Z
x dy dz +
ZZ (Szx )
y dz dx +
ZZ (Sxy )
z dx dy over the
(1 ; x ; y) dy dx = 61 . Both other integrals are (Syz ) calculated similarly. The result is: Q = 1 + 1 + 1 = 1 . Z 6 6 Z6 2 C: Calculate the integral R~ = ~r d~S = (x~i + y~j + z~k) (dy dz ~i + dz dx~j + dx dy ~k) where same plane region as in A:
x dy dz =
(S )
0
0
(S )
~ = ~0. the surface region is the same as in A: Performing the computations we get R
13.3.3 Integral Theorems
13.3.3.1 Integral Theorem and Integral Formula of Gauss 1. Integral Theorem of Gauss or the Divergence Theorem
~ over The integral theorem of Gauss gives the relation between a volume integral of the divergence of V a volume v, and a surface integral over the surface S surrounding this volume. The orientation of the surface (see 8.5.2.1, p. 483) is de ned so that the exterior side is the positive one. The vector function V~ should be continuous, their rst partial derivatives should exist and be continuous. ZZ ZZ Z - V~ d~S = div V~ dv: (13.118a) (S )
(v)
666 13. Vector Analysis and Vector Fields ~ through a closed surface S is equal to the integral of divergence of V ~ over The scalar ow of the eld V the volume v bounded by S . In Cartesian coordinates we get: ZZ ZZ Z @Vx @Vy @Vz ! -(Vx dy dz + Vy dz dx + Vz dx dy) = (13.118b) @x + @y + @z dx dy dz: (S ) (v)
2. Integral Formula of Gauss
In the planar case, the integral theorem of Gauss restricted to the x y plane becomes the integral formula of Gauss. It represents the correspondence between a line integral and the corresponding surface integral: Z Z " @Q(x y) @P (x y) I ; dx dy = P (x y) dx + Q(x y) dy] : (13.119) @x @y (B ) (C )
B denotes a plane region which is bounded by C . P and Q are continuous functions with continuous rst partial derivatives.
3. Sector Formula
The sector formula is an important special case of the Gauss integral formula. We can calculate the area of plane regions with it. For Q = x, P = ;y it follows that ZZ I F = dx dy = 12 x dy ; y dx]: (13.120) (B )
(C )
13.3.3.2 Integral Theorem of Stokes
The integral theorem of Stokes gives the relation between a surface integral over an oriented surface ~ , and the integral along the closed boundary curve C of the surface region S de ned in the vector eld V S . The sense of the curve C is chosen so that the sense of traverse forms a right screw with the surface ~ should be continuous and it should have continuous normal (see 13.3.2.1, p. 663). The vector function V rst partial derivatives. ZZ I ~ d~S = V ~ d~r: rot V (13.121a) (S )
(C )
The vector ow of the rotation through a surface S bounded by the closed curve C is equal to the ~ along the curve C . contour integral of the vector eld V In Cartesian coordinates, we have: I (Vx dx + Vy dy + Vz dz) (C )
=
! ! Z Z @Vz @Vy ! @Vx ; @Vz dz dx + @Vy ; @Vx dx dy : ; dy dz + @y @z @z @x @x @y (S )
(13.121b)
In the planar case, the integral theorem of Stokes, just as that of Gauss, becomes into the integral formula (13.119) of Gauss.
13.3.3.3 Integral Theorems of Green
The Green integral theorems give relations between volume and surface integrals. They are the applica~ = U1 grad U2, where U1 and U2 are scalar eld functions tions of the Gauss theorem for the function V and v ZZ is the volume surrounded by the surface Z ZZ S . (13.122) (U1 %U2 + grad U2 grad U1 ) dv = -U1 grad U2 d~S (v)
(S )
13.4 Evaluation of Fields 667
ZZ Z (v)
ZZ
(U1 %U2 ; U2 %U1 ) dv = -(U1 grad U2 ; U2 grad U1) d~S: (S )
In particular for U1 = 1, we have: ZZ Z ZZ %U dv = -grad U d~S: (v)
(13.123)
(13.124)
(S )
In Cartesian coordinates the third Green theorem has the following form (compare 13.118b): ! ZZ Z @ 2 U @ 2 U @ 2 U ! ZZ + 2 + 2 dv = - @U dy dz + @U dz dx + @U dx dy : (13.125) 2 @x @y @z @x @y @z (v) (S )
I
A: Calculate the line integral I = (x2 y3 dx + dy + z dz) with C as the intersection curve of the (C )
cylinder x2 + y2 = a2 and the plane z = 0. We get with the Stokes theorem: I ZZ ZZ Z2 Za 6 ~ d~r = rot V ~ d~S = ; I= V 3x2 y2 dx dy = ;3 r5 cos2 ' sin2 ' dr d' = ; a8 with '=0 r=0 (C ) (S ) (S ) ~ = ;3x2 y2~k, d~S = ~k dx dy and the circle S : x2 + y2 a2 . rot V I B: Determine the ux I = V~ d~S in the drift space V~ = x3~i + y3~j + z3~k through the surface S (S )
of the sphere x2 + y2 + z2 = a2 . The theorem of Gauss yields: I ZZ Z ZZ Z Z2 Z Za ~ d~S = ~ dv = 3 (x2 + y2 + z2 ) dx dy dz = 3 I= V div V r4 sin dr d d' = 12 a5 . 5 '=0 =0 r=0 (S ) (v) (v)
C: Heat conduction equation: ZZ Z The change in time of the heat Q of a space region v containing no
heat source is given by dQ dt =
(v)
c% @T @t dv (speci c heat-capacity c, density %, temperature T ), while
the corresponding time-dependent change of the heat ow through the surface S of v is given by dQ dt = ZZ grad T d~S (thermal conductivity ). Applying the theorem of Gauss for the surface integral we
(S )
ZZ Z " @T c% @t ; div ( grad T ) dv = 0 the heat conduction equation c @T @t = div ( grad T ), (v ) 2 which has the form @T @t = a %T in the case of a homogeneous solid (c, %, constants). get from
13.4 Evaluation of Fields 13.4.1 Pure Source Fields
~ 1 a pure source eld or an irrotational source eld when its rotation is equal to zero We call a eld V everywhere. If the divergence is q(~r), then we have: ~ 1 = q(~r) ~ 1 = ~0: div V rot V (13.126)
668 13. Vector Analysis and Vector Fields In this case, the eld has a potential U , which is de ned at every point P by the Poisson dierential equation (see 13.5.2, p. 670) V~ 1 = grad U div grad U = %U = q(~r) (13.127a) ~ 1 = ;grad U is used.) The evaluation of U comes where ~r is the position vector of P . (In physics, V from ZZZ div V ~ (~r ) dv(~r ) U (~r) = ; 41 (13.127b) j~r ; ~r j : ~ must be di erenThe integration is taken over the whole of space (Fig. 13.19). The divergence of V tiable and be decreasing quickly enough for large distances. m1 or q1
dv( r*)
m2 or q2
field
er
r-r*
P r
r *
0
Figure 13.19
FN=-g
m1m2 r
2
er
or
qq FC= 1 1 22 er 4pe r
Figure 13.20
13.4.2 Pure Rotation Field or Zero-Divergence Field
~ 2 whose divergence is equal to zero A pure rotation (or curl) eld or a solenoidal eld is a vector eld V ~ (~r), then we have: everywhere. If the rotation is w ~2=0 ~2=w ~ (~r): div V rot V (13.128a) ~ (~r) cannot be arbitrary it must satisfy the equation div w ~ = 0. With the assumptions The rotation w ~ = 0 i.e., rot rot A ~ =w ~ V~ 2(~r) = rot A~ (~r) div A (13.128b) we get according to (13.95) ~ ; %A ~ =w ~ = ;w ~ i.e., %A ~: grad div A (13.128c) ~ So, A(~r) formally satis es the Poisson di erential equation just as the potential U of an irrotational ~ 1 and that is why it is called a vector potential. For every point P : eld V ZZZ w~ (~r ) V~ 2 = rot A~ with A~ = 41 (13.128d) j~r ; ~r j dv(~r ): The meaning of ~r is the same as in (13.127b) the integration is taken over the whole of space.
13.4.3 Vector Fields with Point-Like Sources 13.4.3.1 Coulomb Field of a Point-Like Charge
The Coulomb eld is an important example of an irrotational eld, which is also solenoidal, except at the location of the point charge, the point source (Fig. 13.20). The eld and potential equations are: U = ; re in physics also er : (13.129a) E~ = re3~r
13.5 Dierential Equations of Vector Field Theory 669
The scalar ow is 4e or 0, depending on whether the surface S encloses the point source or not: I S encloses the point source E~ d~S = 40e ifotherwise (13.129b) : (S )
The quantity e is the source intensity or source strength.
13.4.3.2 Gravitational Field of a Point Mass
The eld of gravity of a point mass is the second example of an irrotational and at the same time solenoidal eld, except at the center of mass. We also call it the Newton eld. Every relation valid for the Coulomb eld is valid analogously also for the Newton eld.
13.4.4 Superposition of Fields
13.4.4.1 Discrete Source Distribution
Analogously to the superposition of the elds of physics, the vector elds of mathematics superpose ~ have the potentials U , then the vector eld each other. The superposition law is: If the vector elds V V~ = 1V~ has the potential U = 1U . For n discrete point sources with source strength e ( = 1 2 : : : n), whose elds are superposed, the resulting eld can be determined by the algebraic sum of the potentials U : n X V~ (~r) = ;grad U with U = j~r ;e~r j : (13.130a) =1 Here, the vector ~r is again the position vector of the point under consideration, ~r are the position vectors of the sources. ~ 1 and a zero-divergence eld V ~ 2 together and they are everywhere If there is an irrotational eld V continuous, then "
ZZZ q(~r ) ZZZ w~ (~r ) ~ ~ V~ = V~ 1 + V~ 2 = ; 41 grad dv ( r ) ; rot dv ( r ) : (13.130b) j~r ; ~r j j~r ; ~r j ~ (~r) is unique if jV ~ (~r)j decreases If the vector eld is extended to in nity, then the decomposition of V sucient rapidly for r = j~rj ! 1. The integration is taken over the whole of space.
13.4.4.2 Continuous Source Distribution
If the sources are distributed continuously along lines, surfaces, or in domains of space, then, instead of the nite source strength e , we have in nitesimals corresponding to the density of the source distributions, and instead of the sums, we have integrals over the domain. In the case of a continuous space ~. distribution of source strength, the divergence is q(~r) = div V Similar statements are valid for the potential of a eld de ned by rotation. In the case of a continuous ~. ~ (~r) = rot V space rotation distribution, the \ rotation density " is de ned by w
13.4.4.3 Conclusion
A vector eld is determined uniquely by its sources and rotations in space if all these sources and rotations lie in a nite space.
13.5 Dierential Equations of Vector Field Theory 13.5.1 Laplace Dierential Equation
~ 1 = grad U containing no sources, leads The problem to determine the potential U of a vector eld V to the equation according to (13.126) with q(~r) = 0 ~ 1 = div grad U = %U = 0 div V (13.131a)
670 13. Vector Analysis and Vector Fields i.e., to the Laplace dierential equation. In Cartesian coordinates we have: 2 2 2 %U = @@xU2 + @@yU2 + @@zU2 = 0: (13.131b) Every function satisfying this di erential equation and which is continuous and possesses continuous rst and second order partial derivatives is called a Laplace or harmonic function. We distinguish three basic types of boundary value problems: 1. Boundary value problem (for an interior domain) or Dirichlet problem: We determine a function U (x y z) which is harmonic inside a given space or plane domain and takes the given values at the boundary of this domain. 2. Boundary value problem (for an interior domain) or Neumann problem: We determine a function U (x y z), which is harmonic inside a given domain and whose normal derivative @U @n takes the given values at the boundary of this domain. 3. Boundary value problem (for an interior domain): We determine a function U (x y z), which is harmonic inside a given domain and the expression 2 2 U + @U @n ( const + 6= 0) takes given values at the boundary of this domain.
13.5.2 Poisson Dierential Equation
~ 1 = grad U with given divergence, leads The problem to determine the potential U of a vector eld V to the equation according to (13.126) with q(~r) 6= 0 div V~1 = div grad U = %U = q(~r) 6= 0 (13.132a) i.e., to the Poisson dierential equation. Since in Cartesian coordinates: 2 2 2 %U = @@xU2 + @@yU2 + @@zU2 (13.132b) the Laplace di erential equation (13.131b) is a special case of the Poisson di erential equation (13.132b). The solution is the Newton potential (for point masses) or the Coulomb potential (for point charges) ZZZ q(~r ) dv(~r ) U = ; 41 (13.132c) j~r ; ~r j : The integration is taken over the whole of space. U (~r) tends to zero suciently rapidly for increasing j~rj values. We can discuss the same three boundary value problems for the Poisson di erential equation as for the solution of the Laplace di erential equation. The rst and the third boundary value problems can be solved uniquely for the second one we have to prescribe further special conditions (see 9.5]).
671
14 FunctionTheory
14.1 Functions of Complex Variables
14.1.1 Continuity, Dierentiability
14.1.1.1 Denition of a Complex Function
Analogously to real functions, we can assign complex values to complex values, i.e., to the value z = x + i y we can assign a complex number w = u + i v, where u = u(x y) and v = v(x y) are real functions of two real variables. We then write w = f (z). The function w = f (z) is a mapping from the complex z plane to the complex w plane. The notions of limit, continuity, and derivative of a complex function w = f (z) can be de ned analogously to real functions of real variables.
14.1.1.2 Limit of a Complex Function
The limit of a function f (z) is equal to the complex number w0 if for z approaching z0 the value of the function f (z) approaches w0 : w0 = zlim f (z ): (14.1a) !z0 In other words: For any positive " there is a (real) such that for every z satisfying (14.1b), except maybe z0 itself, the inequality (14.1c) holds: jz0 ; zj < (14.1b) jw0 ; f (z)j < ": (14.1c) The geometrical meaning is as follows: Any point z in the circle with center z0 and radius , except maybe the center z0 itself, is mapped into a point w = f (z) inside a circle with center w0 and radius " in the w plane where f has its range, as shown in Fig. 14.1. The circles with radii and " are also called the neighborhoods U"(w0) and U (z0). y
v
z0
a)
0
d
z plane
w0
x
b)
0
e
w plane
u
Figure 14.1
14.1.1.3 Continuous Complex Functions
A function w = f (z) is continuous at z0 if it has a limit there, a substitution value, and these two are equal, i.e., if for an arbitrarily small given neighborhood U"(w0) of the point w0 = f (z0) in the w plane there exists a neighborhood U (z0) of z0 in the z plane such that w = f (z) 2 U"(w0) for every z 2 U (z0). As represented in Fig. 14.1, U"(w0) is, e.g., a circle with radius " around the point w0 . We then write f (z) = f (z0) or lim f (z0 + ) = f (z0 ): (14.2) zlim !z0 !0
14.1.1.4 Di erentiability of a Complex Function A function w = f (z) is di erentiable at z if the di erence quotient w = f (z + z) ; f (z) z z
(14.3)
672 14. Function Theory has a limit for z ! 0, independently of how z approaches zero. This limit is denoted by f 0(z) and it is called the derivative of f (z). The function f (z) = Re z = x is not di erentiable at any point z = z0, since approaching z0 parallel to the x-axis the limit of the di erence quotient is one, and approaching parallel to the y-axis this value is zero.
14.1.2 Analytic Functions
14.1.2.1 Denition of Analytic Functions
A function f (z) is called analytic, regular or holomorphic on a domain G, if it is di erentiable at every point of G. The boundary points of G, where f 0(z) does not exist, are singular points of f (z). The function f (z) = u(x y)+iv(x y) is di erentiable in G if u and v have continuous partial derivatives in G with respect to x and y and they also satisfy the Cauchy{Riemann dierential equations: @u = @v @u = ; @v : (14.4) @x @y @y @x The real and imaginary parts of an analytic function satisfy the Laplace di erential equation: 2 2 @ 2 v + @ 2 v = 0: u(x y) = @@xu2 + @@yu2 = 0 (14.5a) v(x y) = @x (14.5b) 2 @y 2 The derivatives of the elementary functions of a complex variable can be calculated with the help of the same formulas as the derivative of the corresponding real functions. A: f (z) = z3 f 0(z) = 3z2 B: f (z) = sin z f 0(z) = cos z.
14.1.2.2 Examples of Analytic Functions 1. Elementary Functions
The elementary algebraic and transcendental functions are analytic in the whole z plane except at some isolated singular points. If a function is analytic on a domain, i.e., it is di erentiable, then it is di erentiable arbitrarily many times. A: The function w = z2 with u = x2 ; y2 v = 2xy is everywhere analytic. B: The function w = u + iv, de ned by the equations u = 2x + y v = x + 2y, is not analytic at any point. C: The function f (z) = z3 with f 0(z) = 3z2 is analytic. D: The function f (z) = sin z with f 0(z) = cos z is analytic.
2. Determination of the Functions u and v
If both the functions u and v satisfy the Laplace di erential equation, then they are harmonic functions (see 13.5.1, p. 669). If one of these harmonic functions is known, e.g., u , then the second one, as the conjugate harmonic function v, can be determined up to an additive constant with the Cauchy{ Riemann di erential equations: ! Z @u @u + @ Z @u dy : v = @x dy + '(x) with d' = ; (14.6) dx @y @x @x Analogously u can be determined if v is known.
14.1.2.3 Properties of Analytic Functions
1. Absolute Value or Modulus of an Analytic Function
The absolute value (modulus) of an analytic function is: q (14.7) jwj = jf (z)j = u(x y)]2 + v(x y)]2 = '(x y): The surface jwj = '(x y) is called its relief , i.e., jwj is the third coordinate above every point z = x+i y.
14.1 Functions of Complex Variables 673
q
A: The absolute value of the function sin z = sin x cosh y+i cos x sinh y is j sin zj = sin2 x + sinh2 y . The relief is shown in Fig. 14.2a. B: The relief of the function w = e1=z is shown in Fig. 14.2b. For the reliefs of several analytic functions see 14.10]. |f(z)|
|f(z)| y
2 1 2
sin
sinh y
1
x
cosh a)
b)
x
2. Roots
Figure 14.2
Since the absolute value of a functions is never negative, the relief is always above the z plane, except the points where jf (z)j = 0 holds, so f (z) = 0. The z values, where f (z) = 0 holds, are called the roots of the function f (z ).
3. Boundedness
A function is bounded on a certain domain, if there exists a positive number N such that jf (z)j < N is valid everywhere in this domain. In the opposite case, if no such number N exists, then the function is called unbounded.
4. Theorem about the Maximum Value
If w = f (z) is an analytic function on a closed domain, then the maximum of its absolute value is attained on the boundary of the domain. 5. Theorem about the Constant (Theorem of Liouville) If w = f (z) is analytic in the whole plane and also bounded, then this function is a constant: f (z) = const.
14.1.2.4 Singular Points
If a function w = f (z) is analytic in a neighborhood of z = a, i.e., in a small circle with center a, except a itself, then f has a singularity at a. There exist three types of singularities: 1. f (z) is bounded in the neighborhood. Then there exists w = zlim f (z), and setting f (a) = w the !a function becomes analytic also at a. In this case, f has a removable singularity at a. 2. If zlim jf (z)j = 1, then f has a pole at a. About poles of di erent orders see 14.3.5.1, p. 693. !a 3. If f has neither a removable singularity nor a pole, then f has an essential singularity. In this case, for any complex w there exists a sequence zn ! a such that f (zn) ! w. A: The function w = z ;1 a has a pole at a.
674 14. Function Theory B: The function w = e1=z has an essential singularity at 0 (Fig. 14.2b).
14.1.3 Conformal Mapping
14.1.3.1 Notion and Properties of Conformal Mappings 1. Denition
A mapping from the z plane to the w plane is called a conformal mapping if it is analytic and injective. In this case, w = f (z) = u + iv f 0(z) 6= 0: (14.8) The conformal mapping has the following properties: dx The transformation dw = f 0(z) dz of the line element dz = dy is the composition of a dilatation by = jf 0(z)j and of a rotation by = arg f 0(z). This means that in nitesimal circles are transformed into almost circles, triangles into (almost) similar triangles (Fig. 14.3). The curves keep their angles of intersection, so an orthogonal family of curves is transformed into an orthogonal family (Fig. 14.4). y
v
y
v
2
2 w
z
0
a)
x 0
b) Figure 14.3
1
α z u
0
a)
1
α w x 0
b) Figure 14.4
u
Remark: Conformal mappings can be found in physics, electrotechnics, hydro- and aerodynamics and in other areas of mathematics.
2. The Cauchy{Riemann Equations
The mapping between dz and dw is given by the ane di erential transformation @u @v dx + @v dy dv = @x (14.9a) du = @u @x dx + @y dy @y and in matrix form dw = A dz with A = uvxx uvyy : (14.9b) According to the Cauchy{Riemann di erential equations, A has the form of a rotation-stretching matrix (see 3.5.2.2, 2., p. 191) with as the stretching factor: ;sin A = uvxx ;uvxx = cos (14.10a) sin cos q q ux = vy = cos (14.10b) = jf 0(z)j = u2x + u2y = vx2 + vy2 (14.10c) 0 ;uy = vx = sin (14.10d) = arg f (z) = arg (ux + ivx) : (14.10e)
3. Orthogonal Systems
The coordinate lines x = const and y = const of the z plane are transformed by a conformal mapping into two orthogonal families of curves. In general, we can generate a bunch of orthogonal curvilinear
14.1 Functions of Complex Variables 675
c1=3
c1=2
c1=0 y
c1=1
c1=-1
c1=-3
c1=-2
coordinate systems by analytic functions and conversely, for every conformal mapping there exist an orthogonal net of curves which is transformed into an orthogonal coordinate system. A: In the case of u = 2x + y v = x + 2y (Fig. 14.5), orthogonality does not hold. v
2
1
c=
c=
c=2 c=1 c=0 x c=-1
0
u
c1 =2
c1 =1
b)
2 c1 =1 c1 =0
c=-2
a)
-2
c=
c1 =-
0
0 c= 1 c=
Figure 14.5
B: In the case w = z2 the orthogonality is retained, except at the point z = 0 because here w0 = 0. The coordinate lines are transformed into two confocal families of parabolas (Fig. 14.6), the rst quadrant of the z plane into the upper half of the w plane. y
0
a)
v
x
u
b)
Figure 14.6
14.1.3.2 Simplest Conformal Mappings
In this paragraph, we discuss some transformations with their most important properties, and we give the graph of their isometric net in the z plane, i.e., the net which is transformed into an orthogonal Cartesian net in the w plane. The boundaries of the z regions mapped into the upper half of the w plane are denoted by shading. Black regions are mapped onto the square in the w plane with vertices (0 0), (0 1), (1 0), and (1 1) by the conformal mapping (Fig. 14.7).
1. Linear Function
For the conformal mapping given in the form of a linear function w = az + b the transformation can be done in three steps: a) Rotation of the plane by the angle = arg a according to : w1 = ei z: b) Stretching by the factor jaj: w2 = jajw1: c) Parallel translation by b : w = w2 + b:
(14.11a) (14.11b)
Altogether, every gure is transformed into a similar one. The points z1 = 1 and z2 = b for a 6= 1, 1;a b 6= 0 are transformed into themselves, and they are called xed points. Fig. 14.8 shows the orthogonal
676 14. Function Theory net which is transformed into the orthogonal Cartesian net. y
y
v (1,1)
0
0
x
x
u
Figure 14.7
Figure 14.8
Figure 14.9
2. Inversion
The conformal mapping w = z1 (14.12) represents an inversion with respect to the unit circle and a reection in the real axis, namely, a point z of the z plane with the absolute value r and with the argument ' is transformed into a point w of the w plane with the absolute value 1=r and with the argument ;' (see Fig. 14.10). Circles are transformed into circles, where lines are considered as limiting cases of circles (radius ! 1). Points of the interior of the unit circle become exterior points and conversely (see Fig. 14.11). The point z = 0 is transformed into w = 1. The points z = ;1 and z = 1 are xed points of this conformal mapping. The orthogonal net of the transformation (14.12) is shown in Fig. 14.9. Remark: In general a geometric transformation is called inversion with respect to a circle with radius r, when a point P2 with radius r2 inside the circle is transformed into a point P1 on the elongation of ;! the same radius vector OP2 outside of the circle with radius OP 1 = r1 = r2=r2. Points of the interior of the circle become exterior points and conversely. y
y
v
P1 z P2 0
1
w= 1z
1
x
0
a)
Figure 14.10
1
x
0
u
b)
Figure 14.11
3. Linear Fractional Function
For the conformal mapping given in the form of a linear fractional function +b w = az cz + d
(14.13a)
14.1 Functions of Complex Variables 677
the transformation can be performed in three steps: a) Linear function: w1 = cz + d: b) Inversion: w2 = w1 : (14.13b) 1 c) Linear function: w = ac + bc ;c ad w2: Circles are transformed again into circles (circular transformation ), where straight lines are considered as limiting cases of circles with r ! 1. Fixed points of this conformal mapping are the both points satisfying the quadratic equation +b z = az (14.14) cz + d : If the points z1 and z2 are inverses of each other with respect to a circle K1 of the z plane, then their images w1 and w2 in the w plane are also inversions of each other with respect to the image circle K2 of K1 . The orthogonal net which has the orthogonal Cartesian net as its image is represented in Fig. 14.12. y y y
0
1
x
x
Figure 14.12
x
Figure 14.13
Figure 14.14
4. Quadratic Function
The conformal mapping described by a quadratic function w = z2 (14.15a) has the form in polar coordinates and as a function of x and y: w = 2 ei2' (14.15b) w = u + i v = x2 ; y2 + 2i xy: (14.15c) It is obvious from the polar coordinate representation that the upper half of the z plane is mapped onto the whole w plane, i.e., the whole image of the z plane will cover twice the whole w plane. The representation in Cartesian coordinates shows that the coordinate lines of the w plane, u = const and v = const, come from the orthogonal families of hyperbolas x2 ; y2 = u and 2xy = v of the z plane (Fig. 14.13). Fixed points of this mapping are z = 0 and z = 1. This mapping is not conformal at z = 0.
5. Square Root
The conformal p mapping given in the form as a square root of z, w= z (14.16) transforms the whole z plane whether onto the upper half of the w plane or onto the lower half of it, i.e., the function is double-valued. The coordinate lines of the w plane come from two orthogonal families of confocal parabolas with the focus at the origin of the z plane and with the positive or with the negative
678 14. Function Theory real half-axis as their axis (Fig. 14.14). Fixed points of the mapping are z = 0 and z = 1. The mapping is not conformal at z = 0.
6. Sum of Linear and Fractional Linear Functions
The conformal mapping given by the function w = k2 z + z1 (k a real constant k > 0) (14.17a) i ' can be transformed by the polar coordinate representation of z = e and by separating the real and imaginary parts according to (14.8): ! ! u = k2 + 1 cos ' v = k2 ; 1 sin ': (14.17b) Circles with = 0 = const of the z plane (Fig. 14.15a) are transformed into confocal ellipses ! u2 + v2 = 1 with a = k + 1 k ; 1 b = (14.17c) 0 0 a2 b2 2 0 2 0 in the w plane (Fig. 14.15b). The foci are the points k of the real axis. For the unit circle with = 0 = 1 we get the degenerate ellipse of the w plane, the double segment (;k +k) of the real axis. Both the interior and the exterior of the unit circle are transformed onto the entire w plane with the cut (;k +k), so its inverse function is double-valued: p 2 2 z = w + kw ; k : (14.17d) The lines ' = '0 of the z plane (Fig. 14.15c) become confocal hyperbolas u2 ; v2 = 1 with = k cos ' = k sin ' (14.17e) 0 0 2 2 with foci k (Fig. 14.15d). The hyperbolas corresponding to the coordinate half-axis of the z plane 3 ' = 0 2 2 are degenerate in the axis u = 0 (v axis) and in the intervals (;1 ;k) and (k 1) of the real axis running there and back. y
v
y j=j0
0 a)
7. Logarithm
x
-k
0
k
0
u
x
c)
b)
Figure 14.15
The conformal mapping given in the form of the logarithm function w = Ln z (14.18a) has the form for z given in polar coordinates: u = ln v = ' + 2k (k = 0 1 2 : : :): (14.18b) We can see from this representation that the coordinate lines u = const and v = const come from concentric circles around the origin of the z plane and from rays starting at the origin of the z plane
14.1 Functions of Complex Variables 679 y v
-k
0
k
ϕ=1
x
u
d) Figure 14.15 Figure 14.16 (Fig. 14.16). The isometric net is a polar net. The logarithm function Ln z is in nitely many valued (see (14.74c), p. 699). If we restrict our investigation to the principal value ln z of Ln z (; < v +), then the whole z plane is transformed into a stripe of the w plane bounded by the lines v = , where v = belongs to the stripe.
8. Exponential Function
The conformal mapping given in the form of an exponential function (see also 14.5.2, 1., p. 698) w = ez (14.19a) has the form in polar coordinates: w = ei : (14.19b) We get from z = x + i y: = ex and = y: (14.19c) If y changes from ; to +, and x changes from ;1 to +1, then takes all values from 0 to 1 and from ; to . A 2 wide stripe of the z plane, parallel to the x-axis, will be transformed into the entire w plane (Fig. 14.17). y
v (u,v)
p 0 −p
r y
x 0
a)
u
b)
Figure 14.17
9. The Schwarz{Christo el Formula
By the Schwarz{Christo el formula Zw z = C1 (t ; w ) 1 (t ; dtw ) 2 (t ; w ) n + C2 (14.20a) 1 2 n 0 the interior of a polygon of the z plane can be mapped onto the upper half of the w plane. The polygon has n exterior angles 1 2 : : : n (Fig. 14.18a,b). We denote by wi (i = 1 : : : n) the points of the real axis in the w plane assigned to the vertices of the polygon, and by t the integration variable. The oriented boundary of the polygon is transformed into the oriented real axis of the w plane by this mapping.
680 14. Function Theory y
v 3
z2
zn anp
a)
a3p z4 z
z1
a2p 0 w1 w2 w 3 w 4
a1p
0
x
wn u
b)
Figure 14.18 For large values of t, the integrand behaves as 1=t2 and is regular at in nity. Since the sum of all the exterior angles of an n-gon is equal to 2, we get: n X =1
= 2:
(14.20b)
The complex constants C1 and C2 yield a rotation, a stretching and a translation they do not depend on the form of the polygon, only on the size and the position of the polygon in the z plane. Three arbitrary points w1 w2 w3 of the w plane can be assigned to three points z1 z2 z3 of the polygon in the z plane. If we assign a point at in nity in the w plane, i.e., w1 = 1 to a vertex of the polygon in the z plane, e.g., to z = z1 , then the factor (t ; w1) 1 is omitted. If the polygon is degenerate, e.g., a vertex is at in nity, then the corresponding exterior angle is , so 1 = 1, i.e., the polygon is a half-stripe. C A
b) C
a)
r A1
A
8
d
8
B
A2
B
C
Figure 14.19
A: We want to map a certain region of the z plane. It is the shaded region in z w Fig. 14.19a. Considering P = 2 we choose three points as the table shows A 1 1 ;1 (Fig. 14.19a,b) . The formula of the mapping is: Zw p p p p B 0 ;1=2 0 dt z = C1 (t + 1)t;1=2 = 2C1 w ; arctan w = i 2d w ; arctan w . C 1 3=2 1 0 Z 0 ;1 + ei' 1=2 i ei' d' i ' For the determination of C1 we substitute t = e ; 1: id = C1 lim!0 = ei'
C1, i.e., C1 = i d . We get the constant C2 = 0 considering that the mapping assigns \z = 0 ! w = 0". B: Mapping of a rectangle. Let the vertices of the rectangle be z14 = K z23 = K + iK 0. The points z1 and z2 should be transformed into the points w1 = 1 and w2 = 1=k (0 < k < 1) of the real axis, z4 and z3 are reections of z1 and z2 with respect to the imaginary axis. According to the Schwarz reection principle (see 14.1.3.3, p. 681) they must correspond to the points w4 = ;1 and w3 = ;1=k (Fig. 14.20a,b). So, the mapping formula for a rectangle (1 = 2 = 3 = 4 = 1=2) of the position
14.1 Functions of Complex Variables 681 y
z3
z2
v
z1
z4 0
a)
x
−1 k w b) 3
−1
1
w4 0
w1
1 k w2 u
Figure 14.20 Zw dt dt sketched above is: z = C1 q = C1 s . The 0 0 (t ; w1)(t ; w2)(t ; w3)(t ; w4) 2 (t ; 1) t2 ; k12 point z = 0 has the image w = 0 and the image of z = iK is w = 1. With C1 = 1=k we get Zw Z' dt p d 2 2 = F (' k) (substituting t = sin w = sin '): z= q = 0 0 1 ; k sin (1 ; t2)(1 ; k2t2 ) F (' k) is the elliptic integral of the rst kind (see 8.1.4.3, p. 437). We get the constant C2 = 0 considering that the mapping assigns \ z = 0 ! w = 0 ".
Zw
14.1.3.3 The Schwarz Reection Principle 1. Statement
Suppose f (z) is an analytic complex function in a domain G, and the boundary of G contains a line segment g1. If the function is continuous on g1 and it maps the line g1 into a line g10 , then the points lying symmetric with respect to g1 are transformed into points lying symmetric with respect to g10 (Fig. 14.21). source
P
g1
G
G'
g1'
boundary
P'
f(z)
P
boundary
source
P'
Figure 14.21
2. Application
source
Figure 14.22
sink
Figure 14.23
The application of this principle makes it easier to perform calculations and the representations of plane regions with straight line boundaries: If the line boundary is a stream line (isolating boundary in Fig. 14.22), then the sources are reected as sources, the sinks as sinks and curls as curls with the opposite sense of rotation. If the line boundary is a potential line (heavy conducting boundary in Fig. 14.23), then the sources are reected as sinks, the sinks as sources and curls as curls with the same sense of rotation.
14.1.3.4 Complex Potential
1. Notion of the Complex Potential
~ =V ~ (x y) in the x y plane with continuous and di erentiable components We will consider a eld V ~ for the zero-divergence and the irrotational case. Vx(x y) and Vy (x y) of the vector V x @Vy a) Zero-divergence eld with div V~ = 0, i.e., @V @x + @y = 0: That is the integrability condition for the di erential equation expressed with the eld or stream function " (x y)
682 14. Function Theory Vy = ; @" @x :
d" = ;Vy dx + Vx dy = 0 (14.21a) and then Vx = @" @y
(14.21b)
~ the di erence " (P2) ; " (P1) is a measure of the ux through a For two points P1 , P2 of the eld V curve connecting the points P1 and P2, in the case when the whole curve is in the eld. y @Vx b) Irrotational eld with rot V~ = ~0, i.e., @V @x ; @y = 0: That is the integrability condition for the di erential equation with the potential function (x y) d = Vx dx + Vy dy = 0 (14.22a) and then Vx = @ Vy = @ (14.22b) @x @y : The functions and " satisfy the Cauchy{Riemann di erential equations (see 14.1.2.1, p. 672) and both satisfy the Laplace di erential equation ( = 0 " = 0). We combine the functions and " into the analytic function W = f (z) = (x y) + i " (x y) (14.23) ~. and this function is called the complex potential of the eld V ~ in the sense of the usual notation in physics and Then ;(x y) is the potential of the vector eld V electrotechnics (see 13.3.1.6, 2., p. 662). The level lines of " and form an orthogonal net. We have ~: the following equalities for the derivative of the complex potential and the eld vector V dW = @ ; i @ = V ; iV dW = f 0(z) = V + iV : (14.24) x y dz @x @y x y dz
2. Complex Potential of a Homogeneous Field
The function W = az (14.25) with real a is the complex potential of a eld whose potential lines are parallel to the y-axis and whose direction lines are parallel to the x-axis (Fig. 14.24). A complex a results in a rotation of the eld (Fig. 14.25). y
y Ψ=const
Ψ=const
x Φ=const
Figure 14.24
3. Complex Potential of Source and Sink
Φ=const
x
Figure 14.25
The complex potential of a eld with a strength of source s > 0 at the point z = z0 has the equation: W = 2s ln(z ; z0) : (14.26) A sink with the same intensity has the equation: (14.27) W = ; 2s ln(z ; z0 ) :
14.1 Functions of Complex Variables 683
The direction lines run away radially from z = z0 , while the potential lines are concentric circles around the point z0 (Fig. 14.26).
4. Complexes Potential of a Source{Sink System
We get the complex potential for a source at the point z1 and for a sink at the point z2, both having the same intensity, by superposition ; z1 : (14.28) W = 2s ln zz ; z2 The potential lines = const form Apollonius circles with respect to z1 and z2 the direction lines " = const are circles through z1 and z2 (Fig. 14.27). y
Ψ=const
y z1
Φ=const z0 0
Ψ=const Φ=const
x
Figure 14.26
z2 0
x
Figure 14.27
5. Complex Potential of a Dipole
The complex potential of a dipole with dipole moment M > 0 at the point z0 , whose axis encloses an angle with the real axis (Fig. 14.28), has the equation: i (14.29) W = 2(Me z ; z0 ) :
6. Complex Potential of a Curl
If the intensity of the curl is j; j with real ; and its center is at the point z0, then its equation is: (14.30) W = 2;i ln(z ; z0 ): In comparison with Fig. 14.26, the roles of the direction lines and the potential are interchanged. For complex ; (14.30) gives the potential of a source of curl, whose direction and potential lines form two families of spirals orthogonal to each other (Fig. 14.29).
14.1.3.5 Superposition Principle
1. Superposition of Complex Potentials
A system composed of several sources, sinks, and curls is an additive superposition of single elds, i.e., we get its function by adding their complex potential and stream functions. Mathematically this is possible because of the linearity of the Laplace di erential equations = 0 and " = 0.
2. Composition of Vector Fields
1. Integration A new eld can be constructed from complex potentials not only by addition but also by integration of the weight functions. Let a curl be given with density %(s) on a line segment l. Then we get a Cauchy type integral (see
684 14. Function Theory y
y
z0
Ψ=const
α
z0
Ψ=const x
Φ=const 0
x
Φ=const
0
Figure 14.28 14.2.3, p. 689) for the derivative of the complex potential: dW = 1 Z %(s) ds = 1 Z % ( ) d dz 2i (l) z ; (s) 2i (l) z ;
Figure 14.29 (14.31)
where (s) is the complex parametric representation of the curve l with arclength s as its parameter.
2. Maxwell Diagonal Method If we want to compose the superposition of two elds with the
potentials 1 and 2 , then we draw the potential line gures 1 ]] and 2 ]] so that the value of the potential changes by the same amount h between two neighboring lines in both systems, and we direct the lines so that the higher values are on the left-hand side. The lines lying in the diagonal direction to the net elements formed by 1]] and 2 ]] give the potential lines of the eld ]], whose potential is = 1 + 2 or = 1 ; 2 . We get the gure of 1 + 2]] if the oriented sides of the net elements are added as vectors (Fig. 14.30a), and we get the gure of 1 ; 2 ]] when we subtract them (Fig. 14.30b). The value of the composed potential changes by h at transition from one potential line to the next one. Vector lines and potential lines of a source and a sink with an intensity quotient je1j=je2j = 3=2 (Fig. 14.31a,b). F2+h
F2+h F2
F2
F1+h
F1+h a)
F1
b)
F1
Figure 14.30
14.1.3.6 Arbitrary Mappings of the Complex Plane
A function w = f (z = x + i y) = u(x y) + i v(x y) (14.32a) is de ned if the two functions u = u(x y) and v = v(x y) with real variables are de ned and known. The function f (z) must not be analytic, as it was required in conformal mappings. The function w maps the z plane into the w plane, i.e., it assigns to every point z a corresponding point w .
a) Transformation of the Coordinate Lines y = c ;! u = u(x c) v = v(x c) x is a parameter
14.2 Integration in the Complex Plane 685
z1
z2
z1
z2
b)
a)
Figure 14.31 x = c1 ;! u = u(c1 y) v = v(c1 y) y is a parameter: (14.32b) b) Transformation of Geometric Figures Geometric gures as curves or regions of the z plane are usually transformed into curves or regions of the w plane: x = x(t) y = y(t) ! u = u(x(t) y(t)) v = v(x(t) y(t)) t is a parameter: (14.32c) For u = 2x + y, v = x + 2y, the lines y = c are transformed into u = 2x + c, v = x + 2c, hence into the lines v = u2 + 32 c. The lines x = c1 are transformed into the lines v = 2u ; 3c1 (Fig. 14.5). The shaded region in Fig. 14.5a is transformed into the shaded region in Fig. 14.5b. c) Riemann Surface If we get the same value w for several di erent z for the mapping w = f (z), then the image of the function consists of several planes \lying on each other". If we cut these planes and connect them along a curve, then we get a many-sheeted surface, the so-called many-sheeted Riemann surface (see 14.13]). This correspondence can bepconsidered also in an inverse relation, in the case of multiple-valued functions as, e.g., the functions n z, Ln z, Arcsin z, Arctan z. u
v
0
w = z2 : While z = rei' overruns the entire z plane, i.e., 0 ' < 2, the values of w = %ei = r2ei2', cover twice the w plane. We can imagine that we place two w planes on each other, and we cut and connect them along the negative real axis according to Fig. 14.32. This surface is called the Riemann surface of the function w = z2 .
Figure 14.32 The zero point is called a branch point. The image of the function ez (see (14.69)) is a Riemann surface of in nitely many sheets. (In many cases the planes are cut along the positive real axis. It depends on whether the principal value of the complex number is de ned for the interval (; +] or for the interval 0 2).)
14.2 Integration in the Complex Plane 14.2.1 De nite and Inde nite Integral
14.2.1.1 Denition of the Integral in the Complex Plane 1. Denite Complex Integral
Suppose f (z) is continuous in a domain G, and the curve C is recti able, it connects the points A and B , and the whole curve is in this domain. We decompose the curve C between the points A and B by
686 14. Function Theory arbitrary division points zi into n subarcs (Fig.14.33). We choose a point i on every arc segment and form the sum n X
f (i) zi with i=1 If the limit n X f (i) zi nlim !1
zi = zi ; zi;1 :
y
(14.33a) (14.33b)
i=1 exists for zi ! 0 and n ! 1 independently of the choice of the points i, then we call this limit the denite complex integral Z ZB
A
zi-1
zi
0
B zi
G
C x
Figure 14.33
I=
f (z) dz = (C ) f (z) dz (14.34) _ A AB along the curve C between the points A and B . The value of the integral usually depends on the path of the integral.
2. Indenite Complex Integral
If the de nite integral is independent of the path of the integral (see 14.2.2, p. 688), then we denote: Z F (z) = f (z) dz + C with F 0(z) = f (z): (14.35) Here C is the integration constant which is complex, in general. The function F (z) is called an indenite complex integral. The inde nite integrals of the elementary functions of a complex variable can be calculated with the same formulas as the integrals of the corresponding elementary function of a real variable. Z Z A: sin z dz = ; cos z + C . B: ez dz = ez + C .
3. Relation between Denite and Indenite Complex Integrals
If the function f (z) has an inde nite integral, then the relation between its de nite and inde nite integral is
Z
_ AB
ZB f (z) dz = f (z) dz = F (zB ) ; F (zA): A
(14.36)
14.2.1.2 Properties and Evaluation of Complex Integrals
1. Comparison with the curvilinear integral of the second type
The de nite complex integral has the same properties as the curvilinear integral of the second type (see 8.3.2, p. 464): a) If we reverse the direction of the path of integration, then the integral changes its sign. b) If we decompose the path of integration into several parts, then the value of the total integral is the sum of the integrals on the parts.
2. Estimation of the Value of the Integral
If the absolute value of the function f (z) does not exceed a positive number M at the points z of the _ path of integration AB which has the length s, then:
Z f (z) dz Ms with jf (z)j M: _ AB
(14.37)
14.2 Integration in the Complex Plane 687
3. Evaluation of the Complex Integral in Parametric Representation
_ If the path of integration AB (or the curve C ) is given in the form x = x(t) y = y(t) (14.38) and the t values for the initial and endpoint are tA and tB , then the de nite complex integral can be calculated with two real integrals. We separate the real and the imaginary parts of the integrand and we get:
ZB
ZB
ZB
A
A ZtB
A
(C ) f (z) dz = (u dx ; v dy) + i (v dx + u dy) =
tA
ZtB u(t)x0(t) ; v(t)y0(t)] dt + i v(t)x0(t) + u(t)y0(t)] dt
with f (z) = u(x y) + iv(x y)
ZB
z = x + i y:
tA
(14.39a) (14.39b)
The notation (C ) f (z) dz means that the de nite integral is calculated along the curve C between the A
Z
points A and B . The notation f (z) dz and
Z
f (z) dz is also often used, and has the same meaning. _ AB I = (z ; z0)n dz (n 2 Z). Let the curve C be a circle around the point z0 with radius r0 : (C ) x = x0 + r0 cos t y = y0 + r0 sin t with 0 t 2. For every point z of the curve C : z = x + i y = z0 + r0 (cos t + i sin t) dz = r0(; sin t + i cos t) dt. After Z 2 substituting these values and transforming n +1 according to the de Moivre formula we get: I = r0 (cos nt + i sin nt)(; sin t + i cos t) dt 0 for n0 6= ;1 Z 2 = r0n+1 i cos(n + 1)t ; sin(n + 1)t] dt = 2i for n = ;1: 0
Z
(C )
4. Independence of the Path of Integration
Suppose a function of a complex variable is de ned in a simply connected domain. The integral (14.34) of the function can be independent of the path of integration, which connects the xed points A(zA) and B (zB ). A sucient and necessary condition is that the function is analytic in this domain, i.e., the Cauchy{Riemann di erential equations (14.4) are satis ed. Then also the equality (14.36) is valid. If a domain is bounded by a simple closed curve, then the domain is simply connected.
5. Complex Integral along a Closed Curve
Suppose f (z) is analytic in a simply connected domain. If we integrate the function f (z) along a closed curve C which is the boundary of this domain, the value of the integral according to the Cauchy integral theorem is equal to zero (see 14.2.2, p. 688): I f (z) dz = 0: (14.40) If f (z) has singular points in this domain, then the integral is calculated by using the residue theorem (see 14.3.5.5, p. 694), or by the formula (14.39a). The function f (z) = z ;1 a , with a singular point at z = a has an integral value for the closed curve I dz directed counterclockwise around a (Fig.14.34) z ; a = 2i. (C )
688 14. Function Theory
14.2.2 Cauchy Integral Theorem
14.2.2.1 Cauchy Integral Theorem for Simply Connected Domains
If a function f (z) is analytic in a simply connected domain, then we get two equivalent statements: a) TheI integral is equal to zero along any closed curve C : f (z) dz = 0: (14.41)
b) The value of the integral
ZB A
f (z) dz is independent of the curve connecting the points A and B , i.e.,
it depends only on A and B . This is the Cauchy integral theorem.
14.2.2.2 Cauchy Integral Theorem for Multiply Connected Domains
If C , C1 , C2, : : :, Cn are simple closed curves such that the curve C encloses all the C ( = 1 2 : : : n), none of the curves C encloses or intersects another one, and furthermore the function f (z) is analytic in a domain G which contains the curves and the region between C and C , i.e., at least the region shadedI in Fig.14.35I, then I I f (z) dz = f (z) dz + f (z) dz + : : : + f (z) dz (14.42) (C )
(C1 )
(C2 )
(Cn )
if the curves C , C1 , : : :, Cn are oriented in the same direction, e.g., counterclockwise. This theorem is useful for the calculation of an integral along a closed curve C , if it also encloses singular points of the function f (z) (see 14.3.5.5, p. 694). y
y
C C3
C1 a
Cn
C2
C2
C
0
C C1
-1
0
x
x
Figure 14.34
Figure 14.35
Figure 14.36
I Calculate the integral z(zz;+11) dz, where C is a curve enclosing the origin and the point z = ;1 (C ) (Fig. 14.36). Applying the Cauchy integral theorem, the integral along C is equal to the sum of the integrals along C1 and C2, where C1 is a circle around the origin with radius r1 = 1=2 and C2 is a circle around the point z = ;1 with radius r2 = 1=2. The integrand can be decomposed into partial fractions. I z;1 I 2 dz I 2 dz I dz I dz Then we get: dz = z(z + 1) z+1 + z+1 ; z ; z = 0 + 4i ; 2i ; 0 = 2i. (C )
(C1 )
(C2 )
(C1 )
(Compare the integrals with the example in 14.2.1.2, 3., p. 687.)
(C2 )
14.3 Power Series Expansion of Analytic Functions 689
14.2.3 Cauchy Integral Formulas
14.2.3.1 Analytic Function on the Interior of a Domain
If f (z) is analytic on a simple closed curve C and on the simply connected domain inside it, then the following representation is valid for every interior point z of this domain (Fig. 14.37): I ( ) f (z) = 21i f ; (14.43) z d (Cauchy integral formula) (C )
where traces the curve C counterclockwise. With this formula, the values of an analytic function in the interior of a domain are expressed by the values of the function on the boundary of this domain. The existence and the integral representation of the n-th derivative of the function analytic on the domain G follows from (14.43): I (14.44) f (n)(z) = 2n!i ( ;f (z))n+1 d: (C )
Consequently, if a complex function is di erentiable, i.e., it is analytic, then it is di erentiable arbitrarily many times. In contrast to this, di erentiability does not include repeated di erentiability in the real case. The equations (14.43) and (14.44) are called the Cauchy integral formulas.
14.2.3.2 Analytic Function on the Exterior of a Domain
If a function f (z) is analytic on the entire part of the plane outside of a closed curve of integration C , then the values and the derivatives of the function f (z) at a point z of this domain can be given with the same Cauchy formulas (14.43), (14.44), but the orientation of the curve C is now clockwise (Fig. 14.38). Also certain real integrals can be calculated with the help of the Cauchy integral formulas (see 14.4, p. 694). y
y C
z
z
0
x
Figure 14.37
C
0
x
Figure 14.38
14.3 Power Series Expansion of Analytic Functions 14.3.1 Convergence of Series with Complex Terms
14.3.1.1 Convergence of a Number Sequence with Complex Terms
An in nite sequence of complex numbers z1 z2 : : : zn : : : has a limit z (z = nlim zn) if for arbitrarily !1 given positive " there exists an n0 such that the inequality jz ; znj < " holds for every n > n0 , i.e., from a certain n0 the points representing the numbers zn zn+1 : : : are inside of a circle with radius " and center at z. p If the expression f n ag means the root with the smallest non-negative argument, then the limit p n f ag = 1 is valid for arbitrary a (Fig. 14.39). nlim !1
690 14. Function Theory 2
y
y a = z1
2
2
1
0
a = z2 3 a = z3 4 a = z4 1
3
i+ i + i 2 4
i+ i 2 2
3
i 4
i+ i + i + i 2 4 8 x
Figure 14.39
0
x
Figure 14.40
14.3.1.2 Convergence of an Innite Series with Complex Terms
A series a1 + a2 + + an + with complex terms ai converges to the number s, if s = nlim (a1 + a2 + + an) (14.45) !1 holds. The number s is the sum of the series. If we connect the points corresponding to the numbers sn = a1 + a2 + + an in the z plane by a broken line, then convergence means that the end of the broken line approaches the point s. 2 3 4 2 3 A: i + i2 + i3 + i4 + . B: i + i2 + 2i 2 + (Fig. 14.40) . A series is called absolutely convergent (see B), if the series of absolute values of the terms ja1j + ja2j + ja3j + is also convergent. The series is called conditionally convergent (see A) if the series is convergent but not absolutely convergent. If the terms of a series are functions fi (z), like f1(z) + f2 (z) + + fn(z) + (14.46) then its sum is a function de ned for the values z for which the series of the substitution values is convergent.
14.3.1.3 Power Series with Complex Terms 1. Convergence
A power series with complex coecients has the form P (z ; z0) = a0 + a1 (z ; z0) + a2 (z ; z0 )2 + + an(z ; z0)n + (14.47a) where z0 is a xed point in the complex plane and the coecients a are complex constants (which can also have real values). For z0 = 0 the power series has the form P (z) = a0 + a1 z + a2 z2 + + anzn + : (14.47b) If the power series P (z ; z0 ) is convergent for a value z1, then it is absolutely and uniformly convergent for every z in the interior of the circle with radius r = jz1 ; z0 j and center at z0 .
2. Circle of Convergence
The limit between the domain of convergence and the domain of divergence of a complex power series is a uniquely de ned circle. We determine its radius just as in the real case, if the imits 1 an q r = nlim or r = lim (14.48) !1 n n !1 an+1 janj exist. If the series is divergent everywhere except at z = z0 , then r = 0 if it is convergent everywhere, then r = 1. The behavior of the power series on the boundary circle of the domain of convergence should be investigated point by point. n 1 The power series P (z) = P zn with radius of convergence r = 1 is divergent for z = 1 (harmonic n=1
14.3 Power Series Expansion of Analytic Functions 691
series) and convergent for z = ;1 (according to the Leibniz criteria for alternating series). This power series is convergent for all further points of the unit circle jzj = 1 except the point z = 1.
3. Derivative of Power Series in the Circle of Convergence
Every power series represents an analytic function f (z) inside of the circle of convergence. We get the derivative by a term-by-term di erentiation. The derivative series has the same radius of convergence as the original one.
4. Integral of Power Series in the CircleZ of Convergence z
We get the power series expansion of the integral f ( ) d by a term-by-term integration of the power z0 series of f (z). The radius of convergence remains the same.
14.3.2 Taylor Series
Every function f (z) analytic in a domain G can be expanded uniquely into a power series of the form
f (z) =
1 X
n=0
an(z ; z0)n
(Taylor series)
(14.49a)
for any z0 in G, where the circle of convergence is the greatest circle around z0 which belongs entirely to the domain G (Fig. 14.41). The coecients an are complex numbers in general, and for them we get: (n) an = f n(!z0) : (14.49b) The Taylor series can be written in the form 0 00 (n) f (z) = f (z0 ) + f (1!z0 ) (z ; z0 ) + f 2!(z0 ) (z ; z0 )2 + + f n(!z0 ) (z ; z0 )n + : (14.49c) Every power series is the Taylor expansion of its sum function in the interior of its circle of convergence. Examples of Taylor expansions are the series representations of the functions ez , sin z, cos z, sinh z, and cosh z in 14.5.2, p. 698. y C2
C1
B
i
z0 -1
Figure 14.41
0
C1 z0
C
1 x C0
Figure 14.42
Figure 14.43
14.3.3 Principle of Analytic Continuation
We consider the case when the circles of convergence K0 around z0 and K1 around z1 of the two power series 1 1 X X (14.50a) f0(z) = an(z ; z0 )n and f1 (z) = bn (z ; z1 )n n=0
n=0
have a certain common domain (Fig. 14.42) and in this domain they are equal: f0(z) = f1 (z):
(14.50b)
692 14. Function Theory Then both power series are the Taylor expansions of the same analytic function f (z), belonging to the points z0 and z1 . The function f1(z) is called the analytic continuation into K1 of the function f0(z) de ned only in K0 . The geometric series f0 (z) =
1 X
n=0
zn with the circle of convergence K0 (r0 = 1) around z0 = 0 and
1 1 X z ; i n with the circle of convergence K (r = p2) around z = i have the same f1(z) = 1 ; 1 1 1 i n=0 1 ; i analytic function f (z) = 1=(1 ; z) as their sum in their own circle of convergence, consequently also on the common part of them (doubly shaded region in Fig. 14.42) for z 6= 1. So, f1 (z) is the analytic continuation of f0 (z) from K0 into K1 (and conversely).
14.3.4 Laurent Expansion
Every function f (z), which is analytic in the interior of a circular ring between two concentric circles with center z0 and radii r1 and r2, can be expanded into a generalized power series, into the so-called Laurent series: 1 X a;k + a;k+1 + f (z) = an(z ; z0 )n = + (z ; z0)k (z ; z0 )k;1 n=;1 a + z ;;1z + a0 + a1(z ; z0) + a2(z ; z0)2 + + ak (z ; z0)k + : (14.51a) 0 The coecients an are usually complex and they are uniquely de ned by the formula I an = 21i ( ;f (z ))n+1 d (n = 0 1 2 : : :) (14.51b) 0 (C )
where C denotes an arbitrary closed curve which is in the circular ring r1 < jzj < r2, and the circle with radius r1 is inside of it, and its orientation is counterclockwise (Fig. 14.43). If the domain G of the function f (z) is larger than the circular ring, then the domain of convergence of the Laurent series is the largest circular ring with center z0 lying entirely in G. 1 Determine the Laurent series expansion of the function f (z) = , around z0 = 0 in (z ; 1)(z ; 2) the circular ring 1 < jzj < 2 where f (z) is analytic. First we decompose the function f (z) into partial fractions: f (z) = z ;1 2 ; z ;1 1 . Since j1=zj < 1 and jz=2j < 1 holds in the considered domain, the two terms of this decomposition can be written as the sums of geometric series absolutely convergent in the entire circular ring 1 < jzj < 2. We get: 1 1 n X X z . f (z) = (z ; 1)(1 z ; 2) = ; 1 1 ; 1 z = ; z1n ; 12 2 n| =1{z } 2 1; 2 z 1; z |n=0 {z } jzj > 1 jzj < 2
14.3.5 Isolated Singular Points and the Residue Theorem
14.3.5.1 Isolated Singular Points
If a function f (z) is analytic in the neighborhood of a point z0 but not at the point z0 itself, then z0 is called an isolated singular point of the function f (z). If f (z) can be expanded into a Laurent series in the neighborhood of z0
f (z) =
1 X
n=;1
an(z ; z0 )n
(14.52)
14.3 Power Series Expansion of Analytic Functions 693
then the isolated singular point can be classi ed by the behavior of the Laurent series: 1. If the Laurent series does not contain any term with a negative power of (z ; z0 ), i.e., an = 0 for n < 0 holds, then the Laurent series is a Taylor series with coecients given by the Cauchy integral formula I (n) an = 21i ( ; z0 );n;1f ( ) d = f n(!z0 ) : (14.53) (K )
In this case, the function f (z) itself is either analytic at the point z0 and f (z0) = a0 or z0 is a removable singularity. 2. If the Laurent series contains a nite number of terms with negative powers of (z ; z0 ), i.e., am 6= 0, an = 0 for n < m < 0, then z0 is called a pole, a pole of order m, or a pole of multiplicity m. If we multiply by (z ; z0)m , and not by any lower power, then f (z) is transformed into a function which is analytic at z0 and in its neighborhood. f (z) = 21 z + z1 has a pole of order one at z = 0. 3. If the Laurent series contains an in nite number of terms with negative powers of (z ; z0), then z0 is an essential singularity of the function f (z). Approaching a pole, jf (z)j tends to 1. Approaching an essential singularity, f (z) gets arbitrarily close to any complex number c. 1 X The function f (z) = e1=z , whose Laurent series is f (z) = n1! z1n , has an essential singularity at n=0 z = 0.
14.3.5.2 Meromorphic Functions
If an otherwise holomorphic function has only a nite number of poles as singular points, then it is called meromorphic. A meromorphic function can always be represented as the quotient of two analytic functions. Examples of functions meromorphic on the whole plane are the rational functions which have a nite number of poles, and also transcendental functions such as tan z and cot z.
14.3.5.3 Elliptic Functions
Elliptic functions are double periodic functions whose singularities are poles, i.e., they are meromorphic functions with two independent periods (see 14.6, p. 702). If the two periods are !1 and !2 , which are in a non-real relation, then f (z + m!1 + n!2) = f (z) (m n = 0 1 2 : : : Im !!1 6= 0): (14.54) 2 The range of f (z) is already attained in a primitive period parallelogram with the points 0 !1 !1 + !2 !2 .
14.3.5.4 Residue
The coecient a;1 of the power (z ; z0);1 in the Laurent expansion of f (z) is called the residue of the function f (z ) at the point z0 : I a;1 = Res f (z)jz=z0 = 21i f ( ) d: (14.55a) (K )
The residue belonging to a pole of order m can be calculated by the formula 1 d m;1 f (z)(z ; z )m]: a;1 = Res f (z)jz=z0 = zlim 0 !z0 (m ; 1)! dz m;1
(14.55b)
694 14. Function Theory If the function can be represented as a quotient f (z) = '(z)=(z), where the functions '(z) and (z) are analytic at the point z = z0 and z0 is a simple root of the equation (z) = 0, i.e., (z0 ) = 0, 0(z0 ) 6= 0 holds, then the point z = z0 is a pole of order one of the function f (z). It follows from (14.55b) that " '(z0) Res '(z) (14.55c) (z) z=z0 = 0 (z0) : If z0 is a root of multiplicity m of the equation (z) = 0, i.e., (z0 ) = 0 (z0) = = (m;1) (z0) = 0 (m) (z0 ) 6= 0 holds, then the point z = z0 is a pole of order m of f (z).
14.3.5.5 Residue Theorem
With the help of residues we can calculate the integral of a function along a closed curve enclosing isolated singular points (Fig. 14.44). If the function f (z) is single valued and analytic in a simply connected domain G except at a nite number of points z0 z1 z2 : : : zn, and the domain is bounded by the closed curve C , then the value of the integral of the function along this closed curve in a counterclockwise direction is the product of 2i and the sum of the residues in all these singular points:
I
(K )
f (z) dz = 2i
n X
k=0
Res f (z) jz=zk :
(14.56)
The function f (z) = ez =(z2 + 1) has poles of order one at z12 = i. TheIcorresponding residues z have the sum sin 1. If K is a circle around the origin with radius r > 1, then z2e+ 1 dz = 2i sin 1. (K )
y
y z1
z2
z4
z2
z1
z3
z4
C x
0
Figure 14.44
z3
0
x
Figure 14.45
14.4 Evaluation of Real Integrals by Complex Integrals
14.4.1 Application of Cauchy Integral Formulas
The value of certain real integrals can be calculated with the help of the Cauchy integral formula. The function f (z) = ez (see 14.5.2,1., p. 698), which is analytic in the whole z plane, can be represented with the Cauchy integral formula (14.43), where the path of integration C is a circle with center z and radius r. The equation of the circle is = z + rei'. We get from (14.43) I Z '=2 e(z+rei') Z 2 ez = 2n!i ( ;ez)n+1 d = 2n!i irei' d' = n! n ez+r cos '+ir sin ';in' d', so that n +1 i ' ( n +1) 2r 0 '=0 r e (C ) 2rn = Z 2er cos '+i(r sin ';n') d' = Z 2er cos ' cos(r sin ' ; n')] d' + i Z 2 er cos ' sin(r sin ' ; n')] d'. n! 0 0 0
14.4 Evaluation of Real Integrals by Complex Integrals 695
Since the imaginary part is equal to zero, we get
Z 2 n er cos ' cos(r sin ' ; n') d' = 2r n! . 0
14.4.2 Application of the Residue Theorem
Several de nite integrals of real functions with one variable can be calculated with the help of the residue theorem. If f (z) is a function which is analytic in the whole upper half of the complex plane including the real axis except the singular points z1 z2 : : : zn above the real axis (Fig. 14.45), and if one of the roots of the equation f (1=z) = 0 has multiplicity m 2 (see 1.6.3.1, 1., p. 43), then + Z1
f (x) dx = 2i
;1
n X i=1
Res f (z)jz=zi :
(14.57)
1 dx : The equation f 1 = x6 = 0 has = 3 2 3 2 x (x + 1)3 ;1 (1 + x ) 1 + x12 a root of order six at x = 0. The function w = 1 2 3 has a single singular point z = i in the upper (1 + z ) half-plane, which is a pole of order 3, since the equation (1 + z2 )3 = 0 has two triple roots at i and ;i. The residue is according to (14.55b): "
d2 (z ; i)3 . From d2 z ; i 3 = d2 (z + i);3 = 12(z + i);5 it Res (1 + z12 )3 j = 2!1 dz 2 (1 + z 2 )3 dz2 1 + z2 dz2 z=i z=i Z +1 1 6 3 ;5 follows that Res (1 + z2 )3j = 6(z + i) jz=i = (2i)5 = ; 16 i, and with (14.57): f (x) dx = ;1 z=i 3 3 2i ; 16 i = 8 . For further applications of residue theory see, e.g., 14.12]. Calculation of the integral
Z +1
14.4.3 Application of the Jordan Lemma 14.4.3.1 Jordan Lemma
In many cases, real improper integrals with an in nite domain of integration can be calculated by complex integrals along a closed curve. To avoid the always recurrent estimations, we use the Jordan lemma about improper integrals of the form
Z
(CR )
f (z)ei z dz
(14.58a)
where CR is the half-circle arc with center at the origin and with the radius R in the upper half of the z plane (Fig. 14.46). The Jordan lemma distinguishes the following cases: a) > 0: If f (z) tends to zero uniformly in the upper half-plane and also on the real axis for jzj ! 1 and is a positive number, then for R ! 1
Z
(CR )
f (z)ei z dz ! 0:
(14.58b)
b) = 0: If the expression z f (z) tends to zero uniformly for jzj ! 1, then the above statement is also valid in the case = 0. c) < 0: If the half-circle is now below the real axis, then the corresponding statement is also valid for < 0. d) The statement is also valid if only an arc segment is considered instead of the complete half-circle.
696 14. Function Theory e) TheZ corresponding statement is valid for the integral in the form (CR )
f (z)e z dz
(14.58c)
where CR is a half-circle or an arc segment in the left half-plane with > 0, or in the right one with < 0. y
y
y
CR
0
R x
-R
Figure 14.46
II p 4
r
-r -R
III
CR
0
Figure 14.47
14.4.3.2 Examples of the Jordan Lemma
R x
0
I
R
x
Figure 14.48
Z1 1. Evaluation of the Integral xx2sin+x a2 dx. 0
The following complex integral is assigned to the above real integral: ZR ZR ZR ZR xei x 2i x 2sin x2 dx = i x 2sin x2 dx + x 2cos x2 dx = x +a x +a x +a x2 + a2 dx. 0 | {z } ;R ;R ;R | {z } even function = 0 (odd integrand) I iz The very last of these integrals is part of the complex integral z2ze+ a2 dz. The curve C contains the (C ) CR half-circle de ned above and the part of the real axis between the values ;R and R (R > jaj). The complex integrand has the only singular point in" the upper half-plane z = a i. We obtain from the
I zei z i z ze zei z = ie; a , hence residue theorem: I = dz = 2i zlim (z ; ai) = 2i zlim 2 2 2 2 ! a i ! a i z +a z +a z + ai (C )
I=
Z
(CR )
zei z dz + ZR xei x dx = ie; a . It follows from lim I and from the Jordan lemma that R!1 z2 + a2 x2 + a2 ;R
Z1 x sin x ; 2 + a2 dx = 2 e x 0
a
( > 0 a 0).
Several further integrals can be evaluated in a similar way Table 21.8, p. 1056. 2. Sine Integral (see also (8.95), p. 460) Z1 The integral sinx x dx is called the sine integral or the integral sine (see also 8.2.5, 1., p. 460). Analo0 Z iz gously to the previous example, we investigate the complex integral e dz with the curve C according z C to Fig. 14.47. The integrand of the complex integral has a pole of rst order at z = 0, so
14.4 Evaluation of Real Integrals by Complex Integrals 697
" iz ZR Z2 Z iz I = 2i limz!0 ez z = 2i hence I = 2i sinx x dx + i eir(cos '+i sin ') d' + ez dz = 2i. This r CR limit is evaluated as R ! 1, r ! 0, where the second integral tends to 1 uniformly for r ! 0 with respect to ', i.e., the limiting process r ! 0 can be done behind the integral sign. Then we get with the Jordan lemma: Z1 Z1 sin x 2i sin x dx + i = 2i hence dx = 2 : (14.59) x x 0 0
3. Step Function
Discontinuous real functions can be represented as complex integrals. The so-called step function is an example: 8 1 for t > 0 Z eitz < F (t) = 21i dz = : 1=2 for t = 0 (14.60) z 0 for t < 0 : ;^! The symbol ;^! denotes a path of integration along the real axis (j R j! 1) going round the origin (Fig. 14.47). If t denotes time, then the function (t) = cF (t ; t0) represents a quantity which jumps at time t = t0 from 0 through the value c=2 to the value c. We call it a step function or also a Heaviside function. It is used in the electrotechnics to describe suddenly occurring voltage or current impulses.
4. Rectangular Pulse
A further example of the application of complex integrals and the Jordan lemma is the representation of the rectangular pulse: 8 0 for t < a and t > b Z ei(b;t)z Z ei(a;t)z < 1 1 " (t) = 2i dz ; 2i dz = : 1 for a < t < b (14.61) z z 1=2 for t = a and t = b: ;^! ; ^!
5. Fresnel Fntegrals
To derive the Fresnel integral
Z1 0
Z1 q sin(x2 ) dx = cos(x2 ) dx = 12 =2 0
(14.62)
Z
we investigate the integral I = e;z2 dz on the closed path of integration shown in Fig. 14.48. We K
get, according to the Cauchy integral theorem: I = II + III + IIII = 0 with II =
Z =4
ZR 0
e;x2 dx
e;R2 (cos 2'+i sin 2')+i' d', "ZR
ZR Z0 p IIII = e i4 eir2 dr = 21 2(1 + i) i sin r2 dr ; cos r2 dr . 0 0 R Z =4 2 i Estimation of III: Since jij = je j = 1 ( real) holds, we get: jIIIj < R e;R cos 2' d' 0 Z Z =2 2 Z Z =2 sin ' 2 ;R cos ' = R2 e;R2 cos ' d' + R2 d' e;R cos ' d' < R2 e;R2 cos d' + R2 sin e 0 0 2 ;R cos ;R2 cos + 1 ;2Re sin 0 < < 2 . Performing the limiting process Rlim I we get the < R !1 2e III = iR
0
698 14. Function Theory
p values of the integrals II and III : limR!1 II = 1 Rlim I = 0. We get the given formulas (14.62) !1 II 2 by separating the real and imaginary parts.
14.5 Algebraic and Elementary Transcendental Functions 14.5.1 Algebraic Functions 1. Denition
A function which is the result of nitely many algebraic operations performed with z and maybe also with nitely many constants, is called an algebraic function. In general, a complex algebraic function w(z) can be de ned in an implicit way as a polynomial, just as its real analogue a1zm1 wn1 + a2zm2 wn2 + + ak zmk wnk = 0: (14.63) Such functions cannot always be solved for w.
2. Examples of Algebraic Functions Linear function: w = az + b:
(14.64)
Quadratic function: w = z2 :
(14.66) + i: Fractional linear function: w = zz ; i
Inverse function: w = z1 :
p
Square root function: w = z2 ; a2:
(14.65) (14.67) (14.68)
14.5.2 Elementary Transcendental Functions
The complex transcendental functions have de nitions corresponding to the transcendental real functions, just as in the case of the algebraic functions. For a detailed discussion of them see, e.g., 21.1] or 21.10].
1. Natural Exponential Function 2 3 ez = 1 + 1!z + z2! + z3! + :
(14.69)
The series is absolutely convergent in the whole z plane. a) Pure imaginary exponent iy: This is valid according to the Euler relation (see 1.5.2.4, p. 35): eiy = cos y + i sin y with ei = ;1: (14.70) b) General case z = x + iy: ez = ex+iy = exeiy = ex(cos y + i sin y) (14.71a) Re (ez ) = ex cos y Im (ez ) = ex sin y jez j = ex arg(ez ) = y: (14.71b) c) The function ez is periodic, its period is 2 i: ez = ez+2k i (k = 0 1: 2 : : :) : (14.71c) 0 2 k i (2 k +1) i In particular: e = e = 1 e = ;1: (14.71d) d) Exponential form of a complex number (see 1.5.2.4, p. 35): a + ib = ei': (14.72) e) Euler relation for complex numbers : eiz = cos z + i sin z (14.73a) e;iz = cos z ; i sin z: (14.73b)
2. Natural Logarithmw w = Ln z
if z = e :
(14.74a)
14.5 Algebraic and Elementary Transcendental Functions 699
Since z = ei ', we can write: Ln z = ln + i (' + 2k) and (14.74b) Re (Ln z) = ln Im (Ln z) = ' + 2k (k = 0 1 2 : : :) : (14.74c) Since Ln z is a multiple-valued function (see 2.8.2, p. 84), we usually give only the principal value of the logarithm ln z : ln z = ln + i ' (; < ' +): (14.74d) The function Ln z is de ned for every complex number, except zero.
3. General Exponential Function z zLn a
a =e : az (a 6= 0) is a multiple-valued function (see 2.8.2, p. 84) with principal value az = ez ln a :
4. Trigonometric Functions and Hyperbolic Functions iz ;iz 3 5 sin z = e ;2ie = z ; z3! + z5! ; iz ;iz 2 4 cos z = e +2 e = 1 ; z2! + z4! ;
(14.75a) (14.75b) (14.76a) (14.76b)
z ;z 3 5 sinh z = e ; e = z + z + z + (14.77a) 2 3! 5! z ;z 2 4 cosh z = e +2 e = 1 + z2! + z4! + : (14.77b) All four series are convergent on the entire plane and they are all periodic. The period of the functions (14.76a,b) is 2, the period of the functions (14.77a,b) is 2 i. The relations between these functions for any real or complex z are: sin i z = i sinh z (14.78a) cos i z = cosh z (14.78b) sinh i z = i sin z (14.79a) cosh i z = cos z: (14.79b) The transformation formulas of the real trigonometric and hyperbolic functions (see 2.7.2, p. 79, and 2.9.3, p. 89) are also valid for the complex functions. We calculate the values of the functions sin z, cos z, sinh z, and cosh z for the argument z = x +i y with the help of the formulas sin(a + b), cos(a + b), sinh(a + b), and cosh(a + b) or we can do it by using the Euler relation (see 1.5.2.4, p. 35). cos(x + i y) = cos x cos i y ; sin x sin i y = cos x cosh y ; i sin x sinh y: (14.80) Therefore: Re (cos z) = cos Re (z) cosh Im (z) (14.81a) Im (cos z) = ; sin Re (z) sinh Im (z): (14.81b) The functions tan z, cot z, tanh z, and coth z are de ned by the following formulas: sin z cot z = cos z tan z = cos (14.82a) tanh z = sinh z coth z = cosh z : (14.82b) z sin z cosh z sinh z
5. Inverse Trigonometric Functions and Inverse Hyperbolic Functions
These functions are many-valued functions, and we can express them with the help of the logarithm function: p p Arcsin z = ;i Ln (i z + 1 ; z2 ) (14.83a) Arsinh z = Ln (z + z2 + 1) (14.83b) p2 p2 Arccos z = ;i Ln (z + z ; 1) (14.84a) Arcosh z = Ln (z + z ; 1) (14.84b)
700 14. Function Theory + iz Arctan z = 2i1 Ln 11 ; iz
(14.85a)
+z Artanh z = 12 Ln 11 ; z
(14.85b)
+1: +1 Arcoth z = 12 Ln zz ; (14.86b) Arccot z = ; 2i1 Ln ii zz ; (14.86a) 1 1 The principal values of the inverse trigonometric and the inverse hyperbolic functions can be expressed by the same formulas using the principal value of the logarithm ln z: p p arcsin z = ;i ln(i z + 1 ; z2 ) (14.87a) arsinh z = ln(z + z2 + 1) (14.87b) p2 p2 arccos z = ;i ln(z + z ; 1) (14.88a) arcosh z = ln(z + z ; 1) (14.88b) 1 1 + z 1 1 + i z artanh z = 2 ln 1 ; z (14.89b) arctan z = 2i ln 1 ; iz (14.89a) +1: +1 arcoth z = 21 ln zz ; (14.90b) arccot z = ; 2i1 ln iizz ; (14.90a) 1 1
6. Real and Imaginary Part of the Trigonometric and Hyperbolic Functions (See Table 14.1) 7. Absolute Values and Arguments of the Trigonometric and Hyperbolic Functions (See Table 14.2) Table 14.1 Real and imaginary parts of the trigonometric and hyperbolic functions
Function w = f (x + iy) sin(x iy) cos(x iy) tan(x iy) sinh(x iy) cosh(x iy) tanh(x iy)
Real part Re (w) sin x cosh y cos x cosh y sin 2x cos 2x + cosh 2y sinh x cos y cosh x cos y sinh 2x cosh 2x + cos 2y
Imaginary part Im (w) cos x sinh y sin x sinh y 2y cos 2sinh x + cosh 2y cosh x sin y sinh x sin y 2y cosh 2sin x + cos 2y
Table 14.2 Absolute values and arguments of the trigonometric and hyperbolic functions
Function w = f (x + iy) sin(x iy) cos(x iy) sinh(x iy) cosh(x iy)
Absolute value jwj q 2 2 q sin x + sinh 2 y 2 qcos 2x + sinh2 y q sinh2 x + sin y sinh x + cos2 y
Argument arg w
arctan(cot x tanh y) arctan(tan x tanh y) arctan(coth x tan y) arctan(tanh x tan y)
14.5 Algebraic and Elementary Transcendental Functions 701
14.5.3 Description of Curves in Complex Form
A complex function of one real variable t can be represented in parameter form: z = x(t) + iy(t) = f (t): (14.91) As t changes, the points z draw a curve z(t). In the following, we give the equations and the corresponding graphical representations of the line, circle, hyperbola, ellipse, and logarithmic spiral.
1. Straight Line
a) Line through a point (z1 ')(' is the angle with the x-axis, Fig. 14.49a): z = z1 + tei':
(14.92a)
b) Line through two points z1 z2(Fig. 14.49b): z = z1 + t(z2 ; z1 ):
(14.92b)
2. Circle
a) radius r, center at the point z0 = 0 (Fig. 14.50a): z = reit (jzj = r): b) radius r, center at the point z0 (Fig. 14.50b): z = z0 + reit (jz ; z0j = r): z1
a)
3. Ellipse
(14.93b)
y
y
0
(14.93a)
z1
ϕ
x
0
b) Figure 14.49
z2
r 0 x
a) Normal Form xa2 + yb2 = 1 (Fig. 14.51a): 2
y
y
a)
z0
x 0
b) Figure 14.50
r x
2
z = a cos t + ib sin t (14.94a) or z = c eit + d e;it (14.94b) with c = a + b d = a ; b (14.94c) 2 2 i.e., c and d are arbitrary real numbers. b) General Form (Fig. 14.51b): The center is at z1 , the axes are rotated by an angle. z = z1 + ceit + de;it: (14.95) Here c and d are arbitrary complex numbers, they determine the length of the axis of the ellipse and the angle of rotation. 2 2 4. Hyperbola, Normal Form xa2 ; yb2 = 1 (Fig. 14.52): z = a cosh t + ib sinh t (14.96a) or z = cet + c"e;t (14.96b) where c and c" are conjugate complex numbers: (14.96c) c = a +2 ib c" = a ;2 ib :
702 14. Function Theory y
b 0
y
y
b a
z1
a x
0
0 b)
a)
Figure 14.52
Figure 14.51
5. Logarithmic Spiral (Fig. 14.53):
z = a eibt where a and b are arbitrary complex numbers.
14.6 Elliptic Functions
x
x
(14.97)
y
0
x
14.6.1 Relation to Elliptic Integrals q
Integrals in the form (8.21) with integrands R x P (x) canFigure 14.53 not be integrated in closed form if P (x) is a polynomial of degree three or four, except in some special cases, but they are calculated numerically as elliptic integrals (see 8.1.4.3, p. 437). The inverse functions of elliptic integrals are the elliptic functions. They are similar to the trigonometric functions and they can be considered as their generalization. As an illustration, let us consider the special case
Zu 0
(1 ; t2 ); 21 dt = x (juj 1):
(14.98)
We can tell that a) there is a relation between the trigonometric function u = sin x and the principal value of its inverse function u = sin x , x = arcsin u for ; 2 x 2 ;1 u 1 (14.99)
b) the integral (14.98) is equal to arcsin u. The sine function can be considered as the inverse function of the integral (14.98). Analogies are valid for the elliptic integrals. The period of a mathematical pendulum, with mass m, hanging on a non-elastic weightless thread of length l (Fig. 14.54), can be calculated by a second-order non-linear di erential equation. We get this equation from the balance of the forces acting on the mass of the pendulum: 2 !3 d2 + g sin = 0 with (0) = _ (0) = 0 or d 4 d 25 = 2 g d (cos ): 0 dt2 l dt dt l dt
(14.100a)
The relation between the length l and the amplitude s from the dwell is s = l , so s_ = l _ and s = l hold. The force acting on the mass is F = mg, where g is the acceleration due to gravity, and it is decomposed into a normal component FN and a tangential component FT with respect to its path (Fig. 14.54). The normal component FN = mg cos is balanced by the thread stress. Since it is perpendicular to the direction of motion, it has no e ect to the equation of motion. The tangential component FT yields the acceleration of the motion. FT = ms = ml = ;mg sin . The tangential component always points in the direction of dwell.
14.5 Algebraic and Elementary Transcendental Functions 703
We get by separation of variables: s Z d! t ; t0 = gl q : (14.100b) 0 2(cos ! ; cos 0 ) Here, t0 denotes the time for which the pendulum is in the deepest position for the rst time, i.e., where (t0 ) = 0 holds. ! denotes the integration variable. We get the equation s s Z' t ; t0 = gl q d 2 = gl F (k ') (14.100c) 0 1 ; k2 sin after some transformations and with the substitutions sin !2 = k sin k = sin !2 . Here F (k ') is an elliptic integral of the rst kind (see (8.24a), p. 437). The angle of deection = (t) is a periodic function of period 2T with s s T = gl F k 2 = gl K (14.100d) where K represents a complete elliptic integral of the rst kind (Table 21.9). T denotes the period of the pendulum, i.e., the time between two consecutive extreme positions for which d dt = 0. If the
q
amplitude is small, i.e., sin , then T = 2 l=g holds. ϑ
y
ϑ0
FT mg
Figure 14.54
ω 1+ω 2
y
ω1
m ϑ FN
ω2
1
−ω 1 −ω 2
dnx
K
x
-1
2K cnx
Figure 14.55
3K
x 4K
snx
Figure 14.56
14.6.2 Jacobian Functions 1. Denition
It follows for 0 < k < 1 from the representation (8.23a) and (8.24a), 8.1.4.3, p. 437 for the elliptic integral of the rst kind F (k ') that dF = (1 ; k2 sin2 '); 21 > 0 (14.101) d' i.e., F (k ') is strictly monotone with respect to ', so the inverse function
Z'
d = u(') (14.102b) 1 ; k2 sin2 0 exists. It is called the amplitude function. The so-called Jacobian functions are de ned as: snu = sin ' = sin am(k u) (amplitude sine) (14.103a) cnu = cos ' = cos am(k u) (amplitude cosine) (14.103b) ' = am(k u) = '(u)
(14.102a)
of u =
q
704 14. Function Theory
p
dnu = 1 ; k2sn2u
(amplitude delta):
(14.103c)
2. Meromorphic and Double Periodic Functions
The Jacobian functions can be continued analytically in the z plane. The functions snz, cnz, and dnz are then meromorphic functions (see 14.3.5.2, p. 693), i.e., they have only poles as singularities. Besides, they are double periodic: Each of these functions f (z) has exactly two periods !1 and !2 with f (z + !1 ) = f (z) f (z + !2) = f (z): (14.104) Here, !1 and !2 are two arbitrary complex numbers, whose ratio is not real. The general formula f (z + m!1 + n!2) = f (z) (14.105) follows from (14.104), and here m and n are arbitrary integers. Meromorphic double periodic functions are called elliptic functions. The set fz0 + 1!1 + 2 !2: 0 1 2 < 1g (14.106) with an arbitrary xed z0 2 C, is called the period parallelogram of the elliptic function. If this function (Fig. 14.55) is bounded in the whole period parallelogram, then it is a constant. The Jacobian functions (14.103a) and (14.103b) are elliptic functions. The amplitude function (14.102a) is not an elliptic function.
3. Properties of the Jacobian functions
The properties of the Jacobian functions given in Table 14.3 can be got by the substitutions k02 = 1 ; k2 K 0 = F k0 2 K = F k 2 (14.107) where m and n are arbitrary integers. Table 14.3 Periods, roots and poles of Jacobian functions snz
Periods !1 !2 4K 2iK 0
cnz 4K 2(K + iK ) 0
dnz
2K 4iK 0
Roots
Poles
2mK + 2niK 0 (2m + 1)K + 2niK
9 > = 2mK + (2n + 1)iK 0 > "
0
(2m + 1)K + (2n + 1)iK 0
The shape of snz, cnz, and dnz can be found in Fig. 14.56. The following relations are valid for the Jacobian functions except at the poles: 1. sn2 z + cn2 z = 1 k2sn2z + dn2z = 1 (14.108) v) + (snv)(cnu)(dnu) (14.109a) 2. sn(u + v) = (snu)(cn1v)(dn ; k2(sn2u)(sn2v) u)(dnu)(snv)(dnv) cn(u + v) = (cnu)(cn1v);;k(sn (14.109b) 2 (sn2 u)(sn2 v ) 2 (snu)(cnu)(snv )(cnv ) dn(u + v) = (dnu)(dnv1);;kk2 (sn (14.109c) 2 u)(sn2 v )
3. (snz)0 = (cnz)(dnz) (dnz)0 = ;k2 (snz)(cnz):
(14.110a)
(cnz)0 = ;(snz)(dnz)
(14.110b) (14.110c)
14.5 Algebraic and Elementary Transcendental Functions 705
For further properties of the Jacobian functions and further elliptic functions see 14.8], 14.12].
14.6.3 Theta Function
We apply the theta function to evaluate the Jacobian functions
1 (z q) = 2q 14 2 (z q) = 2q 14
1 X
(;1)nqn(n+1) sin(2n + 1)z
(14.111a)
qn(n+1) cos(2n + 1)z
(14.111b)
n=0 1 X
n=0
3 (z q) = 1 + 2 4 (z q) = 1 + 2
1 X
n=1 1 X
n=1
qn2 cos 2nz
(14.111c)
(;1)n qn2 cos 2nz:
(14.111d)
If jqj < 1 (q complex) holds, then the series (14.111a){(14.111d) are convergent for every complex argument z. We use the brief notation in the case of a constant q k (z) := k (z q) (k = 1 2 3 4): (14.112) Then, the Jacobian functions have the representations: z z 1 2 (0) 2K (14.113a) 2K snz = 2K 04 (0) cnz = 4 (0) (14.113b) 2 (0) 4 z 1 4 2zK 2K z ! ! 3 0 2 (0) 2 (14.113d) dnz = 4 (0) 2zK (14.113c) with q = exp ; K k = 3 (0) 4 K 3 (0) 2K and K K 0 are as in (14.107).
14.6.4 Weierstrass Functions
The functions }(z) = }(z !1 !2) (14.114a) (z) = (z !1 !2) (14.114b) (z) = (z !1 !2) (14.114c) were introduced by Weierstrass, and here !1 and !2 represent two arbitrary complex numbers whose quotient is not real. We substitute !mn = 2(m!1 + n!2) (14.115a) where m and n are arbitrary real numbers, and we de ne i X h ;2 }(z !1 !2) = z;2 + 0 (z ; !mn);2 ; !mn : (14.115b) mn
The accent behind the sum sign denotes that the value pair m = n = 0 is omitted. The function }(z !1 !2) has the following properties: 1. It is an elliptic function with periods !1 and !2. 2. The series (14.115b) is convergent for every z 6= !mn. 3. The function }(z !1 !2) satis es the di erential equation
706 14. Function Theory }02 = 4}3 ; g2} ; g3 (14.116a)
with g2 = 60
X0 mn
;4 !mn
The quantities g2 and g3 are called the invariants of }(z !1 !2). 4. The function u = }(z !1 !2) is the inverse function of the integral Z1 z = p4t3 ;dtg t ; g : 2 3 u
" 0
}0(v) 2 ; }(u) ; }(v): }(u + v) = 14 }}((uu)) ; ; }(v) The Weierstrass functions i X h ;1 ;2 (z) = z;1 + 0 (z ; !mn);1 + !mn + !mn z
5.
mn Z z h
g3 = 140
X0 mn
;6 !mn :
(14.116b)
(14.117) (14.118) (14.119a)
2 ! i Y (t) ; t;1 dt = z 0 1 ; !z exp !z + 2!z2 (14.119b) 0 mn mn mn mn are not double periodic, so they are not elliptic functions. The following relations are valid: (z) = zexp
1. 2. 3. 4. 5.
0(z) = ;}(z) (z) = (ln (z)) (14.120) (;z) = ; (z) (;z) = ;(z) (14.121) (z + 2!1) = (z) + 2 (!1) (z + 2!2) = (z) + 2 (!2) (14.122) 0 }0(v) (u + v) = (u) + (v) + 12 }}((uu)) ; (14.123) ; }(v) : Every elliptic function is a rational function of the Weierstrass functions }(z) and (z).
707
15 IntegralTransformations 15.1 Notion of Integral Transformation
15.1.1 General De nition of Integral Transformations
AnIntegral transformation is a correspondence between two functions f (t) and F (p) in the form
F (p ) =
+ Z1
K (p t)f (t) dt:
(15.1a)
;1
The function f (t) is called the original function, its domain is the original space. The function F (p) is called the transform, its domain is the image space. The function K (p t) is called the kernel of the transformation. In general, t is a real variable, and p = + i! is a complex variable. We may use a shorter notation by introducing the symbol T for the integral transformation with kernel K (p t): F (p) = T ff (t)g: (15.1b) Then, we call it a T transformation.
15.1.2 Special Integral Transformations
We get di erent integral transformations for di erent kernels K (p t) and di erent original spaces. The most widely known transformations are the Laplace transformation, the Laplace{Carson transformation, and the Fourier transformation. We give an overview of the integral transformations of functions of one variable in Table 15.1. More recently, some additional transformations have been introduced for use in pattern recognition and in characterizing signals, such as the Wavelet transformation, the Gabor transformation and the Walsh transformation (see 15.6, p. 740 .).
15.1.3 Inverse Transformations
The inverse transformation of a transform into the original function has special importance in applications. With the symbol T ;1 the inverse integral transformation of (15.1a) is f (t) = T ;1 fF (p)g: (15.2a) The operator T ;1 is called the inverse operator of T , so T ;1 fT ff (t)gg = f (t): (15.2b) The determination of the inverse transformation means the solution of the integral equation (15.1a), where the function F (p) is given and function f (t) is to be determined. If there is a solution, then it can be written in the form f (t) = T ;1 fF (p)g: (15.2c) The explicit determination of inverse operators for di erent integral transformations, i.e., for di erent kernels K (p t), belongs to the fundamental problems of the theory of integral transformations. The user can solve practical problems by using the given correspondences between transforms and original functions in the corresponding tables (Table 21.13, p. 1067, Table 21.14, p. 1072, and Table 21.15, p. 1086).
15.1.4 Linearity of Integral Transformations
If f1(t) and f2(t) are transformable functions, then T fk1f1 (t) + k2f2 (t)g = k1T ff1(t)g + k2T ff2(t)g (15.3) where k1 and k2 are arbitrary numbers. That is, an integral transformation represents a linear operation on the set T of the T -transformable functions.
708 15. Integral Transformations Table 15.1 Overview of integral transformations of functions of one variable
Transformation Kernel K(p,t)
Symbol
Laplace transformation
Lff (t)g = e;ptf (t)dt
Two-sided Laplace transformation
0 for t < 0 e;pt for t > 0 e;pt
Remark
Z1 0
LIIff (t)g =
+ Z1
;pt
e f (t)dt
;1
Finite Laplace transformation
8 0 for t < 0 Za < ;pt 0 < t < a Laff (t)g = e;ptf (t)dt : e0 for for t > a 0
Laplace{Carson transformation
0 for t < 0 pe;pt for t > 0
Fourier transformation One-sided Fourier transformation Finite Fourier transformation Fourier cosine transformation Fourier sine transformation Mellin transformation Hankel transformation of order
Stieltjes transformation
e;i!t
(
0 for t < 0 e;i!t for t > 0
Z1
Cff (t)g = pe;ptf (t)dt 0
Fff (t)g =
+ Z1
p = + i!
LIIff (t)1(t)g = Lff (t)g 0where for t < 0 1(t) = 1 for t>0
The Carson transformation can also be a twosided and nite transformation.
e;i!t f (t)dt
p = + i!
=0
FIff (t)g = e;i!t f (t)dt
p = + i!
=0
;1 1
Z 0
8 0 for t < 0 Za < ;i!t 0 < t < a Faff (t)g = e;i!t f (t)dt p = + i! : e0 for for t > a 0 ( Z1 0 for t < 0 Re ei!t] for t > 0 Fcff (t)g = f (t) cos !t dt p = + i! 0
(
0 for t < 0 F ff (t)g = Z1f (t) sin !t dt p = + i! i !t Im e ] for t > 0 s
0 for t < 0 t p;1 for t > 0 0 for t < 0 tJ (t) for t > 0 8 0 for t < 0 < : p +1 t for t > 0
0
=0 =0 =0
Z1
Mff (t)g = t p;1f (t)dt 0
p = + i! ! = 0
Z (t) is the -th orH ff (t)g = tJ (t)f (t)dt Jder Bessel function of the 0 1
Z1 Sff (t)g = pf +(t)t dt 0
rst kind.
15.1 Notion of Integral Transformation 709
15.1.5 IntegralTransformationsforFunctionsofSeveralVariables Integral transformations for functions of several variables are also called multiple integral transformations (see 15.13]). The best-known ones are the double Laplace transformation, i.e., the Laplace transformation for functions of two variables, the double Laplace{Carson transformation and the double Fourier transformation. The de nition of the double Laplace transformation is
F (p q) = L2ff (x y)g
Z1 Z1
x=0 y=0
e;px;qy f (x y) dx dy:
(15.4)
The symbol L denotes the Laplace transformation for functions of one variable (see Table 15.1).
15.1.6 Applications of Integral Transformations
1. Fields of Applications
Besides the great theoretical importance that integral transformations have in such basic elds of mathematics as the theory of integral equations and the theory of linear operators, they have a large eld of application in the solution of practical problems in physics and engineering. Methods with applications of integral transformations are often called operator methods. They are suitable to solve ordinary and partial di erential equations, integral equations and di erence equations.
2. Scheme of the Operator Method
The general scheme to the use of an operator method with an integral transformation is represented in Fig. 15.1. We get the solution of a problem not directly from the original de ning equation we rst apply an integral transformation. The inverse transformation of the solution of the transformed equation gives the solution of the original problem. Problem
Equation of the problem
Solution of the equation
Transformation Solution by using the transformation
Transformed equation
Result
Inverse transformation Solution of the transformed equation
Figure 15.1 The application of the operator method to solve ordinary di erential equations consists of the following three steps: 1. Transition from a di erential equation of an unknown function to an equation of its transform. 2. Solution of the transformed equation in the image space. The transformed equation is usually no longer a di erential equation, but an algebraic equation. 3. Inverse transformation of the transform with help of T ;1 into the original space, i.e., determination of the solution of the original problem. The major diculty of the operator method is usually not the solution of the transformed equation, but the transform of the function and the inverse transformation.
710 15. Integral Transformations
15.2 Laplace Transformation
15.2.1 Properties of the Laplace Transformation
15.2.1.1 Laplace Transformation, Original and Image Space 1. Denition of the Laplace Transformation The Laplace transformation
Z1
Lff (t)g = e;ptf (t) dt = F (p)
(15.5)
0
assigns a function F (p) of a complex variable p to a function f (t) of a real variable t, if the given improper integral exists. f (t) is called the original function, F (p) is called the transform of f (t). The improper integral exists if the original function f (t) is piecewise smooth in its domain t 0, in the original space, and for t ! 1, suppose jf (t)j Ke t with certain constants K > 0, > 0. The domain of the transform F (p) is called the image space. In the literature one can nd the Laplace transformation also introduced in the Wagner or Laplace{ Carson form
Z1
LW ff (t)g = p e;ptf (t) dt = p F (p):
(15.6)
0
2. Convergence
The Laplace integral Lff (t)g converges in the half-plane Re p > (Fig. 15.2). The transform F (p) is an analytic function with the properties: 1. Relim F (p) = 0: (15.7a) p!1 This property is a necessary condition for F (p) to be a transform. p F (p) = A 2. lim p!0
(15.7b)
(p!1)
if the original function f (t) has a nite limit tlim !1 f (t) = A. (t!0)
Im p
0
f(t)=sin t
α
f(t)=sin(at)
1
Re p
1 2p a
2p 0 a)
Figure 15.2
t
0
t
b)
Figure 15.3
3. Inverse Laplace Transformation
We can retrieve the original function from the transform with the formula cZ+i1 t > 0, L;1fF (p)g = 21i e ptF (p) dp = f0 (t) for for t < 0. c;i1
(15.8)
15.2 Laplace Transformation 711
The path of integration of this complex integral is a line Re p = c parallel to the imaginary axis, where Re p = c > . If the function f (t) has a jump at t = 0, i.e., t lim f (t) 6= 0, then the integral has the ! +0 1 mean value 2 f (+0) there.
15.2.1.2 Rules for the Evaluation of the Laplace Transformation
The rules for evaluation are the mappings of operations in the original domain into operations in the transform space. In the following, we denote the original functions by lowercase letters, the transforms are denoted by the corresponding capital letters.
1. Addition or Linearity Law
The Laplace transform of a linear combination of functions is the same linear combination of the Laplace transforms, if they exist. With constants 1 : : : n we get: Lf1f1(t) + 2 f2(t) + + nfn(t)g = 1F1 (p) + 2F2 (p) + + nFn(p): (15.9)
2. Similarity Laws
The Laplace transform of f (at) (a > 0 a real) is the Laplace transform of the original function divided by a and with the argument p=a: Lff (at)g = a1 F ap (a > 0 real): (15.10a) Analogously for the inverse transformation F (ap) = a1 L f at : (15.10b) Fig. 15.3 shows the application of the similarity laws for a sine function. Determination of the Laplace transform of f (t) = sin(!t). For the correspondence of the sine function we have Lfsin(t)g = F (p) = 1=(p2 + 1). Application of the similarity law gives Lfsin(!t)g = 1 1 1 ! ! F (p=!) = ! (p=!)2 + 1 = p2 + !2 .
3. Translation Law
1. Shifting to the Right The Laplace transform of an original function shifted to the right by a
(a > 0) is equal to the Laplace transform of the non-shifted original function multiplied by the factor e;ap: Lff (t ; a)g = e;ap F (p): (15.11a) 2. Shifting to the Left The Laplace transform of an original function shifted to the left by a is to eap multiplied by the di erence of the transform of the non-shifted function and the integral Requal a f (t) e;pt dt: 0
2
Za
3
Lff (t + a)g = e ap 4F (p) ; e;pt f (t) dt 5 : 0
(15.11b)
Figs. 15.4 and 15.5 show the cosine function shifted to the right and a line shifted to the left.
4. Frequency Shift Theorem
The Laplace transform of an original function multiplied by e;bt is equal to the Laplace transform with the argument p + b ( b is an arbitrary complex number): Lfe;btf (t)g = F (p + b): (15.12)
712 15. Integral Transformations 1
f(t)
1
f(t)
f(t)
f(t)
2π 0
t
0 3
3+2π t
0
Figure 15.4
5. Di erentiation in00 the Original Space 0 (n)
t
-3 0
t
Figure 15.5
If the derivatives f (t) f (t) : : : f (t) exist for t > 0 and the highest derivative of f (t) has a transform, then the lower derivatives of f (t) and also f (t) have a transform, and: 9 Lff 0(t)g = p F (p) ; f (+0) > > > Lff 00(t)g = p2 F (p) ; f (+0) p ; f 0(+0) > > ........................................................ > = (15.13) Lff (n)(t)g = pn F (p) ; f (+0) pn;1 ; f 0(+0) pn;2 ; > > > ; f (n;2) (+0) p ; f (n;1)(+0) with > > " f () (+0) = t lim f () (t): ! +0 Equation (15.13) gives the following representation of the Laplace integral, which can be used for approximating the Laplace integral: f 0(+0) f 00(+0) 1 (n) Lff (t)g = f (+0) (15.14) p + p2 + p3 + + pn Lff (t)g:
6. Di erentiation in the Image Space Lftnf (t)g = (;1)n F (n)(p):
(15.15)
The n-th derivative of the transform is equal to the Laplace transform of the (;t)n -th multiple of the original function f (t): Lf(;1)n tn f (t)g = F (n)(p)
(n = 1 2 : : :) :
7. Integration in the Original Space
(15.16)
The transform of an integral of the original function is equal to 1=pn (n > 0) multiplied by the transform of the original function: 8 9 8 9 Zn;0 1 > F (z) F2(p ; z) dz > > < 2i x1;i1 1 Lff1(t) f2 (t)g = > x2Z+i1 1 > > F1 (p ; z) F2 (p) dz: : 2i x2 ;i1
(15.24)
The integration is performed along a line parallel to the imaginary axis. In the rst integral, x1 and p must be chosen so that z is in the half plane of convergence of Lff1g and p ; z is in the half plane of convergence of Lff2g. The corresponding requirements must be valid for the second integral.
15.2.1.3 Transforms of Special Functions 1. Step Function
The unit jump at t = t0 is called a step function (Fig. 15.7) (see also 14.4.3.2, 3., p. 697) it is also called the Heaviside unit step function: for t > t 0 u(t ; t0 ) = 10 for (15.25) t < t0 (t0 > 0): A: f (t) = u(t ; t0) sin !t F (p) = e;t0 p ! cos !pt20 ++!p2sin ! t0 (Fig. 15.8). (Fig. 15.9). B: f (t) = u(t ; t0) sin ! (t ; t0 ) F (p) = e;t0 p p2 +! !2 f(t) 1
1
u(t-t0)
f(t)
u(t-t0) sin ωt
0 t0 0 t0
t
1
0
f(t) u(t-t0) sin ω(t-t0)
t0
t
t
Figure 15.7
Figure 15.8
Figure 15.9
2. Rectangular Impulse
A rectangular impulse of height 1 and width T (Fig. 15.10) is composed by the superposition of two step functions in the form 8 0 for t < t < 0 uT (t ; t0 ) = u(t ; t0) ; u(t ; t0 ; T ) = : 1 for t0 < t < t0 + T (15.26) 0 for t > t0 + T ;t0 p ;Tp LfuT (t ; t0 )g = e (1 p; e ) : (15.27)
3. Impulse Function (Dirac Function)
(See also 12.9.5.4, p. 641.) The impulse function (t ; t0 ) can obviously be interpreted as a limit of the rectangular impulse of width T and height 1=T at the point t = t0 (Fig. 15.11): 1 u(t ; t ) ; u(t ; t ; T ) ]: (15.28) (t ; t0) = Tlim 0 0 !0 T
15.2 Laplace Transformation 715 f(t)
f(t) 1
0
uT(t-t0)
t0
1 T t
t0+T
0
Figure 15.10
t0
t0+T
t
Figure 15.11
For a continuous function h(t),
Zb a
h(t) (t ; t0 ) dt = h0 (t0 )
if t0 is inside (a b), if t0 is outside (a b).
(15.29)
Relations such as
(t ; t0) = du(tdt; t0)
Lf (t ; t0 )g = e;t0 p
(t0 0)
(15.30)
are investigated generally in distribution theory (see 12.9.5.3, p. 640).
4. Piecewise Di erentiable Functions
The transform of a piecewise di erentiable function can be determined easily with the help of the function: If f (t) is piecewise di erentiable and at the points t ( = 1 2 : : : n) it has jumps a , then its rst derivative can be represented in the form
df (t) = f 0 (t) + a (t ; t ) + a (t ; t ) + + a (t ; t ) (15.31) 1 1 2 2 n n s dt where fs0 (t) is the usual derivative of f (t), where it is di erentiable. If jumps occur rst in the derivative, then similarformulas are valid. In this way, we can easily determine the transform of functions which correspond to curves composed of parabolic arcs of arbitrarily high degree, e.g., curves found empirically. In formal application of (15.13), we should replace the values f (+0) f 0(+0) : : : by zero in the case of a jump. A: + b for 0 < t < t0 (Fig. 15.12) f 0(t) = a u (t) + b (t) ; (at + b) (t ; t ) Lff 0(t)g = f (t) = at t0 0 0 0 otherwise " ! a (1 ; e;t0 p) + b ; (at + b) e;t0 p Lff (t)g = 1 a + b ; e;t0 p a + at + b . 0 0 p p p p
B: 8 2t0 0 for t > 2t0 ;t0 p 2 00 00 f (t) = (t); (t;t0 ); (t;t0 )+ (t;2t0 ) Lff (t)g = 1;2e;t0 p +e;2t0 p Lff (t)g = ( 1 ; pe2 ) . 8 E t=t 0 > < C: f (t) = > E;E (t ; T )=t0 :0
for 0 < t < t0 for t0 < t < T ; t0 (Fig. 15.15) for T ; t0 < t < T otherwise
716 15. Integral Transformations f(t)
f(t)
f’(t) 1 t0
b
2t0
0 t0
0
0
t
t0
2t0
t
t
Figure 15.12 Figure 15.13 Figure 15.14 for 0 < t < t0 for t0 < t < T ; t0 (t > T ) (Fig. 15.16) for T ; t0 < t < T otherwise h i f 00(t) = Et (t); Et (t;t0); Et (t;T +t0)+ Et (t;T ) Lff 00(t)g = Et 1 ; e;t0 p ; e;(T ;t0 )p + e;Tp 0 0 0 0 0 ;t0 p ;(T ;t0 )p E ( 1 ; e )( 1 ; e ) Lff (t)g = t . p2 0
8 E=t0 > < f 0(t) = > 0;E=t0 :0
f(t)
E t0
E
f’(t) t0
T-t0 T
0 0
t0
T-t0 T
t
t
Figure 15.15
Figure 15.16
D: 2 0 < t < 1 (Fig. 15.17) f 0(t) = 1 ; 2t for 0 < t < 1 (Fig. 15.18) f (t) = t0 ; t for 0 otherwise otherwise 00 f (t) = ;2u1 (t) + (t) + (t ; 1) ;p ;p Lff 00(t)g = ; p2 (1 ; e;p) + 1 + e;p Lff (t)g = 1 +p2e ; 2 (1 ;p3 e ) . f(t)
f’(t) 1
1/4
1/2
1
0 0
1/2
1
Figure 15.17
5. Periodic Functions
t
t
Figure 15.18
The transform of a periodic function f (t) with period T , which is a periodic continuation of a function f (t), can be obtained from the Laplace transform of f (t) multiplied by the periodization factor (1 ; e;Tp);1: (15.32)
15.2 Laplace Transformation 717
A: The periodic continuation of f (t) from example B (see above) with period T = 2t0 is f (t) with ;t0 p 2 ;t0 p Lff (t)g = (1 ; pe2 ) 1 ; e1;2t0 p = p21(1;+ee;t0 p) . B: The periodic continuation of f (t) from example C (see above) with period T is f (t) with ;t0 p ;(T ;t0 )p Lff (t)g = E (1 ;te p2 (1) (1; ;e;eTp) ) . 0
15.2.1.4 Dirac Function and Distributions
In describing certain technical systems by linear di erential equations, functions u(t) and (t) often occur as perturbation or input functions, although the conditions required in 15.2.1.1, 1. p. 710, are not satis ed: u(t) is discontinuous, and (t) cannot be de ned in the sense of classical analysis. Distribution theory o ers a solution by introducing so-called generalized functions (distributions), so that with the known continuous real functions (t) can also be examined, where the necessary di erentiability is also guaranteed. Distributions can be represented in di erent ways. One of the best known representations is the continuous real linear form, introduced by L. Schwartz (see 12.9.5, p. 639). We can associate Fourier coecients and Fourier series uniquely to periodic distributions, analogously to real functions (see 7.4, p. 420).
1. Approximations of the Function
Analogously to (15.28), the impulse function (t) can be approximated by a rectangular impulse of width " and height 1=" (" > 0): jtj < "=2 f (t ") = 10=" for (15.33a) for jtj "=2: Further examples of the approximation of (t) are the error curve (see 2.6.3, p. 72) and Lorentz function (see 2.11.2, p. 94): t2 f (t ") = p1 e; 2"2 (" > 0) (15.33b) " 2 f (t ") = t2"= (" > 0): (15.33c) + "2 These functions have the common properties:
1.
Z1
f (t ") dt = 1:
(15.34a)
;1
2. f (;t ") = f (t ") i.e., they are even functions. 1 for t = 0 3. "lim f ( t " ) = 0 for t = 6 0: !0
(15.34b) (15.34c)
2. Properties of the Function
Important properties of the function are:
1.
xZ+a
x;a
f (t) (x ; t) dt = f (x) (f is continuous a > 0):
(15.35)
2. (x) = 1 (x) ( > 0):
(15.36)
3. (g(x)) =
(15.37)
n X
1 (x ; x ) with g(x ) = 0 and g0(x ) 6= 0 (i = 1 2 : : : n): i i i
i=1 jg (xi )j 0
718 15. Integral Transformations Here we consider all roots of g(x) and they must be simple. 4. n-th Derivative of the Function: After n repeated partial integrations of
f (n)(x) =
xZ+a
x;a
f (n)(t) (x ; t) dt
(15.38a)
we obtain a rule for the n-th derivative of the function: (;1)nf (n) (x) =
xZ+a
x;a
f (t) (n)(x ; t) dt:
(15.38b)
15.2.2 Inverse Transformation into the Original Space To perform an inverse transformation, we have the following possibilities:
1. Using a table of correspondences, i.e., a table with the corresponding original functions and transforms (see Table 21.13, p. 1067). 2. Reducing to known correspondences by using some properties of the transformation (see 15.2.2.2, p. 718, and 15.2.2.3, p. 719). 3. Evaluating the inverse formula (see 15.2.2.4, p. 720).
15.2.2.1 Inverse Transformation with the Help of Tables
The use of a table is shown here by an example with Table 21.13, p. 1067. Further tables can be found, e.g., in 15.3]. ( ) F (p) = (p + c)(1p2 + !2) = F1(p) F2 (p), L;1fF1 (p)g = L;1 p2 +1 !2 = !1 sin !t = f1 (t), ( ) L;1fF2(p)g = L;1 p +1 c = e;ct = f2 (t). We have to apply the convolution theorem (15.23): f (t) = L;1fF1(p) F2(p)g Zt Zt = f1( ) f2(t ; ) d = e;c(t; ) sin!! d = c2 +1 !2 c sin !t ;! ! cos !t + e;ct : 0 0
15.2.2.2 Partial Fraction Decomposition 1. Principle
In many applications, we have transforms in the form F (p) = H (p)=G(p), where G(p) is a polynomial of p. If we already have the original functions for H (p) and 1=G(p), then we get the required original function F (p) by applying the convolution theorem.
2. Simple Real Roots of G(p)
If the transform 1=G(p) has only simple poles p ( = 1 2 : : : n), then we get the following partial fraction decomposition: n 1 1 =X (15.39) G(p) =1 G0(p )(p ; p ) : The corresponding original function is ( ) X n q(t) = L;1 G1(p) = G0(1p ) ep t : (15.40) =1
15.2 Laplace Transformation 719
3. The Heaviside Expansion Theorem
If the numerator H (p) is also a polynomial of p with a lower degree than G(p), then we can obtain the original function of F (p) with the help of the Heaviside formula n H (p ) X p t e : (15.41) f (t) = G 0 =1 (p )
4. Complex Roots
Even in cases when the denominator has complex roots, we can use the Heaviside expansion theorem in the same way. We can also collect the terms belonging to complex conjugate roots into one quadratic expression, whose inverse transformation can be found in tables also in the case of roots of higher multiplicity. F (p) = (p + c)(1p2 + !2) , i.e., H (p) = 1, G(p) = (p + c)(p2 + !2), G0(p) = 3p2 + 2pc + !2. The poles p1 = ;c p2 = i! p3 = ;i! are all simple. According to the Heaviside theorem we get 1 e;ct ; 1 f (t) = !2 + ei!t ; 2!(!1+ ic) e;i!t or by using partial fraction decomposition and c2 2!(! "; ic)
1 1 + c;p 1 e;ct + c sin !t ; cos !t . These the table F (p) = !2 + f ( t ) = c2 p + c p2 + !2 !2 + c2 ! expressions are identical.
15.2.2.3 Series Expansion
1 In order to obtain f (t) from F (p), we can try to expand F (p) into a series F (p) = P Fn(p), whose n=0 terms Fn(p) are transforms of known functions, i.e., Fn(p) = Lffn(t)g.
1. F (p) is an Absolutely Convergent Series
If F (p) has an absolutely convergent series 1 X F (p) = pa nn (15.42) n=0 for jpj > R, where the values n form an arbitrary increasing sequences of numbers 0 < 0 < 1 < < n < < ! 1, then a termwise inverse transformation is possible: 1
n ;1 X (15.43) f (t) = an ;t ( ) : n n=0 ; denotes the gamma function (see 8.2.5, 6., p. 461). In particular, for n = n + 1, i.e., for F (p) = 1 an+1 1 P , we get the series f (t) = P ann+1! tn, which is convergent for every real and complex t. Furn=0 pn+1 n=0 thermore, we can have an estimation in the form jf (t)j < C ecjtj (C c real constants). 0 11 !;1=2 X 1 1 1 1 F (p) = p = 1+ 2 = @ ; 2 A 2n1+1 . After a termwise transformation into p p 1 + p2 p 0 1 1n=0 2n n 1 1 X X 1)n t 2n the original space we get f (t) = @ ; 2 A (2tn) ! = ((; = J0 (t) (Bessel function of 2 n=0 n n=0 n ! ) 2 0 order).
2. F (p) is a Meromorphic Function
If F (p) is a meromorphic function, which can be represented as the quotient of two integer functions (of two functions having everywhere convergent power series expansions) which do not have common
720 15. Integral Transformations roots, and so can be rewritten as the sum of an integer function and nitely many partial fractions, then we get the equality c+iy Z n 1 Z netp F (p) dp = X b ep t ; 21i etpF (p) dp: (15.44) 2i c;iyn =1 (K ) n
Here p ( = 1 2 : : : n) are the rst-order poles of the function F (p), b are the corresponding residues (see 14.3.5.4, p. 693), y are certain values and K are certain curves, for example, half circles in the sense represented in Fig. 15.19. We get the solution f (t) in the form Z 1 X (15.45) f (t) = b ep t if 21i etpF (p) dp ! 0 =1 (K ) n
as y ! 1, what is often not easy to verify.
y
yn pn y3 p3 y p2 2 y1 p1
A
yn
jω B
x K K2 1 -y1 K -y2 Kn 3 -y3 -yn
C
E
ε
x
F -jω
D -y n
Figure 15.19 Figure 15.20 In certain cases, e.g., when the rational part of the meromorphic function F (p) is identically zero, the above result is a formal application of the Heaviside expansion theorem to meromorphic functions.
15.2.2.4 Inverse Integral
The inverse formula c+iy 1 Z netp F (p) dp f (t) = ynlim (15.46) !1 2 i c;iyn represents a complex integral of a function analytic in a certain domain. The usual methods of integration for complex functions can be used, e.g., the residue calculation or certain changes of the path of integration according to the Cauchy integral theorem. p F (p) = p2 +p !2 e; p is double valued because of pp. Therefore, we chose the following path of inteI Z Z Z Z Z Z p gration (Fig. 15.20): 21i etp p2 +p !2 e; p dp = + + + + + = _ _ _ (K ) DA BE FC AB CD EF
X
p
p
Res etpF (p) = e; !=2 cos(!t ; !=2). According to the Jordan lemma (see 14.4.3, p. 695), the _ _ integral part over AB and CD vanishes as yn ! 1. The integrand remains bounded on the circular _ arc EF (radius "), and the length of the path of integration tends to zero for " ! 0 so this term of the integral also vanishes. We have to investigate the integrals on the two horizontal segments BE and FC , where we have to consider the upper side (p = rei ) and the lower side (p = re;i ) of the negative real axis:
15.2 Laplace Transformation 721
Z0
Z1 Z ;1 Z1 p F (p)etp dp = ; e;tr r2 +r !2 e;i r dr F (p)etp dp = e;tr r2 +r !2 e i ;1 0 0 0 Finally we get: p r Z1 p r f (t) = e; !=2 cos ! t ; !2 ; 1 e;tr rrsin 2 + ! 2 dr .
p
r dr.
0
15.2.3 Solution of Dierential Equations using Laplace Transformation
We have already noticed from the rules of calculation of the Laplace transformation (see 15.2.1.2, p. 711), that using the Laplace transformation we can replace complicated operations, such as di erentiation or integration in the original space, by simple algebraic operations in the image space. Here, we have to consider some additional conditions, such as initial conditions in using the di erentiation rule. These conditions are necessary for the solution of di erential equations.
15.2.3.1 Ordinary Di erential Equations with Constant Coecients 1. Principle
The n-th order di erential equation of the form y(n)(t) + cn;1 y(n;1)(t) + + c1 y0(t) + c0 y(t) = f (t) (15.47a) (n;1) 0 0 ( n ;1) with the initial values y(+0) = y0, y (+0) = y0 : : : y (+0) = y0 can be transformed by Laplace transformation into the equation n X
k=0
ck pk Y (p) ;
n kX ;1 X ck pk;;1y0() = F (p)
k=1
=0
(cn = 1):
(15.47b)
n Here G(p) = P ck pk = 0 is the characteristic equation of the di erential equation (see 4.5.2.1, p. 279). k=0
2. First-Order Di erential Equations
The original and the transformed equations are: y0(t) + c0y(t) = f (t) y(+0) = y0 (15.48a) where c0 = const. For Y (p) we get Y (p) = F (pp+) +c y0 : 0 Special case: For f (t) = e t ( const) we get Y (p) = (p ; )(p + c ) + p +y0c 0 0 ! t y(t) = + c e + y0 ; + c e;c0t : 0 0
(p + c0 ) Y (p) ; y0 = F (p)
(15.48b) (15.48c) (15.49a) (15.49b) (15.49c)
3. Second-Order Di erential Equations
The original and transformed equations are: y00(t) + 2ay0(t) + by(t) = f (t) y(+0) = y0 (p2 + 2ap + b) Y (p) ; 2ay0 ; (py0 + y00 ) = F (p): We then get for Y (p) (2a + p) y0 + y00 : Y (p) = F (p) + 2 p + 2ap + b
y0(+0) = y00 :
(15.50a) (15.50b) (15.50c)
722 15. Integral Transformations Distinction of Cases: a) b < a2 : G(p) = (p ; 1)(p ; 2) (1 2 real 1 6= 2) q(t) = ;1 (e 1 t ; e 2 t ): 1
b) b = a2 :
2 2 G(p) = (p ; )
(15.52a)
q(t) = t e t :
c) b > a2 : G(p) has complex roots, p q(t) = p 1 2 e;at sin b ; a2t: b;a
(15.51a) (15.51b) (15.52b) (15.53a) (15.53b)
We obtain the solution y(t) as the convolution of the original function of the numerator of Y (p) and q(t). The application of the convolution can be avoided if we can nd a direct transformation of the right-hand side. The transformed equation for the di erential equation y00(t) + 2y0(t) + 10y(t) = 37 cos 3t + 9e;t p 9 with y0 = 1 and y00 = 0 is Y (p) = p2 +p 2+p 2+ 10 + (p2 + 9)(p37 2 + 2p + 10) + (p + 1)(p2 + 2p + 10) . We get the representation Y (p) = p2 + ;2pp+ 10 ; (p2 + 219p + 10) + (p2 p+ 9) + (p218+ 9) + (p +1 1) by partial fraction decomposition of the second and third terms of the right-hand side but not separating the second-order terms into linear ones. We get the solution after termwise transformation (see Table 21.13, p. 1067) y(t) = (; cos 3t ; 6 sin 3t)e;t + cos 3t + 6 sin 3t + e;t.
4. n-th Order Di erential Equations
The characteristic equation G(p) = 0 of this di erential equation has only simple roots 1 2 : : : n, and none of them is equal to zero. We can distinguish two cases for the perturbation function f (t). 1. If the perturbation function f (t) is the jump function u(t) which often occurs in practical problems, then the solution is: n 1 +X 1 t>0 t (15.54b) y ( t ) = u(t) = 10 for (15.54a) for t < 0 G(0) =1 G0( ) e : 2. For a general perturbation function f (t), we get the solution y~(t) from (15.54b) in the form of the Duhamel formula which uses the convolution (see 15.2.1.2, 11., p. 713): Zt y~(t) = dtd y(t ; )f ( ) d = dtd y # f ]: (15.55) 0
15.2.3.2 Ordinary Linear Di erential Equations with Coecients Depending on the Variable
Di erential equations whose coecients are polynomials in t can also be solved by Laplace transformation. Applying (15.16), in the image space we get a di erential equation, whose order can be lower than the original one. If the coecients are rst-order polynomials, then the di erential equation in the image space is a rstorder di erential equation and maybe it can be solved more easily. 2 Bessel di erential equation of 0 order: t ddtf2 + ddtf + tf = 0 (see (9.51a, p. 509) for n = 0). The transformation into the image space results in p ; dpd p2F (p) ; pf (0) ; f 0 (0) ] + pF (p) ; f (0) ; dFdp(p) = 0 or dF dp = ; p2 + 1 F (p) .
15.2 Laplace Transformation 723
Z q Separation of the variables and integration yields log F (p) = ; p2 dp = ; log p2 + 1 + log C , p +1 C p F (p) = p2 + 1 (C is the integration constant) F (t) = CJ0 (t) (see in 15.2.2.3,1., p. 719 with the Bessel function of 0 order).
15.2.3.3 Partial Di erential Equations 1. General Introduction
The solution of a partial di erential equation is a function of at least two variables: u = u(x t). Since the Laplace transformation represents an integration with respect to only one variable, the other variable should be considered as a constant in the transformation:
Z1
Lfu(x t)g = e;ptu(x t) dt = U (x p): 0
(15.56)
x also remains xed in the transformation of derivatives: ( ) @u ( x t L @t ) = p Lfu(x t)g ; u(x +0) ( 2 ) (15.57) L @ u@t(x2 t) = p2Lfu(x t)g ; u(x +0)p ; ut(x +0): For di erentiation with respect to x we suppose that they are interchangeable with the Laplace integral: ( ) @ Lfu(x t)g = @ U (x p): L @u(@tx t) = @x (15.58) @x This way, we get an ordinary di erential equation in the image space. Furthermore, we have to transform the boundary and initial conditions into the image space.
2. Solution of the One-Dimensional Heat Conduction Equation for a Homogeneous Medium
1. Formulation of the Problem Suppose the one-dimensional heat conduction equation with van-
ishing perturbation and for a homogeneous medium is given in the form uxx ; a;2ut = uxx ; uy = 0 (15.59a) in the original space 0 < t < 1, 0 < x < l and with the initial and boundary conditions u(x +0) = u0(x) u(+0 t) = a0 (t) u(l ; 0 t) = a1 (t): (15.59b) The time coordinate is replaced by y = at. (15.59a) is also a parabolic type equation, just as the three-dimensional heat conduction equation (see 9.2.3.3, p. 537). 2. Laplace Transformation The transformed equation is d2U = p U ; u (x) (15.60a) 0 dx2 and the boundary conditions are U (+0 p) = A0(p) U (l ; 0 p) = A1(p): (15.60b) The solution of the transformed equation is p p U (x p) = c1ex p + c2e;x p: (15.60c) It is a good idea to produce two particular solutions U1 and U2 with the properties U1 (0 p) = 1 U1 (l p) = 0 (15.61a) U2 (0 p) = 0 U2 (l p) = 1 i.e., (15.61b)
724 15. Integral Transformations x p e;x p (l;x) p e;(l;x) p U2 (x p) = ee l pp ; (15.61d) (15.61c) U1 (x p) = e e l pp ; ; e;l pp : ; e;l pp The required solution of the transformed equation has the form U (x p) = A0(p) U1(x p) + A1 (p) U2(x p): (15.62) 3. Inverse Transformation The inverse transformation is especially easy in the case of l ! 1: ! p x Z t a (t ; ) x2 (15.63b) U (x p) = a0(p)e;x p (15.63a) u(x t) = 2p 0 3=2 exp ; 4 d : 0 p
p
p
p
15.3 Fourier Transformation
15.3.1 Properties of the Fourier Transformation
15.3.1.1 Fourier Integral
1. Fourier Integral in Complex Representation
The basis of the Fourier transformation is the Fourier integral, also called the integral formula of Fourier: If a non-periodic function f (t) satis es the Dirichlet conditions (see 7.4.1.2, 3., p. 422) in an arbitrary nite interval, and furthermore the integral + Z1
;1
jf (t)j dt (15.64a)
+ Z 1 +Z 1 is convergent, then f (t) = 21 ei!(t; ) f ( ) d! d
(15.64b)
;1 ;1
at every point where the function f (t) is continuous, and f (t + 0) + f (t ; 0) = 1 Z1d! +Z 1f ( ) cos ! (t ; ) d 2 0 ;1 at the points of discontinuity.
(15.64c)
2. Equivalent Representations
Other equivalent forms for the Fourier integral (15.64b) are: + Z 1 +Z 1 f ( ) cos ! (t ; ) ] d! d : 1. f (t) = 21 ;1 ;1
Z1
2. f (t) = a(!) cos !t + b(!) sin !t ] d! 0
+ Z1 a(!) = 1 f (t) cos !t dt
(15.65c)
;1
Z1
3. f (t) = A(!) cos !t + (!) ] d!: 0
Z1
4. f (t) = A(!) sin !t + '(!) ] d!: 0
(15.65a)
with the coecients + Z1 b(!) = 1 f (t) sin !t dt:
(15.65b) (15.65d)
;1
(15.66) (15.67)
15.3 Fourier Transformation 725
The following relations are valid here:
q A(!) = a2 (!) + b2 (!) cos (!) = Aa((!!)) cos '(!) = Ab((!!))
(15.68a) (15.68c) (15.68e)
'(!) = (!) + 2 sin (!) = Ab((!!))
sin '(!) = Aa((!!)) :
(15.68b) (15.68d) (15.68f)
15.3.1.2 Fourier Transformation and Inverse Transformation 1. Denition of the Fourier Transformation
The Fourier transformation is an integral transformation of the form (15.1a), which comes from the Fourier integral (15.64b) if we substitute
F (! ) =
+ Z1
e;i! f ( ) d :
(15.69)
;1
We get the following relation between the real original function f (t) and the usually complex transform F (!): + Z1 f (t) = 21 ei! t F (!) d!:
(15.70)
;1
In the brief notation we use F :
F (!) = Ff f (t) g =
+ Z1
e;i! t f (t) dt:
(15.71)
;1
The original function f (t) is Fourier transformable if the integral (15.69), i.e., an improper integral with the parameter !, exists. If the Fourier integral does not exist as an ordinary improper integral, we consider it as the Cauchy principal value (see 8.2.3.3, 1., p. 457). The transform F (!) is also called the Fourier transform it is bounded, continuous, and it tends to zero for j!j ! 1: lim F (!) = 0: (15.72) j!j!1
The existence and boundedness of F (!) follow directly from the obvious inequality
jF (!)j
+ Z1
;1
je;i! t f (t)j dt
+ Z1
;1
jf (t)j dt:
(15.73)
The existence of the Fourier transform is a sucient condition for the continuity of F (!) and for the properties F (!) ! 0 for j!j ! 1. This statement is often used in the following form: If the function f (t) in (;1 1) is absolutely integrable, then its Fourier transform is a continuous function of !, and (15.72) holds. The following functions are not Fourier transformable: Constant functions, arbitrary periodic functions (e.g., sin ! t cos ! t), power functions, polynomials, exponential functions (e.g., e t , hyperbolic functions).
2. Fourier Cosine and Fourier Sine Transformation
In the Fourier transformation (15.71), the integrand can be decomposed into a sine and a cosine part. So, we get the sine and the cosine Fourier transformation.
726 15. Integral Transformations 1. Fourier Sine Transformation Z1 Fs(!) = Fsf f (t) g = f (t) sin (!t) dt:
(15.74a)
2. Fourier Cosine Transformation Z1 Fc(!) = Fcf f (t) g = f (t) cos (! t) dt:
(15.74b)
0
0
3. Conversion Formulas Between the Fourier sine (15.74a) and the Fourier cosine transformation
(15.74b) on one hand, and the Fourier transformation (15.71) on the other hand, the following relations are valid: (15.75a) F (!) = Ff f (t) g = Fcf f (t) + f (;t) g ; iFsf f (t) ; f (;t) g 1 i Fc(!) = 2 Ff f (t) g: (15.75c) Fs(!) = 2 Ff f (jtj)sign t g (15.75b) For an even or for an odd function f (t) we have the representation f (t) even: Ff f (t) g = 2Fcf f (t) g (15.75d) f (t) odd: Ff f (t) g = ;2iFsf f (t) g:
3. Exponential Fourier Transformation
Di erently from the de nition of F (!) in (15.71), the transform + Z1 Fe(!) = Feff (t)g = 12 ei!t f (t) dt ;1 is called the exponential Fourier transformation, and we have F (!) = 2Fe(;!):
4. Tables of the Fourier Transformation
(15.76) (15.77)
Based on formulas (15.75a,b,c) we either do not need special tables for the corresponding Fourier sine and Fourier cosine transformations, or we have tables for Fourier sine and Fourier cosine transformations and we may calculate F (!) with the help of (15.75a,b,c). In Table 21.14.1 (see 21.14.1, p. 1072) and Table 21.14.2 (see 21.14.2, p. 1078) the Fourier sine transforms Fs(!), the Fourier cosine transforms Fc(!), and for some functions the Fourier transform F (!) in Table 21.14.3 (see 21.14.3, p. 1083) and the exponential transform Fe(!) in Table 21.14.4 (see 21.14.4, p. 1085) are given. The function of the unipolar rectangular impulse f (t) = 1 for jtj < t0 f (t) = 0 for jtj > t0 (A.1) (Fig. 15.21) satis es the assumptions of the de nition Z +t0 of the Fourier 2integral (15.64a). According to (15.65c,d) we get for the coecients a(!) = 1 cos ! t dt = ! sin ! t0 and b(!) = ;t0 Z Z + t 1 1 0 2 sin ! t0 cos ! t d! (A.3). ;t0 sin ! t dt = 0 (A.2) and so from (15.65b), f (t) = 0 !
5. Spectral Interpretation of the Fourier Transformation
Analogously to the Fourier series of a periodic function, the Fourier integral for a non-periodic function has a simple physical interpretation. A function f (t), for which the Fourier integral exists, can be represented according to (15.66) and (15.67) as a sum of sinusoidal vibrations with continuously changing frequency ! in the form A(!) d! sin ! t + '(!) ] (15.78a) A(!) d! cos ! t + (!) ]: (15.78b) The expression A(!) d! gives the amplitude of the wave components and '(!) and (!) are the phases.
15.3 Fourier Transformation 727 F(ω) 2t0 f(t)
2 ω
1 -t0
0
t0
t
-3π
-2π
-π
0
π
2π
3π ωt0
Figure 15.21 Figure 15.22 We have the same interpretation for the complex formulation: The function f (t) is a sum (or integral) of summands depending on ! of the form 1 F (!) d! ei! t (15.79) 2 where the quantity 1 F (!) also determines the amplitude and the phase of all the parts. 2 This spectral interpretation of the Fourier integral and the Fourier transformation has a big advantage in applications in physics and engineering. The transform F (!) = jF (!)jei(!) or jF (!)j ei'(!) (15.80a) is called the spectrum or frequency spectrum of the function f (t), the quantity jF (!)j = A(!) (15.80b) is the amplitude spectrum and '(!) and (!) are the phase spectra of the function f (t). The relation between the spectrum F (!) and the coecients (15.65c,d) is F (!) = a(!) ; ib(!) ] (15.81) from which we get the following statements: 1. If f (t) is a real function, then the amplitude spectrum F (!) is an even function of !, and the phase spectrum is an odd function of !. 2. If f (t) is a real and even function, then its spectrum F (!) is real, and if f (t) is real and odd, then the spectrum F (!) is imaginary. If we substitute the result (A.2) for the unipolar rectangular impulse function on p. 726 into (15.81), then we get for the transform F (!) and for the amplitude spectrum jF (!)j (Fig. 15.22) sin ! t0 sin ! t 0 F (!) = Ff f (t) g = a(!) = 2 ! (A.3), jF (!)j = 2 ! (A.4). The points of contact of the amplitude spectrum jF (!)j with the hyperbola !2 are at !t0 = (2n + 1) 2 (n = 0 1 2 : : :) .
15.3.1.3 Rules of Calculation with the Fourier Transformation
As we have already pointed out for the Laplace transformation, the rules of calculation with integral transformations mean the mappings of certain operations in the original space into operations in the image space. If we suppose that both functions f (t) and g(t) are absolutely integrable in the interval (;1 1) and their Fourier transforms are F (!) = Ff f (t) g and G(!) = Ff g(t) g (15.82) then the following rules are valid.
1. Addition or Linearity Laws
If and are two coecients from (;1 1), then: Ff f (t) + g(t) g = F (!) + G(!):
(15.83)
728 15. Integral Transformations
2. Similarity Law
For real 6= 0, Ff f (t=) g = jj F (!):
3. Shifting Theorem
For real 6= 0 and real , Ff f (t + ) g = (1=) ei != F (!=) or Ff f (t ; t0 ) g = e;i!t0 F (!): If we replace t0 by ;t0 in (15.85b), then we get Ff f (t + t0) g = ei!t0 F (!):
4. Frequency-Shift Theorem
For real > 0 and 2 (;1 1), Ff ei tf (t) g = (1=)F ((! ; )=) or Ff ei!0t f (t) g = F (! ; !0):
5. Di erentiation in the Image Space
If the function tnf (t) is Fourier transformable, then Ff tnf (t)g = inF (n)(!) where F (n) (!) denotes the n-th derivative of F (!).
(15.84) (15.85a) (15.85b) (15.85c) (15.86a) (15.86b)
(15.87)
6. Di erentiation in the Original Space
1. First Derivative If a function f (t) is continuous and absolutely integrable in (;1 1) and it tends to zero for t ! 1, and the derivative f 0(t) exists everywhere except, maybe, at certain points, and this derivative is absolutely integrable in (;1 1), then Ff f 0(t) g = i! Ff f (t) g: (15.88a) 2. n-th Derivative If the requirements of the theorem for the rst derivative are valid for all deriva(n;1)
tives up to f , then Ff f (n)(t) g = (i!)nFf f (t) g: (15.88b) These rules of di erentiation will be used in the solution of di erential equations (see 15.3.2, p. 731).
7. Integration in the Image Space
If the function tnf (t) is absolutely integrable in (;1 1), then the Fourier transform of the function f (t) has n continuous derivatives, which can be determined for k = 1 2 : : : n as
dk F (!) = +Z 1 @ k h e;i!tf (t) i dt = (;1)k +Z 1e;i!ttk f (t) dt d!k ;1 @!k ;1 and we have k lim d F (!) = 0: !!1 d! k With the above assumptions these relations imply that n F (! ) Ff tnf (t) g = in d d! n :
(15.89a) (15.89b) (15.89c)
15.3 Fourier Transformation 729
8. Integration in the Original Space and the Parseval Formula
1. Integration Theorem If the assumption + Z1
f (t) dt = 0
(15.90a)
is ful lled, then
;1
8 Zt 9 < = 1 F : f (t) dt " = i! F (!): ;1
(15.90b)
2. Parseval Formula If the function f (t) and its square are integrable in the interval (;1 1), then
+ Z1 ;1
jf (t)j2 dt = 21
9. Convolution
+ Z1
jF (!)j2 d!:
(15.91)
f1( )f2 (t ; ) d
(15.92)
;1
The two-sided convolution
f1(t) # f2(t) =
+ Z1
;1
is considered in the interval (;1 1) and exists under the assumptions that the functions f1(t) and f2(t) are absolutely integrable in the interval (;1 1). If f1(t) and f2(t) both vanish for t < 0, then we get the one-sided convolution from (15.92)
8 Zt > < f1(t) # f2(t) = > f1( )f2 (t ; ) d :0
for t 0,
(15.93) 0 for t < 0. So, it is a special case of the two-sided convolution. While the Fourier transformation uses the twosided convolution, the Laplace transformation uses the one-sided convolution. For the Fourier transform of a two-sided convolution we have Ff f1(t) # f2(t) g = Ff f1(t) g Ff f2(t) g (15.94) if both integrals + Z1
;1
jf1(t)j2 dt
+ Z1
and
;1
jf2(t)j2 dt
(15.95)
exist, i.e., the functions and their squares are integrable in the interval (;1 1).
Z +1
Calculate the two-sided convolution (t) = f (t) # f (t) = f ( )f (t ; ) d (A.1) for the function ;1 of the unipolarZ rectangular impulse (A.1) in 15.3.1.2, 4. on p. 726. Z t+function t0 t0 Since (t) = f (t ; ) d = f ( ) d (A.2), we get for t < ;2t0 and t > 2t0 , (!) = 0 and ;t0
for ;2t0 t 0, (t) =
Z t+t0
t;t0
d = t + 2t0 : (A.3) ;t0 Z t0 Analogously, we get for 0 < t 2t0 : (t) = d = ;t + 2t0: (A.4) t;t0 Altogether, for this convolution 15.23) we get 8 t + 2t (Fig. for ;2t0 t 0 < 0 (t) = f (t) # f (t) = : ;t + 2t0 for 0 < t 2t0 (A.5) 0 for jtj > 2t0 : With the Fourier transform of the unipolar rectangular impulse function (A.1) (p. 726 and Fig. 15.21)
730 15. Integral Transformations 2 we get " (!) = Ff (t) g = Ff f (t) # f (t) = F (!) ]2 = 4 sin!!2 t0 (A.6) and for the amplitude 2 spectrum of the function f (t) we have jF (!)j = 2 sin!! t0 and jF (!)j2 = 4 sin!!2 t0 : (A.7)
ϕ(t)
ψ(t)
1 -2t0 -2t0
0
2t0
t
Figure 15.23
0 -1
2t0 t
Figure 15.24
10. Comparing the Fourier and Laplace Transformations
There is a strong relation between the Fourier and Laplace transformation, since the Fourier transformation is a special case of the Laplace transformation with p = i!. Consequently, every Fourier transformable function is also Laplace transformable, while the reverse statement is not valid for every f (t). Table 15.2 contains comparisons of several properties of both integral transformations. Table 15.2 Comparison of the properties of the Fourier and the Laplace transformation
Fourier transformation
+R1 F (!) = Ff f (t) g = e;i! tf (t) dt ;1 ! is real, it has a physical meaning, e.g., frequency. One shifting theorem. interval: (;1 +1) Solution of di erential equations, problems described by two-sided domain, e.g., the wave equation. Di erentiation law contains no initial values. Convergence of the Fourier integral depends only on f (t). It satis es the two-sided convolution law.
Laplace transformation R1 F (p) = Lf f (t) p g = e;ptf (t) dt 0 p is complex, p = r + ix.
Two shifting theorems. interval: 0 1) Solution of di erential equations, problems described by one-sided domain, e.g., the heat conduction equation. Di erentiation law contains initial values. Convergence of the Laplace integral can be improved by the factor e;pt . It satis es the one-sided convolution law.
15.3.1.4 Transforms of Special Functions
A: Which image function belongs to the original function f (t) =Z e;ajtj Re a > 0 (A.1)? Z 0 Consider+A ;i! t;ajtj ing that jtj = ;t for t < 0 and jtj = t for t > 0 with (15.71) we get: e dt = e;(i!;a)t dt+ ;A ;A Z +A ;(i!;a)t 0 ;(i!+a)t +A (i!;a)A 1 ; e;(i!+a)A e e ; 1 + e ;(i!+a)t e dt = ; i! ; a ; i! + a = i! ; a + i! + a (A.2). Since je;aAj 0 ;A 0 e;A Re a and Re a > 0, the limit exists for A ! 1, so we get F (!) = Ff e;ajtj g = a2 2+a!2 (A.3). B: Which image function belongs to the original function f (t) = e;at Re a > 0? The function is not Fourier transformable, since the limit A ! 1 does not exist.
15.3 Fourier Transformation 731
C: Determinate Fourier transform of the bipolar rectangular impulse function (Fig. 15.24) 8 1 for ;the 2t0 < t < 0 < '(t) = : ;1 for 0 < t < 2t0 (C.1) 0 for jtj > 2t0
where '(t) can be expressed by using equation (A.1) given for the unipolar rectangular impulse on p. 726, since '(t) = f (t + t0 ) ; f (t ; t0) (C.2). By the Fourier transformation according to (15.85b, 15.85c) we get (!) = Ff '(t) g = ei! t0 F (!) ; e;i! t0 F (!), (C.3) from which, using (A.1), we have: 2 &(!) = (ei! t0 ; e;i! t0 ) 2 sin!! t0 = 4i sin !! t0 (C.4). D: Transform of a damped oscillation: The damped oscillation represented in Fig. 15.25a is given 0 t 0 for ;1 < t < ;t0 > > < 1 h 1 ; e;a(t+t0 ) i for ;t0 t +t0 y(t) = > a > h i > : a1 e;a(t;t0 ) ; e;a(t;t0 ) for t0 < t < 1: Function (15.96f) is represented graphically in Fig. 15.26.
(15.96b) (15.96d) (15.96e)
(15.96f)
15.3.2.2 Partial Di erential Equations 1. General Remarks
The solution of a partial di erential equation is a function of at least two variables: u = u(x t). As the Fourier transformation is an integration with respect to only one variable, the other variable is considered a constant during the transformation. Here we keep as a constant the variable x and transform with respect to t:
Ff u(x t) g =
+ Z1
e;i! t u(x t) dt = U (x !):
(15.97)
;1
During the transformation of the derivatives we also keep the variable x: ( (n) ) F @ @tu(nx t) = (i!)nFf u(x t) g = (i!)nU (x !): (15.98) The di erentiation with respect to x supposes that it is interchangeable with the Fourier integral: ) ( (x t) = @ u(x t) ] = @ U (x !): (15.99) F @u@x @x @x So, we get an ordinary di erential equation in the image space. We also have to transform the boundary and initial conditions into the image space, of course.
2. Solution of the One-Dimensional Wave Equation for a Homogeneous Medium
1. Formulation of the Problem The one-dimensional wave equation with vanishing perturbation term and for a homogeneous medium is: uxx ; utt = 0: (15.100a) Like the three-dimensional wave equation (see 9.2.3.2,p. 536), the equation (15.100a) is a partial di erential equation of hyperbolic type. The Cauchy problem is correctly de ned by the initial conditions u(x 0) = f (x) (;1 < x < 1) ut(x 0) = g(x) (0 t < 1): (15.100b) 2. Fourier Transformation We perform the Fourier transformation with respect to x where the time coordinate is kept constant: Ff u(x t) g = U (! t): (15.101a)
15.4 Z-Transformation 733
We get:
2 (i!)2U (! t) ; d U (!2 t) = 0 with (15.101b) dt Ff u(x 0) g = U (! 0) = Ff f (x) g = F (!) (15.101c) Ff ut(x 0) g = U 0 (! 0) = Ff g(x) g = G(!): (15.101d) !2U + U 00 = 0: (15.101e) The result is an ordinary di erential equation with respect to t with the parameter ! of the transform. The general solution of this known di erential equation with constant coecients is U (! t) = C1ei! t + C2e;i! t: (15.102a) We determine the constants C1 and C2 from the initial values U (! 0) = C1 + C2 = F (!) U 0 (! 0) = i! C1 ; i! C2 = G(!) (15.102b) and we get C1 = 12 F (!) + i1! G(!) ] C2 = 12 F (!) ; i1! G(!) ]: (15.102c) The solution is therefore (15.102d) U (! t) = 12 F (!) + i1! G(!) ]ei! t + 12 F (!) ; i1! G(!) ]e;i! t :
3. Inverse Transformation We use the shifting theorem Ff f (ax + b) g = 1=a eib!=a F (!=a)
(15.103a)
for the inverse transformation of F (!), and we get F ;1f ei! t F (!) g = f (x + t) F ;1 e;i! t F (!) ] = f (x ; t): Applying the integration rule 8 Zx 9 < = F : f ( ) d " = i1! F (!) we get ;1
(15.103b) (15.103c)
xZ+t Zx Zx F ;1 i1! G(!)ei! t = F ;1f G(!)ei! t g d = g ( + t) d = g (z) dz ;1
;1
after substituting s + t = z, analogously to the previous integral we get xZ;t F ;1 ; i1! G(!)e;i! t = ; g (z) dz: ;1 Finally, the solution in the original space is Zx+t u(x t) = 12 f (x + t) + 12 f (x ; t) + g (z) dz: x;t
15.4 Z-Transformation
(15.103d)
;1
(15.103e)
(15.104)
In natural sciences and also in engineering we can distinguish between continuous and discrete processes. While continuous processes can be described by di erential equations, the discrete processes result mostly in dierence equations. The solution of di erential equations mostly uses Fourier and Laplace transformations, however, to solve di erence equations other operator methods have been
734 15. Integral Transformations developed. The best known method is the z-transformation, which is closely related to the Laplace transformation.
15.4.1 Properties of the Z-Transformation
15.4.1.1 Discrete Functions f(t)
f0 f1
f2
f3
0 T 2T 3T
Figure 15.27
t
If a function f (t) (0 t < 1) is known only at discrete values tn = nT (n = 0 1 2 : : : T > 0 is a constant) of the argument, then we write f (nT ) = fn and we form the sequence ffng. Such a sequence is produced, e.g., in electrotechnics by \scanning" a function f (t) at discrete time periods tn. Its representation results in a step function (Fig. 15.27). The sequence ffng and the function f (nT ) de ned only at discrete points of the argument, which is called a discrete function, are equivalent.
15.4.1.2 Denition of the Z-Transformation 1. Original Sequence and Transform
We assign the in nite series n 1 X F (z) = fn 1z (15.105) n=0 to the sequence ffng. If this series is convergent, then we call the sequence ffng z-transformable, and we write F (z) = Zffng: (15.106) ffng is called the original sequence, F (z) is the transform. z denotes a complex variable, and F (z) is a complex-valued function. fn = 1 (n = 0 1 2 : : :) . The corresponding in nite series is 1 n X 1 : F (z ) = (15.107) z 1 n=0 It represents a geometric series with common ratio 1=z, which is convergent if < 1 and its sum is z F (z) = z ;z 1 . It is divergent for 1z 1. Therefore, the sequence f1g is z-transformable for z1 < 1, i.e., for every exterior point of the unit circle jzj = 1 in the z plane.
2. Properties
Since the transform F (z) according to (15.105) is a power series of the complex variable 1=z, the properties of the complex power series (see 14.3.1.3, p. 690) imply the following results: a) For a z-transformable sequence ffng, there exists a real number R such that the series (15.105) is absolutely convergent for jzj > 1=R and divergent for jzj < 1=R. The series is uniformly convergent for jzj 1=R0 > 1=R. R is the radius of convergence of the power series (15.105) of 1=z. If the series is convergent for every jzj > 0, we write R = 1. For non z-transformable sequences we write R = 0. b) If ffng is z-transformable for jzj > 1=R, then the corresponding transform F (z) is an analytic function for jzj > 1=R and it is the unique transform of ffng. Conversely, if F (z) is an analytic function for jzj > 1=R and is regular also at z = 1, then there is a unique original sequence ffng for F (z). Here, F (z) is called regular at z = 1, if F (z) has a power series expansion in the form (15.105) and F (1) = f0 .
3. Limit Theorems
Analogously to the limit properties of the Laplace transformation ((15.7b), p. 710), the following limit theorems are valid for the z-transformation:
15.4 Z-Transformation 735
a) If F (z) = Zffng exists, then
f0 = zlim F (z ): (15.108) !1 Here z can tend to in nity along the real axis or along any other path. Since the series zfF (z) ; f0 g = f1 + f2 z1 + f3 z12 + (15.109) 1 1 1 z2 F (z) ; f0 ; f1 z = f2 + f3 z + f4 z2 + (15.110) ... ... ... are obviously z transforms, analogously to (15.108) we get 2 F (z ) ; f ; f 1 : : : : f1 = zlim z f F ( z ) ; f g f = lim z (15.111) 0 2 0 1 !1 z!1 z We can determine the original function ffng from its transform F (z) in this way. b) If nlim f exists, then !1 n f = z!lim (z ; 1)F (z): (15.112) nlim !1 n 1+0 We can however determine the value nlim f from (15.112) only if its existence is guaranteed, since the !1 n above statement is not reversible. fn = (;1)n (n = 0 1 2 : : :) . Then Zffng = z +z 1 and z!lim (z ; 1) z +1 1 = 0, but nlim (;1)n does !1 1+0 not exist.
15.4.1.3 Rules of Calculations
In applications of the z-transformation it is very important to know how certain operations de ned on the original sequences a ect the transforms, and conversely. For the sake of simplicity we will use the notation F (z) = Zffng for jzj > 1=R.
1. Translation
We distinguish between forward and backward translations. 1. First Shifting Theorem: Zffn;k g = z;k F (z) (k = 0 1 2 : : :) (15.113) here fn;k = 0 is de ned for n ; k < 0. " kX ;1 2. Second Shifting Theorem: Zffn+k g = zk F (z) ; f 1z (k = 1 2 : : :) : (15.114) =0
2. Summation
For jzj > max 1 R1
3. Di erences
holds
Z
(nX ) ;1 f = z ;1 1 F (z) : =0
For the dierences %fn = fn+1 ; fn %m fn = %(%m;1 fn) (m = 1 2 : : : %0fn = fn) the following holds: Zf%fng = (z ; 1)F (z) ; zf0 Zf%2fng = (z ; 1)2F (z) ; z(z ; 1)f0 ; z%f0 ... ... =
Zf%k fng
=
k ;1 (z ; 1)k F (z) ; z P (z ; 1)k;;1% f0 : =0
(15.115)
(15.116)
(15.117)
736 15. Integral Transformations
4. Damping
For an arbitrary complex number 6= 0 and jzj > jRj : Zfnfng = F z :
(15.118)
5. Convolution
The convolution of two sequences ffng and fgng is the operation
f n # gn =
n X
=0
f gn; :
(15.119)
If the z-transformed functions Zffng = F (z) for jj > 1=R1 and Zfgng = G(z) for jzj > 1=R2 exist, then Zffn # gng = F (z)G(z) (15.120) 1 1 for jzj > max R1 R2 . Relation (15.120) is called the convolution theorem of the z-transformation. It corresponds to the rules of multiplying two power series.
6. Di erentiation of the Transform Zfnfng = ;z dFdz(z) :
(15.121)
We can determine higher-order derivatives of F (z) by the repeated application of (15.121).
7. Integration of the Transform
Under the assumption f0 = 0, ( ) Z1 Z fnn = F ( ) d : z
(15.122)
15.4.1.4 Relation to the Laplace Transformation
If we describe a discrete function f (t) (see 15.4.1.1, p. 734) as a step function, then f (t) = f (nT ) = fn for nT t < (n + 1) T (n = 0 1 2 : : : T > 0 T const): (15.123) We can use the Laplace transformation (see 15.2.1.1, 1., p. 710) for this piecewise constant function, and for T = 1 we get: Z+1 1 n 1 1 ;np ;(n+1)p ;p X X X Lff (t)g = F (p) = fne;pt dt = fn e ; pe = 1;e fne;np: (15.124) p n=0 n n=0 n=0 The in nite series in (15.124) is called the discrete Laplace transformation and it is denoted by D:
Dff (t)g = Dffng =
1 X
n=0
fne;np:
(15.125)
If we substitute ep = z in (15.125), then Dffng represents a series with increasing powers of 1=z, which is a so-called Laurent series (see 14.3.4, p. 692). The substitution ep = z suggested the name of the z transformation. With this substitution from (15.124) we nally get the following relations between the Laplace and z-transformation in the case of step functions: or pLff (t)g = 1 ; 1z Zffng: (15.126b) pF (p) = 1 ; z1 F (z) (15.126a) In this way, we can transform the relations of z-transforms of step functions (see Table 21.15, p. 1086)
15.4 Z-Transformation 737
into relations of Laplace transforms (see Table 21.13, p. 1067) of step functions, and conversely.
15.4.1.5 Inverse of the Z-Transformation
The inverse of the z-transformation is to nd the corresponding unique original sequence ffng from its transform F (z). We write
Z ;1fF (z)g = ffng:
(15.127)
There are di erent possibilities for the inverse transformation.
1. Using Tables
If the function F (z) is not given in tables, then we can try to transform it to a function which is given in Table 21.15.
2. Laurent Series of F (z)
We get the inverse transform directly from the de nition (15.105), p. 734, if a series expansion of F (z) with respect to 1=z is known or if it can be determined.
! 3. Taylor Series of F z1
Since F 1 is a series of increasing powers of z, from (15.105) and the Taylor formula we get z dn F 1 fn = n1! dz (15.128) n z z=0 (n = 0 1 2 : : :):
4. Application of Limit Theorems
Using the limits (15.108) and (15.111), p. 735, we can directly determine the original sequence ffng from its transform F (z). F (z) = (z ; 2)(2zz ; 1)2 . We use the previous four methods. 1. By the partial fraction decomposition (see 1.1.7.3, p. 15) of F (z)=z we obtain functions which are contained in Table 21.15. F (z) = 2 A B C z (z ; 2)(z ; 1)2 = z ; 2 + (z ; 1)2 + z ; 1 : So F (z) = z 2;z 2 ; (z ;2z1)2 ; z 2;z 1 and therefore fn = 2(2n ; n ; 1) for n 0: 2. F (z) will be a series with decreasing powers of z by division: F (z) = z3 ; 4z22z+ 5z ; 2 = 2 z12 + 8 z13 + 22 z14 + 52 z15 + 114 z16 + : : : : (15.129) From this expression we get f0 = f1 = 0 f2 = 2 f3 = 8 f4 = 22 f5 = 52 f6 = 114 : : :, but we do not obtain a closed expression for the general term fn. 3. For formulating F 1z and its required derivatives, (see (15.128)) we consider the partial fraction
738 15. Integral Transformations decomposition of F (z), and get: F z1 = 1 ;2 2z ; (1 ;2zz)2 ; 1 ;2 z i.e., F z1 = 0 for z = 0 1 1 dF z dF z 4 ; 4z ; 4 = i.e., 2 3 2 dz (1 ; 2z) (1 ; z) (1 ; z) dz = 0 for z = 0 1 2 2 d F 1z dF z 16 ; 12z ; 12 = i.e., dz2 (1 ; 2z)3 (1 ; z)4 (1 ; z)3 dz2 = 4 for z = 0 d3 F 1z d3 F 1z 96 ; 48z ; 48 = i.e., dz3 (1 ; 2z)4 (1 ; z)5 (1 ; z)4 dz3 = 48 for z = 0 ... ... ... ... from which we can easily obtain f0 f1 f2 f3 : : : considering also the factorials in (15.128). 4. Application of the limit theorems (see 15.4.1.2, 3., p. 734) gives: 2z f0 = zlim F (z) = zlim =0 !1 !1 z 3 ; 4z 2 + 5z ; 2 2 2z =0 f1 = zlim z(F (z) ; f0) = zlim !1 !1 z 3 ; 4z 2 + 5z ; 2
2z 3 f2 = zlim z2 F (z) ; f0 ; f1 1z = zlim =2 !1 !1 z 3 ; 4z 2 + 5z ; 2 f3 = zlim z3 F (z) ; f0 ; f1 1z ; f2 z12 = zlim z3 z3 ; 4z22z+ 5z ; 2 ; z22 = 8 : : : !1 !1 where the Bernoulli{l'Hospital rule is applied (see 2.1.4.8, 2., p. 54). We can determine the original sequence ffng successively.
15.4.2 Applications of the Z-Transformation
15.4.2.1 General Solution of Linear Di erence Equations
A linear di erence equation of order k with constant coecients has the form ak yn+k + ak;1yn+k;1 + + a2 yn+2 + a1 yn+1 + a0 yn = gn (n = 0 1 2 : : :): (15.130) Here k is a natural number. The coecients ai (i = 0 1 : : : k) are given real or complex numbers and they do not depend on n. Here a0 and ak are non-zero numbers. The sequence fgng is given, and the sequence fyng is to be determined. To determine a particular solution of (15.130) the values y0 y1 : : : yk;1 have to be previously given. Then we can determine the next value yk for n = 0 from (15.130). We then get yk+1 for n = 1 from y1 y2 : : : yk and from (15.130). In this way, we can calculate recursively all values yn. We can give however a general solution for the values yn with the z-transformation. We use the second shifting theorem (15.114) applied for (15.130) to get: h i ak zk Y (z) ; y0 ; y1z;1 ; ; yk;1z;(k;1) + + a1 z Y (z) ; y0] + a0 Y (z) = G(z): (15.131) Here we denote Y (z) = Zfyng and G(z) = Zfgng. If we substitute ak zk + ak;1zk;1 + + a1z + a0 = p(z), then the solution of the so-called transformed equation (15.131) is k kX ;1 X Y (z) = p(1z) G(z) + p(1z) yi aj zj;i: (15.132) i=0 j =i+1
15.4 Z-Transformation 739
As in the case of solving linear di erential equations with the Laplace transformation, we have the similar advantage of the z-transformation that initial values are included in the transformed equation, so the solution contains them automatically. We get the required solution fyng = Z ;1 f Y (z)g from (15.132) by the inverse transformation discussed in 15.4.1.5, p. 737.
15.4.2.2 Second-Order Di erence Equations (Initial Value Problem)
The second-order di erence equation is yn+2 + a1yn+1 + a0 yn = gn (15.133) where y0 and y1 are given as initial values. Using the second shifting theorem for (15.133) we get the transformed equation
z2 Y (z) ; y0 ; y1 1z + a1z Y (z) ; y0] + a0Y (z) = G(z): (15.134) 2 If we substitute z + a1 z + a0 = p(z), then the transform is Y (z) = p(1z) G(z) + y0 z(zp(+z)a1) + y1 p(zz) : (15.135) If the roots of the polynomial p(z) are 1 and 2 , then 1 6= 0 and 2 6= 0, otherwise a0 is to be zero, and then the di erence equation could be reduced to a rst-order one. By partial fraction decomposition and applying Table 21.15 for the z-transformation we get the following: 8 1 z z for 6= > ; < 1 2 z 1 ; 2 z ;z1 z ; 2 p(z) = > for 1 = 2 : (z ; 1)2 8 ( ) > < 1n ; 2n Z ;1 p(zz) = fpng = >: 1 ;n;12 for 1 6= 2 (15.136a) n1 for 1 = 2 : Since p0 = 0, by the second shifting theorem ( 2) ( ) Z ;1 pz(z) = Z ;1 z p(zz) = fpn+1g (15.136b) and by the rst shifting theorem ( ) ( ) Z ;1 p(1z) = Z ;1 z1 p(zz) = fpn;1g: (15.136c) Here we substitute p;1 = 0. Based on the convolution theorem we get the original sequence with
yn =
n X
=0
pn;1qn; + y0(pn+1 + a1 pn) + y1p1:
(15.136d)
Since p;1 = p0 = 0, this relation and (15.136a) imply that in the case of 1 6= 2 n+1 n+1 ;1 ; ;1 n n n! n n X 2 ; y 1 ; 2 + a 1 ; 2 + y 1 ; 2 : (15.136e) yn = gn; 1 ; 0 1 1 ; ; ; 1 2 1 2 1 2 1 2 =2 This form can be further simpli ed, since a1 = ;(1 + 2) and a0 = 12 (see the root theorems of Vieta, 1.6.3.1, 3., p. 44), so ;1 ; ;1 n;1 n;1 n n n X 2 ; y a 1 ; 2 + y 1 ; 2 : yn = gn; 1 ; (15.136f) 0 0 1 ; ; 1 2 1 2 1 2 =2
740 15. Integral Transformations In the case of 1 = 2 similarly
yn =
n X
=2
gn; ( ; 1)1;2 ; y0a0(n ; 1)1n;2 + y1n1n;1:
(15.136g)
In the case of second-order di erence equations the inverse transformation of the transform Y (z) can be performed without partial fraction decomposition if we use correspondences such as, e.g., z n;1 sinh bn Z ;1 z2 ; 2az cosh (15.137) b + a2 = a sinh n 2 and the second shifting theorem. By substituting a1 = ;2a cosh b, and a0 = a the original sequence of (15.135) becomes: "n
1 X ;2 n n;1 yn = sinh g (15.138) n; a sinh( ; 1)b ; y0 a sinh(n ; 1)b + y1 a sinh nb : b =2 This formula is useful in numerical computations especially if a0 and a1 are complex numbers. Remark: Notice that the hyperbolic functions are also de ned for complex variables.
15.4.2.3 Second-Order Di erence Equations (Boundary Value Problem)
It often happens in applications that the values yn of a di erence equation are needed only for a nite number of indices 0 n N . In the case of a second-order di erence equation (15.133) both boundary values y0 and yN are usually given. To solve this boundary value problem we start with the solution (15.136f) of the corresponding initial value problem, where instead of the unknown value y1 we have to introduce yN . If we substitute n = N into (15.136f), we can get y1 depending on y0 and yN : "
N X y1 = N ;1 N y0a0(1N ;1 ; 2N ;1) + yN (1 ; 2 ) ; (1;1 ; 2;1)gN ; : (15.139) 1 2 =2 If we substitute this value into (15.136f) then we get n N n n X X 2 yn = ;1 (1;1 ; 2;1)gn; ; ;1 N1 ; (1;1 ; 2;1)gN ; N ; 1 2 =2 1 2 1 2 =2 (15.140) + N ;1 N y0(1N 2n ; 1n2N ) + yN (1n ; 2n)]: 1 2 The solution (15.140) makes sense only if 1N ; 2N 6= 0. Otherwise, the boundary value problem has no general solution, but analogously to the boundary value problems of di erential equations, we have to solve the eigenvalue problem and to determine the eigenfunctions.
15.5 Wavelet Transformation 15.5.1 Signals
If a physical object emits an e ect which spreads out and can be described mathematically, e.g., by a function or a number sequence, then we call it a signal . Signal analysis means that we characterize a signal by a quantity that is typical for the signal. This means mathematically: The function or the number sequence, which describes the signal, will be mapped into another function or number sequence, from which the typical properties of the signal can be clearly seen. For such mappings, of course, some informations can also be lost. The reverse operation of signal analysis, i.e., the reconstruction of the original signal, is called signal synthesis. The connection between signal analysis and signal synthesis can be well represented by an example of Fourier transformation: A signal f (t) (t denotes time) is characterized by the frequency !. Then, formula (15.141a) describes the signal analysis, and formula (15.141b) describes the signal synthesis:
15.5 Wavelet Transformation 741
Z1
F (! ) =
e;i!t f (t) dt
;1
Z1 (15.141a) and f (t) = 1 ei!t F (!) d!: 2 ;1
(15.141b)
15.5.2 Wavelets
The Fourier transformation has no localization property, i.e., if a signal changes at one position, then the transform changes everywhere without the possibility that the position of the change could be recognized \at a glance". The basis of this fact is that the Fourier transformation decomposes a signal into plane waves. These are described by trigonometric functions, which have arbitrary long oscillations with the same period. However, for wavelet transformations there is an almost freely chosen function , the wavelet (small localized wave), that is shifted and compressed for analysing a signal. Examples are the Haar wavelet (Fig. 15.28a) and the Mexican hat (Fig. 15.28b).
A Haar8 wavelet:
ψ(t)
> 0 x < 21 < 1 if 1 = > ;1 if : 0 otherwise. 2 x 1
1 1
(15.142)
B Mexican hat: d2 e;x2 =2 (15.143) (x) = ; dx 2 = (1 ; x2 )e;x2=2 : (15.144)
ψ(t)
1 0
½
t
0
-1 a)
1
t
b)
Figure 15.28 Every function comes into consideration as a wavelet if it is quadratically integrable and its Fourier transform 2(!) according to (15.141a) results in a positive nite integral Z1 j2(!)j (15.145) j!j d!: ;1 Concerning wavelets, the following properties and de nitions can be mentioned: 1. For the mean value of the wavelet:
Z1
(t) dt = 0:
(15.146)
;1
2. The following integral is called the k-th moment of a wavelet : k =
Z1
tk (t) dt:
(15.147)
;1
The smallest positive integer n such that n 6= 0, is called the order of the wavelet . For the Haar wavelet (15.142), n = 1, and for the Mexican hat (15.144), n = 2. 3. When k = 0 for every k, has in nite order. Wavelets with bounded support always have nite order. 4. A wavelet of order n is orthogonal to every polynomial of degree n ; 1.
15.5.3 Wavelet Transformation
For a wavelet (t) we can form a family of curves with parameter a: a (t) = q1 at (a 6= 0): jaj
(15.148)
742 15. Integral Transformations In the case of jaj > 0 theqinitial function (t) is compressed. In the case of a < 0 there is an additional reection. The factor 1= jaj is a scaling factor. The functions a (t) can also be shifted with a second parameter b. Then we get a two-parameter family of curves: ! ab = q1 t ;a b (a b real a 6= 0): (15.149) jaj The real shifting parameter b characterizes the rst moment, while parameter a gives the deviation of the function ab (t). The function ab (t) is called a basis function in connection to the wavelet transformation. The wavelet transformation of a function f (t) is de ned as:
! Z1 Z1 L f (a b) = c f (t)ab (t) dt = qc f (t) t ;a b dt: jaj ;1 ;1
(15.150a)
For the inverse transformation:
f (t) = c
Z1 Z1
;1 ;1
L f (t)ab (t) a12 da db:
(15.150b)
Here c is a constant dependent on the special wavelet . Using the Haar wavelets (15.144) we get ! 8 < 1 if b t < b + a=2 t ; b a = : ;1 if b + a=2 t < b + a 0 otherwise and therefore ! Z b+a=2 Z b+a L f (a b) = q1 b f (t) dt ; b+a=2 f (t) dt jaj q ! Z b+a jaj Z b+a=2 = 2 a2 f (t) dt ; a2 f (t) dt : (15.151) b b+a=2 The value L f (a b) given in (15.151) represents the di erence of the mean values of a function f (t) over two neighboring intervals of length ja2j , connected at the point b.
Remarks: 1. The dyadic wavelet transformation has an important role in applications. As a basis function we
select the functions i ! ij (t) = p1 i t ;2i2 j (15.152) 2 i.e., we get di erent basis functions from one wavelet (t) by doubling or halving the width and shifting by an integer multiple of the width. 2. A wavelet (t) , where the basis functions given in (15.152) form an orthogonal system, is called an orthogonal wavelet. 3. The Daubechies wavelets have especially good numerical properties. They are orthogonal wavelets with compact support, i.e., they are di erent from zero only on a bounded subset of the time scale.
15.5 Wavelet Transformation 743
They do not have a closed form representation (see 15.9]).
15.5.4 Discrete Wavelet Transformation 15.5.4.1 Fast Wavelet Transformation
The integral representation (15.150b) is very redundant, and so the double integral can be replaced by a double sum without loss of information. We consider this idea at the concrete application of the wavelet transformation. We need 1. an ecient algorithm of the transformation, which leads to the concept of multi-scale analysis, and 2. an ecient algorithm of the inverse transformation, i.e., an ecient way to reconstruct signals from their wavelet transformations, which leads us to the concept of frames. For more details about these concepts see 15.9], 15.1]. Remark: The great success of wavelets in many di erent applications, such as calculation of physical quantities from measured sequences pattern and voice recognition data compression in news transmission is based on \fast algorithms". Analogously to the FFT (Fast Fourier Transformation, see 19.6.4.2, p. 928) we talk here about FWT (Fast Wavelet Transformation).
15.5.4.2 Discrete Haar Wavelet Transformation
As an example of a discrete wavelet transformation we describe the Haar wavelet transformation: The values fi (i = 1 2 : : : N ) are given from a signal. The detailed values di (i = 1 2 : : : N=2) are calculated as: si = p1 (f2i;1 + f2i) di = p1 (f2i;1 ; f2i): (15.153) 2 2 The values di are to be stored while the rule (15.153) is applied to the values si, i.e., in (15.153) the values fi are replaced by the values si. This procedure is continued, sequentially so that nally from s(in+1) = p1 s(2ni;) 1 + s(2ni ) di(n+1) = p1 s(2ni;) 1 ; s(2ni ) (15.154) 2 2 a sequence of detailed vectors is formed with components di(n) . Every detailed vector contains information about the properties of the signals. Remark: For large values of N the discrete wavelet transformation converges to the integral wavelet transformation (15.150a).
15.5.5 Gabor Transformation
Time-frequency analysis is the characterization of a signal with respect to the contained frequencies and time periods when these frequencies occur. Therefore, the signal is divided into time segments (windows) and a Fourier transform is used. We call it a Windowed Fourier Transformation (WFT). The window function should be chosen so that a signal is cong(t) sidered only in the window. Gabor applied the window function 0.04 0 -0.04 -30
0
Figure 15.29
30 t
t2 ; 1 g(t) = p e 22 (15.155) 2 (Fig. 15.29). This choice can be explained as g(t), with the \total unit mass", is concentrated at the point t = 0 and the width of the window can be considered as a constant (about 2).
744 15. Integral Transformations The Gabor transformation of a function f (t) then has the form
G f (! s) =
Z1
;1
f (t)g(t ; s)e;i! t dt:
(15.156)
This determines, with which complex amplitude the dominant wave (fundamental harmonic) ei! t occurs during the time interval s ; s + ] in f , i.e., if the frequency ! occurs in this interval, then it has the amplitude jG f (! s)j.
15.6 Walsh Functions 15.6.1 Step Functions
In the approximation theory of functions orthogonal function systems have an important role. For instance, special polynomials or trigonometric functions are used since they are smooth, i.e., they are di erentiable suciently many times in the considered interval. However, there are problems, e.g., the transition of points of a rough picture, when smooth functions are not suitable for the mathematical description, but step functions, piecewise constant functions are more appropriate. Walsh functions are very simple step functions. They take only two function values +1 and ;1. These two function values correspond to two states, so the Walsh functions can be implemented by computers very easily.
15.6.2 Walsh Systems
Analogously to trigonometric functions we can consider periodic step functions. We apply the interval I = 0 1) as a period interval and we divide it into 2n equally long subintervals. Suppose Sn is the set of periodic step functions with period 1 over such an interval. We can consider the di erent step functions belonging to Sn as vectors of a nite dimensional vector space, since every function g 2 Sn is de ned by its values g0 g1 g2 : : : g2n;1 in the subintervals and it can be considered as a vector: gT = (g0 g1 g2 : : : g2n ;1): (15.157) The Walsh functions belonging to Sn form an orthogonal basis with respect to a suitable scalar product in this space. The basis vectors can be enumerated in many di erent ways, so we can get many di erent Walsh systems, which actually contain the same functions. There are three of them which should be mentioned: Walsh{Kronecker functions, Walsh{Kaczmarz functions and Walsh{Paley functions. The Walsh transformation is constructed analogously to the Fourier transformation, where the role of the trigonometric functions is taken by the Walsh functions. We get, e.g., Walsh series, Walsh polynomials, Walsh sine and Walsh cosine transformations, Walsh integral, and analogously to the fast Fourier transformation there is a Fast Walsh Transformation. For an introduction in the theory and applications of Walsh functions see 15.6].
745
16 ProbabilityTheoryand MathematicalStatistics
When experiments or observations are made, various outcomes are possible even under the same conditions. Probability theory and statistics deal with regularity of random outcomes of certain results with respect to given experiments or observations. (In probability theory and statistics, observations are also called experiments, since they have certain outcomes.) We suppose, at least theoretically, that these experiments can be repeated arbitrarily many times under the same circumstances namely, these disciplines deal with the statistics of mass phenomena. The term stochastics is used for the mathematical handling of random phenomena.
16.1 Combinatorics
We often compose new sets, systems, or sequences from the elements of a given set, in a certain way. Depending on the way we do it, we get the notion of permutation, combination, and arrangement. The basic problem of combinatorics is to determine how many di erent choices or arrangements are possible with the given elements.
16.1.1 Permutations 1. Denition
A permutation of n elements is an ordering of the n elements.
2. Number of Permutations without Repetition
The number of di erent permutations of n di erent elements is Pn = n! : (16.1) In a classroom 16 students are seated on 16 places. There are 16! di erent possible arrangements.
3. Number of Permutations with Repetitions
The number Pn(k) of di erent permutations of n elements containing k identical elements (k n) is Pn(k) = nk!! : (16.2) In a classroom 16 schoolbags of 16 students are placed on 16 chairs. Four of them are identical. There are 16!=4! di erent arrangements of the schoolbags.
4. Generalization (k k :::k
The number Pn 1 2 m) of di erent permutations of n elements containing m di erent types of elements with multiplicities k1 k2 : : : km respectively (k1 + k2 + : : : + km = n) is Pn(k1k2:::km) = k !k !n:!: : k ! : (16.3) 1 2 m 5! = 10 Suppose we compose ve-digit numbers from the digits 4, 4, 5, 5, 5. We can have P5(23) = 2!3! di erent numbers.
16.1.2 Combinations 1. Denition
A combination is a choice of k elements from n di erent elements not considering the order of them. We call it a combination of k-th order and we distinguish between combinations with and without repetition.
746 16. Probability Theory and Mathematical Statistics
2. Number of Combinations without Repetition
The number Cn(k) of di erent possibilities to choose k elements from n di erent elements not considering the order is ! Cn(k) = nk with 0 k n (see binomial coecient in 1.1.6.4, 3., p. 13) (16.4) if we choose any element at most once. We call this a combination without repetition. ! There are 30 = 27405 possibilities to choose an electoral board of four persons from 30 partici4 pants.
3. Number of Combinations with Repetition
The number of possibilities to choose k elements from n di erent ones, repeating each element arbitrarily times and not considering the order is ! Cn(k) = n + kk ; 1 : (16.5) In other words, we consider the number of di erent selections of k elements chosen from n di erent elements, where the selected ones must not be di erent. ! Rolling k dice, we can get C6 (k) = k + k6 ; 1 di erent results. Consequently, we can get C6(2) = ! 7 = 21 di erent results with two dice. 2
16.1.3 Arrangements
1. Denition
An arrangement is an ordering of k elements selected from n di erent ones, i.e., arrangements are combinations considering the order.
2. Number of Arrangements without Repetition
The number Vn(k) of di erent orderings of k di erent elements selected from n di erent ones is ! Vn(k) = k! nk = n(n ; 1)(n ; 2) : : : (n ; k + 1) (0 k n): (16.6) How many di erent ways are there to choose a chairman, his deputy, and!a rst and a second assistant for them from 30 participants at an election meeting? The answer is 30 4! = 657720. 4
3. Number of Arrangements with Repetition
An ordering of k elements selected from n di erent ones, where any of the elements can be selected arbitrarily many times, is called an arrangement with repetition. Their number is Vn(k) = nk : (16.7) A: In a soccer-toto with 12 games there are 312 di erent outcomes. B: We can represent 28 = 256 di erent symbols with the digital unit called a byte which contains 8 bits, see for example the well-known ASCII table.
16.2 Probability Theory 747
16.1.4 CollectionoftheFormulasofCombinatorics(seeTable16.1) Table 16.1 Collection of the formulas of combinatorics
Type of choice or Number of possibilities selection of kfrom without repetition with repetition n elements (k n) (k n) Permutations Combinations Arrangements
Pn = n! (n = k) Pn(k) = nk!! ! ! Cn(k) = nk Cn(k) = n + kk ; 1 ! Vn(k) = k! nk Vn(k) = nk
16.2 Probability Theory
16.2.1 Event, Frequency and Probability 16.2.1.1 Events
1. Di erent Types of Events
All the possible outcomes of an experiment are called events in probability theory, and they form the fundamental probability set A. We distinguish the certain event, the impossible event and random events. The certain event occurs every time when the experiment is performed, the impossible event never occurs a random event sometimes occurs, sometimes does not. All possible outcomes of the experiment excluding each other are called elementary events (see also Table 16.2). We denote the events of the fundamental probability set A by A B C : : : , the certain event by I , the impossible event by O. We de ne some operations and relations between the events they are given in Table 16.2.
2. Properties of the Operations
The fundamental probability set forms a Boolean algebra with complement, addition, and multiplication de ned in Table 16.2, and it is called the eld of events. The following rules are valid: 1. a) A + B = B + A (16.8) 1. b) AB = BA: (16.9)
2. a) 3. a) 4. a) 5. a) 6. a) 7. a) 8. a)
A+A=A
(16.10)
A + (B + C ) = (A + B ) + C (16.12) A+A=I
(16.14)
A(B + C ) = AB + AC
(16.16)
A+B =AB
(16.18)
B ; A = BA
(16.20)
A(B ; C ) = AB ; AC
(16.22)
2. b) 3. b) 4. b) 5. b) 6. b) 7. b) 8. b)
AA = A:
(16.11)
A(BC ) = (AB )C:
(16.13)
AA = O:
(16.15)
A + BC = (A + B )(A + C ): (16.17) AB = A + B:
(16.19)
A = I ; A:
(16.21)
AB ; C = (A ; C )(B ; C ): (16.23)
748 16. Probability Theory and Mathematical Statistics 9. a) O A (16.24) 9. b) A I: (16.25) 10. From A B follows a) A = AB (16.26) and b) B = A + BA and conversely: (16.27) 11. Complete System of Events: A system of events A ( 2 ', ' is a nite or in nite set of
indices) is called a complete system of events if the following is valid: X and 11. b) A = I: 11. a) A A = O for 6= (16.28) 2
(16.29)
Table 16.2 Relations between events
Name 1. 2. 3. 4. 5. 6. 7. 8.
Notation A A+B AB
Complementary event of A: Sum of events A and B : Product of the events A and B : Di erence of the events A A ; B and B : Event as a consequence of A B the other: Elementary or simple E event: Compound event: Disjoint or exclusive events AB = O A and B :
De nition A occurs exactly if A does not. A + B is the event which occurs if A or B or both occur. AB is the event which occurs exactly if both A and B occur. A ; B occurs exactly if A occurs and B does not. A B means that from the occurrence of A follows the occurrence of B . From E = A + B it follows that E = A or E = B . Event, which is not elementary. The events A and B cannot occur at the same time.
A: Tossing two coins: Elementary events for the separate tossings: See the table on the right. Head Tail 1. Elementary event for tossing both coins, e.g.: First coin shows 1. Coin A11 A12 head, second shows tail: A11 A22 . 2. Coin A21 A22 Compound event for tossing both coins: First coin shows head: A11 = A11 A21 + A11 A22 2. Compound event for tossing one coin, e.g., the rst one: First coin shows head or tail: A11 + A12 = I . Head and tail on the same coin are disjoint events: A11 A12 = O. B: Lifetime of light-bulbs. We can de ne the elementary events An: the lifetime t satis es the inequalities (n ; 1)%t < t n%t (n = 1 2 : : :, and %t > 0, arbitrary unit of time). X n Compound event A: The lifetime is at most n%t, i.e., A = A . =1
16.2.1.2 Frequencies and Probabilities 1. Frequencies
Let A be an event belonging to the eld of events A of an experiment. If event A occurred nA times while we repeated the experiment n times, then nA is called the frequency, and nA =n = hA is called the relative frequency of the event A. The relative frequency satis es certain properties which can be used to built up an axiomatic de nition of the notion of the probability P (A) of event A in the eld of events A.
16.2 Probability Theory 749
2. Denition of the Probability
A real function P de ned on the eld of events is called a probability if it satis es the following properties: 1. For every event A 2 A we have 0 P (A) 1 and 0 hA 1: (16.30) 2. For the impossible event O and the certain event I , we have P (O) = 0 P (I ) = 1 and hO = 0 hI = 1: (16.31) 3. If the events Ai 2 A (i = 1 2 : : :) are nite or countably many mutually exclusive events (AiAk = O for i 6= k), then P (A1 + A2 + : : :) = P (A1) + P (A2) + : : : and hA1 +A2+::: = hA1 + hA2 + : : : : (16.32)
3. Rules for Probabilities
1: B A yields P (B ) P (A): 2: P (A) + P (A) = 1: 3.a) For n mutually exclusive events Ai (i = 1 : : : n AiAk = O i 6= k), we have
(16.33) (16.34)
P (A1 + A2 + : : : An) = P (A1) + P (A2) + : : : + P (An):
(16.35a)
P (A + B ) = P (A) + P (B ):
(16.35b)
3.b) In particular for n = 2 we have
4. a) For arbitrary events Ai (i = 1 : : : n), we have P (A1 + + An) = P (A1) + + P (An) ; P (A1A2 ) ; ; P (A1An) ;P (A2A3 ) ; ; P (A2An) ; ; P (An;1An) +P (A1A2 A3 ) + + P (A1A2An ) + + P (An;2An;1An) ;
... +(;1)n;1P (A1A2 : : : An): (16.36a) 4:b) In particular for n = 2 we have P (A1 + A2 ) = P (A1) + P (A2) ; P (A1A2): (16.36b) 5. Equally likely events: If every event Ai (i = 1 2 : : : n) of a nite complete system of events occurs with the same probability, then P (Ai) = n1 : (16.37) If A is a sum of m (m n) events with the same probability Ai (i = 1 2 : : : n) of a complete system, then (16.38) P (A) = m n:
4. Examples of Probabilities
A: The probability P (A) to get a 2 rolling a fair die is: P (A) = 61 . B: What is the probability of guessing four numbers for the lotto \6 from 49", i.e., 6 numbers are
to be chosen from the numbers 1 2 : : : 49. If 6 numbers are drawn, then there are 64 possibilities to choose 4. On the other hand there are 49;6 43 49 6;4 = 2 possibilities for the false numbers. Altogether, there are 6 di erent possibilities to draw 6 numbers. Therefore, the probability P (A4) is: 6 43 645 = 0:0968 %: P (A4) = 449 2 = 665896 6
750 16. Probability Theory and Mathematical Statistics Similarly, the probability P (A6) to get a direct hit is: P (A ) = 1 = 0:715 10;7 = 7:15 10;6 %: 6
49 6
C: What is the probability P (A) that at least two persons have birthdays on the same day among k persons? (The years of birth must not be identical, and we suppose that every day has the same probability of being a birthday.) It is easier to consider the complementary event A: All the k persons have di erent birthdays. We get: P (A) = 365 365 ; 1 365365; 2 : : : 365 ;365k + 1 : 365 365 From this it follows that : : : (365 ; k + 1) : P (A) = 1 ; P (A) = 1 ; 365 364 363 365 k k 10 20 23 30 60 Some numerical results: P(A) 0.117 0.411 0.507 0.706 0.994 We can see that the probability that among 23 and more persons at least two have the same birthday is greater than 50 %.
16.2.1.3 Conditional Probability, Bayes Theorem 1. Conditional Probability
The probability of an event B , when it is known that some event A has already occurred, is called a conditional probability and it is denoted by P (B jA) or PA (B ) (read: The probability that B occurs given that A has occurred). It is de ned by ) P (A) 6= 0: P (B jA) = PP(AB (16.39) (A) The conditional probability satis es the following properties: a) If P (A) 6= 0 and P (B ) 6= 0 holds, then P (B jA) = P (AjB ) : (16.40a) P (B ) P (A) b) If P (A1A2 A3 : : : An) 6= 0 holds, then P (A1A2 : : : An) = P (A1)P (A2jA1) : : : P (AnjA1 A2 : : : An;1): (16.40b)
2. Independent Events
The events A and B are independent events if P (AjB ) = P (A) and P (B jA) = P (B ) holds. In this case, we have P (AB ) = P (A)P (B ):
3. Events in a Complete System of Events
(16.41a) (16.41b)
If A is a eld of events and the events Bi 2 A with P (Bi) > 0 (i = 1 2 : : :) form a complete system of events, then for an arbitrary event A 2 A the following formulas are valid:
a) Total Probability Theorem X P (A) = P (AjBi)P (Bi): i
(16.42)
16.2 Probability Theory 751
b) Bayes Theorem with P (A) > 0 P (Bk jA) = XP (AjBk )P (Bk ) : P (AjB )P (B ) i
i
i
(16.43)
Three machines produce the same type of product in a factory. The rst one gives 20 % of the total production, the second one gives 30 % and the third one 50 %. It is known from past experience that 5 %, 4 %, and 2 % of the product made by each machine, respectively, are defective. Two types of questions often arise: a) What is the probability that an article selected randomly from the total production is defective? b) If the randomly selected article is defective, what is the probability that it was made, e.g., by the rst machine? We use the following notation: Ai denotes the event that the randomly selected article is made by the i-th machine (i = 1 2 3) with P (A1) = 0:2, P (A2) = 0:3, P (A3) = 0:5. The events Ai form a complete system of events: AiAj = 0 A1 + A2 + A3 = I . A denotes the event that the chosen article is defected. P (AjA1) = 0:05 gives the probability that an article produced by the rst machine is defective analogously P (AjA2) = 0:04 and P (AjA3) = 0:02 hold. Now, we can answer the questions: a) P (A) = P (A1)P (AjA1) + P (A2)P (AjA2) + P (A3)P (AjA3) = 0:2 0:05 + 0:3 0:04 + 0:5 0:02 = 0:032. 05 = 0:31. b) P (A1jA) = P (A1) P P(A(AjA)1) = 0:2 00::032
16.2.2 Random Variables, Distribution Functions
To apply the methods of analysis in probability theory, we introduce the notions of variable and function.
16.2.2.1 Random Variable
If we assign numbers to the elementary events, then we de ne a random variable X . Then every random event can be described by this variable X . The random variable X can be considered as a quantity which takes its values x randomly from a subset R of the real numbers. If R contains nite or countably many di erent values, then we call X a discrete random variable In the case of a continuous random variable, R can be the whole real axis or it may contain subintervals. For the precise de nition see 16.2.2.2, 2., p. 752. There are also mixed random variables. A: If we assign the values 1 2 3 4 to the elementary events A11 A12 A21 A22 , respectively, in example A, p. 748, then we de ne a discrete random variable X . B: The lifetime T of a randomly selected light-bulb is a continuous random variable. The elementary event T = t occurs if the lifetime T is equal to t.
16.2.2.2 Distribution Function
1. Distribution Function and its Properties
A random variable X can be de ned by its distribution function F (x) = P (X x) for ; 1 < x < 1: (16.44) It determines the probability that the random variable X takes a value between ;1 and x. Its domain is the whole real axis. The distribution function has the following properties: a) F (;1) = 0 F (+1) = 1.
752 16. Probability Theory and Mathematical Statistics b) F (x) is a non-decreasing function of x. c) F (x) is continuous on the right. Remarks: 1. From the de nition it follows that P (X = a) = F (a) ; x!lima;0 F (x). 2. In the literature, also the de nition F (x) = P (X < x) is often used. In this case P (X = a) = x!lim F (x) ; F (a). a+0
2. Distribution Function of Discrete and Continuous Random Variables
a) Discrete Random Variable: A discrete random variable X , which takes the values xi (i = 1 2 : : :) with probabilities P (X = xi) = pi (i = 1 2 : : :), has the distribution function X F (x ) = p i :
(16.45)
x i x
b) Continuous Random Variable: A random variable is called continuous if there exists R a nonnegative function f (x) such that the probability P (X 2 S ) can be expressed as P (X 2 S ) = S f (x)dx
for any domain S such that it is possible to consider an integral over it. This function is the so-called density function. A continuous random variable takes any given value xi with 0 probability, so we rather consider the probability that X takes its value from a nite interval a b]:
Zb P (a X b) = f (t) dt:
(16.46)
a
A continuous random variable has an everywhere continuous distribution function:
F (x) = P (X x) =
Zx
f (t) dt:
(16.47)
;1
F 0(x) = f (x) holds at the points where f (x) is continuous. Remark: When there is no confusion about the upper integration limit, often the integration variable is denoted by x instead of t.
3. Area Interpretation of the Probability
By introducing the distribution function and density function in (16.47), we can represent the probability P (X x) = F (x) as an area between the density function f (t) and the x-axis on the interval ;1 < t x (Fig. 16.1a). f(t)
f(t) F(x) α
a)
0
x
t
b) Figure 16.1
0
xα
t
Often there is given a probability value . If P (X > x) = (16.48) holds, we call the corresponding value of the abscissa x = x the quantile or the fractile of order (Fig. 16.1b). This means the area under the density function f (t) to the right of x is equal to .
16.2 Probability Theory 753
Remark: In the literature, the area to the left of x is also used for the de nition of quantile.
In mathematical statistics, for small values of , e.g., = 5% or = 1%, we also use the notion of signicance level or type 1 error rate. The most often used quantiles for the most important distributions in practice are given in tables (Table 21.16, p. 1089, to Table 21.20, p. 1096).
16.2.2.3 Expected Value and Variance, Chebyshev Inequality
For a global characterization of a distribution, we mostly use the parameters expected value, denoted by , and the variance 2 of a random variable X . The expected value can be interpreted with the terminology of mechanics as the abscissa of the center of gravity of a surface bounded by the curve of the density function f (x) and the x-axis. The variance represents a measure of deviation of the random variable X from its expected value .
1. Expected Value
If g(X ) is a function of the random variable X , then g(X ) is also a random variable. Its expected value or expectation is de ned as:
a) Discrete Case:
E (g(X )) =
b) Continuous Case: E (g(X )) =
X
g(xk )pk if the series
k + Z1
g(x)f (x) dx if
;1
Z +1 ;1
1 X
jg(xk )jpk exists.
(16.49a)
jg(x)jf (x) dx exists.
(16.49b)
k=1
The expected value of the random variable X is de ned as
X = E (X ) =
X k
xk pk or
+ Z1
xf (x) dx
(16.50a)
;1
if the corresponding sum or integral with the absolute values exists. We note that (16.49a,b) yields that E (aX + b) = a X + b (a b const) (16.50b) is also valid. Of course, it is possible that a random variable does not have any expected value.
2. Moments of Order n
We introduce: a) Moment of Order n: E (X n) b) Central Moment of Order n: E ((X ; X )n):
3. Variance and Standard Deviation
(16.51a) (16.51b)
In particular, for n = 2, the central moment is called the variance or dispersion: 8X > > (xk ; X )2 pk or
> +Z 1 (16.52) 2 f (x) dx > ( x ; ) X > :;1 if the expected values occurring in the formula exist. The quantity X is called the standard deviation. The following relations are valid: (16.53) D2(X ) = X2 = E (X 2) ; 2X D2(aX + b) = a2 D2(X ):
4. Weighted and Arithmetical Mean
In the discrete case, the expected value is obviously the weighted mean E (X ) = p1x1 + : : : + pnxn
(16.54)
754 16. Probability Theory and Mathematical Statistics of the values x1 : : : xn with the probabilities pk as weights (k = 1 : : : n). The probabilities for the uniform distribution are p1 = p2 = : : : = pn = 1=n, and E (X ) is the arithmetical mean of the values xk : E (X ) = x1 + x2 +n : : : + xn : (16.55) In the continuous case, the density function of the continuous uniform distribution on the nite interval a b] is 8 1 < f (x) = : b ; a for a x b, (16.56) 0 otherwise, and it follows that Zb 2 E (X ) = b ;1 a x dx = a +2 b X2 = (b ;12a) : (16.57) a
5. Chebyshev Inequality
If the random variable X has the expected value and standard deviation , then for arbitrary > 0 the Chebyschev inequality is valid: P (jX ; j ) 12 : (16.58) That is, it is very unlikely that the values of the random variable X are farther from the expected value than a multiple of the standard deviation ( large).
16.2.2.4 Multidimensional Random Variable
If the elementary events mean that n random variables X1 : : : Xn take n real values x1 : : : xn, then a random vector X = (X1 X2 : : : Xn) is de ned (see also random vector, 16.3.1.1, 4., p. 770). The corresponding distribution function is de ned by F (x1 : : : xn) = P (X1 x1 : : : Xn xn ): (16.59) The random vector is called continuous if there is a function f (t1 : : : tn) such that
F (x 1 : : : x n ) =
Zx1
;1
Zxn
f (t1 : : : tn) dt1 : : : dtn
(16.60)
;1
holds. The function f (t1 : : : tn) is called the density function. It is non-negative. If some of the variables x1 : : : xn tend to in nity, then we get the so-called marginal distributions. Further investigations and examples can be found in the literature. The random variables X1 : : : Xn are independent random variables if F (x1 : : : xn) = F1 (x1 )F2(x2 ) : : : Fn(xn) f (t1 : : : tn) = f1(t1 ) : : : fn(tn ): (16.61)
16.2.3 Discrete Distributions
1. Two-Stage Population and Urn Model
Suppose we have a two-stage population with N elements, i.e., the population we consider has two classes of elements. One class has M elements with a property A, the other one has N ; M elements which do not have the property A. If we investigate the probabilities P (A) = p and P (A) = 1 ; p for randomly chosen elements, then we distinguish between two cases: When we select n elements one after the other, we either replace the previously selected element before selecting the next one, or we do not replace it. The selected n elements, which contain k elements with the property A, is called the sample, n being the size of the sample. This can be illustrated by the urn model.
16.2 Probability Theory 755
2. Urn model
Suppose there are a lot of black balls and white balls in a container. The question is: What is the probability that among n randomly selected balls there are k black ones. If we put every chosen ball back into the container after we have determined its color, then the number k of black ones among the chosen n balls has a binomial distribution. If we do not put back the chosen balls and n M and n N ; M , then the number of black ones has a hypergeometric distribution.
16.2.3.1 Binomial Distribution
Suppose we observe only the two events A and A in an experiment, and we do n independent experiments. If P (A) = p and P (A) = 1 ; p holds every time, then the probability that A takes place exactly k times is ! Wpn(k) = nk pk (1 ; p)n;k (k = 0 1 2 : : : n): (16.62) For every choice of an independent element from the population, the probabilities are N ;M P (A) = M (16.63) N P (A) = N = 1 ; p = q: The probability of getting an element with property A for the rst k choices, then an element with the remaining property A for the n ; k choices is pk (1 ; p)n;k , because the results of choices are independent of each other. We get the same result assigning the k places any other way. We can assign these places ! n = n! (16.64) k k!(n ; k)! ! di erent ways, and these events are mutually exclusive, so we add nk equal numbers to get the required probability. A random variable Xn, for which P (Xn = k) = Wpn(k) holds, is called binomially distributed with parameters n and p.
1. Expected Value and Variance E (Xn) = = n p
(16.65a)
D2(Xn) = 2 = n p(1 ; p):
(16.65b)
2. Approximation of the Binomial Distribution by the Normal Disribution
If Xn has a binomial distribution, then ! ! Xn ; E (Xn) = p1 Z exp ;t2 dt: lim P (16.65c) n!1 D(Xn) 2 2 ;1 This means that, if n is large, the binomial distribution can be well approximated by a normal distribution (see 16.2.4.1, p. 758) with parameters X = E (Xn) and 2 = D2(Xn), if p or 1 ; p are not too small. The approximation is the more accurate the closer p is to 0:5 and the larger n is, but acceptable if np > 4 and n(1 ; p) > 4 hold. For very small p or 1 ; p, the approximation by the Poisson distribution (see (16.68) in 16.2.3.3) is useful.
3. Recursion Formula
The following recursion formula is recommended for practical calculations with the binomial distribution: k p W n(k): Wpn(k + 1) = nk ; (16.65d) +1 q p
4. Sum of Binomially Distributed Random Variables
If Xn and Xm are both binomially distributed random variables with parameters n, p and m, p, then the random variable X = Xn + Xm is also binomially distributed with parameters n + m, p.
756 16. Probability Theory and Mathematical Statistics Fig. 16.2a,b,c represents the distributions of three binomially distributed random variables with parameters n = 5 p = 0:5, 0:25, and 0:1. Since the binomial coecients are symmetric, the distribution is symmetric for p = q = 0:5, and the farther p is from 0:5 the less symmetric the distribution is.
16.2.3.2 Hypergeometric Distribution
Just as with the binomial distribution, we suppose that we have a two-stage population with N elements, i.e., the population we consider has two classes of elements. One class has M elements with a property A, the other one has N ; M elements which do not have the property A. In contrast to the case of binomial distribution, we do not replace the chosen ball of the urn model. The probability that among the n chosen balls there are k black ones is ! ! M N ;M n!; k n (k) = k P (X = k) = WMN with (16.66a) N n 0 k n k M n ; k N ; M: (16.66b) If also n M and n N ; M hold, then the random variable X with the distribution (16.66a) is said to be hypergeometrically distributed. 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
a)
p=0.50
0 1 2 3 4 5
k
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
b)
p=0.25
0 1 2 3 4 5
k
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
c)
p=0.10
0 1 2 3 4 5
k
Figure 16.2
1. Expected Value and! Variance ! of the Hypergeometric Distribution M N ;M = E (X ) = k k Nn!; k = n M N k=0 k n X
! ! M N ;M 2 2 = D2(X ) = E (X 2) ; E (X )]2 = k2 k Nn!; k ; n M N k=0 k M N ;n = nM N 1; N N ;1 :
(16.67a)
n X
2. Recursion Formula
(n ; k)(M ; k) n n (k + 1) = WMN (k + 1)(N ; M ; n + k + 1) WMN (k):
(16.67b) (16.67c)
16.2 Probability Theory 757
In Fig. 16.3a,b,c we represent three hypergeometric distributions for the cases N = 100, M = 50, 25 and 10, for n = 5. These cases correspond to the cases p = 0:5 0:25, and 0:1 of Fig. 16.2a,b,c. There is no signi cant di erence between the binomial and hypergeometric distributions in these examples. If also M and N ; M are much larger than n, then the hypergeometric distribution can be well approximated by a binomial one with parameters as in (16.63). 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
a)
p=0.50
0 1 2 3 4 5
k
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
b)
16.2.3.3 Poisson Distribution
p=0.25
0 1 2 3 4 5
k
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
c)
p=0.10
0 1 2 3 4 5
k
Figure 16.3
If the possible values of a random variable X are the non-negative integers with probabilities k P (X = k) = k! e; (k = 0 1 2 : : : > 0) (16.68) then it has a Poisson distribution with parameter .
1. Expected Value and Variance of the Poisson Distribution E (X ) =
(16.69a)
D2(X ) = :
(16.69b)
2. Sum of Independent Poisson Distributed Random Variables
If X1 and X2 are independent Poisson distributed random variables with parameters 1 and 2, then the random variable X = X1 + X2 also has a Poisson distribution with parameter = 1 + 2.
3. Recursion! Formula ! k k+1 P
= k + 1P :
(16.69c)
4. Connection between Poisson and Binomial Distribution
We can get the Poisson distribution as a limit of binomial distributions with parameters n and p if n ! 1, and p (p ! 0) changes with n so that np = = const, i.e., the Poisson distribution is a good approximation for a binomial distribution for large n and small p with = np. In practice, we use it if p 0:08 and n 1500p hold, because the calculations are easier with a Poisson distribution. Table 21.16, p. 1089, contains numerical values for the Poisson distribution. Fig. 16.4a,b,c represents three Poisson distributions with = np = 2:5, 1:25 and 0:5, i.e., with parameters corresponding to Figs. 16.2 and 16.3.
5. Application
The number of independently occurring point-like discontinuities in a continuous medium can usually be described by a Poisson distribution, e.g., number of clients arriving in a store during a certain time interval number of misprints in a book, etc.
758 16. Probability Theory and Mathematical Statistics 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
a)
λ=2.5
0 1 2 3 4 5
k
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
b)
λ=1.25
0 1 2 3 4 5
k
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
c)
λ=0.50
0 1 2 3 4
k
Figure 16.4
16.2.4 Continuous Distributions 16.2.4.1 Normal Distribution
1. Distribution Function and Density Function
A random variable X has a normal distribution if its distribution function is Zx ; (t; )2 P (X x) = F (x) = p1 e 22 dt: (16.70) 2 ;1 Then it is also called a normal variable, and the distribution is called a ( , ) normal distribution. The function (t; )2 f (t) = p1 e; 22 (16.71) 2 is the density function of the normal distribution. It takes its maximum at t = and it has inection points at (see (2.59), p. 72, and Fig. 16.5a). j(t)
f(t)
F(x)
a)
0 m−s
m
m+s
2. Expected Value 2and Variance
t
b) Figure 16.5
0
x
t
The parameters and of the normal distribution are its expected value and variance, respectively, i.e. + Z 1 ; (x; )2 E (X ) = p1 (16.72a) xe 22 dx = 2 ;1 + Z1 (x; )2 D2(X ) = E (X ; )2 ] = p1 (x ; )2e; 22 dx = 2 : 2 ;1
(16.72b)
16.2 Probability Theory 759
If the normal random variables X1 and X2 are independent with parameters 1, 1 and 2, 2 , resp., then the random variable X = k1Xq1 + k2X2 (k1, k2 real constants) also has a normal distribution with parameters = k1 1 + k2 2, = k1 21 2 + k222 2. If we perform the substitution = t ; in (16.70), then the calculation of the substitution values of the distribution function of any normal distribution is reduced to the calculation of the substitution values of the distribution function of the (0 1) normal distribution, which is called the standard normal distribution. Consequently, the probability P (a X b) of a normal variable can be expressed by the distribution function (x) of the standard normal distribution: ! P (a X b) = b ; ; a ; : (16.73)
16.2.4.2 Standard Normal Distribution, Gaussian Error Function 1. Distribution Function and Density Function
We get from (16.70) with = 0 and 2 = 1 the distribution function Zx ; t2 Zx P (X x) = (x) = p1 e 2 dt = '(t) dt (16.74a) 2 ;1 ;1 of the so-called standard normal distribution. Its density function is t2 '(t) = p1 e; 2 (16.74b) 2 it is called the Gaussian error curve (Fig. 16.5b). The values of the distribution function (x) of the (0 1) normal distribution are given in Table 21.17, p. 1091. Only the values for the positive arguments x are given, while we get the values for the negative arguments from the relation (;x) = 1 ; (x): (16.75)
2. Probability Integral
The integral (x) is also called the probability integral or Gaussian error integral. We can also nd in the literature the following de nitions: Zx t2 Zx p 0(x) = p1 e; 2 dt = (x) ; 21 (16.76a) erf (x) = p2 e;t2 dt = 2 0 ( 2x): (16.76b) 2 0 0
erf denotes the error function.
16.2.4.3 Logarithmic Normal Distribution
1. Density Function and Distribution Function
The continuous random variable X has a logarithmic normal distribution, or lognormal distribution with parameters L and L2 if it can take all positive values, and if the random variable Y , de ned by Y = log X (16.77) has a normal distribution with parameters L and L2 . Consequently, the random variable X has the density function 80 > < ! for t 0 f (t) = > logpe exp ; (log t ; L)2 for t > 0 (16.78) : tL 2 2L2
760 16. Probability Theory and Mathematical Statistics and the distribution function 80 for x 0 > < ! log x F (x) = > p1 Z exp ; (t ; L)2 dt for x > 0: : L 2 2L2 ;1 We can use either the natural or the decimal logarithm in practical applications.
(16.79)
2. Expected Value and Variance
Using the natural logarithm we get the expected value and variance of the lognormal distribution: 2! = exp L + 2L 2 = exp L2 ; 1 exp 2 L + L2 : (16.80)
3. Remarks
a) The density function of the lognormal distribution is continuous everywhere and it has positive values only for positive arguments. Fig. 16.6 shows the density functions of lognormal distributions
for di erent L and L. Here we used the natural logarithm. b) Here the values L and L2 are not the expected value and variance of the lognormal random variable itself, but of the variable Y = log X . c) The values of the distribution function F (x) of the lognormal distribution can be calculated by the distribution function (x) of the standard normal distribution (see (16.74a)), in the following way: ! F (x) = log x; L : (16.81) L d) The lognormal distribution is often applied in lifetime analysis of economical, technical, and biological processes. e) The normal distribution can be used in additive superposition of a large number of independent random variables, and the lognormal distribution is used for multiplicative superposition of a large number of independent random variables. f(t) 2.0
f(t) 1.5 1.0
l=2 1.0
mL=0 ; sL=1
0.5
0
1.5
mL=ln 0.5 ; sL=0.5
0.5
1.0
1.5
2.0
l=1
0.5 2.5 t
0
Figure 16.6
0.5
1.0
1.5
2.0
t
Figure 16.7
16.2.4.4 Exponential Distribution
1. Density Function and Distribution Function
A continuous random variable X has an exponential distribution with parameter ( > 0) if its density function is (Fig. 16.7) ( t 0), if its density function is 8 > for t < 0 1 ; exp ; x (16.86) for x 0: :
2. Expected Value and Variance
= ; 1 + 1 2 = 2 ; 1 + 2 ; ; 2 1 + 1 : Here ; (x) denotes the gamma function (see 8.2.5, 6., p. 461): Z1 ; (x) = tx;1e;t dt for x > 0:
(16.87) (16.88)
0
In (16.85), is the shape parameter and is the scale parameter (Fig. 16.8, Fig. 16.9). f(t) 1.0
f(t) 1.0
b=1
a=2 b=1
a=2
a=1 0.5
0.5
b=2
b=4
a=0.5 0
0.5
1.0
1.5
Figure 16.8
2.0
2.5
t
0
1
2
3
Figure 16.9
4
5
t
762 16. Probability Theory and Mathematical Statistics Remarks: a) The Weibull distribution becomes an exponential distribution for = 1 with = 1 . b) The Weibull distribution also has a three-parameter form if we introduce a position parameter .
Then the distribution "function is:! F (x) = 1 ; exp ; x ; : (16.89) c) The Weibull distribution is especially useful in life expectancy theory, because, e.g., it describes the functional lifetime of building elements with great exibility.
16.2.4.6
2 (Chi-Square) Distribution
1. Density Function and Distribution Function
Let X1 , X2,. . . , Xn be n independent (0 1) normal random variables. Then the distribution of the random variable 2 = X12 + X22 + + Xn2 (16.90) is called the 2 distribution with n degrees of freedom. Its distribution function is denoted by F 2 (x), and the corresponding density function by f 2 (t). 8 1 t n2 ; 1e; 2t for (t > 0) > < f 2 (t) = 2n=2; n (16.91a) 2 > : 0 for t 0. Z 1 x t n2 ; 1e; 2t dt (x > 0): F 2 (x) = P (2 x) = (16.91b) 2n=2 ; n2 0
2. Expected Value and Variance E (2) = n
(16.92a)
3. Sum of Independent Random Variables
D2(2 ) = 2n:
(16.92b)
If X1 and X2 are independent random variables both having a 2 distribution with n and m degrees of freedom, then the random variable X = X1 + X2 has a 2 distribution with n + m degrees of freedom.
4. Sum of Independent Normal Random Variables
If X1, X2,. . . , Xn are independent, (0 ) normal random variables, then n X X = Xi2 has the density function f (t) = 12 f 2 t2 i=1 nt n X 1 n X = n Xi 2 has the density function f (t) = 2 f 2 2 i=1 v u n 2! u X X = t n1 Xi2 has the density function f (t) = 2t2 f 2 t 2 : i=1
5. Quantile
(16.93) (16.94) (16.95)
For the quantile (see 16.2.2.2, 3., p. 752) 2 m of the 2 distribution with m degrees of freedom (Fig. 16.10), (16.96) P (X > 2 m ) = :
16.2 Probability Theory 763 fX2(t)
fF(t) a
a
2
0
X
t
a, m
0
fa, m1,m2
t
Figure 16.10 Figure 16.11 Quantiles of the 2 distribution can be found in Table 21.18, p. 1093.
16.2.4.7 Fisher F Distribution
1. Density Function and Distribution Function
If X1 and X2 are independent random variables both having 2 distribution with m1 and m2 degrees of freedom, then the distribution of the random variable X1 & X2 Fm1 m2 = m (16.97) 1 m2 is a Fisher distribution or F distribution with m1 , m2 degrees of freedom. m1 m2 8 m1 > ; + > m = 2 m = 2 1 2 m m t 2 ;1 2 > < 1 m12 2m2 m1 m2 for t > 0 2 fF (t) = > 2 (16.98a) ; 2 ; 2 m1 t + m2 2 + 2 > 2 2 > : 0 for t 0. For x 0 we have FF (x) = P (Fm1 m2 x) = 0, for x > 0: FF (x) = P (Fm1 m2 x) m1 m1 m1 =2 m2 m2 =2 ; m1 + m2 Zx t 2 ; 1 dt m12 2m2 = (16.98b) m1 m2 2 2 ; 2 ; 2 0 m1 t + m2 2 + 2 2 2
2. Expected Value and Variance E (Fm1 m2 ) = mm;2 2 2
3. Quantile
(16.99a)
The quantiles (see 16.2.2.2, 3., p. 752) t Table 21.19, p. 1094.
m1 m2
2 2 (m1 + m2 ; 2) D2(Fm1 m2 ) = m2m(m ; 2)2(m ; 4) : 1
2
2
(16.99b)
of the Fisher distribution (Fig. 16.11) can be found in
16.2.4.8 Student t Distribution
1. Density Function and Distribution Function
If X is a (0 1) normal random variable and Y is a random variable independent from X and it has a 2 distribution with m = n ; 1 degrees of freedom, then the distribution of the random variable T = qX (16.100) Y=m
764 16. Probability Theory and Mathematical Statistics is called a Student t distribution or t distribution with m degrees of freedom. The distribution function is denoted by FS (x), and the corresponding density function by fS (t). m + 1 ; 1 1 2 fS (t) = pm (16.101a) m+1 2! 2 ; m2 t 1+ m
; m 2+ 1 Zx 1 dt FS (x) = P (T x) = fS (t) dt = pm m m+1 : 2! 2 ; 2 ;1 ;1 t 1+ m Zx
fs(t)
fs(t)
0
ta, m
2. Expected Value and Variance E (T ) = 0 (m > 1)
a 2
a 2
a
a)
(16.101b)
t
−ta/2, m 0
b)
+ta/2, m
t
Figure 16.12
(16.102a)
3. Quantile
D2(T ) = mm; 2 (m > 2):
(16.102b)
The quantiles t m and t =2m of the t distribution (Fig. 16.12a,b), for which P ( jT j > t =2m ) = (16.103b) P (T > t m ) = (16.103a) or holds, are given in Table 21.20, p. 1096. The Student t distribution, introduced by Gosset under the name Student, is used in the case of samples with small sample size n, when only estimations can be given for the mean and for the standard deviation. The standard deviation (16.102b) nolonger depends on the deviation of the population from where the sample is taken.
16.2.5 Law of Large Numbers, Limit Theorems
The law of large numbers gives a relation between the probability P (A) of a random event A and its relative frequency nA=n with a large number of repeated experiments.
1. Law of Large Numbers of Bernoulli
The following inequality holds for arbitrary given numbers " > 0 and > 0 1 (16.104b) (16.104a) if n 4"2 : P nnA ; P (A) < " 1 ; For other similar theorems see 16.5]. How many times should we roll a not necessarily fair die if the relative frequency of the 6 should be closer to its probability than 0.1 with a probability of at least 95 % ?
16.2 Probability Theory 765
Now, " = 0:01 and = 0:05, so 4"2 = 2 10;5, and according to the law of large numbers of Bernoulli n 5 104 must hold. This is an extremely large number. We can reduce n, if we know the distribution function.
2. Central Limit Theorem of Lindeberg{Levy
If the independent random variables X1 ,. . . , Xn all have the same distribution with an expected value and a variance 2 , then the distribution of the random variable n 1X n i=1 Xi ; Yn = =pn (16.105) tends to the (0 1) normal distribution for n ! 1, i.e., for its distribution function Fn(y) we get y 1 Z e; t22 dt: p (16.106) lim F ( y ) = n n!1 2 ;1 If n > 30 holds, then Fn(y) can be replaced by the (0 1) normal distribution. Further limit theorems can be found in 16.5], 16.7]. We take a sample of 100 items from a production of resistors. We suppose that their actual resistance values are independent and they have the same distribution with deviation 2 = 150. The mean value for these 100 resistors is x = 1050 ). In which domain is the true expected value with a probability of 99 % ? We are looking for an " such that P (jX ; j ") = 0:99 holds. We can suppose (see (16.105)) that ;p the random variable Y = X = n has a (0:1) normal distribution. From P (jY j ) = P (; Y ) = P (Y ) ; P (Y < ;), and from P (Y ;) = 1 ; P (Y ) it follows that P (jY j ) = 2P (Y ) ; 1 = 0:99. p So, P (Y ) = () = 0:995 and from Table 21.17, p. 1091, we get = 2:58. Since = 100 = 1:225, we get with a 99 % probability: j1050 ; j < 2:58 1:225, i.e., 1046:8 ) < < 1053:2 ).
16.2.6 Stochastic Processes and Stochastic Chains
Many processes occurring in nature and those being studied in engineering and economics can be realistically described only by time-dependent random variables. The electric consumption of a city at a certain time t has a random uctuation that is dependent on the actual demand of the households and industry. The electric consumption can be considered as a continuous random variable X . When the observation time t changes, electric consumption is a continuous random variable at every moment, so it is a function of time. The stochastic analysis of time-dependent random variables leads to the concept of stochastic processes, which has a huge literature of its own (see, e.g., 16.7], 16.8]). Some introductory notions will be given next.
16.2.6.1 Basic Notions, Markov Chains 1. Stochastic Processes
A set of random variables depending on one parameter is called a stochastic process. The parameter, in general, can be considered as time t, so the random variable can be denoted by Xt and the stochastic process is given by the set fXtjt 2 T g: (16.107) The set of parameter values is called the parameter space T , the set of values of the random variables is the state space Z .
766 16. Probability Theory and Mathematical Statistics
2. Stochastic Chains
If both the parameter space and the state space are discrete, i.e., the state variable Xt and the parameter t can have only nite or countably in nite di erent values, then the stochastic process is called a stochastic chain. In this case the di erent states and di erent parameter values can be numbered: Z = f1 2 : : : i i + 1 : : : g (16.108) T = ft0 t1 : : : tm tm+1 : : :g with 0 t0 < t1 < : : : < tm < tm+1 < : : : : (16.109) The times t0 t1 : : : are not necessary equally spaced.
3. Markov Chains, Transition Probabilities
If the probability of the di erent values of Xtm+1 in a stochastic process depends only on the state at time tm , then the process is called a Markov chain. The Markov property is de ned precisely by the requirement that P (Xtm+1 = im+1 jXt0 = i0 Xt1 = i1 : : : Xtm = im ) = P (Xtm+1 = im+1 jXtm = im ) for all m 2 f0 1 2 : : :g and for all i0 i1 : : : im+1 2 Z: (16.110) Consider a Markov chain and times tm and tm+1 . The conditional probabilities P (Xtm+1 = j jXtm = i) = pij (tm tm+1 ) (16.111) are called the transition probabilities of the chain. The transition probability determines the probability by which the system changes from the state Xtm = i at tm into the state Xtm+1 = j at tm+1 . If the state space of a Markov chain is nite, i.e., Z = f1 2 : : : N g, then the transition probabilities pij (t1 t2) between the states at times t1 and t2 can be represented by a quadratic matrix P(t1 t2), by the so-called transition matrix: 0 1 p11(t1 t2) p12(t1 t2) : : : p1N (t1 t2) B p (t t ) p (t t ) : : : p2N (t1 t2) C C C P(t1 t2) = BBB@ 21 ..1 2 22 1 2 : (16.112) C A . pN 1(t1 t2) pN 2(t1 t2) : : : pNN (t1 t2) The times t1 and t2 are not necessarily consecutive.
4. Time-Homogeneous (Stationary) Markov Chains
If the transition probabilities of a Markov chain (16.111) do not depend on time, i.e., pij (tm tm+1 ) = pij (16.113) then the Markov chain is called time-homogeneous or stationary. A stationary Markov chain with a nite state space Z = f1 2 : : : N g has the transition matrix 0 p11 p12 : : : p1N 1 B p p : : : p2N C C P = BB@ ..21 22 C (16.114a) A . pN 1 pN 2 : : : pNN where a) pij 0 for all i j and (16.114b)
b)
N X
j =1
pij = 1 for all i:
(16.114c)
Being independent of time pij gives the transition probability from the state i into the state j during time unit. The number of busy lines in a telephone exchange can be modeled by a stationary Markov chain. For the sake of simplicity we suppose that we have only two lines. Hence, the states are i = 0 1 2 . Let
16.2 Probability Theory 767
the time unit be, e.g., 1 minute. Suppose the transition matrix pij is: 0 0:7 0:3 0:0 1 (pij ) = @ 0:2 0:5 0:3 A (i j = 0 1 2): 0:1 0:4 0:5 In the matrix (pij ) the rst row corresponds to the state i = 0. The matrix element p12 = 0 3 (second row, third column) shows the probability that two lines are busy at time tm given that one was busy at tm;1 . Remark: Every quadratic matrix P = (pij ) of size N N satisfying the properties (16.114b) and (16.114c) is called a stochastic matrix. Their row vectors are called stochastic vectors. Although the transition probabilities of a stationary Markov chain do not depend on time, the distribution of the random variable Xt is given at a given time by the probabilities
P (Xt = i) = pi (t) (i = 1 2 : : : N ) (16.115a)
with
N X i=1
pi(t) = 1
(16.115b)
since the process is in one of the states with probability one at any time t. Probabilities (16.115a) can be written in the form of a probability vector
p = p1(t) p2(t) : : : pN (t)T : (16.116) The probability vector p is a stochastic vector. It determines the distribution of the states of a stationary Markov chain at time period t.
5. Probability Vector and Transition Matrix
Let the transition matrix P of a stationary Markov chain be given (according to (16.114a,b,c)). Starting with the probability distribution at time period t determine the probability distribution at t + 1, that is, calculate p(t + 1) from P and p(t):
p(t + 1) = p(t) P and furthermore p(t + k) = p(t) Pk :
Remarks: 1. For t = 0 it follows from (16.118) that p(k) = p(0)Pk
(16.117) (16.118)
(16.119) that is, a stationary Markov chain is uniquely determined by the initial distribution p(0) and the transition matrix P. 2. If matrices A and B are stochastic matrices, then Ck = AB is a stochastic matrix, as well. Consequently, if P is a stochastic matrix, then the powers P are also stochastic matrices. A particle changes its position (state) Xt (1 x 5) along a line in time periods t = 1 2 3 : : : according to the following rules: a) If the particle is at x = 2 3 4, then it moves to the right by a unit during the next time unit with probability p = 0:6 and to the left with probability 1 ; p = 0:4. b) At points x = 1 and x = 5 the particle is absorbed, i.e., it stays there with probability 1. c) At time t = 0 the position of the particle is x = 2. Determine the probability distribution p(3) at time period t = 3. By (16.119) the probability distribution p(3) = p(0) P3 holds with p(0) = (0 1 0 0 0) and with the
768 16. Probability Theory and Mathematical Statistics transition matrix 01 0 0 0 01 0 1 0 0 0 0 1 B C B C 0 : 4 0 0 : 6 0 0 0 : 496 0 0 : 288 0 0 : 216 P = BBB@ 0 0:4 0 0:6 0 CCCA : We get P3 = BBB@ 0:160 0:192 0 0:288 0:360 CCCA 0 0 0:4 0 0:6 0:064 0 0:192 0 0:744 0 0 0 0 1 0 0 0 0 1 and nally p(3) = (0:496 0 0:288 0 0:216) .
16.2.6.2 Poisson Process 1. The Poisson Process
In the case of a stochastic chain both the state space Z and the parameter space T are discrete, that is, the stochastic process is observed only at discrete time periods t0 t1 t2 : : : : Now, we study a process with continuous parameter space T , and it is called a Poisson process. 1. Mathematical Formulation Mathematical formulation of the Poisson process: a) Let the random variable Xt be the number of signals in the time interval 0,t) b) let the probability pX (t) = P (Xt = x) be the probability of x signals during the time interval 0 t). Additionally, the following assumptions are required, which hold in the process of radioactive decay and many other random processes (at least approximately): c) The probablity P (Xt = x) of x signals in a time interval of length t depends only on x and t, and does not depend on the position of the time interval on the time axis. d) The numbers of signals in disjoint time intervals are independent random variables. e) The probability to get at least one signal in a very short interval of length %t is approximately proportional to this length. The proportionality factor is denoted by ( > 0). 2. Distribution Function By properties a){e) the distribution of the random variable Xt is determined. We get: )x ; t P (Xt = x) = (t (16.120) x! e 2 where = t is the expected value and = t the variance.
3. Remarks 1. From (16.120) we get the Poisson distribution as a special case for t = 1 (see 16.2.3.3, p.757). 2. To interpret the parameter or to estimate its value from observed data the following properties
are useful: is the average number of signals during a time unit, 1 is the average distance (in time) between two signals in a Poisson process. 3. The Poisson process can be interpreted as the random motion of a particle in the state space Z = f0 1 2 : : :g. The particle starts in the state 0, and at every sign it jumps from state i into the next state i + 1. Furthermore, for a small interval %t the transition probability pii+1 from state i into the state i + 1 should be: pii+1 %t: (16.121) is called the transition rate.
4. Examples of Poisson Processes
Radioactive decay is a typical example of a Poisson process: The number of decays (signals) are registered with a counter and marked on the time axis. The observation interval should be relatively small with respect to the half-period of the radiating matter. Consider the number of calls registered in a telephone exchange until time t and calculate, e.g., the probability that at most x calls are registered until time t with the assumption that the average number
16.3 Mathematical Statistics 769
of calls during a time unit is . In reliability testing, the number of failures of a reparable system is counted during a period of duty. In queuing theory we consider the number of customers arriving at the counter of a department store, to a booking oce or to a gasoline station.
2. Birth and Death Processes
One of the generalizations of the Poisson process is the following: We assume that the transition rate i in (16.121) depends on the state i. Another generalization is when the transition from state i into state i ; 1 is allowed. The corresponding transition rate is denoted by i. The state i can be considered, e.g., as the number of individuals in a population. It increases by one at transition from state i into state i + 1, and decreases by one at transition from i into i ; 1. These stochastic processes are called birth and death processes. Let p(Xt = i) = pi (t) be the probability that the process is in state i at time t. Analogously to the Poisson process: from i ; 1 into i : pi;1i i;1%t from i + 1 into i : pi+1i i+1%t (16.122) from i into i : pii 1 ; (i + i)%t: Remark: The Poisson process is a pure birth process with a constant transition rate.
3. Queuing
The simplest queuing system is considered as a counter where customers are served one by one in the order of their arrival time. The waiting room is suciently large, so no one needs to leave because it becomes full. The customers arrive according to a Poisson process, that is, the interarrival time between two clients is exponentially distributed with parameter , and these interarrival times are independent. In many cases also the serving time has an exponential distribution with parameter . The parameters and have the following meanings: : average number of arrivals per time unit, 1 : average interarrival time, : average number of served clients per time unit, 1 : average serving time.
Remarks: 1. If the number of clients standing in the queue is considered as the state of this stochastic process, then
the above simple queuing model is a birth and death process with constant birth rate and constant death rate . 2. The above queuing model can be modi ed and generalized in many di erent ways, e.g., there can be several counters where the clients are served and/or the arrival times and serving times follow di erent distributions (see 16.8], 16.17]).
16.3 Mathematical Statistics
Mathematical statistics provides an application of probability theory for given mass phenomena. Its theorems allow us to make statements with certain probability about properties of given sets, which statements are based on the results of some experiments whose number should be kept low for economical reasons.
16.3.1 Statistic Function or Sample Function 16.3.1.1 Population, Sample, Random Vector 1. Population
The Population is the set of all elements of interest in a particular study. We can consider any set of things having the same property in a certain sense, e.g., every article of a certain production process
770 16. Probability Theory and Mathematical Statistics or all the values of a measuring sequence occurring in a permanent repetition of an experiment. The number N of the elements of a population can be very large, even practically in nite. We often use the word population to denote also the set of numerical values assigned to the elements.
2. Sample
In order not to check the total population about the considered property, data are collected only from a subset, from a so-called sample of size n (n N ). We talk about a random choice if every element of the population has the same chance of being chosen. A random sample of size n from a nite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected. A random sample from an in nite population is a sample selected such that each element is selected independently. The random choice can be made by so-called random numbers. We often use the word sample for the set of values assigned to the selected elements.
3. Random Choice with Random Numbers
It often happens that a random selection is physically impossible on the spot, e.g., in the case of piled material, like concrete stabs. Then we apply random numbers for a random selection (see Table 21.21, p. 1097). Most calculators can generate uniformly distributed random numbers from the interval 0 1]. Pushing the button RAN or RAND we get a number between 0:00 : : : 0 and 0:99 : : : 9. The digits after the decimal point form a sequence of random numbers. We often take random numbers from tables. Two-digit random numbers are given in Table 21.21, p. 1097. If we need larger ones, then we can compose several-digit numbers from them by writing them after each other. A random sample is to be examined from a transport of 70 piled pipes. The sample size is supposed to be 10. We number the pipes from 00 to 69. A two-digit table of random numbers is applied to select the numbers. We x the way we choose the numbers, e.g., horizontally, vertically or diagonally. If during this process random numbers occur repeatedly, or they are larger than 69, then they are simply omitted. The pipes corresponding to the chosen random numbers are the elements of the sample. If we have a several-digit table of random numbers, we can decompose them into two-digit numbers.
4. Random Vector
A random variable X can be characterized by its distribution function, by its parameters, where the distribution function itself is determined completely by the properties of the population. These are unknown at the beginning of a statistical investigation, so we want to collect as much information as possible with the help of samples. Usually we do not restrict our investigation to one sample but we apply more samples (with same size n if it is possible, for practical reasons). The elements of a sample are chosen randomly, so the realizations take their values randomly, i.e., the rst value of the rst sample is usually di erent from the rst value of the second sample. Consequently, the rst value of a sample is a random variable itself denoted by X1 . Analogously, we can introduce the random variables X2 X3 : : : Xn for the second, third,. . . , n-th sample values, and they are called sample variables. Together, they form the random vector X = (X1 X2 : : : Xn): Every sample of size n with elements xi can be considered as a vector x = (x1 x2 : : : xn) as a realization of the random vector.
16.3.1.2 Statistic Function or Sample Function
Since the samples are di erent from each other, their arithmetic means x are also di erent. We can consider them as realizations of a new random variable denoted by X which depends on the sample
16.3 Mathematical Statistics 771 variables X1 X2 : : : Xn. 1: sample: x11 x12 : : : x1n with mean x1: 2: sample: x21 x22 : : : x2n with mean x2: (16.123) ... ... ... ... ... ... m-th sample: xm1 xm2 : : : xnn with mean xm : We denote the realization of the j -th sample variable in the i-th sample by xij (i = 1 2 : : : m j = 1 2 : : : n). A function of the random vector X = (X1 X2 : : : Xn) is again a random variable, and it is called a statistic or sample function. The most important sample functions are the mean, variance, median and range.
1. Mean
The mean X of the random variables Xi is: n X (16.124a) X = n1 Xi: i=1 The mean x of the sample (x1 x2 : : : xn ) is n X x = n1 xi: (16.124b) i=1 It is often useful to introduce an estimate x0 in the calculations of the mean. It can be chosen arbitrarily but possibly close to the mean x. If, e.g., xi, (i = 1 2 : : :) are several-digit numbers in a long measuring sequence, and they di er only in the last few digits, it is simpler to do the calculations only with the smaller numbers zi = xi ; x0 : (16.124c) Then we get n X x = x0 + n1 zi = x0 + z: (16.124d) i=1
2. Variance
The variance S 2 of the random variables Xi with mean X is de ned by: n X S 2 = n ;1 1 (Xi ; X )2 : (16.125a) i=1 The realization of the variance with the help of the sample (x1 x2 : : : xn) is n X s2 = n ;1 1 (xi ; x)2 : (16.125b) i=1 It is proven that in the estimation of the variance of the original population we get a more accurate estimation by dividing n ; 1 than by dividing n. With the estimated value x0 we get n n X X zi2 ; n(x ; x0 )2 zi2 ; z zi i =1 i =1 i =1 2 = : s = n;1 n;1 n X For x = x0 the correction is z zi = 0 because z = 0 holds. n X
i=1
(16.125c)
772 16. Probability Theory and Mathematical Statistics
3. Median
Let the n elements of the sample be arranged in ascending (or descending) order. If n is odd, then the n + 1 -th item if n is even, then the median is the average value of the median X~ is the value of the 2 n -th and n + 1 -th items, the two items on the middle. 2 2 The median x~ in a particular sample (x1 x2 : : : xn), whose elements are arranged in ascending (or descending) order, is 8 < xm+1 if n = 2m + 1 x~ = : xm+1 + xm if n = 2m: (16.126) 2
4. Range
R = max Xi ; min Xi i i
(i = 1 2 : : : n):
(16.127a)
The range R of a particular sample (x1 x2 : : : xn) is R = xmax ; xmin: (16.127b) Every particular realization of a sample function is denoted by a lowercase letter, except the range R, i.e., for a particular sample (x1 x2 : : : xn) we calculate the particular values x, s2, x~, and R. We take a sample of 15 loudspeakers from a running proi Xi i Xi i Xi duction of loudspeakers. The interesting quantity X is the 1 1.01 6 1.00 11 1.00 air gap induction B , measured in Tesla. We get from these 2 1.02 7 0.99 12 1.00 data: 3 1.00 8 1.01 13 1.02 x = 1:0027 or x = 1:0027 with x0 = 1:00 4 0.98 9 1.01 14 1.00 s2 = 1:2095 10;4 or s2 = 1:2076 10;4 with x0 = 1:00 5 0.99 10 1.00 15 1.01 x~ = 1:00 R = 0:04.
16.3.2 Descriptive Statistics
16.3.2.1 Statistical Summarization and Analysis of Given Data
In order to describe statistically the properties of certain elements, we characterize them by a random variable X . Usually, the n measured or observed values xi of the property X form the starting point of a statistical investigation, which is made to nd some parameters of the distribution or the distribution itself of X . Every measured sequence of size n can be considered as a random sample from an in nite population, if the experiment or the measurement could be repeated in nitely many times under the same conditions. Since the size n of a measuring sequence can be very large, we proceed as follows: 1. Protocol, Prime Notation The measured or observed values xi are recorded in a protocol list. 2. Intervals or Classes We consider an interval, which contains the n data xi (i = 1 2 : : : n) of the sample, and divide it into k subintervals, so-called classes or class intervals. Usually 10{20 classes are selected with equal length h, and their boundaries are called class boundaries. The endpoints of the total interval are not uniquely de ned in general, we choose them approximately symmetricly with respect to the smallest and largest value of the sample, and class boundaries should be di erent from any sample value. 3. Frequencies and Frequency Distribution The absolute frequencies hj (j = 1 2 : : : k) are the numbers hj of data (occupancy number) belonging to a given interval %xj . The ratios hj =n (in %) are called relative frequencies. If the values hj =n are represented over the classes as rectangles, then we get a graphical representation of the given frequency distribution, and this representation is called a histogram (Fig. 16.13a). The values hj =n can be considered as the empirical values of the probabilities or the density function f (x).
16.3 Mathematical Statistics 773
4. Cumulative Frequency Adding the absolute or relative frequencies, we get the cumulative absolute or relative frequency
Table 16.3 Frequency table
Class
50 71 91 111 131 151 171 191 211 231 251
{ 70 { 90 { 110 { 130 { 150 { 170 { 190 { 210 { 230 { 250 { 270
24
hj hj =n (%) Fj (%) 1 1 2 9 15 22 30 27 9 6 3
0.8 0.8 1.6 7.2 12.0 17.6 24.0 21.6 7.2 4.8 2.4
0.8 1.6 3.2 10.4 22.4 40.0 64.0 85.6 92.8 97.6 100.0
Fj = h1 + h2 +n + hj % (j = 1 2 : : : k): (16.128) If we represent the value Fj at the upper boundary and draw a horizontal line until the next boundary, then we get a graphical representation of the empirical distribution function, which can be considered as an approximation of the unknown underlying distribution function F (x) (Fig. 16.13b). Suppose we perform n = 125 measurements during a study. The results spread in the interval from 50 to 270, so it is reasonable to divide this interval into k = 11 classes with a length h = 20. We get the frequency table Table 16.3.
hj n /%
Fj/% 100
20
80
16
60
12
40
8
20
4 a)
70
110 150
190 230
270 x
16.3.2.2 Statistical Parameters
b)
70
110
150
190
230
270 x
Figure 16.13
After summarizing and analyzing the data of the sample as given in 16.3.2.1, p. 772, we can approximate the parameters of the random variable by the following parameters:
1. Mean
If we use directly the data of the sample, then the sample mean is n X x = n1 xi: i=1 If we use the means xj and frequencies hj of the classes, then we get k X x = n1 hj xj : j =1
(16.129a) (16.129b)
2. Variance
If we use directly the data of the sample, then the sample variance is n X s2 = n ;1 1 (xi ; x)2 : i=1
(16.130a)
774 16. Probability Theory and Mathematical Statistics If we use the means xj and frequencies hj of the classes, then we get k X (16.130b) s2 = n ;1 1 hj (xj ; x)2: j =1 The class midpoint uj (the midpoint of the corresponding interval) is also often used instead of xj .
3. Median
The median x~ of a distribution is de ned by P (X < x~) = 12 : The median may not be a uniquely determined point. The median of a sample is 8x if n = 2m + 1, < m+1 x~ = : xm+1 + xm if n = 2m. 2
4. Range
R = xmax ; xmin:
(16.131a) (16.131b) (16.132)
5. Mode or Modal Value
is the data value that occurs with greatest frequency. It is denoted by D.
16.3.3 Important Tests
One of the fundamental problems of mathematical statistics is to draw conclusions about the population from the sample. There are two types of the most important questions: 1. The type of the distribution is known, and we want to get some estimate2 for its parameters. A distribution can be characterized mostly quite well by the parameters and (here is the exact value of the expected value, and 2 is the exact variance), consequently one of the most important questions is how good an estimation can we give for them, based on the samples. 2. Some hypotheses are known about these parameters, and we want to check if they are true. The most often occurring questions are: a) Is the expected value equal to a given number or not? b) Are the expected values for two populations equal or not? c) Does the distribution of the random variable with and 2 t a given distribution or not? etc. Because in observations and measurements, the normal distribution has a very important role, we discuss the goodness of t test for a normal distribution. The basic idea can be used for other distributions, too.
16.3.3.1 Goodness of Fit Test for a Normal Distribution
There are di erent tests in mathematical statistics to decide if the data of a sample come from a normal distribution. We discuss a graphical one based on normal probability paper, and a numerical one based on the use of the chi-square distribution (\2 test").
1. Goodness of Fit Test with Probability Paper
a) Principle of Probability Paper The x-axis in a right-angled coordinate system is scaled equidis-
tantly, while the y-axis is scaled on the following way: It is divided equidistantly with respect to Z , but scaled by ZZ ; t2 y = (Z ) = p1 e 2 dt: (16.133) 2 ;1
16.3 Mathematical Statistics 775
If a random variable X has a normal distribution with expected value and variance 2 , then for its distribution function (see 16.2.4.2, p. 759) F (x) = x ; = (Z ) (16.134a) x Z 0 (16.134c) holds, i.e., 1 + x ; ; 1 ; Z= (16.134b) must be valid, and so there is a linear relation between x and Z and (16.134c).
b) Application of Probability Paper
y We consider the data of the sample, we calz F(t) culate the cumulative relative frequencies ac3 99.86 cording to (16.128), and sketch these onto the 2 97.72 probability paper as the ordinates of the points with abscissae the upper class boundaries. If 1 84.13 these points are approximately on a straight line, 0 50.00 (with small deviations) then the random variable −1 15.87 can be considered as a normal random variable −2 2.28 (Fig. 16.14). As we see from Fig. 16.14, the distribution to −3 0.14 which the data of Table 16.3 belong, can be considered as a normal distribution. We can also see that 176 37:5 (from the x values be70 110 150 190 230 270 x longing to the 0 and 1 values of Z ). m−s m m+s Remark: The values Fi of the relative cumulative frequencies can be plotted more easily on the Figure 16.14 probability paper, if its scaling is equidistant with respect to y, which means a non-equidistant scaling for the ordinates.
2. 2 Test for Goodness of Fit
We want to check if a random variable X can be considered normal. We divide the range of X into k classes and we denote the upper limit of the j -th class (j = 1 2 : : : k) by j . Let pj be the \theoretical" probability that X is in the j -th class, i.e., pj = F ( j ) ; F ( j;1) (16.135a) where F (X ) is the distribution function of X (j = 1 2 : : : k 0 is the lower limit of the rst class with F ( 0) = 0). Because X is supposed to be normal, then ! F ( j ) = j ; (16.135b) must hold, where (x) is the distribution function of the standard normal distribution (see 16.2.4.2, p. 759). The parameters and 2 of the population are usually not known. We use x and s2 as an approximation of them. We have to make the decomposition of the range so that the expected frequencies for every class should exceed 5, i.e., if the size of the sample is n, then npj 5. Now, we consider the sample (x1 x2 : : : xn) of size n and calculate the corresponding frequencies hj (for the classes given above). Then the random variable k 2 X 2S = (hj ;npnpj ) (16.135c) j j =1 has approximately a 2 distribution with m = k ; 1 degrees of freedom if we know and 2 , m = k ; 2 if we estimated one of them from the sample, and m = k ; 3 if we estimated both by x and s2.
776 16. Probability Theory and Mathematical Statistics Now, we determine a level , which is called the signicance level, and determine the quantile 2 k;i (i depends on the number of unknown parameters) of the corresponding 2 distribution, e.g., from Table 21.18, p. 1093. This means P (2 2 k;i) = holds. Then we compare the value 2S we got in (16.135c) and this quantile, and if 2S < 2 k;i (16.135d) holds, we accept the assumption that the sample came from a normal distribution. This test is called the 2 test for goodness of t. The following 2 test is based on the example on p. 773. The sample size is n = 125, with the mean x = 176:32 and variance s2 = 36:70. These values are used as approximations of the unknown parameters and 2 of the population. We can determine the test statistic 2S according to (16.135c) after performing the calculations according to (16.135a) and (16.135b), as shown in Table 16.4, p. 776. Table 16.4 2 test
j 70 90 110 130 150 170 190 210 230 250 270
hj
19 > 1 = 13 2> 9" 15 22 30 27 9 6 9 3
j ; j ; ! ;2:90 ;2:35 ;1:81 ;1:26 ;0.72 ;0.17
0.37 0.92 1.46 2:01 2:55
0:0019 0:0094 0:0351 0:1038 0.2358 0.4325 0.6443 0.8212 0.9279 0:9778 0:9946
pj 0:0019 0:0075 0:0257 0:0687 0.1320 0.1967 0.2118 0.1769 0.1067 0:0499 0:0168
npj
0:2375 9 > 0:9375 = 12:9750 3:2125 > 8:5857 " 16.5000 24.5875 26.4750 22.1125 13.3375 6:2375 8:3375 2:1000
(hj ; npj )2
npj
0.00005 0.1635 0.2723 0.4693 1.0803 1.4106 0.0526
2S = 3:4486
It follows from the last column that 2S = 3:4486. Because of the requirement npj 5, the number of classes is reduced from k = 11 to k = k ; 4 = 7. We calculated the theoretical frequencies npj with the estimated values x and s2 of the sample instead of and 2 of the population, so the degrees of freedom of the corresponding 2 distribution is reduced by 2. The critical value is the quantile 2 k;1;2. For = 0:05 we get 20:054 = 9:5 from Table 21.18, p. 1093, so because of the inequality 2S < 20:054 there is no contradiction to the assumption that the sample is from a population with a normal distribution.
16.3.3.2 Distribution of the Sample Mean
Let X be a continuous random variable. Suppose we can take arbitrarily many samples of size n from the corresponding population. Then the sample mean is also a random variable X , and it is also continuous.
1. Condence Probability of the Sample Mean
If X has a normal distribution with parameters and 2, then X is also a normal random variable with parameters and 2=n, i.e., the density function f (x) of X is concentrated more around than the density function f (x) of the population. For any value " > 0: p ! (16.136) P ( jX ; j ") = 2 " ; 1 P ( jX ; j ") = 2 " n ; 1:
16.3 Mathematical Statistics 777
Table 16.5 Con dence level for the sample mean
n P jX ; j 12
1 4 16 25 49
38.29 % 68.27 % 95.45 % 98.76 % 99.96 %
!
It follows from this that with increasing sample size n, the probability that the sample mean is a good approximation of is also increasing. We get for " = 12 from (16.136) P jX ; j 12 = p 2 12 n ; 1, and for di erent values of n we get the values listed in Table 16.5. We see from Table 16.5, e.g., that with a sample size n = 49, the probability that the sample mean x di ers from by less than 12 is 99:95 %.
2. Sample Mean Distribution for Arbitrary Distribution of the Population
The random variable X has an approximately normal distribution with parameters and 2=n for any distribution of the population with expected value and variance 2. This fact is based on the central limit theorem (see 16.2.5,2., p. 765).
16.3.3.3 Condence Limits for the Mean
1. Condence Interval for the Mean with a Known Variance 2
If X is a random variable with parameters and 2 , then according to 16.3.3.2, p. 776, X is approximately a normal random variable with parameters and 2 =n. Then substitution of p Z = X ; n (16.137) yields a random variable Z which has approximately a standard normal distribution, therefore
Z" P ( jZ j ") = '(x) dx = 2(") ; 1: ;"
(16.138)
If the given signi cance level is , namely, P ( jZ j ") = 1 ; (16.139) is required, then " = "() can be determined from (16.138), e.g., from Table 21.17, p. 1091, for the standard normal distribution. From jZ j "() and from (16.137) we get the relation = x pn "(): (16.140) The values x pn "() in (16.140) are called condence limits for the expected value and the interval between them is called a condence interval for the expected value with a known 2 and given significance level . In other words: The expected value is between the con dence limits (16.140) with a probability 1 ; . Remark: If the sample size is large enough, then we can use s2 instead of 2 in (16.140). The sample size is considered to be large, if n > 100, but in practice, depending on the actual problem, it is considered to be suciently large if n > 30. If n is not large enough, then we have to apply the t distribution to determine the con dence limits as in (16.143).
2. Condence Interval for the Expected Value with an Unknown Variance 2
If the variance 2 of the population is unknown, then we replace it by the sample variance s2 and instead of (16.137) we get the random variable p T = X s; n (16.141)
778 16. Probability Theory and Mathematical Statistics which has a t distribution (see 16.2.4.8, p. 763) with m = n ; 1 degrees of freedom. Here n is the size of the sample. If n is large, e.g., n > 100 holds, then T can be considered as a normal random variable as Z in (16.137). We get for a given signi cance level ! Z" j pn " = 1 ; : P ( jT j ") = ft(x) dx = P jX ; (16.142) s ;" From (16.142) it follows that " = "( n) = t =2n;1 , where t =2n;1 is the quantile of the t distribution (with n ; 1 degrees of freedom) for the signi cance level (Table 21.20, p. 1096). It follows from jT j = t =2n;1 that = x psn t =2n;1 : (16.143) s The values x pn t =2n;1 are called the condence limits for the expected value of the distribution of the population with an unknown variance 2 and with a given signi cance level . The interval between these limits is the condence interval. A sample contains the following 6 measured values: 0:842 0:846 0:835 0:839 0:843 0:838: We get from this x = 0:8405 and s = 0:00394. What is the maximum deviation of the sample mean x from the expected value of the population distribution, if the signi cance level is given as 5 % or 1 % ? 1. = 0:05: We read frompTable 21.20, p. 1096, that t =25 = 2:57, and we get jX ; j 2:57 0:00394= 6 = 0:0042. Thus, the sample mean x = 0:8405 di ers from the expected value by less than 0:0042 with a probability 95 %.p 2. = 0:01: t =25 = 4:03 jX ; j 4:03 0:00394= 6 = 0:0065, i.e., the sample mean x di ers from by less than 0:0065 with a probability 99 %.
16.3.3.4 Condence Interval for the Variance
If the random variable X has a normal distribution with parameters and 2 , then the random variable 2 2 = (n ; 1) s 2
(16.144)
fX2 (x)
has a 2 distribution with m = n ; 1 degrees of freedom, where n is the sample size and s2 is the α/2 α/2 sample deviation. f 2 (x) denotes the density func1-α tion of the 2 distribution in Fig. 16.15, and we see that 2 2 0 x Xu Xo P (2 < 2u) = P (2 > 2o ) = 2 : (16.145) Thus, using the quantiles of the 2 distribution Figure 16.15 (see Table 21.18, p. 1093) we obtain that 2u = 21; =2n;1 2o = 2 =2n;1 : (16.146) Considering (16.144) we get the following estimation for the unknown variance 2 of the population distribution with a signi cance level : (n ; 1)s2 2 (n ; 1)s2 : (16.147) 2 =2n;1 21; =2n;1 The con dence interval given in (16.147) for 2 is fairly large for small sample sizes. For the numerical data of the example on p. 778 and for = 5% we get from Table 21.18, p. 1093,
16.3 Mathematical Statistics 779
20:0255 = 0:831 and 20:9755 = 12:8, so it follows from (16.147) that 0:625 s 2:453 s with s = 0:00394.
16.3.3.5 Structure of Hypothesis Testing
A statistical hypothesis testing has the following structure: 1. First we develop a hypothesis H that the sample belongs to a population with some given properties, e.g., H : The population distribution has a normal distribution with parameters and 2 (or another given distribution), or H : The expected value is equal to a given value 0, or H : Two populations have the same expected value, 1 ; 2 = 0, etc. 2. We determine a con dence interval B , based on our hypothesis (in general with tables). The value of the sample function should be in this interval with the given probability, e.g., with probability 99% for = 0:01). 3. We calculate the value of the sample function and we accept the hypothesis if this value is in the given interval B , otherwise we reject it. Test the hypotesis H : = 0 with a signi cance level . p The random variable T = X ;s 0 n has a t distribution with m = n ; 1 degrees of freedom according to 16.3.3.3, p. 777. It follows from this that we have to reject this hypothesis, if x is not in the con dence interval given by (16.143), i.e., if (16.148) jx ; 0j psn t =2n;1 holds. We say that there is a signicant dierence. For further problems about tests see 16.14].
16.3.4 Correlation and Regression
Correlation analysis is used to determine some kind of dependence between two or more properties of the population from the experimental data. The form of this dependence between these properties is determined with the help of regression analysis.
16.3.4.1 Linear Correlation of two Measurable Characters 1. Two-Dimensional Random Variable
In the following, we mostly use the formulas for continuous random variables, but it is easy to replace them by the corresponding formulas for discrete variables. Suppose that X and Y , as a two-dimensional random variable (X Y ), have the joint distribution function
F (x y) = P (X x Y y) =
Zx Zy
f (x y) dx dy
(16.149a)
;1;1
F1(x) = P (X x Y < 1) F2 (y) = P (X < 1 Y y): (16.149b) The random variables X and Y are said to be independent of each other if F (x y) = F1(x) F2(y) (16.150) holds. We can determine the fundamental quantities assigned to X and Y from their joint density function f (x y):
a) Expected Values X = E ( X ) =
Z1 Z1 Z1 Z1 x f (x y) dx dy (16.151a) Y = E (Y ) = y f (x y) dx dy
;1;1
;1;1
(16.151b)
780 16. Probability Theory and Mathematical Statistics b) Variances
X2 = E ((X ; X )2 )
c) Covariance
Y2 = E ((Y ; Y )2):
(16.152a)
d) Correlation Coecient
XY = E ((X ; X )(Y ; Y )) : (16.153)
(16.152b)
%XY = XY : (16.154) X Y We assume that every expected value above exists. The covariance can also be calculated by the formula Z1 Z1 XY = E (XY ) ; X Y where E (XY ) = xy f (x y) dx dy: (16.155) ;1;1
The correlation coecient is a measure of the linear dependence of X and Y , because of the following facts: All points (X Y ) are on one line with probability 1 if %2XY = 1 holds. If X and Y are independent random variables, then their covariance is equal to zero, %XY = 0. From %XY = 0, it does not follow that X and Y are independent, but it does if they have a two-dimensional normal distribution which is de ned by the density function
(x ; X )2 1 q1 f (x y) = exp ; 2 2(1 ; %XY ) X2 2X Y 1 ; %2XY 2 (16.156) ;2 %XY (x ; X)(y ; Y ) + (y ;2 Y ) : X Y Y
2. Test for Independence of two Variable
We often have the question of whether the variables X and Y can be considered independent with %XY = 0, if the sample with size n and with the measured values (xi yi) (i = 12 : : : n) comes from a two-dimensional normal distributed population. The test is performed in the following way: a) We have the hypothesis H : %XY = 0 . b) We determine a signi cance level and determine the quantile t m of the t distribution from Table 21.20, p. 1096, for m = n ; 2. c) We calculate the empirical correlation coecients rxy and calculate the test statistics (sample function) n X (xi ; x)(yi ; y) p i=1 : (16.157b) with rxy = v (16.157a) t = rqxy n ;2 2 u n n u 1 ; rxy tX(xi ; x)2 X(yi ; y)2 i=1 i=1 d) We reject the hypothesis if jtj t m holds.
16.3.4.2 Linear Regression for two Measurable Characters 1. Determination of the Regression Line
If we have detected a certain dependence between the variables X and Y by the correlation coecient, then the next problem is to nd the functional dependence Y = f (X ). We consider mostly linear dependence. The simplest case is linear regression, when we suppose that for any xed value of x the random variable Y in the population has a normal distribution with the expected value E (Y ) = a + bx (16.158) and a variance 2 independent of x. The relation (16.158) means that the mean value of the random variable Y depends linearly on the xed value of x. The values of the parameters a, b and 2 of the
16.3 Mathematical Statistics 781
population are usually unknown, and we estimate them approximately by the least squares method from a sample with values (xi yi) (i = 1 2 : : : n). The least squares method requires that n X i=1
yi ; (a + bxi )]2 = min!
(16.159)
and we get the estimates n X
(xi ; x)(yi ; y) ~b = i=1 n X (xi ; x)2
1 2 2 a~ = y ; ~bx ~ 2 = nn ; ; 2 sy (1 ; rxy ) with
(16.160a)
i=1
n X x = n1 xi i=1
n X y = n1 yi
n X s2y = n ;1 1 (yi ; y)2
i=1
(16.160b)
i=1
and the empirical correlation coecient rxy is given in (16.157b). The coecients a~ and ~b are called regression coecients. The line y (x) = a~ + ~bx is called the regression line.
2. Condence Intervals for the Regression Coe cients
Our next question is, after the determination of the regression coecients a~ and ~b, how well do the estimates approximate the theoretical values a and b. We form the test statistics p p p s n;2 v n : (16.161b) tb = (~b ; b) sxq n ; 22 (16.161a) and ta = (~a ; a) xq n 2 u sy 1 ; rxy sy 1 ; rxy u tX xi 2 i=1
These are realizations of random variables having a t distribution with m = n ; 2 degrees of freedom. We can determine the quantile t =2m taken from Table 21.20, p. 1096, for a given signi cance level v and because P ( jtj < t =2m ) = 1 ; holds for t = ta and t = tb : q uX n 2 u t xi 2 q 2 sy 1 ; rxy s 1;r j~b ; bj < t =2n;2 sy pn ; xy2 (16.162a) ja~ ; aj < t =2n;2 s pn ; 2 pi=1n : (16.162b) x x We can determine a condence region for the regression line y = a + bx with the con dence interval given in (16.162a,b) for a and b.
16.3.4.3 Multidimensional Regression 1. Functional Dependence
Suppose that there is a functional dependence between the characters X1 , X2 ,. . . , Xn, and Y , which is described by the theoretical regression function
y = f (x1 x2 : : : xn) =
s X
j =0
aj gj (x1 x2 : : : xn):
(16.163)
The functions gj (x1 x2 : : : xn) are known functions of n independent variables. The coecients aj in (16.163) are constant multipliers in this linear combination. We also call expression (16.163) linear regression, although the functions gj can be arbitrary. The function f (x1 x2 ) = a0 + a1 x1 + a2x2 + a3x1 2 + a4 x2 2 + a5 x1 x2 , which is a complete quadratic polynomial of two variables with g0 = 1, g1 = x1, g2 = x2 , g3 = x1 2, g4 = x2 2 , and g5 = x1 x2 , is an example for a theoretical linear regression function.
782 16. Probability Theory and Mathematical Statistics
2. Writing in Vector Form
It is useful to write formulas in vector form in the multidimensional case x = (x1 x2 : : : xn)T so, (16.163) now has the form:
y = f (x) =
s X
j =0
aj gj (x):
(16.164) (16.165)
3. Solution and Normal Equation System
The theoretical dependence (16.163) cannot be determined by the measured values (x(i) fi) (i = 1 2 : : : N ) because of random measuring errors. We are looking for the solution in the form s X y = f~(x) = a~j gj (x) j =0
(16.166a) (16.166b)
and using the least squares method (see 16.3.4.2, 1., p. 781) we determine the coecients a~j as the estimations of the theoretical coecients aj , from the equation N h X i=1
i2 fi ; f~ x(i) = min! :
(16.166c)
Introducing the notation
0 (1) 1 0 a~0 1 0 f1 1 g0 x g1 x(1) : : : gs x(1) C B B B a~ C Bf C B (2) g1 x(2) : : : gs x(2) C C C a~ = BB@ ..1 CCA f = BB@ ..2 CCA G = BBB g0 x. C C . . . . . B . . . . C . . . . @ A a~s fN g0 x(N ) g1 x(N ) : : : gs x(N )
(16.166d)
we get from (16.166c) the so-called normal equation system GTGa~ = GTf (16.166e) to determine ~a. The matrix GTG is symmetric, so the Cholesky method (see 19.2.1.2, p. 893) is especially good to solve (16.166e). Consider the sample whose result is given in the next table. Determine the coecients of the regression function (16.167): x1 5 3 5 3 x2 0.5 0.5 0.3 0.3 f~(x1 x2 ) = a0 + a1 x1 + a2x2 : (16.167) f (x1 x2) 1.5 3.5 6.2 3.2 From (16.166d) it follows that 0 1:5 1 0 1 5 0:5 1 0 a~ 1 0 B C 3 : 5 a~ = @ a~1 A f = B@ 6:2 CA G = BB@ 11 35 00::53 CCA (16.168) a~2 3:2 1 3 0:3 and (16.166e) is a~0 = 7:0 4~a0 + 16 a~1 + 1:6 a~2 = 14:4 a~1 = 0:25 16~a0 + 68 a~1 + 6:4 a~2 = 58:6 (16.169) i.e., a~2 = ;11: 1:6~a0 + 6:4~a1 + 0:68~a2 = 5:32
16.3 Mathematical Statistics 783
4. Remarks
1. To determine the regression coecients we start with the interpolation f~ x(i) = fi (i = 1 2 : : :
N ), i.e., with G~a = f : (16.170) In the case s < N , (16.170) is an overdetermined system of equations which can be solved by the Householder method (see 19.6.2.2, p. 921). The multiplication of (16.170) by GT to get (16.166e) is also called Gauss transformation. If the columns of the matrix G are linearly independent, i.e., rank G = s + 1 holds, then the normal equation system (16.166e) has a unique solution, which coincides with the result of (16.170) got by the Householder method. 2. Also in the multidimensional case, it is possible to determine con dence intervals for the regression coecients with the t distribution, analogously to (16.162a,b). 3. We can analyse the assumption (16.166b) by the help of the F distribution (see 16.2.4.7, p. 763), if there are some superuous variables in it.
16.3.5 Monte Carlo Methods 16.3.5.1 Simulation
Simulation methods are based on constructing equivalent mathematical models. These models are then easily analysed by computer. In such cases, we talk about digital simulation. A special case is given by Monte Carlo methods when certain quantities of the model are randomly selected. These random elements are selected by using random numbers.
16.3.5.2 Random Numbers
Random numbers are realizations of certain random quantities (see 16.2.2, p. 751) satisfying given distributions.
1. Uniformly Distributed Random Numbers
These numbers are uniformly distributed in the interval 0 1], they are realizations of the random variable X with the following density function f0 (x) and distribution function F0 (x): 8 0 for 0 x < 0x1 f0(x) = 10 for F ( x ) = (16.171) for 0 < x 1 0 : x1 for otherwise x 1:
1. Method of the Inner Digits of Squares A simple method to generate random numbers is suggested by J. v. Neumann. It is also called the method of the inner digits of squares, and it starts from a decimal number z 2 (0 1) which has 2n digits. Then we form z2 , so we get a decimal number
which has 4n digits. We erase its rst and its last n digits, so we again have a number with 2n digits. To get further numbers, we repeat this procedure. In this way we get 2n digit decimal numbers from the interval 0 1] which can be considered random numbers with a uniform distribution. The value of 2n is selected according to the largest number representable in the computer. For example, we may choose 2n = 10. This procedure is seldom recommended, since it produces more smaller numbers than it should. Several other di erent methods have been developed. 2n = 4 : z = z0 = 0 1234 , z02 = 0 01 5227 56 , z = z1 = 0 5227 , z12 = 0 27 3215 29 , z = z2 = 0 3215 usw. Th rst three random numbers are z0 z1 and z2 . 2. Congruence Method The so-called congruence method is widely used: A sequence of integers zi (i = 0 1 2 : : :) is formed by the recursion formula zi+1 c zi mod m: (16.172)
784 16. Probability Theory and Mathematical Statistics Here z0 is an arbitrary positive number and c and m denote positive integers, which must be suitably chosen. For zi+1 we take the smallest non-negative integer satisfying the congruence (16.172). The numbers zi=m are between 0 and 1 and can be used for uniformly distributed random numbers.
3. Remarks a) Wepchoose m = 2r , where r is the number of bits in a computer word, e.g., r = 40. Then the order of c is m. b) Random number generators using certain algorithms produce so-called pseudorandom numbers. c) On calculators and also in computers, \ran" or \rand" is used for generating random numbers.
2. Random Numbers with other Distributions
To get random numbers with an arbitrary distribution function F (x) we adopt the following procedure: Consider a sequence of uniformly distributed random numbers 1 2 : : : from 0 1]. With these numbers we form the numbers i = F ;1( i) for i = 1 2 : : : : Here F ;1(x) is the inverse function of the distribution function F (x). Then we get:
P (i x) = P (F ;1( i) x) = P ( i F (x)) =
FZ(x) 0
f0 (t) dt = F (x)
(16.173)
i.e., the random numbers 1 2 : : : satisfy a distribution with the distribution function F (x).
3. Tables and Application of Random Numbers
1. Construction Random number tables can be constructed in the following way. We index ten identical chips by the numbers 0 1 2 : : : 9. We place them into a box and shu3e them. One of them is then selected, and its index is written into the table. Then we replace the chip into the box, shu3e again, and choose the next one. In this way a sequence of random numbers is produced, which is written in groups (for easier usage) into the table. In Table 21.21, p. 1097, four random digits form a group. In the procedure, we have to guarantee that the digits 0 1 2 : : : 9 always have equal probability. 2. Application of Random Numbers The use of a table of random numbers is demonstrated with an example. Suppose we choose randomly n = 20 items from a population of N = 250 items. We renumber the objects from 000 to 249. We choose a number in an arbitrary column or row in Table 21.21, p. 1097, and we determine a rule of how the remaining 19 random numbers should be chosen, e.g., vertically, horizontally or diagonally. We consider only the rst three digits from these random numbers, and we use them only if they are smaller than 250.
16.3.5.3 Example of a Monte Carlo Simulation We consider the approximate evaluation of the integral
Z1 I = g(x) dx
(16.174)
0
as an example of the use of uniformly distributed random numbers in a simulation. We discuss two solution methods.
1. Applying the Relative Frequency
We suppose 0 g(x) 1 holds. We can always guarantee this condition by a transformation (see (16.179), p. 785). Then the integral I is an area inside the unit square E (Fig. 16.16). If we consider the numbers of a sequence of uniformly distributed random numbers from the interval 0 1] in pairs as the coordinates of points of the unit square E , then we get n points Pi (i = 1 2 : : : n). If we denote by m the number of points inside the area A, then considering the notion of the relative frequency (see 16.2.1.2, p. 748):
y 1
g(x)
E A
0
1
Figure 16.16
x
16.3 Mathematical Statistics 785
Z1 0
g(x) dx m n:
(16.175)
To achieve relatively good accuracy with the ratio in (16.175), we need a very large number of random numbers. This is the reason why we are looking for possibilities to improve the accuracy. One of these methods is the following Monte Carlo method. Some others can be found in the literature.
2. Approximation by the Mean Value
To determine (16.174), we start with n uniformly distributed random numbers 1 2 : : : n as the realization of the uniformly distributed random variable X . Then the values gi = g( i) (i = 1 2 : : : n) are realizations of the random variable g(X ), whose expectation according to formula (16.49a,b), p. 753, is: Z1 Z1 n X E (g(X )) = g(x)f0(x)dx = g(x)dx n1 gi: (16.176) i=1 ;1 0 This method, which uses a sample to obtain the mean value, is also called the usual Monte Carlo method.
16.3.5.4 Application of the Monte Carlo Method in Numerical Mathematics 1. Evaluation of Multiple Integrals
First, we have to show how to transform a de nite integral of one variable
Zb I = h(x) dx
(16.177)
a
into an expression which contains the integral
Z1 I = g(x) dx with 0 g(x) 1:
(16.178)
0
Then we can apply the Monte Carlo method given in 16.3.5.3. Introduce the following notation: x = a + (b ; a)u m = min h(x) M = max h(x): (16.179) x2ab]
Then (16.177) becomes
x2ab]
Z1 I = (M ; m)(b ; a) h(a + (Mb ;;am)u) ; m du + (b ; a)m 0
(16.180)
where the integrand h(a + (Mb ;;am)u) ; m = g(u) satis es the relation 0 g(u) 1. The approximate evaluation of multiple integrals with Monte Carlo methods is demonstrated by an example of a double integral ZZ V = h(x y) dx dy with h(x y) 0: (16.181) S
S denotes a plane surface domain given by the inequalities a x b and '1 (x) y '2 (x), where '1(x) and '2(x) denote given functions. Then V can be considered as the volume of a cylindrical solid K , which stands perpendicular to the x y plane and its upper surface is given by h(x y). If h(x y) e holds, then this solid is in a block Q given by the inequalities a x b c y d 0 z
786 16. Probability Theory and Mathematical Statistics e (a b c d e const). After a transformation similar to (16.179), we get from (16.181) an expression containing the integral ZZ V = g(u v) du dv with 0 g(u v) 1 (16.182) S
where V can be considered as the volume of a solid K in the three-dimensional unit cube. The integral (16.182) is approximated by the Monte Carlo method in the following way: We consider the numbers of a sequence of uniformly distributed random numbers from the interval 0 1] in triplets as the coordinates of points Pi (i = 1 2 : : : n) of the unit cube, and count how many points among Pi belong to the solid K . If m point belong to K , then analogously to (16.175) V m (16.183) n: Remark: In de nite integrals with one integration variable we should use the methods given in 19.3, p. 898. For the evaluation of multiple integrals, the Monte Carlo method is still often recommended.
2. Solution of Partial Di erential Equations with the Random Walk Process
The Monte Carlo method can be used for the approximate solution of partial di erential equations with the random walk process. a) Example of a Boundary Value Problem: Consider y the following boundary value problem as an example: G @ 2 u + @ 2 u = 0 for (x y) 2 G y+1 %u = @x (16.184a) 2 @y 2 y G u(x y) = f (x y) for (x y) 2 ;: (16.184b) y-1 Here G is a simply connected domain in the x y plane ; denotes the boundary of G (Fig.16.17). Similarly to the di erence method in paragraph 19.5.1, G is covered by a 0 quadratic lattice, where we can assume, without loss of genx x-1 x x+1 erality, that the step size can be chosen as h = 1. Figure 16.17 This way we get interior lattice points P (x y) and boundary points Ri. The boundary points Ri, which are at the same time also lattice points, are considered in the following as points of the boundary ; of G, i.e.: u(Ri) = f (Ri) (i = 1 2 : : : N ) (16.185) b) Solution Principle: We imagine that a particle starts a random walk from an interior point P (x y). That is: 1. The particle moves randomly from P (x y) to one of the four neighboring points. We assign to each of these four grid points 1=4, the probability to move into them. 2. If the particle reaches a boundary point Ri , then the random walk terminates there with probability one. It can be proven that a particle starting at any interior point P reaches a boundary point Ri after a nite number of steps with probability one. We denote by p(P Ri) = p((x y) Ri) (16.186) the probability that a random walk starting at P (x y) will terminate at the boundary point Ri. Then we get p(Ri Ri) = 1 p(Ri Rj ) = 0 for i 6= j and (16.187) 1 p((x y) Ri) = 4 p((x ; 1 y) Ri)+ p((x +1 y) Ri)+ p((x y ; 1) Ri)+ p((x y +1) Ri)]: (16.188)
16.4 Calculus of Errors 787
The equation (16.188) is a di erence equation for p((x y) Ri). If we start n random walks from the point P (x y), from which mi terminates at Ri (mi n), then we get p((x y) Ri)) mni : (16.189) The equation (16.189) gives an approximate solution of the di erential equation (16.184a) with the boundary condition (16.185). The boundary condition (16.184b) will be ful lled if we substitute
v(P ) = v(x y) =
N X i=1
f (Ri)p((x y) Ri)
(16.190)
because of (16.188), v(Rj ) = P f (Ri)p(Rj Ri) = f (Rj ). i=1 To calculate v(x y) we multiply (16.188) by f (Ri). After summation we get the following di erence equation for v(x y): v(x y) = 14 v(x ; 1 y) + v(x + 1 y) + v(x y ; 1) + v(x y + 1)]: (16.191) If we start n random walks from an interior point P (x y), and among them mj terminate at the boundary point Ri (i = 1 2 : : : N ), then we get an approximate value at the point P (x y) of the boundary value problem (16.184a,b) by n X v(x y) n1 mif (Ri): (16.192) i=1 N
16.3.5.5 Further Applications of the Monte Carlo Method
Monte Carlo methods as stochastic simulation, sometimes called methods of statistical experiments, are used in many di erent areas. For example we mention: Nuclear techniques: Neutrons passing through material layers. Communication: Separating signals and noise. Operations research: Queueing systems, process design, inventory control, service. For further details of these problem areas see for example 16.13].
16.4 Calculus of Errors
Every scienti c measurement, giving certain numerical quantities { regardless of the care which with the measurements are made { is always subject to errors and uncertainties. There are observational errors, errors of the measuring method, instrumental errors and often errors arising from the inherent random nature of the phenomena being measured. Together they compose the measurment error. All measurement errors arising during the measuring process we call deviations. As a consequence a measured quantity represented by a number of signi cant digits can be given only with a rounding error, i.e., with a certain statistical error, which we call the uncertainty of the result. 1. The deviations of the measuring process should be kept as small as possible. On this basis we have to evaluate the possible best approximation, what can be done with the help of smoothing methods which have their origin in the Gaussian least squares method. 2. The uncertainties have to be estimated as well as possible, what can be done with the help of methods of mathematical statistics. Because of the random character of the measuring results we can consider them as statistical samples (see 16.2.3, 1., p. 754) with its probability distribution, whose parameters contain the desired information. In this sence, measurement errors can be seen as sampling errors.
788 16. Probability Theory and Mathematical Statistics
16.4.1 Measurement Error and its Distribution
16.4.1.1 Qualitative Characterization of Measurement Errors
If we qualify the measurement errors by their causes, we can distinguish between the following three types of errors: 1. Rough errors are caused by inaccurate readings or confusion they are excludable. 2. Systematic measurement errors are caused by inaccurately scaled measuring devices and by the method of measuring, where the method of reading the data and also the measured error of the measurement system can play a role. They are not always avoidable. 3. Statistical or random measurement errors can arise from random changes of the measuring conditions that are dicult or impossible to control and also by certain random properties of the events observed. In the theory of measurement errors the rough errors and the systematic measurement errors are excluded and we deal only with the statistical properties and with the random measurement errors in the calculation of the rounding errors.
16.4.1.2 Density Function of the Measurement Error 1. Measurement Protocol
We suppose that in the characterization of the uncertainty we have the measured results listed in a measurement record as a prime notation and we have the relative frequencies or the density function f (x), or the cumulative frequencies or the distribution function F (x) (see 16.3.2.1, p. 772) of the uncertain values. By x we denote the realization of the random variable X , which is under consideration.
2. Error Density Function
Special assumptions about the properties of the measurement error result in certain special properties of the density function of the error distribution: 1. Continuous Density Function Since the random measurement errors can take any value in a certain interval, they are described by a continuous density function f (x). 2. Even Density Function If measurement errors with the same absolute value but with di erent signs are equally likely, then the density function is an even function: f (;x) = f (x). 3. Monotonically Decreasing Density Function If a measuring error with larger absolute value is less likely than an error with smaller absolute value, then the density function f (x) is monotonically decreasing for x > 0. 4. Finite Expected Value The expected value of the absolute value of the error must be nite:
E (jX j) =
Z1
;1
jxjf (x) dx < 1:
(16.193)
Di erent properties of the errors result in di erent types of density functions.
3. Normal Distribution of the Error
1. Density Function and Distribution Function In most practical cases we can suppose that the
distribution of the measurement error is a normal distribution with expected value = 0 and variance 2, i.e., the density function f (x) and the distribution function F (x) of the measurement error are: 2 2 ; x 1 Zx e; 2t2 dt = x : (16.194b) f (x) = p1 e 22 (16.194a) p and F ( x ) = 2 2 ;1 Here (x) is the distribution function of the standard normal distribution (see (16.74a), p. 759, and Table 21.17, p. 1091). In the case of (16.194a,b) we speak about normal errors. 2. Geometrical Representation The density function (16.194a) is represented in Fig. 16.18a with inection points and points at the center of gravity, and its behavior is shown in Fig. 16.18b when the
16.4 Calculus of Errors 789
variance changes. The inection points are at the abscissa values thepcenters of gravity of the halfareas are at . The maximum of the function is at x = 0 and it is 1=( 2). The curve widens as 2 increases, the area under the curve always equals one. This distribution shows that small errors occur often, large errors only seldom. f(x)
f(x) f(x)= 1 e s 2p
x
2
2s
h1 ,s1
2
h2 ,s2 h3 ,s3
a)
−s −g −h
0 +g +s +h
x b)
−3
−2
−1
0
1
2
3
x
Figure 16.18
4. Parameters to2 Characterize the Normally Distributed Error
Beside the variance or the standard deviation which is also called the mean square error or standard error, there are other parameters to characterize the normally distributed error, such as the measure of accuracy h, the average error or mean error , and the probable error . 1. Measure of Accuracy Beside the variance 2, the measure of accuracy h = p1 (16.195) 2 is used to characterize the width of the normal distribution. A narrower Gauss curve results in better accuracy (Fig. 16.18b). If we replace by the experimental value of ~ or ~x obtained from the measured values, the measure of accuracy characteizes the accuracy of the measurement method. 2. Average or Mean Error The expected value of the absolute value of the error is de ned as
Z1 = E (jX j) = 2 xf (x) dx: 0
3. Probable Error The bound of the absolute value of the error with the property P (jX j ) = 12
(16.196) (16.197a)
is called the probable error. It implies that Z+ f (x) dx = 2 ; 1 = 12 (16.197b) ; where (x) is the distribution function of the standard normal distribution. 4. Given Error Bounds If an upper bound a > 0 of an error is given, then we can calculate the probability that the error is in the interval ;a a] by the formula P (jX j a) = 2 a ; 1: (16.198)
5. Relations between Standard Deviation, Average Error, Probable Error, and Accuracy If the error has a normal distribution, then the following relations hold with the
790 16. Probability Theory and Mathematical Statistics constant factor % = 0:4769 : s = p1h = 2 = %p p p = h% = % 2 = % and p (% 2) = 12 :
r = p1 = 2 = p 2h 2% 1 1 % h = p = p = 2
(16.199a) (16.199b) (16.200)
16.4.1.3 Quantitative Characterization of the Measurement Error 1. True Value and its Approximations
The true value xw of a measurable quantity is usually unknown. We choose the expected value of the random variables, whose realizations are the measured values xi (i = 1 2 : : : n), as an estimated value of xw . Consequently, the following means can be considered as an approximation of xw :
1. Arithmetical Mean n X x = n1 xi
(16.201a)
i=1
or
x=
k X j =1
hj xj
(16.201b)
if the measured values are distributed into k classes with absolute frequencies hj and class means xj (j = 1 2 : : : k).
2. Weighted Mean x(g) =
n X i=1
gixi
&X n i=1
gi :
(16.202)
Here the single measured values are weighted by the weighting factors gi (gi > 0) (see 16.4.1.6, 1., p. 793).
2. Error of a Single Measurement in a Measurement Sequence
1. True Error of a Single Measurement in a Measurement Sequence is the di erence between
the true value xw and the measuring result. Because this is usually unknown, the true error "i of the i-th measurement with the result xi is also unknown: "i = xw ; xi: (16.203a) 2. Mean Error of a Single Measurement in a Measurement Sequence is the di erence of the arithmetical mean and the measurement result xi : vi = x ; xi: (16.203b)
3. Mean Square Error of a Single Measurement or Standard Error of a Single Measurement Since the expected value of the sum of the true errors "i and the expected value of
the sum of the mean errors vi of n measurement is zero (independently of how large they are), we also consider the sum of the error squares:
"2 =
n X i=1
"i 2
(16.204a)
v2 =
n X i=1
vi2 :
(16.204b)
From a practical point of view only the value of (16.204b) is interesting, since only the values of vi can be determined from the measuring process. Therefore, the mean square error of a single measurement of a measurement sequence is de ned by
v u & n uX ~ = t vi 2 (n ; 1): i=1
(16.205)
16.4 Calculus of Errors 791
The value ~ is an approximation of the standard deviation of the error distribution. We get for ~ = in the case of normally distributed error: P (j"j ~ ) = 2(1) ; 1 = 0:68: (16.206) That is: The probability that the absolute value of the true value does not exceed , is about 68%. 4. Probable Error is the number , for which P (j"j ) = 12 : (16.207) That is: The probability that the absolute value of the error does not exceed , is 50%. The abscissae divide the area of the left and the right parts under the density function into two equal parts (Fig. 16.18a). The relation between ~ and ~ in the case of a normally distributed error is v u & n uX ~ = 0:6745~ 23 ~ = 23 t vi2 (n ; 1): (16.208) i=1 5. Average Error is the number , which is the expected value of the absolute value of the error:
= E (j"j) =
Z1
;1
jxjf (x) dx:
(16.209)
In the case of a normally distributed error we get = 0:798. It follows from the relation P (j"j ) = 2 ; 1 = 0:576 (16.210) that the probability that the error does not exceed the value is about 57:6 %. The centers of gravity of the left and right areas under the density function (Fig. 16.18a) are at abscissae . We also get: v s u & n uX ~ = 2 ~ = 0:7978~ 0:8~ = 0:8t vi 2 (n ; 1): (16.211) i=1
3. Error of the Arithmetical Mean of a Measurement Sequence
The error of the arithmetical mean x of a measurement sequence is given by the errors of the single measurement:
1. Mean Square Error or Standard Deviation v u & n uX ~AM = t vi2 n(n ; 1)] = p~n : i=1
2. Probable Error v u & n uX ~AM 23 t vi2 n(n ; 1)] = 23 p~n : i=1
(16.212) (16.213)
3. Average Error
v u & n uX ~AM 0:8t vi 2 n(n ; 1)] = 0:8 p~n : i=1
(16.214)
4. Accessible Level of Error Since the three types of errors de ned above (16.212){(16.214) are directly proportional to the corresponding error of the single measurement (16.205), (16.208) and (16.211) and they are proportional to the reciprocal of the square root of n, it is not reasonable to increase the number of the measurements after a certain value. It is more ecient to improve the accuracy h of the measuring method (16.195).
792 16. Probability Theory and Mathematical Statistics
4. Absolute and Relative Errors
1. Absolute Uncertainty, Absolute Error The uncertainty of the results of measurement is char-
acterized by errors "i, vi , i , i, i , or ", v, , , , which measure the reliability of the method of measurement. The notion of the absolute uncertainity, given as the absolute error, is meaningful for all types of errors and for the calculation of error propagation (see 16.4.2, p. 794). They have the same dimension as the measured quantity. The word \absolute" error is introduced to avoid confusion with the notion of relative error. We often use the notation %xi or %x. The word \absolute" has a di erent meaning from the notion of absolute value: It refers to the numerical value of the measured quantity (e.g., length, weight, energy), without restriction of its sign. 2. Relative Uncertainty, Relative Error The relative uncertainty, given by the relative error, is a measure of the quality of the method of measurement with respect to the numerical value of the measured quantity. In contrast to the absolute error, the relative error has no dimension, because it is the quotient of the absolute error and the numerical value of the measured quantity. If this value is not known, we replace it by the mean value of the quantity x: xi = %xxi %xxi : (16.215a) The relative error is given mostly as a percentage and it is also called the percentage error : xi=% = xi 100 %: (16.215b)
5. Absolute and Relative Maximum Error
1. Absolute Maximum Error If the quantity z we want to determine is a function of the measured
quantities x1 x2 : : : xn, i.e., z = f (x1 x2 : : : xn ), then the resulting error must be calculated taking also the function f into consideration. There are two di erent ways to examine errors. The rst approach is that Pstatistical error analysis is applied by smoothing the data values using the least squares method (min (zi ; z)2 ), and in the second approach, an upper bound %zmax is determined for the absolute error of the quantities. If we have n independent variables xi , then: n @ X %zmax = @x f (x1 x2 : : : xn) %xi (16.216) i i=1 where we should substitute the mean value xi for xi . 2. Relative Maximum Error We can get the relative maximum error if we divide the absolute maximum error by the numerical value of the measured value (mostly by the mean of z): (16.217) zmax = %zzmax %zzmax :
16.4.1.4 Determining the Result of a Measurement with Bounds on the Error
A realistic interpretation of a measurement result is possible only if the expected error is also given error estimations and bounds are components of measurement results. It must be clear from the data what is the type of the error, what is the con dence interval and what is the signi cance level. 1. De ning the Error The result of a single measurement is required to be given in the form x = xi %x xi ~ (16.218a) and the result of the mean has the form x = x %xAM x ~AM : (16.218b) Here %x is the most often used distance, the standard deviation. ~ and ~ could also be used. 2. Prescription of Arbitrary Con dence Limits The quantity T = X~; xw has a t distribution AM (16.101b) with f = n ; 1 degrees of freedom in the case of a population with a distribution N ( 2)
16.4 Calculus of Errors 793
according to (16.100). For a required signi cance level or for an acceptance probability S = 1 ; we get the con dence limits for the unknown quantity xw = with the t quantile t =2f = x t =2f ~AM : (16.219) That is, the true value xw is in the interval given by these limits with a probability S = 1 ; . We are mostly interested in keeping the size n of the measurement sequence at its lowest possible level. The length 2t =2f ~AM of the con dence interval decreases by a smaller value p of 1 ; and also by a larger number n of measurements. Since ~AM decreases proportionallypto 1= n and the quantile t =2f with f = n ; 1 degrees of freedom also decreases proportionally to 1= n for values of n between 5 and 10 (see Table 21.20, p. 1096) the length of the con dence interval decreases proportionally to 1=n for such values of n.
16.4.1.5 Error Estimation for Direct Measurements with the Same Accuracy
If we can achieve the same variance i for all n measurements, we talk about measurements with the same accuracy h = const. In this case, the least squares method results in the error quantities given in (16.205), (16.208), and (16.210). Determine the nal result for the measurement sequence given in the following table which contains n = 10 direct measurements with the same accuracy. xi 1.592 1.581 1.574 1.566 1.603 1.580 1.591 1.583 1.571 1.559 vi 103 ; 12 ; 1 + 6 + 14 ; 23 0 ; 11 ; 3 + 9 + 21 vi2 106 144 1 36 196 529 0 121 9 81 441
v u n uX p x = 1:580, ~ = t vi2 /(n ; 1) = 0:0131, ~AM = ~ n = 0:041. i=1
Final result: x = x ~AM = 1:580 0:041.
16.4.1.6 Error Estimation for Direct Measurements with Di erent Accuracy 1. Weighted Measurements
If the direct measurement results xi are obtained from di erent measuring methods or they represent means of single measurements, which belong to the same mean x with di erent variances ~i 2 , we calculate a weighted mean
x(g) =
n X i=1
gixi
&X n i=1
gi
(16.220)
where gi is de ned as 2 (16.221) gi = ~~ 2 : i Here ~ is an arbitrary positive value, mostly the smallest ~i . It serves as a weight unit of the deviations, i.e., for ~i = ~ it is gi = 1. It follows from (16.219) that a larger weight of a measurement results in a smaller deviation ~i .
2. Standard Deviations
The standard deviation of the weight unit is estimated as
~ (g)
v u & n uX = t givi2 (n ; 1) : i=1
(16.222)
794 16. Probability Theory and Mathematical Statistics We have to be sure that ~(g) < ~ . In the opposite case, if ~(g) > ~ , then there are xi values which have systematic deviations. The standard deviation of the single measurement is ~ (g) = ~ (g) ~ (16.223) ~i(g) = p gi ~ i where ~i (g) < ~i can be expected. The standard deviation of the weighted mean is: (g ) ~AM
= ~ (g)
v ,v uX u & n n n ! u t givi2 (n ; 1) X gi : tX gi = u i=1
i=1
3. Error Description
(16.224)
i=1
The error can be described as it is represented in 16.4.1.4, p. 792, either by the de nition of the error or by the t quantile with f degrees of freedom. The nal results of measurement sequences (n = 5) with di erent means xi (i = 1 2 : : : 5) and with di erent standard deviations ~AM i are given in Table 16.6. We calculate (xi)m = 1:5830 and we choose x0 = 1:585 and ~ = 0:009. With zi =vxi ; x0 , gi = ~ 2=~i2
u & n uX g v 2 (n ; 1) =
we get z = ;0:0036 and x = x0 + z = 1:582. The standard deviation is ~ (g) = t
i=1
i i
0:0088 < ~ and ~x = ~AM = 0:0027. The nal result is x = x ~x = 1:585 0:0027. Table 16.6 Error description of a measurement sequence
xi
~AM
(xi)m
~
1.573 1.580 1.582 1.589 1.591
i
0.010 0.004 0.005 0.009 0.011
2 ~AM
i
1.010;4 1.610;5 2.510;5 8.110;5 1.2110;4
= 1.583 = 0.009
gi
0.81 5.06 3.24 1.00 0.66 n X i=1
zi
;1.210;;23 ;5.010;3 ;3.010;3 +4.010 +6.010;3
gi
= 10.7
gizi
;9.710;;32 ;2.510;3 ;9.710;3 4.010 3.910;3 n X i=1
gi zi
= 3.610;2
zi2
1.4410;4 2.5010;5 9.010;6 1.610;5 3.610;5
gizi2
1.1610;4 1.2610;4 2.9110;5 1.610;5 2.3710;5 n X i=1
gizi2
= 3.110;4
16.4.2 Error Propagation and Error Analysis
Measured quantities appear in nal results often after a functional transformation. If the error is small, we can use a linear Taylor expansion with respect to the error. Then we talk about error propagation.
16.4.2.1 Gauss Error Propagation Law 1. Problem Formulation
Suppose we have to determine the numerical value and the error of a quantity z given by the function z = f (x1 x2 : : : xk ) of the independent variables xj (j = 1 2 : : : k). The mean value xj obtained from nj measured values is considered as realizations of the random variable xj , with variance j 2 . We wish to examine how the errors of the variables a ect the function value f (x1 x2 : : : xk ). It is assumed that
16.4 Calculus of Errors 795
the function f (x1 x2 : : : xk ) is di erentiable, its variables are stochastically independent. However they may follow any type of distribution with di erent variances j 2.
2. Taylor Expansion
Since the error represents relatively small changes of the independent variables, the function f (x1 x2 : : : xk ) can be approximated in the neighborhood of the mean xj by the linear part of its Taylor expansion with the coecients aj , so for its error %f we have: %f = f (x1 x2 : : : xk ) ; f (x1 x2 : : : xk ) (16.225a) k k @f dx + @f dx + + @f dx = X @f dx = X a dx %f df = @x (16.225b) 1 @x2 2 @xk k j=1 @xj j j=1 j j 1 where the partial derivatives @f=@xj are taken at (x1 x2 : : : xk ). The variance of the function is
f 2 = a1 2x1 2 + a2 2x2 2 + + ak 2xk 2 =
3. Approximation of the Variance f2
k X
j =1
aj 2 xj 2 :
(16.226)
Since the variances of the independent variables xj are unknown, we approximate them by the variance of their mean, which is determined from the measured values xjl (l = 1 2 : : : nl ) of the single variables as follows: nj X
(xjl ; xj )2 ~x2j = l=1n (n ; 1) : j j
(16.227)
With these values we form an approximation of f 2:
~f2 =
k X
j =1
aj 2~x2j :
(16.228)
The formula (16.228) is called the Gauss error propagation law.
4. Special Cases
1. Linear Case An often occurring case is when we add the absolute values of the errors of sequentially
occurring error quantities with aj = 1: q ~f = ~12 + ~22 + + ~k2 : (16.229) The pulse length is to be measured at the output of a pulse ampli er of a detector channel for spectrometry of radiation, whose error can be deduced for three components: 1. statistical energy distribution of the radiation of the part passing through the spectometer with an energy E0, which is characterized by ~Str , 2. statistical interference processes in the detector with ~Det , 3. electronic noice of the ampli er of the detector impulse ~el. The total pulse length has the error q 2 2 + ~f = ~Str + ~Det ~el2 : (16.230) 2. Power Rule The variables xj often occur in the following form: z = f (x1 x2 : : : xk ) = ax1 b1 x2 b2 : : : xk bk : (16.231) By logarithmic di erentiation we get the relative error df = b dx1 + b dx2 + + b dxk (16.232) k f 1 x1 2 x2 xk
796 16. Probability Theory and Mathematical Statistics from which by the error propagation law we get for the mean relative error:
v uk !2 X ~f = u u t bj ~xj : (16.233) f xj j =1 Suppose that the function f (x1 x2 x3 ) has the form f (x1 x2 x3) = px1 x22 x3 3, and the standard deviations are x1 , x2 and x3 . The relative errorvis then u 2 2 2 u z = ~ff = t 12 ~xx1 + 2 ~xx2 + 3 ~xx3 : 1 2 3
5. Di erence to the Maximum Error
Declaring the absolute or relative maximal error (16.216), (16.217) means that we do not use smoothing for the values of the measurement. For the determination of the relative and absolute error with the error propagation laws (16.228) or (16.231), smoothing between the measurement values xj means that we determine for them a con dence interval for a previously given level. This procedure is given in 16.4.1.4, p. 792.
16.4.2.2 Error Analysis
The general analysis of error propagation in the calculations of a function '(xi ), when quantities of higher order are neglected, is called error analysis. In the framework of the theory of error analysis we investigate using an algorithm, how an input error %xi a ects the value of '(xi ). In this relation we also talk about di erential error analysis. In numerical mathematics, error analysis means the investigation of the e ect of errors of methods, of roundings, and of input errors to the nal result (see 19.24]).
797
17 DynamicalSystemsandChaos
17.1 Ordinary Dierential Equations and Mappings 17.1.1 Dynamical Systems 17.1.1.1 Basic Notions
1. The Notion of Dynamical Systems and Orbits
A dynamical system is a mathematical object to describe the development of a physical, biological or another system from real life depending on time. It is de ned by a phase space M , and by a oneparameter family of mappings 't : M ! M , where t is the parameter (the time). In the following, the phase space is often IRn, a subset of it, or a metric space. The time parameter t is from IR (time continuous system) or from Z or from Z+ (time discrete system). Furthermore, it is required for arbitrary x 2 M that a) '0(x) = x and b) 't('s(x)) = 't+s(x) for all t s. The mapping '1 is denoted briey by '. In the following, the time set is denoted by ; , hence, ; = IR ; = IR+ ; = Z or ; = Z+. If ; = IR, then the dynamical system is also called a ow if ; = Z or ; = Z+, then the dynamical system is discrete. In case ; = IR and ; = Z, the properties a) and b) are satis ed for every t 2 ; , so the inverse mapping ('t );1 = ';t also exists, and these systems are called invertible dynamical systems. If the dynamical system is not invertible, then ';t (A) means the pre-image of A with respect to 't for an arbitrary set A M and arbitrary t > 0, i.e., ';t (A) = fx 2 M : 't(x) 2 Ag: If the mapping 't : M ! M is continuous or k times continuously di erentiable for every t 2 ; (here M IRn), then the dynamical system is called continuous or C k -smooth, respectively. For an arbitrary xed x 2 M , the mapping t 7;! 't (x) t 2 ; de nes a motion of the dynamical system starting from x at time t = 0: The image (x) of a motion starting at x is called the orbit (or the trajectory) through x, namely (x) = f't(x)gt2; . Analogously, the positive semiorbit through x is de ned by +(x) = f't(x)gt 0 and, if ; 6= IR+ or ; 6= Z+, then the negative semiorbit through x is de ned by ;(x) = f't (x)gt 0 . The orbit (x) is a steady state (also equilibrium point or stationary point) if (x) = fxg, and it is T-periodic if there exists a T 2 ; , T > 0, such that 't+T (x) = 't (x) for all t 2 ; , and T 2 ; is the smallest positive number with this property. The number T is called the period.
2. Flow of a Di erential Equation
Consider the ordinary linear planar dierential equation x_ = f (x) (17.1) where f : M ! IRn (vector eld ) is an r -times continuously di erentiable mapping and M = IRn or M is an open subset of IRn. In the following, thevEuclidean norm k k is used in IRn, i.e., for arbitrary
u n uX x 2 IRn, x = (x1 : : : xn), its norm is kxk = t x2i . If the mapping f is written componentwise i=1
f = (f1 : : : fn), then (17.1) is a system of n scalar di erential equations x_ i = fi (x1 : : : xn) i = 1 2 : : : n: The Picard{Lindelof theorem on the local existence and uniqueness of solutions of di erential equations locally and the theorem on the r-times dierentiability of solutions with respect to the initial values (see 17.11]) guarantee that for every x0 2 M , there exist a number " > 0, a sphere B (x0 ) = fx: kx ; x0 k < g in M and a mapping ': (;" ") B (x0 ) ! M such that: 1. '( ) is (r + 1)-times continuously di erentiable with respect to its rst argument (time) and rtimes continuously di erentiable with respect to its second argument (phase variable)
798 17. Dynamical Systems and Chaos 2. for every xed x 2 B (x0 ), ' ( x) is the locally unique solution of (17.1) in the time interval (;" ") which starts from x at time t = 0 i.e., @' (t x) = '_ (t x) = f ('(t x)) holds for every t 2 (;" "), @t
'(0 x) = x, and every other solution with initial point x at time t = 0 coincides with '(t x) for all small jtj. Suppose that every local solution of (17.1) can be extended uniquely to the whole of IR. Then there exists a mapping ': IR M ! M with the following properties: 1. '(0 x) = x for all x 2 M . 2. '(t + s x) = '(t '(s x)) for all t s 2 IR and all x 2 M . 3. '( ) is continuously di erentiable (r + 1) times with respect to its rst argument and r times with respect to the second one. 4. For every xed x 2 M , '( x) is a solution of (17.1) on the whole of IR. Then the C r -smooth ow generated by (17.1) can be de ned by 't : = '(t ). The motions '( x) : IR ! M of a ow of (17.1) are called integral curves. The equation x_ = (y ; x) y_ = rx ; y ; xz z_ = xy ; bz (17.2) is called a Lorenz system of convective turbulence (see also 17.2.4.3, p. 827). Here > 0 r > 0 and b > 0 are parameters. The Lorenz system corresponds to a C 1 ow on M = IR3.
3. Discrete Dynamical System
Consider the di erence equation xt+1 = '(xt ) (17.3) which can also be written as an assignment x 7;! '(x). Here ' : M !n M is a continuous or r times continuously di erentiable mapping, where in the second case M IR . If ' is invertible, then (17.3) de nes an invertible discrete dynamical system through the iteration of ', namely, 't = '| {z '} for t > 0 't = |';1 {z ';1} for t < 0 '0 = id: (17.4) t times ;t times If ' is not invertible, then the mappings 't are de ned only for t 0. For the realization of 't see (5.86), p. 296. A: The di erence equation xt+1 = xt (1 ; xt ) t = 0 1 : : : (17.5) with parameter 2 (0 4] is called a logistic equation . Here M = 0 1], and ' : 0 1] ! 0 1] is, for a xed , the function '(x) = x(1 ; x). Obviously, ' is in nitely many times di erentiable, but not invertible. Hence (17.5) de nes a non-invertible dynamical system. B: The di erence equation xt+1 = yt + 1 ; ax2t yt+1 = bxt t = 0 1 : : : (17.6) with parameters a > 0 and b 6= 0 is called a H enon mapping . The mapping ': IR2 ! IR2 corresponding to (17.6) is de ned by '(x y) = (y + 1 ; ax2 bx), is in nitely often di erentiable and invertible.
4. Volume Contracting and Volume Preserving Systems
The invertible dynamical system f'tgt2; on M IRn is called dissipative (respectively volume-preserving or conservative), if the relation vol('t(A)) < vol(A) (respective vol('t(A)) = vol(A)) holds for every set A M with a positive n-dimensional volume vol(A) and every t > 0 (t 2 ; ). A: Let ' in (17.3) be a C r -dieomorphism (i.e.: ' : M ! M is invertible, M IRn open, ' and ';1 are C r -smooth mappings) and let D'(x) be the Jacobi matrix of ' in x 2 M . The discrete system (17.3) is dissipative if j det D'(x)j < 1 for all x 2 M , and conservative if j det D'(x) j 1 in M .
17.1 Ordinary Dierential Equations and Mappings 799
B: For the system (17.6) D'(x y) = ;2bax 10 and so j det D'(x y)j b. Hence, (17.6) is dissipative if jbj < 1, and conservative if jbj = 1. The Henon mapping can be decomposed into three mappings (Fig. 17.1): First, the initial domain is stretched and bent by the mapping x0 = x y0 = y + 1 ; ax2 in a area-preserving way, then it is contracted in the direction of the x0 -axis by x00 = bx0 y00 = y0 (at jbj < 1), and nally it is reected with 00 00 000 00 000 00 respect to the line y = x by x = y y = x . y
y'
x
17.1.1.2 Invariant Sets
y'''
y''
x'
x''
x'''
Figure 17.1
1. -t and ! -Limit Set, Absorbing Sets
Let f' gt2; be a dynamical system on M . The set A M is invariant under f't g, if 't (A) = A holds for all t 2 ; , and positively invariant under f'tg, if 't(A) A holds for all t 0 from ; . For every x 2 M , the !-limit set of the orbit passing through x is the set !(x) = fy 2 M : 9 tn 2 ; tn ! +1 'tn (x) ! y as n ! +1g: (17.7) The elements of !(x) are called !-limit points of the orbit. If the dynamical system is invertible, then for every x 2 M , the set (x) = fy 2 M : 9 tn 2 ; tn ! ;1 'tn (x) ! y as n ! +1g (17.8) is called the -limit set of the orbit passing through x the elements of (x) are called the -limit points of the orbit. For many systems which are volume decreasing under the ow there exists a bounded set in phase space such that every orbit reaching it stays there as time increases. A bounded, open and connected set U M is called absorbing with respect to f't gt2; , if 't(U ) U holds for all positive t from ; . (U is the closure of U .) Consider the system of di erential equations x_ = ;y + x (1 ; x2 ; y2) y_ = x + y (1 ; x2 ; y2) (17.9a) in the plane. Using the polar coordinates x = r cos y = r sin , the solution of (17.9a) with initial state (r0 0) at time t = 0 has the form r(t r0) = 1 + (r0;2 ; 1) e;2t];1=2 (t 0 ) = t + 0 : (17.9b) This representation of the solution shows that the ow of (17.9a) has a periodic orbit with period 2, which can be given in the form ((1 0)) = f(cos t sin t) t 2 0 2]g. The limit sets of an orbit through p are: 8 > < (0 0) k p k < 1 0)) p 6= (0 0) (p) = > ((1 0)) k p k = 1 and !(p) = ((1 (0 0) p = (0 0): : kpk >1 Every open sphere Br = f(x y): x2 + y2 < r2g with r > 1 is an absorbing set for (17.9a).
2. Stability of Invariant Sets
Let A be an invariant set of the dynamical system f'tgt2; de ned on (M ). The set A is called stable , if every neighborhood U of A contains another neighborhood U1 U of A such that 't (U1) U holds
800 17. Dynamical Systems and Chaos for all t > 0. The set A, which is invariant under f'tg, is called asymptotically stable if it is stable and the following relations are satis ed: x2M t 9 % > 0 8dist( (17.10) x A) < % : dist(' (x) A) ;! 0 for t ! +1: Here, dist(x A) = yinf (x y): 2A
3. Compact Sets
Let (M ) be a metric space. A system fUigi2I of open sets is called an open covering of M if every point of M belongs to at least one Ui . The metric space (M ) is called compact if it is possible to choose nitely many Ui1 : : : Uir from every open covering fUigi2I of M such that M = Ui1 Uir holds. The set K M is called compact if it is compact as a subspace.
4. Attractor, Domain of Attraction t
Let f' gt2; be a dynamical system on (M ) and A an invariant set for f'tg. Then W (A) = fx 2 M : !(x) Ag is called the domain of attraction of A. A compact set 4 M is called an attractor of f'tgt2; on M if 4 is invariant under f't g and there is an open neighborhood U of 4 such that !(x) = 4 for almost every (in the sense of Lebesgue measure) x 2 U . 4 = ((1 0)) is an attractor of the ow of (17.9a). Here W (4) = IR2 n f(0 0)g: For some dynamical systems, a more general notion of an attractor makes sense. So, there are invariant sets 4 which have periodic orbits in every neighborhood of 4 which are not attracted by 4, e.g., the Feigenbaum attractor. The set 4 may not be generated by a single limit set !. A compact set 4 is called an attractor in the sense of Milnor of the dynamical system f't gt2; on M if 4 is invariant under f't g and the domain of attraction of 4 contains a set with positive Lebesgue measure.
17.1.2 Qualitative Theory of Ordinary Dierential Equations 17.1.2.1 Existence of Flows, Phase Space Structure 1. Extensibility of Solutions
Besides the di erential equation (17.1), which is called autonomous , there are di erential equations whose right-hand side depends explicitly on the time and they are called non-autonomous : x_ = f (t x): (17.11) Let f : IR M ! M be a C r -mapping with M IRn. By the new variable xn+1 := t, (17.11) can be interpreted as the autonomous di erential equation x_ = f (xn+1 x) x_ n+1 = 1. The solution of (17.11) starting from x0 at time t0 is denoted by '( t0 x0). In order to show the global existence of the solutions and with this the existence of the ow of (17.1), the following theorems are useful. 1. Criterion of Wintner and Conti If M = IRn in (17.1) and thereZ exists a continuous function +1 1 dr = +1 holds, then ! : 0 +1) ! 1 +1) such that kf (x)k ! (kxk) for all x 2 IRn and 0 ! (r ) every solution of (17.1) can be extended onto the whole of IR+. For example, the following functions satisfy the criterion of Wintner and Conti: !(r) = Cr + 1 or !(r) = C rj ln rj + 1, where C > 0 is a constant. 2. Extension Principle If a solution of (17.1) stays bounded as time increases, then it can be extended to the whole of IR+. Assumption: In the following, the existence of the ow f'tgt2IR of (17.1) is always assumed.
2. Phase Portrait
a) If '(t) is a solution of (17.1), then the function '(t + c) with an arbitrary constant c is also a solution. b) Two arbitrary orbits of (17.1) have no common point or they coincide. Hence, the phase space of (17.1) is decomposed into disjoint orbits. The decomposition of the phase space into disjoint orbits is called a phase portrait .
17.1 Ordinary Dierential Equations and Mappings 801
c) Every orbit, di erent from a steady state, is a regular smooth curve, which can be closed or not closed.
3. Liouville's Theorem t
Let f' gt2IR be the ow of (17.1), D M IRn be an arbitrary bounded and measurable set, Dt : = 't(D) and Vt : = vol(Dt ) be the n-dimensional volume of Dt (Fig. 17.2a). Then the relation d V = Z divf (x) dx holds for arbitrary t 2 IR . For n = 3, Liouville's theorem states: dt t Dt
d V = Z Z Z divf (x x x ) dx dx dx : 1 2 3 1 2 3 dt t Dt
(17.12) 2
T Dt
a)
D
M b)
Figure 17.2
Corollary: If divf (x) < 0 in M holds for (17.1), then the ow of (17.1) is volume contracting. If divf (x) 0 in M holds, then the ow of (17.1) is volume preserving. A: For the Lorenz system (17.2), divf (x Zy ZzZ) ;( +1+ b). Since > 0 and b > 0, divf (x y z) < dV = 0 holds. With Liouville's theorem, dt ;( + 1 + b) dx1 dx2 dx3 = ;( + 1 + b) Vt obviously t Dt
holds for any arbitrary bounded and measurable set D IR3. The solution of the linear di erential equation V_ t = ;( + 1 + b) Vt is Vt = V0 e;(+1+b)t , so that Vt ! 0 follows for t ! +1. B: Let U IRn IRn be an open subset and H : U ! IR a C 2-function. Then, x_ i = @H @yi (x y) y_ i = ; @H @xi (x y) (i = 1 2 : : : n) is called a Hamiltonian dierential equation . The function H is called the Hamiltonian of the system. If f denotes the right-hand side of this di erential equation, then obviously
n " @2H 2H X @ divf (x y) = (x y ) ; @yi@xi (x y) 0. Hence, the Hamiltonian di erential equations are i=1 @xi @yi volume preserving.
17.1.2.2 Linear Di erential Equations 1. General Statements n
Let A(t) = aij (t)]ij=1 be a matrix function on IR , where every component aij : IR ! IR is a continuous function, and let b: IR ! IRn be a continuous vector function on IR . Then x_ = A(t)x + b(t) (17.13a) is called an inhomogeneous linear rst-order dierential equation in IRn, and x_ = A(t)x (17.13b) is the corresponding homogeneous linear rst-order dierential equation . 1. Fundamental Theorem for Homogeneous Linear Di erential Equations Every solution of (17.13a) exists on the whole of IR. The set of all solutions of (17.13b) forms an n-dimensional vector subspace LH of the C 1-smooth vector functions over IR .
802 17. Dynamical Systems and Chaos 2. Fundamental Theorem for Inhomogeneous Linear Di erential 1Equations The set of all
solutions LI of (17.13a) is an n-dimensional ane vector subspace of the C -smooth vector functions over IR of the form LI = '0 + LH , where '0 is an arbitrary solution of (17.13a). Let '1 : : : 'n be arbitrary solutions of (17.13b) and = '1 : : : 'n] the corresponding solution matrix . Then satis es the matrix dierential equation Z_ (t) = A(t)Z (t), on IR, where Z 2 IRnn. If the solutions '1 : : : 'n form a basis of LH , then = '1 : : : 'n] is called the fundamental matrix of (17.13b). W (t) = det (t) is the Wronskian determinant with respect to the solution matrix of (17.13b). The formula of Liouville states that: W_ (t) = Sp A(t) W (t) (t 2 IR): (17.13c) For a solution matrix, either W (t) 0 on IR or W (t) 6= 0 for all t 2 IR . The system '1 : : : 'n is a basis of LH , if and only if det '1 (t) : : : 'n(t)] 6= 0 for a t (and so for all t). 3. Theorem (Variation of Constants Formula) Let be an arbitrary fundamental matrix of (17.13b). Then the solution ' of (17.13a) with initial point p at time t = can be represented in the form
Zt '(t) = (t)( );1 p + (t)(s);1 b(s) ds (t 2 IR):
(17.13d)
2. Autonomous Linear Di erential Equations
Consider the di erential equation x_ = A x (17.14) where A is a constant matrix of type (n nn). The operator norm (see also 12.5.1.1, p. 619) of a matrix A is given by kAk = maxfkA xk x 2 IR kxk 1g, where for the vectors of IRn the Euclidean norm is again considered. Let A and B be two arbitrary matrices of type (n n). Then a) kA + B k kAk + kB k b) kAk = jj kAk ( 2 IR) n c) kAxk kAkkxk x 2 IR ) d) kAB k kAkkB k e) kAk = pmax where max is the greatest eigenvalue of AT A . The fundamental matrix with initial value En at time t = 0 of (17.14) is the matrix exponential function 1 X A2 t2 Aiti eAt = En + At (17.15) 1 ! + 2 ! + = i=0 i ! with the following properties: a) The series of eAt is uniformily convergent with respect to t on an arbitrary compact time interval and absolutely convergent for every xed t b) keAtk ekAkt (t 0) c) dtd (eAt ) = (eAt) = AeAt = eAt A (t 2 IR) d) e(t+s)A = etA esA (s t 2 IR) e) eAt is regular for all t and (eAt);1 = e;At f) ifA+ABand BA are commutative matrices of type (n n), i.e., AB = BA holds, then B eA = eA B and e = e eB g) if A and B are matrices of type (n n) and B is regular, then eBAB;1 = B eA B ;1 .
17.1 Ordinary Dierential Equations and Mappings 803
3. Linear Di erential Equations with Periodic Coe cients
We consider the homogeneous linear di erential equation (17.13b), where A(t) = aij (t)]nij=1 is a Tperiodic matrix function, i.e., aij (t) = aij (t + T ) (8t 2 IR i j = 1 : : : n). In this case we call (17.13b) a linear T-periodic dierential equation. Then every fundamental matrix of (17.13b) can be written in the form (t) = G(t)etR , where G(t) is a smooth, regular T -periodic matrix function and R is a constant matrix of type (n n) (Floquet's theorem). Let (t) be the fundamental matrix of the T -periodic di erential equation (17.13b), normed at t = 0, i.e., (0) = En, and let (t) = G(t)etR be a representation of it according to Floquet's theorem. The matrix (T ) = eRT is called the monodromy matrix of (17.13b) the eigenvalues j of (T ) are the multipliers of (17.13b). A number 2 C is a multiplier of (17.13b) if and only if there exists a solution ' 6 0 of (17.13b) such that '(t + T ) = '(t) (t 2 IR) holds.
17.1.2.3 Stability Theory
1. Ljapunov Stability and Orbital Stability
Consider the non-autonomous di erential equation (17.11). The solution '(t t0 x0) of (17.11) is said to be stable in the sense of Lyapunov if: 8 t1 t0 8 " > 0 9 = (" t1) 8x1 2 M : k'(t t1 x1) ; '(t t0 x0 )k < " kx1 ; '(t1 t0 x0 )k < (17.16a) 8 t t1 : The solution '(t t0 x0 ) is called asymptotically stable in the sense of Lyapunov, if it is stable and: 8 t1 t0 9 % = %(t1 ) 8 x1 2 M : k'(t t1 x1 ) ; '(t t0 x0 )k ! 0 kx1 ; '(t1 t0 x0)k < % (17.16b) for t ! +1: For the autonomous di erential equation (17.1), there are other important notions of stability besides the Lyapunov stability. The solution '(t x0 ) of (17.1) is called orbitally stable (asymptotically orbitally stable), if the orbit (x0 ) = f'(t x0 ) t 2 IRg is stable (asymptotically stable) as an invariant set. A solution of (17.1) which represents an equilibrium point is Lyapunov stable exactly if it is orbitally stable. The two types of stability can be di erent for periodic solutions of (17.1). Let a ow be given in IR3 , whose invariant set is the torus T 2. Locally, let the ow be described in a rectangular coordinate system by !_ 1 = 0 !_ 2 = f2 (!1), where f2 : IR ! IR is a 2 periodic smooth function, for which: 8!1 2 IR 9 U1 (neighborhood of !1) 8 1 2 2 U1 : f2 ( 1) 6= f2 ( 2): 1 6= 2 An arbitrary solution satisfying the initial conditions (!1(0) !2(0)) can be given on the torus by !1(t) !1(0) !2(t) = !2(0) + f2(!1(0)t (t 2 IR): From this representation it can be seen that every solution is orbitally stable but not Lyapunov stable (Fig. 17.2b).
2. Asymptotical Stability, Theorem of Lyapunov
A scalar-valued function V is called positive denite in a neighborhood U of a point p 2 M IRn, if: 1. V : U M ! IR is continuous. 2. V (x) > 0 for all x 2 U n fpg and V (p) = 0 . Let U M be an open subset and V : U ! IR a continuous function. The function V is called a Lyapunov function of (17.1) in U , if V ('(t)) does not increase while for the solution '(t) 2 U holds. Let V : U ! IR be a Lyapunov function of (17.1) and let V be positive de nite in a neighborhood U of p. Then p is stable. If the condition V ('(t x0 )) = constant (t 0) always yields '(t x0) p for
804 17. Dynamical Systems and Chaos a solution ' of (17.1) with '(t x) 2 U (t 0), i.e., if the Lyapunov function is constant along a complete trajectory, then this trajectory can only be an equilibrium point, and the equilibrium point p is also asymptotically stable. The point (0 0) is a steady point of the planar dierential equation x_ = y y_ = ;x ; x2 y. The function V (x y) = x2 + y2 is positive de nite in every neighborhood of (0 0) and for its derivative d 2 2 dt V (x(t) y(t)) = ;2x(t) y(t) < 0 holds along an arbitrary solution for x(t)y(t) 6= 0. Hence, (0 0) is asymptotically stable.
3. Classication and Stability of Steady States
Let x0 be an equilibrium point of (17.1). In the neighborhood of x0 the local behavior of the orbits of (17.1) can be described under certain assumptions by the variational equation y_ = D f (x0)y, where D f (x0 ) is the Jacobian matrix of f in x0 . If D f (x0 ) does not have an eigenvalue j with Re j = 0, then the equilibrium point x0 is called hyperbolic. The hyperbolic equilibrium point x0 is of type (m k) if D f (x0 ) has exactly m eigenvalues with negative real parts and k = n ; m eigenvalues with positive real parts. The hyperbolic equilibrium point of type (m k) is called a sink, if m = n, a source, if k = n, and a saddle point, if m 6= 0 and k 6= 0 (Fig. 17.3). A sink is asymptotically stable sources and saddles are unstable (theorem on stability in the rst approximation). Within the three topological basic types of hyperbolic equilibrium points (sink, source, saddle point) further algebraic distinctions can be made. A sink (source) is called a stable node (unstable node) if every eigenvalue of the Jacobian matrix is real, and a stable focus (unstable focus ) if there are eigenvalues with non-vanishing imaginary parts. For n = 3, we get a classi cation of saddle points as saddle nodes and saddle foci.
Type of equilibrium point
Sink
Source
Saddle point
Eigenvalues of the Jacobian matrix Phase portrait
4. Stability of Periodic Orbits
Figure 17.3
Let '(t x0 ) be a T -periodic solution of (17.1) and (x0 ) = f'(t x0 ) t 2 0 T ]g its orbit. Under certain assumptions, the phase portrait in a neighborhood of (x0 ) can be described by the variational equation y_ = D f ('(t x0 )) y. Since A(t) = D f ('(t x0)) is a T -periodic continuous matrix function of type (n n), it follows from the Floquet theorem (see 17.1.2.2,3., p. 803) that the fundamental matrix x0 (t) of the variational equation can be written in the form x0 (t) = G(t)eRt , where G is a T -periodic regular smooth matrix function with G(0) = En, and R represents a constant matrix of type (n n) which is not uniquely given. The matrix x0 (T ) = eRT is called the monodromy matrix of the periodic orbit (x0 ), and the eigenvalues 1 : : : n of eRT are called multipliers of the periodic orbit (x0 ). If the orbit (x0 ) is represented by another solution '(t x1 ), i.e., if (x0) = (x1), then the multipliers of (x0) and (x1 ) coincide. One of the multipliers of a periodic orbit is always equal to one (Andronov{Witt theorem). Let 1 : : : n;1 n = 1 be the multipliers of the periodic orbit (x0 ) and let x0 (T ) be the monodromy matrix of (x0). Then n X
j =1
j = Sp x0 (T ) and
n Y
j =1
j = det x0 (T ) = exp
ZT 0
!
SpD f ('(t x0 ))dt
17.1 Ordinary Dierential Equations and Mappings 805
= exp
ZT 0
!
divf ('(t x0)) dt :
R T
(17.17)
Hence, if n = 2, then 2 = 1 and 1 = exp 0 divf ('(t x0 )) dt . Let '(t (1 0)) = (cos t sin t) be a 2-periodic solution of (17.9a). The matrix A(t) of the variational equation with respect to this solution is ! 2 cos2 t ;1 ; sin 2t : A(t) = D f ('(t (1 0))) = ; 2 1 ; sin 2t ;2 sin t The fundamental matrix (10) (t) normed at t = 0 is given by ! ;2t t ; sin t = cos t ; sin t e;2t 0 (10) (t) = ee;2t cos sin t cos t 0 1 sin t cos t where the last product is a Floquet representation of (10) (t). Thus, 1 = e;4 and 2 = 1. The multipliers can be determined without the Floquet representation. For system (17.9a) divf (x y) = 2 ; 4x 2 ; R 4y2 holds, and hence divf (cos t sin t) ;2. According to the formula above, 1 = exp 02 ;2dt = exp(;4).
5. Classication of Periodic Orbits
If the periodic orbit of (17.1) has no further multiplier on the complex unit circle besides n = 1, then is called hyperbolic . The hyperbolic periodic orbit is of type (m k) if there are m multipliers inside and k = n ; 1 multipliers outside the unit circle. If m > 0 and k > 0, then the periodic orbit of type (m k) is called a saddle point. According to the Andronov{Witt theorem a hyperbolic periodic orbit of (17.1) of type (n ; 1 0) is asymptotically stable. Hyperbolic periodic orbits of type (m k) with k > 0 are unstable. A: A periodic orbit = f'Z (t) t 2 0 T ]g in the plane with multipliers 1 and 2 = 1 is asymptotiT cally stable if j 1j < 1, i.e., if divf ('(t)) dt < 0. 0 B: If there is a further multiplier besides n = 1 on the complex unit circle, then the Andronov{ Witt theorem cannot be applied. The information about the multipliers is not sucient for the stability analysis of the periodic orbit. C: As an example, let the planar system x_ = ;y + x f (x2 + y2) y_ = x + y f (x2 + y2) be given by the smooth function f : (0 +1) ! IR, which additionally satis es the properties f (1) = f 0(1) = 0 and f (r)(r ; 1) < 0 forall r 6= 1 r >0. Obviously, '(t) = (cos t sin t) is a 2-periodic solution of the t ; sin t 1 0 is the Floquet representation of the fundamental matrix. system and (10) (t) = cos sin t cos t 0 1 It follows that 1 = 2 = 1. The use of polar coordinates results in the system r_ = r f (r2) _ = 1. This representation yields that the periodic orbit ((1 0)) is asymptotically stable.
6. Properties of Limit Sets, Limit Cycles
The - and ! -limit sets de nned in 17.1.1.2, p. 799, have with respect to the ow of the di erential equation (17.1) with M IR the following properties. Let x 2 M be an arbitrary point. Then: a) The sets (x) and !(x) are closed. b) If +(x) (respectively ;(x)) is bounded, then !(x) 6= (respectively (x) 6= ) holds. Furthermore, !(x) (respectively (x)) are in this case invariant under the ow (17.1) and connected. If for instance, +(x) is not bounded, then !(x) is not necessarily connected (Fig. 17.4a). For a planar autonomous di erential equation (17.1), (i.e., M IR2 ) the Poincare{Bendixson theorem is valid. Poincar e{Bendixson Theorem: Let '( p) be a non-periodic solution of (17.1), for which +(p) is
806 17. Dynamical Systems and Chaos y
y x
w(x)
a)
x b)
x c)
Figure 17.4 bounded. If !(p) contains no equilibrium point of (17.1), then !(p) is a periodic orbit of (17.1). Hence, for autonomous di erential equations in the plane, attractors more complicated than an equilibrium point or a periodic orbit are not possible. A periodic orbit of (17.1) is called a limit cycle, if there exists an x 2= such that either !(x) or (x) holds. A limit cycle is called a stable limit cycle if there exists a neighborhood U of such that = !(x) holds for all x 2 U , and an unstable limit cycle if there exists a neighborhood U of such that = (x) holds for all x 2 U . A: For the ow of (17.9a), the property = !(p) for all p 6= (0 0) is valid for the periodic orbit = f(cos t sin t) t 2 0 2)g. Hence, U = IR2nf(0 0)g is a neighborhood of such that with it, is a stable limit cycle (Fig. 17.4b). B: In contrast, for the linear di erential equation x_ = ;y y_ = x, the orbit = f(cos t sin t) t 2 0 2]g is a periodic orbit, but not a limit cycle (Fig. 17.4c).
7. m-Dimensional Embedded Tori as Invariant Sets
A di erential equation (17.1) can have an m-dimensional torus as an invariant set. An m-dimensional torus T m embedded into the phase space M IRn is de ned by a di erentiable mapping g : IRm ! IRn, which is supposed to be 2-periodic in every coordinate !i as a function (!1 : : : !m) 7! g(!1 : : : !m). In simple cases, the motion of the system (17.1) on the torus can be described in a rightangular coordinate system by the di erential equations !_ i = !i (i = 1 2 : : : m). The solution of this system with initial values (!1(0) : : : !m(0)) at time t = 0 is !i(t) = !it + !i(0) (i = 1 2 : : : m t 2 IR). A continuous function f : IR ! IRn is called quasiperiodic if f has a representation of the form f (t) = g(!1t !2t : : : !nt), where g is also a di erentiable function as above, which is 2-periodic in every component, and the frequencies !i are incommensurable , i.e., there are no such integers ni with m 2 P ni > 0 for which n1!1 + + nm !m = 0 holds. i=1
17.1.2.4 Invariant Manifolds
1. Denition, Separatrix Surfaces
Let be a hyperbolic equilibrium point or a hyperbolic periodic orbit of (17.1). The stable manifold W s( ) (respectively unstable manifold W u( )) of is the set of all points of the phase space such that the orbits tending to as t ! +1 (resepctively t ! ;1) pass through these points: (17.18) W s( ) = fx 2 M : !(x) = g and W u( ) = fx 2 M : (x) = g: Stable and unstable manifolds are also called separatrix surfaces . In the plane, the di erential equation x_ = ;x y_ = y + x2 (17.19a) is considered. The solution of (17.19a) with initial state (x0 y0) at time t = 0 is explicitly given by 2 (17.19b) '(t x0 y0) = (e;t x0 ety0 + x30 (et ; e;2t )) : For the stable and unstable manifolds of the equilibrium point (0 0) of (17.19a) we get: 2 W s((0 0)) = f(x0 y0): t!lim '(t x0 y0) = (0 0)g = f(x0 y0): y0 + x30 = 0g, +1
17.1 Ordinary Dierential Equations and Mappings 807
W u((0 0)) = f(x0 y0): t!;1 lim '(t x0 y0) = (0 0)g = f(x0 y0): x0 = 0 y0 2 IRg (Fig. 17.5a). y
u
W ((0,0))
LxN
x s
M
W ((0,0)) a)
N LxM
b)
Figure 17.5 Let M x M and Lx N be the corresponding tangent planes to M and N through x. The surfaces M and N are transversal to each other if for all x 2 M \ N the following relation holds: dim LxM + dim Lx N ; n = dim (LxM \ LxN ): For the section represented in Fig. 17.5b we have dim LxM = 2 dim LxN = 1 and dim(LxM \ LxN ) = 0. Hence, the section represented in Fig. 17.5b is transversal. and N be two smooth surfaces in IRn, and let L
2. Theorem of Hadamard and Perron
Important properties of separatrix surfaces are given by the Theorem of Hadamard and Perron : Let be a hyperbolic equilibrium point or a hyperbolic periodic orbit of (17.1). a) The manifolds W s( ) and W u( ) are generalized C r -surfaces, which locally look like C r -smooth elementary surfaces. Every orbit of (17.1), which does not tend to for t ! +1 or t ! ;1, respectively, leaves a suciently small neighborhood of for t ! +1 or t ! ;1, respectively. b) If = x0 is an equilibrium point of stype (m k), uthen W s(x0 ) and W u(x0 ) are surfaces of dimension m and k, respectively. The surfaces W (x0 ) and W (x0 ) are tangent at x0 to the stable vector subspace
E s = fy 2 IRn : e D f (x0 )t y ! 0 for t ! +1g of equation y_ = D f (x0 )y (17.20a) and the unstable vector subspace E u = fy 2 IRn : e D f (x0 )t y ! 0 for t ! ;1g of equation y_ = D f (x0 )y respectively: (17.20b) c) If is a hyperbolic periodic orbit of type (m k) then W s( ) and W u( ) are surfaces of dimension m + 1 and k + 1, respectively, and they intersect each other transversally along (Fig. 17.6a). A: To determine a local stable manifold through the steady state (0 0) of the di erential equation s ((0 0)) has the following form: (17.19a) we suppose that Wloc s Wloc((0 0)) = f(x y): y = h(x) jxj < % h: (;% %) ! IR di erentiableg. s ((0 0)). Based on the invariance, for times s near Let (x(t) y(t)) be a solution of (17.19a) lying in Wloc to t we get y(s) = h(x(s)). By di erentiation and representation of x_ and y_ from the system (17.19a) we get the initial value problem h0(x) (;x) = h(x) + x2 h(0) = 0 for the unknown function h(x). If we are looking for the solution in the form of a series expansion h(x) = a22 x2 + a3!3 x3 + , where h0(0) = 0 is taken under consideration, then we get by comparing the coecients a2 = ; 23 and ak = 0 for k 3. B: For the system x_ = ;y + x(1 ; x2 ; y2) y_ = x + y(1 ; x2 ; y2) z_ = z (17.21) with a parameter > 0, the orbit = f(cos t sin t 0) t 2 0 2]g is a periodic orbit with multipliers 1 = e;4 2 = e 2 and 3 = 1. In cylindrical coordinates x = r cos y = r sin z = z, with initial values (r0 0 z0) at time t = 0,
808 17. Dynamical Systems and Chaos the solution of (17.21) has the representation (r(t r0) (t 0 ) e t z0 ), where r(t r0) and (t 0 ) is the solution of (17.9a) in polar coordinates. Consequently, W s( ) = f(x y z): z = 0g n f(0 0 0)g and W u( ) = f(x y z): x2 + y2 = 1g (cylinder). Both separatrix surfaces are shown in Fig. 17.6b. u
W (g)
u
W (g)
s
W (g)
s
W (g) g a)
b)
Figure 17.6
3. Local Phase Portraits Near Steady States for n = 3
We consider now the di erential equation (17.1) with the hyperbolic equilibrium point 0 for n = 3. Set A = D f (0) and let det E ; A] = 3 + p2 + q + r be the characteristic polynomial of A. With the notation = p q ; r and % = ;p2 q2 + 4p3r + 4q3 ; 18pqr + 27r2 (discriminant of the characteristic polynomial), the di erent equilibrium point types are characterized in Table 17.1.
4. Homoclinic and Heteroclinic Orbits
Suppose 1 and 2 are two hyperbolic equilibrium points or periodic orbits of (17.1). If the separatrix surfaces W s(1) and W u(2) intersect each other, then the intersection consists of complete orbits. For two equilibrium points or periodic orbits, the orbit W s(1) \ W u(2) is called heteroclinic if 1 6= 2 (Fig. 17.7a), and homoclinic if 1 = 2. Homoclinic orbits of equilibrium points are also called separatrix loops (Fig. 17.7b). W (g2) s
u
W (g1)
g2
g1 a)
u
W (x0)
W (g1) s
u
W (g2)
x0 b)
s
W (x0)
Figure 17.7 Consider the Lorenz system (17.2) with xed parameters = 10 b = 8=3 and with variable r. The equilibrium point (0 0 0) of (17.2) is a saddle for 1 < r < 13:926 : : : , which is characterized by a twodimensional stable manifold W s and a one-dimensional unstable manifold W u. If r = 13:926 : : : , then there are two separatrix loops at (0 0 0), i.e., as t ! +1 branches of the unstable manifold return (over the stable manifold) to the origin (see 17.9]).
17.1.2.5 Poincare Mapping
1. Poincare Mapping for Autonomous Di erential Equations P
Let = f'(t x0 ) t 2 0 T ]g be a T -periodic orbit of (17.1) and a (n ; 1)-dimensional smooth hypersurface, which intersects the orbit transversally in x0 (Fig. 17.8a). Then, there is a neighborhood U of x0 and a smooth function : U ! IR such that (x0 ) = T and '( (x) x) 2 P for all x 2 U . P P The mapping P : U \ ! with P (x) = '( (x) x) is called the Poincar e mapping of at x0 . If the right-hand side f of (17.1) is r times continuously di erentiable, then P is also r times continuously di erentiable . The eigenvalues of the Jacobi matrix DP (x0) are the multipliers 1 : : : n;1 of the periodic orbit. They do not depend on the choice of x0 on and on the choice of the transversal surface.
17.1 Ordinary Dierential Equations and Mappings 809
Parameter domain > 0 r > 0 q>0
Table 17.1 Steady state types in three{dimensional phase spaces
%0
Type of equili- Roots of the charac- Dimension of brium point teristic polynomial W s and W u stable node stable focus
% < 0:
Parameter domain < 0 r < 0 q>0
> 0 r < 0 q 0 or r0
%0
Type of equili- Roots of the charac- Dimension of brium point teristic polynomial W s and W u unstable node unstable focus
< 0 r > 0 q 0 or r>0 q>0 % < 0:
Imj = 0 j > 0 j = 1 2 3 Re12 > 0 3 > 0
dim W s = 0 dim W u = 3
% > 0:
%0
Type of equili- Roots of the charac- Dimension of brium point teristic polynomial W s and W u saddle node saddle focus
% < 0:
Parameter domain
dim W s = 3 dim W u = 0
% > 0:
% < 0:
Parameter domain
Imj = 0 j < 0 j = 1 2 3 Re12 < 0 3 < 0
Imj = 0 12 < 0 3 > 0 Re12 < 0 3 > 0
dim W s = 2 dim W u = 1
% > 0:
%0
Type of equili- Roots of the charac- Dimension of brium point teristic polynomial W s and W u saddle node saddle focus
Imj = 0 12 > 0 3 < 0 Re12 > 0 3 < 0 % > 0:
dim W s = 1 dim W u = 2
810 17. Dynamical Systems and Chaos A system (17.3) in M = U can be connected with the Poincare mapping, which makes sense until the iterates stay in U . The periodic orbits of (17.1) correspond to the equilibrium points of this discrete system, and the stability of these equilibrium points corresponds to the stability of the periodic orbits of (17.1). s x x0 j(t(x)x) a)
x1 b)
x0 steady state
M
x2
periodic orbit
Figure 17.8 We consider for the system (17.9a) the transversal hyperplanes X = f(r ): r > 0 = 0 g in polar coordinate form. For these planes U = P can be chosen. Obviously, (r) = 2 (8r > 0) and so P (r) = 1 + (r;2 ; 1) e;4 ];1=2 where the solution representation of (17.9a) is used. It is also valid that P (P) = P P (1) = 1 and P 0(1) = e;4 < 1.
2. Poincare Mapping for Non-Autonomous Time-Periodic Di erential Equations
A non-autonomous di erential equation (17.11), whose right-hand side f has period T with respect to t, i.e., for which f (t + T x) = f (t x) (8 t 2 IR 8 x 2 M ) holds, is interpreted as an autonomous di erential equation x_ =Pf (s x) s_ = 1 with cylindrical phase space M fs mod T g. Let s0 2 fs mod T g be arbitrary. Then, = M fs0 g is a transversal plane (Fig. 17.8b). The Poincare mapping is given globally as P : P ! P over x0 7;! '(s0 + T s0 x0 ), where '(t s0 x0 ) is the solution of (17.11) with the initial state x0 at time s0 .
17.1.2.6 Topological Equivalence of Di erential Equations 1. Denition
Suppose, besides (17.1) with the corresponding ow f'tgt2IR, that a further autonomous di erential equation x_ = g(x) (17.22) is given, where g : N ! IRn is a C r -mapping on the open set N IRn. Of course, the ow ft gt2IR of (17.22) should exist. The di erential equations (17.1) and (17.22) (or their ows) are called topologically equivalent if there exists a homeomorphism h : M ! N (i.e., h is bijective, h and h;1 are continuous), which transforms each orbit of (17.1) to an orbit of (17.22) preserving the orientation, but not necessarily preserving the parametrization. The systems (17.1) and (17.22) are topologically equivalent if there also exists a continuous mapping : IR M ! IR , besides the homeomorphism h : M ! N , such that is strictly monotonically increasing at every xed x 2 M , maps IR onto IR, with (0 x) = 0 for all x 2 M and asatis es the relation h('t (x)) = (tx) (h(x)) for all x 2 M and t 2 IR . In the case of topological equivalence, the equilibrium points of (17.1) go over into steady states of (17.22) and periodic orbits of (17.1) go over into periodic orbits of (17.22), where the periods are not necessarily coincident. Hence, if two systems (17.1) and (17.22) are topologically equivalent, then the topological structure of the decomposition of the phase spaces into orbits is the same. If two systems (17.1) and (17.22) are topologically equivalent with the homeomorphism h: M ! N and if h preserves the parametrization, i.e., h('t(x)) = t (h(x)) holds for every t x, then (17.1) and (17.22) are called
17.1 Ordinary Dierential Equations and Mappings 811 topologically conjugate. Topological equivalence or conjugacy can also refer to subsets of the phase spaces M and N . Suppose, e.g., (17.1) is de ned on U1 M and (17.22) on U2 N . We say that (17.1) on U1 is topologically equivalent to (17.22) on U2 if there exists a homeomorphism h : U1 ! U2 which transforms the intersection of the orbits of (17.1) with U1 into the intersection of the orbits of (17.22) with U2 preserving the orientation. A: Homeomorphisms for (17.1) and (17.22) are mappings where, e.g., stretching and shrinking of the orbits are allowed cutting and closing are not. The ows corresponding to phase portraits of Fig. 17.9a and Fig. 17.9b are topologically equivalent the ows shown in Fig. 17.9a and Fig. 17.9c are not.
b)
a)
c)
Figure 17.9 B: Consider the two linear planar equations (see 17.11]) 1 dierential ;3 and B = 4 0 . The phase portraits of these systems x_ = Ax and x_ = Bx with A = ; ;3 ;1 0 ;8 close to (0 0) are shown in Fig. 17.10a and Fig. 17.10b. The homeomorphism h : IR2 ! IR2 with h(x) = Rx , where R = p1 11 ;11 , and the function 2 1 2 : IR IR ! IR with (t x) = 2 t transform the orbits of the rst system into the orbits of the second one. Hence, the two systems are topologically equivalent. x2
x2 x1
x1 b)
a)
Figure 17.10
2. Theorem of Grobman and Hartman
Let p be a hyperbolic equilibrium point of (17.1). Then, in a neighborhood of p the di erential equation (17.1) is topologically equivalent to its linearization y_ = Df (p)y.
17.1.3 Discrete Dynamical Systems
17.1.3.1 Steady States, Periodic Orbits and Limit Sets 1. Types of Steady State Points
Let x0 be an equilibrium point of (17.3) with M IRn. The local behavior of the iteration (17.3) close to x0 is given, under certain assumptions, by the variational equation yt+1 = D'(x0)yt t 2 ; . If D'(x0) has no eigenvalue i with jij = 1, then the steady state point x0 , analogously to the di erential equation case, is called hyperbolic. The hyperbolic equilibrium point x0 is of type (m k) if Df (x0) has exactly m eigenvalues inside and k = n ; m eigenvalues outside the complex unit circle. The hyperbolic equilibrium point of type (m k) is called a sink for m = n, a source for k = n and a saddle point for
812 17. Dynamical Systems and Chaos m > 0 and k > 0. A sink is asymptotically stable sources and saddles are unstable (theorem on stability in the rst approximation for discrete systems).
2. Periodic Orbits
Let (x0 ) = f'k (x0) k = 0 T ; 1g be a T -periodic orbit (T 2) of (17.3). If x0 is a hyperbolic equilibrium point of the mapping 'T , then (x0) is called hyperbolic . The matrix D'T (x0) = D'('T ;1(x0 )) D'(x0 ) is called the monodromy matrix the eigenvalues i of D'T (x0 ) are the multipliers of (x0 ). If all multipliers i of (x0 ) have an absolute value less than one, then the periodic orbit (x0 ) is asymptotically stable.
3. Properties of ! -Limit Set
Every ! -limit set !(x) of (17.3) with M = IRn is closed, and !('(x)) = !(x). If the semiorbit + (x) is bounded, then !(x) 6= and !(x) is invariant under '. Analogous properties are valid for -limit sets. Suppose the di erence equation xt+1 = ;xt t = 0 1 is given on IR with '(x) = ;x. Obviously, the relations !(1) = f1 ;1g !('(1)) = !(;1) = !(1), and '(!(1)) = !(1) are satis ed for x = 1. We mention that !(1) is not connected, is di erent from the case of di erential equations.
17.1.3.2 Invariant Manifolds 1. Separatrix Surfaces
Let x0 be an equilibrium point of (17.3). Then W s(x0) = fy 2 M : 'i(y) ! x0 for i ! +1g is called a stable manifold and W u(x0 ) = fy 2 M : 'i(y) ! x0 for i ! ;1g an unstable manifold of x0 . Stable and unstable manifolds are also called separatrix surfaces .
2. Theorem of Hadamard and Perron
The theorem of nHadamard and Perron describes the properties of separatrix surfaces for discrete systems in M IR : If x0 is a hyperbolic equilibrium point of (17.3) of type (m k), then W s(x0 ) and W u(x0 ) are generalized C r -smooth surfaces of dimension m and k, respectively, which locally look like C r -smooth elementary surfaces. The orbits of (17.3), which do not tend to x0 for i ! +1 or i ! ;1, leave a suciently small neighborhood of x0 for i ! +1 or i ! ;1, respectively. The surfaces W s(x0 ) and W u(x0 ) are tangent at x0 to the stable vector subspace E s = fy 2 IRn : nD'(x0 )]i y ! 0 for i ! ;1g of yi+1 = D'(x0)yi and the unstable vector subspace E u = fy 2 IR : D'(x0 )]iy ! 0 for i ! ;1g, respectively. We consider the following time discrete dynamical system from the family of Henon mappings: xi+1 = x2i + yi ; 2 yi+1 = xi i 2 Z: (17.23) p p p p Both hyperbolic equilibrium points of (17.23) are P1 = ( 2 2) and P2 = (; 2 ; 2). Determination of the p p local stable and unstable manifolds of P1: The variableptransformation xi = i + 2 yi = i + 2 transforms system (17.23) into the psystemp i+1 = i2 + 2 2p i + pi i+1 = i with the equilibrium point (0 0). The eigenvectors a1 = ( 2 + 3p1) andp a2 = ( 2 ; 3 1) of the Jacobian matrix Df ((0 0)) correspond to the eigenvalues 12 = 2 3, so E s = fta2 t 2 IRg u ((0 0)) = f( ) : = ( ) j j < % : (;% %) ! and E u = fta1 t 2 IRg. Supposing that Wloc p p IR di erentiableg, we are looking for in the form of a power series ( ) = ( 3; 2) +k 2 + . From u u ( i i) 2 Wloc((0 0)), ( i+1 i+1) 2 Wloc((0 0)) follows. This leads to an equation for the coecients of the decomposition of , where k < 0. The theoretical shape of the stable and unstable manifolds is shown in Fig. 17.11a (see 17.12]).
3. Transverse Homoclinic Points
The separatrix surfaces W s(x0 ) and W u(x0 ) of a hyperbolic equilibrium point x0 of (17.3) can intersect each other. If the intersection W s(x0 ) \ W u(x0 ) is transversal, then every point y 2 W s(x0 ) \ W u(x0 ) is called a transversal homoclinic point.
17.1 Ordinary Dierential Equations and Mappings 813 y
u
P1+E
s
W
s
W
P1 Wu
W
u
x s
P1 +E
a)
b)
Figure 17.11 Fact: If y is a transversal homoclinic point, then the orbit f'i(y)g of the invertible system (17.3) consists only of transversal homoclinic points (Fig. 17.11b).
17.1.3.3 Topological Conjugacy of Discrete Systems 1. Denition
Suppose, besides (17.3), a further discrete system xt+1 = (xt ) (17.24) with : N ! N is given, where N IRn is an arbitrary set and is continuous (M and N can be general metric spaces). The discrete systems (17.3) and (17.24) (or the mappings ' and ) are called topologically conjugate if there exists a homeomorphism h : M ! N such that ' = h;1 h. If (17.3) and (17.24) are topologically conjugated, then the homeomorphism h transforms the orbits of (17.3) into orbits of (17.24).
2. Theorem of Grobman and Hartman n n
If ' in (17.3) is a di eomorphism ' : IR ! IR , and x0 a hyperbolic equilibrium point of (17.3), then in a neighborhood of x0 (17.3) is topologically conjugate to the linearization yt+1 = D'(x0)yt.
17.1.4 Structural Stability (Robustness)
17.1.4.1 Structurally Stable Di erential Equations 1. Denition
The di erential equation (17.1), i.e., the vector eld f : M ! IRn, is called structurally stable or robust , if small perturbations of f result in topologically equivalent di erential equations. The precise de nition of robustness requires the notion of distance between two vector elds de ned on M . We restrict our investigations to smooth vector elds on M , which have a common open connected absorbing set U M . Let the boundary @U of U ben a smooth (n ; 1)-dimensional hypersurface and suppose that it can be represented as @U = fx 2 IR : h(x) = 0g, where h : IRn ! IR is a C 1-function with grad h(x) 6= 0 in a neighborhood of @U . Let X1(U ) be the metric space of all smooth vector elds on M with the C 1 metric (f g) = sup k f (x) ; g(x)k + sup k D f (x) ; D g(x)k: (17.25) x2U
x2U
(In the rst term of the right-hand side k k means the Euclidean vector norm, in the second one the operator norm.) The smooth vector elds f intersecting transversally the boundary @U in the direction U , i.e., for which grad h(x)T f (x) 6= 0 (x 2 @U ) and 't (x) 2 U (x 2 @U t > 0) hold, form the set X1+(U ) X1 (U ). The vector eld f 2 X1+(U ) is called structurally stable if there is a > 0 such that every other vector eld g 2 X1+(U ) with (f g) < is topologically equivalent to f . Consider the planar di erential equation g ( ) x_ = ;y + x ( ; x2 ; y2) y_ = x + y ( ; x2 ; y2) (17.26)
814 17. Dynamical Systems and Chaos 1 with parameter , where jj < 1. The di erential equation g belongs, p e.g., to X+(U ) with U = f(x y): x2 + y2 < 2g (Fig. 17.12a). Obviously, (g ( 0) g ( )) = jj ( 2 + 1). The vector eld g ( 0) is structurally unstable there exist vector elds arbitrarily close to g ( 0), which are not topologically equivalent to g ( 0) (Fig. 17.12b,c). This is clear if we consider the polar coordinate representation p r_ = ;r3 + r _ = 1 of (17.26). For > 0 there always exists a stable limit cycle r = . y
y
δU x
a)
U
y
x
x b)
a< =0
c)
a>0
Figure 17.12
2. Structurally Stable Systems in the Plane
Suppose the planar di erential equation (17.1) with f 2 X1+(U ) is structurally stable. Then: a) (17.1) has only a nite number of equilibrium points and periodic orbits. b) All !-limit sets !(x) with x 2 U of (17.1) consist of equilibrium points and periodic orbits only. Theorem of Andronov and Pontryagin: The planar di erential equation (17.1) with f 2 X1+(U ) is structurally stable if and only if: a) All equilibrium points and periodic orbits in U are hyperbolic. b) There are no separatrices, i.e., no heteroclinic or homoclinic orbits, coming from a saddle and tending to a saddle point.
17.1.4.2 Structurally Stable Discrete Systems
In the case of discrete systems (17.3), i.e., of mappings ' : M ! M , let U M IRn be a bounded, open, and connected set with a smooth boundary. Let Di 1(U ) be the metric space of all di eomorphisms on M with the corresponding U de ned C 1 -metric. Suppose the set Di 1+(U ) Di (U ) consists of the di eomorphisms ', for which '(U ) U is valid. The mapping ' 2 Di 1+(U ) (and the corresponding dynamical system (17.3)) is called structurally stable if there exists a > 0 such that every other mapping 2 Di 1+(U ) with (' ) < is topologically conjugate to '.
17.1.4.3 Generic Properties 1. Denition
A property of elements of a metric space (M ) is called generic (or typical) if the set of the elements B Tof M with this property form a set of the second Baire category , i.e., it can be represented as B = Bm , where every set Bm is open and dense in M . m=12:::
A: The sets IR and I IR (irrational numbers) are sets of second Baire category, but Q IR is not. B: Density alone as a property of \typical" is not enough: Q IR and I IR are both dense, but
they cannot be typical at the same time. C: There is no connection between the Lebesgue measure (see 12.9.1, 636) of a setfrom IR 2., p. 1 T S and the Baire category of this set. The set B = Bk with Bk = an ; k 2n an + k 12n , where n 0 k=12::: Q = fang1 n=0 represents the rational numbers, is a set of second Baire category (see 17.5], 17.10]).
17.1 Ordinary Dierential Equations and Mappings 815
2 1 =0 On the other hand, since Bk % Bk+1 and (Bk ) < +1 also (B ) = klim (Bk ) klim !1 !1 k 1 ; 1=2 holds.
2. Generic Properties of Planar Systems, Hamiltonian Systems1
For planar di erential equations the set of all structurally stable systems from X+(U ) is open and dense in X1+(U ). Hence, structurally stable systems are typical for the plane. It is also typical that every orbit of a planar system from X+1 (U ) for increasing time tends to one of a nite number of equilibrium points and periodic orbits. Quasiperiodic orbits are not typical. Under certain assumptions, in the case of Hamiltonian systems, the quasiperiodic orbits of di erential equation are preserved in the case of small perturbations. Hence, Hamiltonian systems are not typical systems. Given in IR4 a Hamiltonian system in action-angle variables j_1 = 0 j_2 = 0 !_ 1 = @H0 !_ 2 = @H0 @j1 @j2 where the Hamiltonian H0(j1 j2) is analytical. Obviously, this system has the solutions j1 = c1 j2 = c2 !1 = !1t + c3 !2 = !2t + c4 with constants c1 : : : c4 , where !1 and !2 can depend on c1 and c2 . The relation (j1 j2 ) = (c1 c2) de nes an invariant torus T 2. Consider now the perturbed Hamiltonian H0(j1 j2) + "H1(j1 j2 !1 !2) instead of H0, where H1 is analytical and " > 0 is a small parameter. The Kolmogorov{Arnold{Moser theorem (KAM theorem) says in this case that if H0 is non-degenera2H ! @ 0 te, i.e., det @j 2 6= 0, then in the perturbed Hamiltonian system most of the invariant non-resonant k tori will not vanish for suciently small " > 0 but will be only slightly deformed. \Most of the tori" means that the Lebesgue measure of the complement set with respect to the tori tends to zero if " tends to 0. A torus, de ned as above and characterized by !1 und !2, is called non-resonant if there exists a ! 1 p constant c > 0 such that the inequality ! ; q q2c:5 holds for all positive integers p and q. 2
3. Non-Wandering points, Morse{Smale Systems t
Let f' gt2IR be a dynamical system on the n-dimensional compact orientable manifold M . The point p 2 M is called non-wandering with respect to f'tg if 8 T > 0 9 t jtj T : 't(Up) \ Up 6= (17.27) holds for an arbitrary neighborhood Up M of p. Steady states and periodic orbits consist only of non-wandering points. The set )('t ) of all non-wandering points of the dynamical systems generated by (17.1) is closed, invariant under f't g and contains all periodic orbits and all ! -limit sets of points from M . The dynamical system f'tgt2IR on M generated by a smooth vector eld is called a Morse{Smale system if the following conditions are ful lled: 1. The system has nitely many equilibrium points and periodic orbits and they are all hyperbolic. 2. All stable and unstable manifolds of equilibrium points and periodic orbits are transversal to each other. 3. The set of all non-wandering points consists only of equilibrium points and periodic orbits. Theorem of Palis and Smale: Morse{Smale systems are structurally stable. The converse statement of the theorem of Palis and Smale is not true: In the case of n 3, there exist structurally stable systems with in nitely many periodic orbits. For n 3, structurally stable systems are not typical.
816 17. Dynamical Systems and Chaos
17.2 Quantitative Description of Attractors 17.2.1 Probability Measures on Attractors 17.2.1.1 Invariant Measure
1. Denition, Measure Concentrated on the Attractor
Let f'tgt2; be a dynamical system on (M ). Let B be the -algebra of Borel sets on M (12.9.1, 2., p. 636) and let : B ! 0 +1] be a measure on B. Every mapping 't is supposed to be measurable. The measure is called invariant under f'tgt2; if (';t(A)) = (A) holds for all A 2 B and t > 0. If the dynamical system f't gt2; is invertible, then the property of the measure being invariant under the dynamical system can be expressed as ('t (A)) = (A) (A 2 B t > 0). The measure is said to be concentrated on the Borel set A M if (M n A) = 0. If 4 is also an attractor of f'tgt2; and is an invariant measure under f't g, then it is concentrated at 4, if (B ) = 0 for every Borel set B with 4 \ B = . The support of a measure : B ! 0 +1], denoted by supp , is the smallest closed subset of M on which the measure is concentrated. A: We consider the Bernoulli shift mapping on M = 0 1]: xt+1 = 2xt (mod1): (17.28a) In this case the map ': 0 1] ! 0 1] is de ned as '(x) = 22xx ; 1 01=2 <x x 1=12: (17.28b) The de nition yields that the Lebesgue measure is invariant under the Bernoulli shift mapping. If a 1 X number x 2 0 1) is written in dyadic form x = an 2;n (an = 0 or 1), then this representation can n=1
be identi ed with x = :a1 a2 a3 : : : . The result of the operation 2x( mod 1) can be written as :a01 a02 a03 : : : with a0i = ai+1 , i.e., all digits ak are shifted to the left by one position and the rst digit is omitted. B: The mapping " : 0 1] ! 0 1] with ( y 0 y < 1=2 " (y) = 22(1 (17.29) ; y ) 1 =2 y 1 is called a tent mapping and the Lebesgue measure is an invariant measure. The homeomorphism p h : 0 1) ! 0 1) with y = 2 arcsin x transforms the mapping ' from (17.5) into (17.29). Hence, in the case of = 4, (17.5) has an invariant measure which is absolutely continuous. For the density 1 (y) 1 of (17.29) and (x) of (17.5) at = 4 it is valid that 1 (y) = (h;1 (y)) j(h;1)0 (y)j. It follows directly that (x) = q 1 . x(1 ; x) C: If x0 is a stable periodic point of period T of the invertible discrete dynamical system f'ig, TX ;1 then = T1 'i (x0 ) is a probability measure for f'ig. Here, x0 is the Dirac measure concentrated i=0 at x0 .
2. Natural Measure
Let 4 be an attractor of f'tgt2; in M with domain of attraction W . For an arbitrary Borel set A W and an arbitrary point x0 2 W de ne the number: t(T A x0 ) : (17.30) (A x0) := Tlim !1 T
17.2 Quantitative Description of Attractors 817
Here, t(T A x0 ) is that part of the time T > 0 for which the orbit portion f't (x0 )gTt=0 lies in the set A. If (A x0) = for -a.e. x0 from W , then let (A) := (A x0 ). Since almost all orbits with initial points x0 2 W tend to 4 for t ! +1, is a probability measure concentrated at 4.
17.2.1.2 Elements of Ergodic Theory 1. Ergodic Dynamical Systems t
A dynamical system f' gt2; on (M ) with invariant measure is called ergodic (we also say that the measure is ergodic) if either (A) = 0 or (M n A) = 0 for every Borel set A with ';t (A) = A (8 t > 0). If f'tg is a discrete dynamical system (17.3), ' : M ! M is a homeomorphism and M is a compact metric space, then there always exists an invariant ergodic measure. A: Suppose there is given with a parameter 2 0 2] the rotation mapping of the circle S 1 xt+1 = xt + (mod 2) t = 0 1 : : : (17.31) with ': 0 2) ! 0 2), de ned by '(x) = x + (mod2). The Lebesgue measure is invariant under '. If 2 is irrational, then (17.31) is ergodic if 2 is rational, then (17.31) is not ergodic. B: Dynamical systems with stable equilibrium points or stable periodic orbits as attractors are ergodic with respect to the natural measure. Birkho Ergodic Theorem: Suppose that the dynamical system f'tgt2; is ergodic with respect to the invariant probability measure . Then, for every integrable function h 2 L1 (M B ), the time 1 Z T h ('t(x )) dt for ows and average along the positive semiorbits f't(x0 )g1 lim 0 t=0 , i.e. h(x0 ) = T ! +1 T 0 Z nX ;1 1 i (x )) for discrete systems, coincide with the space average h d for -a.e. h(x0 ) = nlim h ( ' 0 !1 n i=0 M points x0 2 M .
2. Physical or SBR Measure
The statement of the ergodic theorem is useful only if the support of the measure is large. Let ' : M ! M be a continuous mapping, and : B ! IR be an invariant measure. We call (see 17.9]) an SBR measure (according to Sinai, Bowen and Ruelle) if for any continuous function h : M ! IR the set of all points x0 2 M , for which Z ;1 1 nX i (x )) = h d lim h ( ' (17.32a) 0 n!1 n i=0 M holds, has a positive Lebesgue measure for this. It is sucient that the sequence of measures nX ;1 (17.32b) n := n1 'i (x) i=0 where x Zis the DiracZ measure, weakly converges to for almost all x 2 M , i.e., for every continuous function h d n ! hd as n ! +1. M
M
For some important attractors, such as the Henon attractor, the existence of an SBR measure is proven.
3. Mixing Dynamical Systems t
A dynamical system f' gt2; on (M ) with invariant probability measure is called mixing if lim (A \ ';t (B )) = (A) (B ) holds for arbitrary Borel sets A B M . For a mixing system, the t!+1 measure of the set of all points which are at t = 0 in A and under 't for large t in B , depends only on Here and in the following a.e. is an abbreviation for \almost everywhere".
818 17. Dynamical Systems and Chaos the product (A) (B ). A mixing system is also ergodic: Let f'tg be a mixing system and A be a Borel set with ';t (A) = A (t > 0). Then (A)2 = tlim (';t(A) \ A) = (A) holds and (A) is 0 or 1. !1 t A ow f' g of (17.1) is mixing if and only if the relation Z lim g('t(x)) ; g ] h(x) ; h ] d = 0 (17.33) t!+1 M
holds for arbitrary quadratically integrable functions g h 2 L2(M B ). Here, g and h denote the space average, which is replaced by the time average. The modulo mapping (17.28a) is mixing. The rotation mapping (17.31) is not mixing with respect to the probability measure 2 .
4. Autocorrelation Functiont
Suppose the dynamical system f' gt2; on M with invariant measure is ergodic. Let h : M ! IR be an arbitrary continuous function, f't(x)gt 0 be an arbitrary semiorbit and let the space average 1 ZT h ('t(x)) dt in the time-continuous case and by h be replaced by the time average, i.e., by Tlim !1 T 0 n ;1 X 1 i h (' (x)) in the time-discrete case. With respect to h the autocorrelation function along nlim !1 n i=0 the semiorbit f't(x)gt 0 to a time point 0 is de ned for a ow by 1 ZT h ('t+ (x)) h ('t(x)) dt ; h 2 Ch( ) = Tlim (17.34a) !1 T 0 and for a discrete system by ;1 1 nX Ch( ) = nlim h ('i+ (x)) h ('i(x)) ; h 2 : (17.34b) !1 n i=0 The autocorrelation function is de ned also for negative time, where Ch() is considered as an even function on IR or Z. Periodic or quasiperiodic orbits lead to periodic or quasiperiodic behavior of Ch. A quicker descent of Ch( ) for increasing and arbitrary test function h refers to chaotic behavior. If Ch( ) decreases for increasing with an exponential speed, then it means mixed behavior.
5. Power Spectrum
The Fourier transform of Ch( ) is called a power spectrum (see alsoZ15.3.1.2, 5., p. 726) and is denoted +1 by Ph(!). In the time-continuous case, under the assumption that jCh( )jd < 1, we have
Ph(!) =
+ Z1
Z1
Ch( )e;i! d = 2 Ch( ) cos(! ) d : 0
;1
;1
(17.35a)
+1 In the time-discrete case, if P jCh(k)j < +1 holds, then
k=;1
Ph(!) = Ch(0) + 2
1 X
k=1
Ch(k) cos !k:
(17.35b)
If the absolute integrability or summability of Ch() does not hold, then, in the most important cases, Ph can be considered as a distribution. The power spectrum corresponding to the periodic motions of
17.2 Quantitative Description of Attractors 819
a dynamical system is characterized by equidistant impulses. For quasiperiodic motions, there occur impulses in the power spectrum, which are linear combinations with integer coecients of the basic impulses of the quasiperiodic motion. A \wide-band spectrum with singular peaks" can be considered as an indicator of chaotic behavior. A: Let ' be a T -periodic orbit of (17.1), h be a test function such that the time average of h('(t)) is zero, and suppose h('(t)) has the Fourier representation + X1 ik!0t h('(t)) = k e with !0 = 2T : k=;1 Then we have, with as the distribution,
Ch ( ) =
+ X1
k=;1
jk j2 cos(k!0 ) and Ph(!) = 2
+ X1
k=;1
jk j2 (! ; k!0 ):
B: Suppose ' is a quasiperiodic orbit of (17.1), h is a test function such that the time average is zero along ', and let h('(t)) be the representation (double Fourier series) h('(t)) = Then,
Ch ( ) =
+ X1
+ X1
k1 =;1 k2 =;1 + X1
+ X1
k1 k2 ei(k1 !1+k2!2 )t :
jk1k2 j2 cos(k1!1 + k2!2)
k1 =;1 k2 =;1 + X1 +X1
Ph(!) = 2
k1 =;1 k2 =;1
jk1k2 j2 (! ; k1 !1 ; k2!2):
17.2.2 Entropies
17.2.2.1 Topological Entropy
Let (M ) be a compact metric space and f'k gk2; be a continuous dynamical system with discrete time on M . A distance function n on M for arbitrary n 2 IN is de ned by n(x y) := 0max ('i(x) 'i(y)): (17.36)
i n Furthermore, let N (" n) be the largest number of points from M which have in the metric n a distance at least " from each other. The topological entropy of the discrete dynamical system (17.3) or of the mapping ' is h(') = "lim lim sup 1 ln N (" n). The topological entropy is a measure for the complexity !0 n!1 n of the mapping. Let (M1 1) be a further compact metric space and '1 : M1 ! M1 be a continuous mapping. If both mappings ' and '1 are topologically conjugate, then their topological entropies coincide. In particular, the topological entropy does not depend on the metric. For arbitrary n 2 IN, h ('n) = n h (') holds. If ' is a homeomorphism, then h ('k ) = jkj h (') for all k 2 Z. Based on the last property, we de ne the topological entropy h('t) := h ('1) for a ow 't = '(t ) of (17.1) on M IRn.
17.2.2.2 Metric Entropy
Let f'tgt2; be a dynamical system on M with attractor 4 and with an invariant probability measure concentrated on 4. For an arbitrary " > 0 consider the cubes Q1 (") : : : Qn(") (") of the form f(x1 : : : xn) : ki" xi < (ki + 1)" (i = 1 2 : : : n)g with ki 2 Z, for which (Qi) > 0. For arbitrary x from a Qi the semiorbit f't(x)g1 t=0 is followed for increasing t. In time-distances of > 0 ( = 1 in discrete systems), the N cubes, in which the semiorbit is found is denoted by i1 : : : iN after each other. Let Ei1 :::iN be the set of all starting points in the neighborhood of 4 whose semiorbits at the
820 17. Dynamical Systems and Chaos times ti = i (i = 1 2 : : : N ) are always in Qi1 : : : QiN and let p(i1 iN ) = (Ei1 iN ) be the probability that a (typical) starting point is in Ei1 iN . The entropy gives the increment of the information on average by an experiment which shows that among a nite number of disjoint events which one has really happened. In the above situation this is X HN = ; p (i1 iN ) ln p(i1 iN ) (17.37) (i1 iN )
where the summation is over all symbol sequences (i1 iN ) with length N , which are realized by the orbits described above. The metric entropy or Kolmogorov{Sinai entropy h of the attractor 4 of f'tg with respect to the invariant measure is the quantity h = "lim lim HNN . For discrete systems, the limit as " ! 0 is !0 N !1 omitted. For the topological entropy h(') of ' : 4 ! 4 the inequality h h(') holds. In several cases h(') = supfh : invariant probability measure on 4g: A: Suppose 4 = fx0g is a stable equilibrium point of (17.1) as an attractor, with the natural measure concentrated on x0 . For these attractors h = 0. B: For the shift mapping (17.28a), h(') = h = ln 2, where is the invariant Lebesgue measure.
17.2.3 Lyapunov Exponents 1. Singular Values of a Matrix
Let L be an arbitrary matrix of type (n n). The singular values 1 2 n of L are the non-negative roots of the eigenvalues 1 n 0 of the positive semide nite matrix LT L. The eigenvalues i are enumerated according to their multiplicity. The singular values can be interpreted geometrically. If K" is a sphere with center at 0 and with radius " > 0, then the image L(K") is an ellipsoid with semi-axis lengths i " (i = 1 2 : : : n) (Fig. 17.13a). t
e
L
s2e
s1e
j (x0)
x0
y(t,x0,ev) ev
b)
a)
t
j (x0+ev)
x0+ev
Figure 17.13
2. Denition of Lyapunov Exponents t
Let f' gt2; be a smooth dynamical system on M IRn, which has an attractor 4 with an invariant ergodic probability measure concentrated on 4. Let 1 (t x) n (t x) be the singular values of the Jacobian matrix D 't(x) of 't at the point x for arbitrary t 0 and x 2 4. Then there exists a sequence of numbers 1 n, the Lyapunov exponents , such that 1t ln i(t x) ! i for t ! +1 -a.e. innthe sense of L1. According to the theorem of Oseledec there exists -a.e. a sequence of subspaces of IR IRn = Esx1 % Esx2 % % Esxr+1 = f0g (17.38) 1 t such that for -a.e. x the quantity ln kD ' (x)vk tends to an element sj 2 f1 : : : ng uniformly t with respect to v 2 Esxj n Esxj+1 .
17.2 Quantitative Description of Attractors 821
3. Calculation of Lyapunov Exponents
Suppose i(t x) are the semi-axis lengths of the ellipsoid got by deformation of the unit sphere with center at x by D 't (x). The formula i(x) = tlim sup 1t ln i(t x) can be used to calculate the Lya!1 punov exponents, if additionally a reorthonormalization method, such as Housholder, is used. The function y(t x v) = D 't (x)v is the solution of the variational equation with v at t = 0 associated to the semiorbit +(x) of the ow f'tg. Actually, f't gt2IR is the ow of (17.1), so the variational equation is y_ = D f ('t(x)) y. The solution of this equation with initial v at time t = 0 can be represented as y(t x v) = x(t)v, where x(t) is the normed fundamental matrix of the variational equation at t = 0, which is a solution of the matrix di erential equation Z_ = D f ('t(x)) Z with initial Z (0) = En according to the theorem about di erentiability with respect to the initial state (see 17.1.1.1, 2., p. 797). The number (x v) = tlim sup 1t ln kD 't(x)vk describes the behavior of the orbit (x + "v) 0 < " * 1 !1 with initial x + "v with respect to the initial orbit (x) in the direction v. If (x v) < 0, then the orbits move nearer to x for increasing t in the direction v. If, on the contrary, (x v) > 0, then the orbits move away (Fig. 17.13b). Let 4 be the attractor of the dynamical system f't gt2; and the invariant ergodic measure concentrated on it. Then, the sum of all Lyapunov exponents -a.e. x 2 4 is n X 1 Zt divf ('s(x)) ds i = tlim (17.39a) !1 t i=1 0 in the case of ows (17.1) and for a discrete system (17.3), it is n ;1 X 1 kX i = klim ln j det D '('i(x))j: (17.39b) !1 k i=1 i=0 Hence, in dissipative systems P i < 0 holds. Considering that one of the Lyapunov exponents is i=1 equal to zero if the attractor is not an equilibrium point, the calculation of Lyapunov exponents can be simpli ed (see 17.9]). A: Let be x0 an equilibrium point of the ow of (17.1) and let i be the eigenvalues of the Jacobian matrix at x0 . With the measure concentrated on x0 , the following holds for the Lyapunov exponents: i = Re i (i = 1 2 : : : n). B: Let (x0) = f't (x0) t 2 0 T ]g be a T -periodic orbit of (17.1) and let i be the multipliers of (x0 ). With the measure concentrated on (x0 ) we have that i = T1 ln j ij for i = 1 2 : : : n. n
4. Metric Entropy and Lyapunov Exponents n t
If f' gt2; is a dynamical system on M IR X with attractor 4 and an ergodic probability measure concentrated on 4, then the inequality h i holds for the metric entropy h , where in the sum
i >0
the Lyapunov exponents are repeated according to their multiplicity. The equality X h = i (Pesin's formula)
i >0
(17.40)
is not valid in general. If the measure is absolutely continuous with respect to the Lebesgue measure and ': M ! M is a C 2 -di eomorphism, then Pesin's formula is valid (see also 17.2.4.4, B, p. 827).
822 17. Dynamical Systems and Chaos
17.2.4 Dimensions
17.2.4.1 Metric Dimensions 1. Fractals
Attractors or other invariant sets of dynamical systems can be geometrically more complicated than point, line or torus. Fractals, independently of dynamics, are sets which distinguish themselves by one or several characteristics such as fraying, porousity, complexity, and self-similarity. Since the usual notion of dimension used for smooth surfaces and curves cannot be applied to fractals, we must give a generalized de nition of the dimension. For more details see 17.2], 17.12]. The interval G0 = 0 1] is divided into three subintervals with the same length and the middle open third is removed, so we obtain the set G1 = 0 13 ] 23 1]. hTheni from h both i subintervals h i h ofi G1 the open middle third ones are removed, which yields the set G2 = 0 19 29 13 23 97 89 1 . Continuing this procedure, Gk is obtained from Gk;1 by removing the open middle thirds from the subintervals. So, we get a sequence of sets G0 % G1 % % Gn % , where every Gn consists of 2n intervals of length 1n . 3 1 The Cantor set C is de ned as the set of all points belonging to all Gn, i.e., C = T Gn. The set C is n=1 compact, uncountable, its Lebesgue measure is zero and it is perfect, i.e., C is closed and every point is an accumulation point. The Cantor set can be an example of a fractal.
2. Hausdor Dimension
The motivation for this dimension comes from volume calculation based on Lebesgue measure. If we suppose that a bounded set A IR3 is covered by a nite number of spheres Bri with radius ri ", X4 3 so that Si Bri % A holds, we get for A the \rough volume" ri . Now, we de ne the quantity i 3 X "(A) = inf f 34 ri3g over all nite coverings of A by spheres with radius ri ". If " tends to zero, i then we get the Lebesgue outer measure (A) of A. If A is measurable, the outer measure is equal to the volume vol(A). Suppose M is the Euclidean space IRn or, more generally, a separable metric space with metric and let A M be a subset of it. For arbitrary parameters d 0 and " > 0, the quantity
d"(A) = inf
(X i
(diamBi)d : A
#
)
Bi diamBi "
is determined, where Bi M are arbitrary subsets with diameter diamBi = sup (x y). xy2Bi The Hausdor outer measure of dimension d of A is de ned by d(A) = "lim d"(A) = sup d"(A) !0 ">0
(17.41a)
(17.41b)
and it can be either nite or in nite. The Hausdor dimension dH (A) of the set A is then the (unique) critical value ofthe Hausdor measure: 1 if d(A) 6= 0 for all d 0 dH (A) = + (17.41c) inf fd 0: d(A) = 0g:
Remark:n The quantities d"(A) can be de ned with coverings of spheres with radius ri " or, in the case of IR , of cubes with side length ". Important Properties of the Hausdor Dimension: (HD1) dH ( ) = 0. (HD2) If A IRn, then 0 dH (A) n.
17.2 Quantitative Description of Attractors 823
(HD3) From A B , it follows that dH (A) dH (B ). 1 (HD4) If A = iS=1 Ai , then dH (A) = sup dH (Ai ). i (HD5) If A is nite or countable, then dH (A) = 0. (HD6) If ': M ! M is Lipschitz continuous, i.e., there exists a constant L;1> 0 such that ('(x) '(y)) L (x y) 8x y 2 M , then dH ('(A)) dH (A). If the inverse mapping ' exists as well, and it is also Lipschitz continuous, then dH (A) = dH ('(A)). For the set Q of all rational numbers dH (Q) = 0 because of (HD5). The dimension of the Cantor 2 set C is dH (C ) = ln ln 3 0:6309 : : : .
3. Box-Counting Dimension or Capacity
Let A be a compact set of the metric space (M ) and let N"(A) be the minimal number of sets of diameter ", necessary to cover A. The quantity " (A) dB (A) = lim sup ln N (17.42a) ln 1" "!0 is called the upper box-counting dimension or upper capacity , the quantity " (A) dB (A) = lim inf ln N (17.42b) "!0 ln 1" is called the lower box-counting dimension or lower capacity (then dnC ) of A. If dB (A) = dB (A) := dB (A) holds, then dB (A) is called the box-counting dimension of A. In IR the box-counting dimension can be considered also for bounded sets which are not closed. For a bounded set A IRn, the number N"(A) in the above de nition can also be de ned in the following way: Let IRn be covered by a grid from n-dimensional cubes with side length ". Then, N"(A) can be the number of cubes of the grid having a non-empty intersecting A. Important Properties of the Box-Counting Dimension: (BD1) dH (A) dB (A) always holds. (BD2) For m-dimensional surfaces F IRn holds dH (F ) = dB (F ) = m. (BD3) dB (A) = dB (A) holds for the closure A of A, while often dH (A) < dH (A) is valid. (BD4) If A = Sn An, then, in general, the formula dB (A) = sup dB (An) does not hold for the boxn counting dimension. Suppose A = f0 1 12 13 : : :g . Then dH (A) = 0 and dB (A) = 12 . If A is the set of all rational points in 0 1], then because of BD2 and BD3 dB (A) = 1 holds. On the other hand dH (A) = 0.
4. Self-Similarity
Several geometric gures, which are called self-similar, can be derived by the following procedure: An initial gure is replaced by a new one which is composed of p copies of the original, any of them scaled linearly by a factor q > 1. All gures that are k times scalings of the initial gure in the k-th step are handled as the in the rst step. A: Cantor set: p = 2 q = 3. B: Koch curve: p = 4 q = 3. The rst three steps are shown in Fig. 17.14. Figure 17.14
824 17. Dynamical Systems and Chaos C: Sierpinski gasket: p = 3 q = 2. The rst three steps are shown in Fig. 17.15. (The white triangles are always removed.) Figure 17.15
Figure 17.16
D: Sierpinski carpet: p = 8 q = 3. The rst three steps are shown in Fig. 17.16. (The white squares are always removed.) For the sets in the examples A{D: p: dB = dH = ln ln q
17.2.4.2 Dimensions Dened by Invariant Measures 1. Dimension of a Measure
Let be a probability measure in (M ), concentrated on 4. If x 2 4 is an arbitrary point and B (x) is a sphere with radius and center at x, then d (x) = lim sup ln (lnB (x)) (17.43a) !0 denotes the upper and d (x) = lim (17.43b) inf ln (B (x)) !0 ln denotes the lower pointwise dimension of in x. If d (x) = d (x) := d (x), then d (x) is called the dimension of the measure in x. Young Theorem 1: If the relation d (x) = holds for -a.e. , x 2 4, then = dH ( ) := X inf fd (X )g: (17.44) (X )=1 H The quantity dH ( ) is called the Hausdor dimension of the measure . Suppose M = IRn and let 4 IRn be a compact sphere with Lebesgue measure (4) > 0. The . Then restriction of to 4 is = (4)
(B (x)) ! n and dH ( ) = n:
2. Information Dimension
Suppose, the attractor 4 of f't gt2; is covered by cubes Q1 (") : : : Qn(")(") of side length " as in 17.2.2.2, p. 819. Let be an invariant probability measure on 4. The entropy of the covering Q1 (") : : : Qn(") (") is
H (") = ;
nX (") i=1
pi(") ln pi(") with pi (") = (Qi(")) (i = 1 : : : n(")):
(17.45)
H (") exists, then this quantity has the property of a dimension and is called If the limit dI ( ) = ; "lim !0 ln " the information dimension . Here and in the following a.e. is an abbreviation for \almost everywhere".
17.2 Quantitative Description of Attractors 825
Young Theorem 2: If the relation d (x) = holds for -a.e. x 2 4, then = dH ( ) = dI ( ):
(17.46)
A: Let the measure be concentrated at the equilibrium point x0 of f'tg. Since H"( ) = ;1 ln 1 =
0 always holds for " > 0, so dI ( ) = 0. B: Suppose the measure is concentrated on the limit cycle of f'tg. For " > 0, H"( ) = ; ln " holds and so dI ( ) = 1.
3. Correlation Dimension
n t Let fyig1 i=1 be a typical sequence of points of the attractor 4 IR of f' gt2; , an invariant probability measure on 4 and let m 2 IN be arbitrary. For the vectors xi := (yi : : : yi+m) let the distance be de ned as dist(xi xj ) := 0max kyi+s ; yj+sk, where k k is the Euclidean vector norm. If ! denotes
s m 0 x 0 the Heaviside function !(x) = 1 x > 0 then the expression C m(") = lim sup N12 cardf(xi xj ): dist(xi xj ) < "g N !+1 N X = lim sup N12 ! (" ; dist(xi xj )) (17.47a) N !1 ij =1 is called the correlation integral. The quantity ln C m (") dK = "lim (17.47b) !0 ln " (if it exists) is the correlation dimension .
4. Generalized Dimension
Let the attractor 4 of f't gt2; on M with invariant probability measure be covered by cubes with side length " as in 17.2.2.2, p. 819. For an arbitrary parameter q 2 IR q 6= 1, the sum nX (") Hq (") = 1 ;1 q ln pi(")q where pi(") = (Qi("))
(17.48a)
i=1
is called the generalized entropy of q-th order with respect to the covering Q1 (") : : : Qn(") ("). The R enyi dimension of q-th order is Hq (") dq = ; "lim (17.48b) !0 ln " if this limit exists.
Special Cases of the R enyi Dimension: a) q = 0: d0 = dC (supp ): b) q = 1: d1 := qlim dq = dI ( ): !1 c) q = 2: d2 = dK :
(17.49a) (17.49b) (17.49c)
5. Lyapunov Dimension
Let f'tgt2; be a smooth dynamical system on M IRn with attractor 4 (or invariant set) and with the invariant ergodic probability measure concentrated on 4. If 1 2 n are the Lyapunov k+1 k exponents with respect to and if k is the greatest index for which P i 0 and P i < 0 hold, then i=1
i=1
826 17. Dynamical Systems and Chaos k X
the value
i dL( ) = k + ji=1 j k+1 is called the Lyapunov dimension of the measure . n If P i 0, then dL( ) = n if 1 < 0, then dL( ) = 0.
(17.50)
i=1
Ledrappier Theorem: Let f'tg be a discrete system (17.3) on M IRn with a C 2-function ' and , as above, an invariant ergodic probability measure concentrated on the attractor 4 of f'tg. Then dH ( ) dL( ) holds. A: Suppose the attractor 4 IR2 of a smooth dynamical system f'tg is covered by N" squares with side length ". Let 1 > 1 > 2 be the singular values of D '. Then for the dB -dimensional volume of the attractor mdB ' N" "dB holds. Every square of side length " is transformed by ' approximately
into a parallelogram with side length 2 " and 1 ". If the covering is made by rhombi with side length 2 ", then N2 " ' N" 1 holds. From the relation N""dB ' N2 "("2)dB we get directly 2 ln 1 dB ' 1 ; ln = 1 + j1j : (17.51) 2 2 This heuristic calculation gives a hint of the origin of the formula for the Lyapunov dimension. B: Suppose the Henon system (17.6) is given with a = 1:4 and b = 0:3. The system (17.6) has an attractor 4 (H enon attractor) with a complicated structure for these parameter values. The numerically determined box-counting dimension is dB (4) ' 1:26. It can be shown that there exists an SBR measure for the Henon attractor 4. For the Lyapunov exponents 1 and 2 the formula 1 + 2 = ln j det D '(x)j = ln b = ln 0:3 ' ;1:204 holds. With the numerically determined value 1 ' 0:42 we get 2 ' ;1:62. So, 42 ' 1:26: dL( ) ' 1 + 10::62 (17.52)
17.2.4.3 Local Hausdor Dimension According to Douady and Oesterle n
Let f'tgt2; be a smooth dynamical system on M IR and 4 a compact invariant set. Suppose that an arbitrary t0 0 is xed and let := 't0 . Theorem of Douady and Oesterl e: Let 1 (x) n (x) be the singular values of D (x) and let d 2 (0 n] be a number written as d = d0 + s with d0 2 f0 1 : : : n ; 1g and s 2 0 1]. If sup 1 (x)2 (x) : : : d0 (x)ds0 +1 (x)] < 1 holds, then dH (4) < d. x2
Special Version for Di erential Equations: Let f't gt2IR be the ow of (17.1), 4 be a compact invariant set and let 1 (x) n(x) be the eigenvalues of the symmetrized Jacobian matrix 1 D f (x)T + D f (x)] at an arbitarary point x 2 4. If d 2 (0 n] is a number of the form d = d + s 0 2 where d0 2 f0 : : : n ; 1g and s 2 0 1], and sup 1(x) + + d0 (x) + sd0 +1(x)] < 0 holds, then x2
dH (4) < d. The quantity dDO (x) = 0supiffd1: (0x) 1 > 0 and r > 0 we get the estimation dH (4) 3 ; + b + 1 where (17.54a) v 2 3 ! u u = 12 4 + b + t( ; b)2 + p b + 2 r5 : (17.54b) b;1
20
-10 40
20
17.2.4.4 Examples of Attractors
0 -10
0
A: The horseshoe mapping ' occurs in connection with Poincare mappings containing the transversale intersections of stable and unstable manifolds. The unit square M = 0 1] 0 1] is stretched linearly in one coordinate direction and contracted in the other direction. Finally, this rectangle is bent at the middle (Fig. 17.18). Repeating this procedure in nitely many times, we get a sequence of sets M % '(M ) % , for which
10
Figure 17.17
4=
1 \
k=0
'k (M )
(17.55)
is a compact set and an invariant set with respect to '. 4 attracts all points of M . Apart from one point, 4 can be described Figure 17.18 locally as a product \line Cantor set". B: Let 2 (0 1=2) be a parameter and M = 0 1] 0 1] be the unit square. The mapping ': M ! M given by 8 > < (2x y) if 0 x 21 y 2 0 1] '(x y) = > (17.56a) : (2x ; 1 y + 1 if 1 < x 1 y 2 0 1] 2 2 is called the dissipative baker's mapping. Two iterations of the baker's mapping are shown in Fig. 17.19. The \aky pastry structure" is recognizy able. The set y y 1 1 \ 1 1 4 = 'k (M ) (17.56b) ϕ(Μ)
Μ
ϕ 2( Μ )
k=0
1x
1x
1x
is invariant under ' and all points from M are attracted by 4. The value of the Hausdor dimension is (17.56c) dH (4) = 1 + ;lnln2 :
Figure 17.19 For the dynamical system f'k g there exists an invariant measure on M , which is di erent from the Lebesgue! measure. At the points where the derivative exists, the Jacobian matrix is D 'k ((x y)) = 2k 0 . Hence, the singular values are (k (x y)) = 2k (k (x y)) = k and, consequently, 1 2 0 k
828 17. Dynamical Systems and Chaos the Lyapunov exponents are 1 = ln 2 2 = ln (with respect to the invariant measure ). For the Lyapunov dimension we get (17.56d) dL( ) = 1 + ;lnln2 = dH (4): P Pesin's formula for the metric entropy is valid here, i.e., h = i = ln 2.
i >0 C: Let T be the whole torus with local coordinates (! x y), as is shown in Fig. 17.20a. j(T)
y x
a)
Q
D(Q)
b)
T
c)
Figure 17.20 Let a mapping ': T ! T be de ned by ! ! ! !k + xk (k = 0 1 : : :) !k+1 = 2!k xyk+1 = 12 cos (17.57) sin !k yk k+1 with parameter 2 (0 1=2). The image '(T ), with the intersections '(T ) \ D(!) and '2 (T ) \ D(!), is shown in Fig. 17.20b and Fig. 17.20c. The result of in nitely many intersections is the set 4 = 1 T 'k (T ), which is called a solenoid. The attractor 4 consists of a continuum of curves in the length k=0 direction, and each of them is dense in 4, and unstable. The cross-section of the 4 transversal to these curves is a Cantor set. ln 2 . The set 4 has a neighborhood which is a domain of atThe Hausdor dimension is dH (4) = 1 ; ln traction. Furthermore, the attractor 4 is structurally stable, i.e., the qualitative properties formulated 1 above do not change for C -small perturbations of '. D: The solenoid is an example of a hyperbolic attractor.
17.2.5 Strange Attractors and Chaos
1. Chaotic Attractor t
Let f' gt2; be a dynamical system in the metric space (M ). The attractor 4 of this system is called chaotic if there is a sensitive dependence on the initial condition in 4. The property \ sensitive dependence on the initial conditions " will be made more precise in di erent ways. It is given, e.g., if one of the two following conditions is ful lled: a) All motions of f't g on 4 are unstable in a certain sense. b) The greatest Lyapunov exponent of f'tg is positive with respect to an invariant ergodic probability measure concentrated on 4. Sensitive dependence in the sense of a) occurs for the solenoid. Property b) can be found, e.g., for Henon attractors.
2. Fractals and Strange Attractors
An attractor 4 of f'tgt2; is called fractal if it represents neither a nite number of points or a piecewise di erentiable curve or surface nor a set which is bounded by some closed piecewise di erentiable surface. An attractor is called strange if it is chaotic, fractal or both. The notions chaotic, fractal and strange are used for compact invariant sets analogously even if they are not attractors. A dynamical system is called chaotic if it has a compact invariant chaotic set.
17.3 Bifurcation Theory and Routes to Chaos 829
The mapping xn+1 = 2xn + yn (mod 1) yn+1 = xn + yn (mod 1) (17.58) (Anosov dieomorphism) is considered on the unit square. The adequate phase space for this system is the torus T 2. It is conservative, has the Lebesgue measure as invariant measure, has a countable number of periodic orbits whose union is dense and is mixing. Otherwise, 4 = T 2 is an invariant set with integer dimension 2.
3. Systems Chaotic in the sense of Devaney t
Let f' gt2; be a dynamical system in the metric space (M ) with a compact invariant set 4. The system f'tgt2; (or the set 4) is called chaotic in the sense of Devaney, if: a) f'tgt2; is topologically transitive on 4, i.e., there is a positive semiorbit, which is dense in 4. b) The periodic orbits of f'tgt2; are dense in 4. c) f'tgt2; is sensitive with respect to the initial conditions in the sense of Guckenheimer on 4, i.e., 9 " > 0 8 x 2 4 8 > 0 9 y 2 4 \ U (x) 9 t 0 : ('t(x) 't (y)) " (17.59) where U (x) = fz 2 M : (x z) < g. Consider the space of the 0{1-sequences X = fs = s0s1 s2 : : : si 2 f0 1g (i = 0 1 : : :)g: For two sequences s = s0s1s2 : : : and s0 = s00s01 s02 : : :, their distance is de ned by s0 (s s0) = 20;j ifif ss = 6= s0 where j is the smallest index for which sj 6= s0j . So, (P ) is a complete metric space which is also compact. The mapping : s = s0 s1s2 : : : 7;! (s) = s0 = s1s2 s3 : : : is called a Bernoulli shift . The Bernoulli shift is chaotic in the sense of Devaney.
17.2.6 Chaos in One-Dimensional Mappings
For continuous mappings of a compact interval into itself, there exist several sucient conditions for the existence of chaotic invariant sets. We mention three examples. Shinai Theorem: Let ' : I !k I be a continuous mapping of a compact interval I , e.g., I = 0 1] into itself. Then the system f' g on I is chaotic in the sense of Devaney if and only if the topological entropy of ' on I , i.e., h('), is positive. Sharkovsky Theorem: Consider the following ordering of positive integer numbers: 3 / 5 / 7 / : : : / 2 3 / 2 5 / : : : / 22 3 / 22 5 / : : : : : : / 23 / 22 / 2 / 1: (17.60) Let ' : I ! I be a continuous mapping of a compact interval into itself and suppose f'k g has an nperiodic orbit on I . Then f'k g also has an m-periodic orbit if n / m. Block, Guckenheimer and Misiuriewicz Theorem: Let ' : I ! I be a continuous mapping of the compact interval I into itself such that f'k g has a 2nm-periodic orbit (m > 1, odd). Then h(') 2lnn+12 holds.
17.3 Bifurcation Theory and Routes to Chaos 17.3.1t Bifurcations in Morse{Smale Systems
Let f'"gt2; be a dynamical system generated by a di erential equation or by a mapping on M IRn, which additionally depends on a parameter " 2 V IRl . Every change of the topological structure of the phase portrait of the dynamical system for small changes of the parameter is called bifurcation. The parameter " = 0 2 V is called the bifurcation value if there exist parameter values " 2 V in every
830 17. Dynamical Systems and Chaos neighborhood of 0 such that the dynamical systems f't"g and f't0g are not topologically equivalent or conjugated on M . The smallest dimension of the parameter space for which a bifurcation can be observed is called the codimension of the bifurcation. We distinguish between local bifurcations, which occur in the neighborhood of a single orbit of the dynamical system, and global bifurcations, which a ect a large part of the phase space.
17.3.1.1 Local Bifurcations in Neighborhoods of Steady States 1. Center Manifold Theorem
Consider a parameter-dependent di erential equation x_ = f (x ") or x_i = fi (x1 : : : xn "1 : : : "l) (i = 1 2 : : : n) (17.61) with f : M V ! IRn, where M IRn and V IRl are open sets and f is supposed to be r times continuously di erentiable. Equation (17.61) can be interpreted as a parameter-free di erential equation x_ = f (x ") "_ = 0 in the phase space M V . From the Picard{Lindelof theorem and the theorem on di erentiability with respect to the initial values (see 17.1.1.1, 2., p. 797) it follows that (17.61) has a locally unique solution '( p ") with initial point p at time t = 0 for arbitrary p 2 M and " 2 V , which is r times continuously di erentiable with respect to p and ". Suppose all solutions exist on the whole of IR. Furthermore, we suppose that system (17.61) has the equilibrium point "
n x = 0 at " = 0, i.e., f (0 0) = 0 @f i holds. Let 1 : : : s be the eigenvalues of Dxf (0 0) = @x (0 0) with Rej = 0. Furthermore, j
ij =1
suppose, Dxf (0 0) has exactly m eigenvalues with negative real part and k = n ; s ; m eigenvalues with positive real part. According to the center manifold theorem for di erential equations (theorem of Shoshitaishvili, see 17.12]), the di erential equation (17.61), for " with a suciently small norm k"k in the neighborhood of 0, is topologically equivalent to the system x_ = F (x ") Ax + g(x ") y_ = ;y z_ = z (17.62) s m k with x 2 IR y 2 IR and z 2 IR , where A is a matrix of type (s s) with eigenvalues 1 : : : s, and g represents a C r -function with g(0 0) = 0 and Dxg(0 0) = 0. It follows from representation (17.62) that the bifurcations of (17.61) in a neighborhood of 0 are uniquely described by the di erential equation x_ = F (x "): (17.63) c = fx y z : Equation (17.63) represents the reduced dierential equation to the local center manifold Wloc y = 0 z = 0g of (17.62). The reduced di erential equation (17.63) can often be transformed into a relatively simple form, e.g., with polynomials on the right-hand side, by a non-linear parameterdependent coordinate transformation so that the topological structure of its phase portrait does not change close to the considered equilibrium point. This form is a so-called normal form. A normal form cannot be determined uniquely in general, a bifurcation can be described equivalently by di erent normal forms.
2. Saddle-Node Bifurcation and Transcritical Bifurcation
Suppose (17.61) is given with l = 1, where f is continuously di erentiable at least twice and Dxf (0 0) has the eigenvalue 1 = 0 and n ; 1 eigenvalues j with Rej 6= 0. According to the center manifold theorem, in this case, all bifurcations (17.61) near 0 are described by a one-dimensional reduced di erential equation (17.63). Obviously, here F (0 0) = @F @x (0 0) = 0. If, additionally, it is supposed that 2 @F @ @x2 F (0 0) 6= 0, @" (0 0) 6= 0 and the right-hand side of (17.63) is expanded according to the Taylor formula, then this representation can be transformed by coordinate transformation (see 17.6]) into
17.3 Bifurcation Theory and Routes to Chaos 831
the normal form x_ = + x2 + (17.64) ! ! 2F 2F @ @ for @x2 (0 0) > 0 or x_ = ; x2 + for @x2 (0 0) < 0 , where = (") is a di erentiable function with (0) = 0 and the points indicate higher-order terms. For < 0, (17.64) has two equilibrium points close to x = 0, among which one is stable, the other is unstable. For = 0, these equilibrium points fuse into x = 0, which is unstable. For > 0, (17.64) has no equilibrium point near to 0 (Fig. 17.21b). The multidimensional case results in a saddle-node bifurcation in a neighborhood of 0 in (17.61). This bifurcation is represented in Fig. 17.22 for n = 2 and 1 = 0 2 < 0. The representation of the saddle-node bifurcation in the extended phase space is shown in Fig. 17.21a. For suciently smooth vector elds (17.61), the saddle-node bifurcations are generic. x a
a=0
a0
b)
a)
Figure 17.21
01
0
02 a=0
a0
Figure 17.22
If among the conditions which yield a saddle-node bifurcation for F , the condition @F @" (0 0) 6= 0 is 2F @F @ replaced by the conditions @" (0 0) = 0 and @x@" (0 0) 6= 0, then we get from (17.63) the truncated normal form (without higher-order terms) x_ = x ; x2 of a transcritical bifurcation. For n = 2 and 2 < 0, the transcritical bifurcation together with the bifurcation diagram is shown in Fig. 17.23. Saddle-node and transcritical bifurcations have codimension 1{bifurcations. x
a a0
Figure 17.23
Consider (17.61) with n 2 l = 1 and r 4. Suppose that f (0 ") = 0 is valid for all " with j"j "0 ("0 > 0 suciently small). Suppose the Jacobian matrix Dxf (0 0) has the eigenvalues 1 = 2 = i!
832 17. Dynamical Systems and Chaos with ! 6= 0 and n ; 2 eigenvalues j with Rej 6= 0. According to the center manifold theorm, the bifurcation is described by a two-dimensional reduced di erential equation (17.63) of the form x_ = (")x ; !(")y + g1(x y ") y_ = !(")x + (")y + g2(x y ") (17.65) where ! g1 and g2 are di erentiable functions and !(0) = ! and also (0) = 0 hold. By a non-linear complex coordinate transformation and by the introduction of polar coordinates (r ), (17.65) can be written in the normal form r_ = (")r + a(")r3 + _ = !(") + b(")r2 + (17.66) where dots denote the terms of higher order. The Taylor expansion of the coecient functions of (17.66) yields the truncated normal form r_ = 0(0)"r + a(0)r3 _ = !(0) + !0(0)" + b(0)r2: (17.67) The theorem of Andronov and Hopf guarantees that (17.67) describes the bifurcations of (17.66) in a neighborhood of the equilibrium point for " = 0. The following cases occur for (17.67) under the assumption 0(0) > 0: 1. a(0) < 0 (Fig. 17.24a): : 2. a(0) > 0 (Fig. 17.24b) : a) " > 0: Stable limit cycle and a) " < 0: Unstable limit cycle: unstable equilibrium point: b) " = 0: Cycle and equilibrium point fuse b) " = 0: Cycle and equilibrium point fuse into a stable equilibrium point. into an unstable equilibrium point: c) " < 0: All orbits close to (0 0) tend c) " > 0: Spiral type unstable as in b) for t ! +1 equilibrium point as in b). spirally to the equilibrium point (0 0):
a) e>0
e< 0
b)
e 0
Figure 17.24 The interpretation of the above cases for the initial system (17.61) shows the bifurcation of a limit cycle of a compound equilibrium point (compound focus point of multiplicity 1), which is called a Hopf bifurcation (or Andronov{Hopf bifurcation). The case a(0) < 0 is also called supercritical, the case a(0) > 0 subcritical (supposing that 0(0) > 0). The case n = 3 1 = 2 = i 3 < 0 0(0) > 0 and a(0) < 0 is shown in Fig. 17.25.
e>0
e< 0
Figure 17.25 Hopf bifurcations are generic and have codimension 1. The cases above illustrate the fact that a supercritical Hopf bifurcation under the above assumptions can be recognized by the stability of a focus:
17.3 Bifurcation Theory and Routes to Chaos 833
Suppose the eigenvalues 1(") and 2(") of the Jacobian matrix on the right-hand side of (17.61) at 0 are pure imaginary for " = 0, and for the other eigenvalues j Rej 6= 0 holds. Suppose furthermore d Re (") > 0 and let 0 be an asymptotically stable focus for (17.61) at " = 0. Then there is that d" 1 j"=0 a supercritical Hopf bifurcation in (17.61) at " = 0. The van der Pol di erential equation x + "(x2 ; 1)x_ + x = 0 with parameter " can be written as a planar di erential equation x_ = y y_ = ;"(x2 ; 1)y ; x: (17.68) For " = 0, (17.68) becomes the harmonic oscillator equation and it has only periodic solutions and an p" equilibrium p point, which is stable but not asymptotically stable. With the transformation u = x v = " y for " > 0 (17.68) is transformed into the planar di erential equation u_ = v v_ = ;u ; (u2 ; ") v: (17.69) For the eigenvalues of the Jacobian matrix at the equilibrium point (0 0) of (17.69): s2 d Re (") = 1 > 0. 12(") = 2" "4 ; 1 and so 12(0) = i and d" 1 j"=0 2 As shown in the example of 17.1.2.3, 1., p. 803, (0 0) is an asymptotically stable equilibrium point of (17.69) for " = 0: There is a supercritical Hopf bifurcation for " = 0, and for small " > 0, (0 0) is an unstable focus surrounded by a limit cycle whose amplitude is increasing as " increases.
4. Bifurcations in Two-Parameter Di erential Equations
1. Cusp Bifurcation Suppose the di erential equation (17.61) is given with r 4 and l = 2. Let the Jacobian matrix Dxf (0 0) have the eigenvalue 1 = 0 and n ; 1 eigenvalues j with Rej = 6 0 and
2 suppose that for the reduced di erential equation (17.63) F (0 0) = @F (0 0) = @ F2 (0 0) = 0 und @x @x 3 l3 := @@xF3 (0 0) 6= 0. Then the Taylor espansion of F close to (0 0) leads to the truncated normal form (without higher-order terms see, 17.1]) x_ = 1 + 2 x + sign l3 x3 (17.70) with the parameters 1 and 2. The set f(1 2 x): 1 + 2x + sign l3 x3 = 0g represents a surface in extended phase space and this surface is called a cusp (Fig. 17.26a). a1
x
1
2 b) a1 a)
2
S1
S2
a2 c)
d)
S2 S1
1
a2
e)
Figure 17.26 In the following, we suppose l3 < 0. The non-hyperbolic equilibrium points of (17.70) are de ned by the system 1 + 2 x ; x3 = 0 2 ; 3x2 = 0 and thus they lie on the curves S1 and S2 , which are
834 17. Dynamical Systems and Chaos determined by the set f(1 2 ): 2712 ; 423 = 0g and form a cusp (Fig. 17.26b). If (1 2) = (0 0) then the equilibrium point 0 of (17.70) is stable. The phase portrait of (17.61) in a neighborhood of 0, e.g., for n = 2 l3 < 0 and 1 = 0 is shown in Fig. 17.26c for 2 < 0 (triple node) and in Fig. 17.26d for 2 > 0 (triple saddle) (see 17.6]). At transition from (1 2) = (0 0) into the interior of domain 1 (Fig. 17.26b) the compound nodetype non-hyperbolic equilibrium point 0 of (17.61) splits into three hyperbolic equilibrium points (two stable nodes and a saddle) (supercritical pitchfork bifurcation). In the case of the two-dimensional phase space of (17.61) the phase portraits are shown in Fig. 17.26c,e. When the parameter pair of Si n f(0 0)g (i = 1 2) traverse from 1 into 2 then a double saddle nodetype equilibrium point is formed which nally vanishes. A stable hyperbolic equilibrium point remains. a2
S1 1
2
A
S3
3
2
1
A
a1
S2
S3
3
4
Figure 17.27
2. Bogdanov{Takens Bifurcation Suppose, for (17.61), n 2 l = 2 r 2 hold and the matrix Dxf (0 0) has two eigenvalues 1 = 2 = 0 and n ; 2 eigenvalues j with Rej 6= 0. Let the reduced
two-dimensional di erential equation (17.63) be topologically equivalent to the planar system x_ = y y_ = 1 + 2x + x2 ; xy: (17.71) Then there is a saddle-node bifurcation on the curve S1 = f(1 2): 22 ; 41 = 0g. At the transition on the curve S2 = f(1 2): 1 = 0 2 < 0g from the domain 1 < 0 into the domain 1 > 0 a stable limit cycle arises by a Hopf bifurcation and on the curve S3 = f(1 2): 1 = ;k22 + g (k > 0, constant) there exists a separatrix loop for the original system (Fig. 17.27), which bifurcates into a stable limit cycle in domain 3 (see 17.6], 17.9]). This bifurcation is of a global nature and we say that a single periodic orbit arises from the homoclinic orbit of a saddle or a separatrix loop disappears. 3. Generalized Hopf Bifurcation Suppose that the assumptions of the Hopf bifurcation with r 6 are ful lled for (17.61), and the two-dimensional reduced di erential equation after a coordinate transformation into polar coordinates has the normal form r_ = "1r + "2r3 ; r5 + _ = 1 + . The bifurcation diagram (Fig. 17.28) of this system contains the line S1 = f("1 "2): "1 = 0 "2 6= 0g, whose points represent a Hopf bifurcation (see 17.4], 17.6]). There exist two periodic orbits in domain 3, among which one is stable, the other one is unstable. On the curve S2 = f("1 "2): "22 +4"2 > 0 "1 < 0g, these non-hyperbolic cycles fuse into a compound cycle which disappears in domain 2.
5. Breaking Symmetry
Some di erential equations (17.61) have symmetries in the following sense: There exists a linear transformation T (or a group of transformations) such that f (Tx ") = T f (x ") holds for all x 2 M and " 2 V . An orbit of (17.61) is called symmetric with respect to T if T = . We talk about a symmetry breaking bifurcation at " = 0, e.g., in (17.61) (for l = 1), if there is a stable equilibrium point or a stable limit cycle for " < 0, which is always symmetric with respect to T , and for " = 0 two further stable steady states or limit cycles arise, which are nolonger symmetric with respect to T .
17.3 Bifurcation Theory and Routes to Chaos 835 e1 1
1 2
1
S1 3 e2
S2
2
3
Figure 17.28
") = "x ; x3 the transformation T
For system (17.61) with f (x de ned as T : x 7! ;x is a symmetry, since f (;x ") = ;f (x ") (x 2 IR " 2 IR). For " < 0 the point x1 = 0 is a stable p equilibrium point. For " > 0, besides x1 = 0, there exist the two other equilibrium points x23 = " both are nonsymmetric.
17.3.1.2 Local Bifurcations in a Neighborhood of a Periodic Orbit 1. Center Manifold Theorem for Mappings
Let be periodic orbit of (17.61) for " = 0 with multipliers 1 : : : n;1 n = 1. A bifurcation close to is possible, if when changing ", at least one of the multipliers lies on the complex unit circle. The use of a surface transversal to leads to the parameter-dependent Poincare mapping x 7;! P (x "): (17.72) Then, with open sets E IRn;1 and V IRl let P : E V ! IRn;1 be a C r -mapping where the mapping P~ : E V ! IRn;1 IRl with P~ (x ") = (P (x ") ") should be a C r -di eomorphism. Furthermore, let P (0 0) = 0 and suppose the Jacobian matrix DxP (0 0) has s eigenvalues 1 : : : s with j ij = 1, m eigenvalues s+1 : : : s+m with j ij < 1 and k = n ; s ; m ; 1 eigenvalues s+m+1 : : : n;1 with j ij > 1. Then, according to the the center manifold theorem for mappings (see 17.4]), close to (0 0) 2 E V , the mapping P~ is topologically conjugate to the mapping (x y z ") 7;! (F (x ") Asy Auz ") (17.73) near (0 0) 2 IRn;1 IRl with F (x ") = Acx + g(x "). Here g is a C r -di erentiable mapping satisfying the relations g(0 0) = 0 and Dxg(0 0) = 0. The matrices Ac As and Au are of type (s s) (m m) and (k k), respectively. It follows from (17.73) that bifurcations of (17.72) close to (0 0) are described only by the reduced mapping x 7;! F (x ") (17.74) c = f(x y z ): y = 0 z = 0g. on the local center manifold Wloc F(. ,a)
F(. ,a)
F(. ,a)
x
x
x
a) b)
a0
Figure 17.29
2. Bifurcation of Double Semistable Periodic Orbits
Let the system (17.61) be given with n 2 r 3 and l = 1. Suppose, at " = 0, the system (17.61) has periodic orbit with multipliers 1 = +1 j i j 6= 1 (i = 2 3 : : : n ; 1) and n = 1. According to the
836 17. Dynamical Systems and Chaos center manifold theorem for mappings, the bifurcations of the Poincare mapping (17.72) are described 2 by the one-dimensional reduced mapping (17.74) with Ac = 1. If @@xF2 (0 0) 6= 0 and @F @" (0 0) 6= 0 is supposed, then it leads to the normal forms ! 2 x 7;! F~ (x ) = + x + x2 for @@xF2 (0 0) > 0 or (17.75a) ! 2 x 7;! + x ; x2 for @@xF2 (0 0) < 0 : (17.75b) The iterations from (17.75a) close to 0 and the corresponding phase portraits are represented in Fig. 17.29a and in Fig. 17.29b for di erent (see 17.6]). Close to x = 0 there are for < 0 a stable and an unstable equilibrium point, which fuse for = 0 into the unstable steady state x = 0 . For > 0 there exists no equilibrium point close to x = 0. The bifurcation described by (17.75a) in (17.74) is called a subcritical saddle node bifurcation for mappings . In the case of the di erential equation (17.61), the properties of the mapping (17.75a) describe the bifurcation of a double semistable periodic orbit : For < 0 there exists a stable periodic orbit 1 and an unstable periodic orbit 2, which fuse for = 0 into a semistable orbit , which disappears for > 0 (Fig. 17.30a,b).
a
a0 c)
Figure 17.30
3. Period Doubling or Flip Bifurcation
Let system (17.61) be given with n 2 r 4 and l = 1. We consider a periodic orbit of (17.61) at " = 0 with the multipliers 1 = ;1 j ij 6= 1 (i = 2 : : : n ; 1), and n = 1. The bifurcation behavior of the Poincare mapping in the neighborhood of 0 is described by the one-dimensional mapping (17.74) with Ac = ;1, if we suppose the normal form x 7;! F~ (x ) = (;1 + )x + x3 : (17.76) The steady state x = 0 of (17.76) is stable for small 0 and unstable for < 0. The second iterated p mapping F~ 2 has for < 0 two further stable xed points besides x = 0 for x12 = ; + o(jj), which are not xed points of F~ . Consequently, they must be points of period 2 of (17.76). In general, for a C 4 -mapping (17.74) there is a two-periodic orbit at " = 0, if the following conditions are ful lled (see 17.2]): @F (0 0) = ;1 @F 2 (0 0) = 0 F (0 0) = 0 @x @" (17.77) @ 2 F 2 (0 0) 6= 0 @ 2 F 2 (0 0) = 0 @ 3 F 2 (0 0) 6= 0: @x@" @x2 @x3 2 @F @F Since @x (0 0) = +1 holds (because of @x (0 0) = ;1), the conditions for a pitchfork bifurcation are formulated for the mapping F 2. For the di erential equation (17.61) the properties of the mapping (17.76) imply that at = 0 a stable
17.3 Bifurcation Theory and Routes to Chaos 837
periodic orbit splits from with approximately a double period (period doubling), where loses its stability (Fig. 17.30c). The logistic mapping ' : 0 1] ! 0 1] is given for 0 < 4 by ' (x) = x(1 ; x), i.e., by the discrete dynamical system xt+1 = xt (1 ; xt ): (17.78) The mapping has the following bifurcation behavior (see 17.10]): For 0 < 1 system (17.78) has the equilibrium point 0 with domain of attraction 0 1]. For 1 < < 3, (17.78) has the unstable equilibrium point 0 and the stable equilibrium point 1 ; 1 , where this last one has the domain of attraction (0 1). For 1 = 3 the equilibrium point 1 ; 1 is unstable and leads to a stable two-periodic orbit. p At the value 2 = 1 + 6, the two-periodic orbit is also unstable and leads to a stable 22-periodic orbit. The period doubling continues, and stable 2q -periodic orbits arise at = q . Numerical evidence shows that q ! 1 3:570 : : : as q ! +1. For = 1, there is an attractor F (the Feigenbaum attractor), which has a structure similar to the Cantor set. There are points arbitrarily close to the attractor which are not iterated towards the attractor, but towards an unstable periodic orbit. The attractor F has dense orbits and the Hausdor dimension is dH (F ) 0:538 : : : . On the other hand, the dependence on initial conditions is not sensitive. In the domain 1 < < 4, there exists a parameter set A with positive Lebesgue measure such that system (17.78) has a chaotic attractor of positive Lebesgue measure for 2 A. The set A is interspersed with windows in which period doublings occur. The bifurcation behavior of the logistic mapping can also be found in a class of unimodal mappings, i.e., of mappings of the interval I into itself, which has a single maximum in I . Although the parameter values i , for which period doubling occurs, are di erent from each other for di erent unimodal mappings, the rate of convergence by which these parameters tend to 1, is equal: k ; 1 C ;k , where = 4:6692 : : : is the Feigenbaum constant (C depends on the concrete mapping). The Hausdor dimension is the same for all attractors F at = 1 : dH (F ) 0:538 : : : :
4. Creation of a Torus
Consider system (17.61) with n 3 r 6 and l = 1. Suppose that for all "2close to 0 system (17.61) has i a periodic orbit ". Let the multipliers of 0 be 12 = e with " 62 0 2 3 , j (j = 3 : : : n;1) with j j j 6= 1 and n = 1. According to the center manifold theorem, in this case there exists a two-dimensional reduced C 6mapping x 7;! F (x ") (17.79) with F (0 ") = 0 for " close to 0. If the Jacobian matrix DxF (0 ") has the conjugate complex eigenvalues (") and (") with j (0)j = 1 d j (")j > 0 holds and (0) is not a q-th root of 1 for q = 1 2 3 4, for all " near 0, if d := d" j"=0 then (17.79) can be transformed by a smooth " dependent coordinate transformation into the form x 7! F~ (x ") = F~o(x ") + O(kxk5) (O Landau symbol), where F~o is given in polar coordinates by ! ! r 7;! j (")jr + a(")r3 : (17.80) 2 + !(") + b(")r Here ! and b are di erentiable functions. Suppose a(0) < 0 holds. Then, the equilibrium point r = 0 of (17.80) is asymptotically stable for all " < 0 and unstable for " > 0. Furthermore, for " > 0 there
838 17. Dynamical Systems and Chaos s
exists the circle r = ; ad" (0) , which is invariant under the mapping (17.80) and asymptotically stable (Fig. 17.31a).
< a) e 0
e>0
b)
Figure 17.31
The Neimark{Sacker Theorem (see 17.10], 17.1]) states that the bifurcation behavior of (17.80)
is similar to that of F~ (supercritical Hopf bifurcation for mappings). In mapping (17.79), given by ! ! x 7;! p1 (1 + ")x + y + x2 ; 2y2 y 2 ;x + (1 + ")y + x2 ; x3 there is a supercritical Hopf bifurcation at " = 0. With respect to the di erential equation (17.61), the existence of a closed invariant curve of mapping (17.79) means that the periodic orbit 0 is unstable for a(0) < 0 and for " > 0 a torus arises which is invariant with respect to (17.61) (Fig. 17.31b).
17.3.1.3 Global Bifurcation
Besides the periodic creation orbit which arises if a separatrix loop disapears, (17.61) can have further global bifurcations. Two of them are shown in 17.12] by examples.
1. Emergence of a Periodic Orbit due to the Disappearance of a Saddle-Node
The parameter-dependent system x_ = x(1 ; x2 ; y2) + y(1 + x + ) y_ = ;x(1 + x + ) + y(1 ; x2 ; y2) has in polar coordinates x = r cos y = r sin the following form: r_ = r(1 ; r2) _ = ;(1 + + r cos ): (17.81) Obviously, the circle r = 1 is invariant under (17.81) for an arbitrary parameter , and all orbits (except the equilibrium point (0 0)) tend to this circle for t ! +1. For < 0 there are a saddle and a stable node on the circle, which fuse into a compound saddle-node type equilibrium point at = 0. For > 0, there is no equilibrium point on the circle, which is a periodic orbit (Fig. 17.32).
a0
Figure 17.32
2. Disappearance of a Saddle-Saddle Separatrix in the Plane Consider the parameter-dependent planar di erential equation x_ = + 2xy y_ = 1 + x2 ; y2:
(17.82)
17.3 Bifurcation Theory and Routes to Chaos 839
For = 0, equation (17.82) has the two saddles (0 1) and (0 ;1) and the y-axis as invariant sets. The heteroclinic orbit is part of this invariant set. For small jj 6= 0, the saddle-points are retained while the heteroclinic orbit disappears (Fig. 17.33).
a0
Figure 17.33
Often a strange attractor does not arise suddenly but as the result of a sequence of bifurcations, from which the typical ones are represented in Section 17.3.1. The most important ways to create strange attractors or strange invariant sets are described in the following.
17.3.2.1 Cascade of Period Doublings
Analogously to the logistic equation (17.78), a cascade of period doublings can also occur in timecontinuous systems in the following way. Suppose system (17.61) has the stable periodic orbit "(1) for " < "1. For " = "1 a period doubling occurs near "(1)1 , at which the periodic orbit "(1) loses its stability for " > "1. A periodic orbit "(2)1 with approximately double period splits from it. At " = "2, there is a new period doubling, where "(2)2 loses its stability and a stable orbit "(4)2 with an approximately double period arises. For important classes of systems (17.61) this procedure of period doubling continues, so a sequence of parameter values f"j g arises. Numerical calculations for certain di erential equations (17.61), e.g., for hydrodynamical di erential equations such as the Lorenz system, show the existence of the limit lim "j+1 ; "j = where (17.83) j !+1 " ; " j +2
j +1
is again the Feigenbaum constant. For " = jlim "j , the cycle with in nite period loses its stability, and a strange attractor appears. !1 The geometric background for this strange attractor in (17.61) by a cascade of period doubling is shown in Fig. 17.34a. The Poincare section shows approximately a baker mapping, which suggests that a Cantor set-like structure is created. F(. ,a)
x a)
17.3.2.2 Intermittency
b)
Figure 17.34
Consider a stable periodic orbit of (17.61), which loses its stability for " = 0 when exactly one of the multipliers, for " < 0 inside the unit circle takes the value +1. According to the center manifold theorem, the corresponding saddle-node bifurcation of the Poincare mapping can be described by a one-dimensional mapping in the normal form x 7! F~ (x ) = + x + x2 + . Here is a parameter
840 17. Dynamical Systems and Chaos depending on ", i.e., = (") with (0) = 0. The graph of F~ ( ) for positive is represented in Fig. 17.34b. As can be seen in Fig. 17.34b, the iterates of F~ ( ) stay for a relatively long time in the tunnel zone for !> 0. For equation (17.61), this means that the corresponding orbits stay relatively long in the neighborhood of the original periodic orbit. During this time, the behavior of (17.62) is approximately periodic (laminar phase). After getting through the tunnel zone the considered orbit escapes, which results in irregular motion (turbulent phase). After a certain time the orbit is recovered and a new laminar phase begins. A strange attractor arises in this situation if the periodic orbit vanishes and its stability goes over to the chaotic set. The saddle-node bifurcation is only one of the typical local bifurcations playing a role in the intermittence scenario. Two further ones are period doubling and the creation of a torus.
17.3.2.3 Global Homoclinic Bifurcations 1. Smale's Theorem
Let the invariant manifolds of the Poincare mapping of the di erential equation (17.61) in IR3 near the periodic orbit be as in Fig. 17.11b, p. 17.11. The transversal homoclinic points P j (x0 ) correspond to a homoclinic orbit of (17.61) to . The existence of such a homoclinic orbit in (17.61) leads to a sensitive dependence on initial conditions. In connection with the considered Poincare mapping, horseshoe mappings, introduced by Smale, can be constructed. This leads to the following statements: a) In every neighborhood of a transversal homoclinic point of the Poincare mapping (17.74) there exists a periodic point of this mapping (Smale's theorm). Hence, in every neighborhood of a transversal homoclinic point there exists an invariant set of P m(m 2 IN), 4, which is of Cantor type. The restriction of P m to 4 is topologically conjugate to a Bernoulli shift, i.e., to a mixing system. b) The invariant set of the di erential equation (17.61) close to the homoclinic orbit is like a product of a Cantor set with the unit circle. If this invariant set is attracting, then it represents a strange attractor of (17.61).
2. Shilnikov's Theorem
Consider the di erential equation (17.61) in IR3 with scalar parameter ". Suppose that the system (17.61) has a saddle-node type hyperbolic steady state 0 at " = 0, which exists so long as j"j remains small. Suppose, that the Jacobian matrix Dxf (0 0) has the eigenvalue 3 > 0 and a pair of conjugate complex eigenvalues 12 = a i ! with a < 0. Suppose, additionally, that (17.61) has a separatrix loop 0 for " = 0, i.e., a homoclinic orbit which tends to 0 for t ! ;1 and t ! +1 (Fig. 17.35a). Then, in a neighborhood of a separatrix loop (17.61) has the following phase portrait : a) Let 3 + a < 0. If the separatrix loop breaks at " 6= 0 according to the variant denoted by A in (Fig. 17.35a), then there is exactly one periodic orbit of (17.61) for " = 0. If the separatrix loop breaks at " 6= 0 according to the variant denoted by B in (Fig. 17.35a), then there is no periodic orbit. b) Let 3 + a > 0. Then there exist countably many saddle-type periodic orbits at " = 0 (respectively, for small j"j) close to the separatrix loop 0 (respectively, close to the destroid loop 0). The Poincare mapping with respect to a transversal to the 0 plane generates a countable set of horseshoe mappings at " = 0, from which there remain nitely many for small j"j 6= 0. A
u
g0
W (pe) pe
..
B
0 s
a)
b) W (pe)
Figure 17.35
d(t0)
17.3 Bifurcation Theory and Routes to Chaos 841
3. Melnikov's Method
Consider the planar di erential equation x_ = f (x) + "g(t x) (17.84) where " is a small parameter. For " = 0, let (17.84) be a Hamiltonian system (see 17.1.4.3, 2., p. 815), @H and f = ; @H hold, where H : U IR2 ! IR is supposed to be a C 3{ i.e., for f = (f1 f2) f1 = @x 2 @x1 2 function. Suppose the time-dependent vector eld g : IR U ! IR2 is twice continuously di erentiable, and T -periodic with respect to the rst argument. Furthermore, let f and g be bounded on bounded sets. Suppose that forP " = 0 there exists a homoclinic orbit with respect to the saddle point 0, and the Poincare section t0 of (17.84) in the phase space f(x1 x2 t)g for t = t0 looks as in Fig. 17.35b. The Poincare mapping P"t0 : Pt0 ! Pt0 , for small j"j, has a saddle point p" close to x = 0 with the invariant manifolds W s(p") and W u(p"). If the homoclinic orbit of the unperturbed system is given by '(t ; t0), then the distance between the manifold W s(p") and W u(p"), measured along the line passing through '(0) and perpendicular to f ('(0)), can be calculated by the formula t0 ) + O("2): d(t0) = " k fM('((0)) (17.85a) k Here, M () is the Melnikov function which is de ned by
M (t0 ) =
+ Z1
;1
f ('(t ; t0 )) ^ g(t '(t ; t0 )) dt:
(17.85b)
(For a = (a1 a2) and b = (b1 b2), ^ means a ^ b = a1b2 ; a2 b1 :) If the Melnikov function M has a simple root at t0 , i.e., M (t0 ) = 0 and M 0 (t0 ) 6= 0 hold, then the manifolds W s(p") and W u(p") intersect each other transversally for suciently small " > 0. If M has no root, then W s(p") \ W u(p") = , i.e., there is no homoclinic point. Remark: Suppose the unperturbed system1 (17.84)2 has a heteroclinic orbit given by '(t ; t0 ), running from a saddle point 01 in a saddle 02 . Let p" and p" be the saddle points of the Poincare mapping P"t0 for small j"j. If M , calculated as above, has a simple root at t0 , then W s(p1" ) and W u(p2" ) intersect each other transversally for small " > 0. Consider the periodically perturbated pendulum equation x + sin x = " sin !t, i.e., the system x_ = y y_ = ; sin x + " sin !t, in which " is a small parameter and ! is a further parameter. The unperturbed system x_ = y y_ = ; sin x is a Hamiltonian system with H (x y) = 12 y2 ; cos x. It has (among others) a pair of heteroclinic orbits through (; 0) and ( 0) (in the cylindrical phase space 1 1 S IR these are homoclinic orbits) given by ' (t) = 2 arctan(sinh t) 2 cosh t (t 2 IR). The 2 sin !t0 . Since M has a simple root direct calculation of the Melnikov function yields M (t0 ) = cosh( !=2) at t0 = 0, the Poincare mapping of the perturbed system has transversal homoclinic points for small " > 0.
17.3.2.4 Destruction of a Torus 1. From Torus to Chaos
1. Hopf{Landau Model of Turbulence The problem of transition from regular (laminar) behavior to irregular (turbulent) behavior is especially interesting for systems with distributed parameters, which are described, e.g., by partial di erential equations. From this viewpoint, chaos can be interpreted as behavior irregular in time but ordered in space. On the other hand, turbulence is the behavior of the system, that is irregular in time and in space. The Hopf{Landau model explains the arising of turbulence by an in nite cascade of Hopf bifurcations: For
842 17. Dynamical Systems and Chaos " = "1 a steady state befurcates in a limit cycle, which becomes unstable for "2 > "1 and leads to a torus T 2. At the k-th bifurcation of this type a k-dimensional torus arises, generated by non-closed orbits which wind on it. The Hopf{Landau model does not lead in general to an attractor which is characterized by sensitive dependence on the initial conditions and mixing. 2. Ruelle{Takens{Newhouse Scenario Suppose that in system (17.61) we have n 4 and l = 1. Suppose also that changing the parameter ", the bifurcation sequence \equilibrium point ! periodic orbit ! torus T 2 ! torus T 3" is achieved by three consecutive Hopf bifurcations. Let the quasiperiodic ow on T 3 be structurally unstable. Then, certain small perturbation of (17.61) can lead to the destruction of T 3 and to the creation of a structurally stable strange attractor.
3. Theorem of Afraimovich and2 Shilnikov on the Loss of Smoothness and the Destruction of the Torus T Let the suciently smooth system (17.61) be given with n 3
and l = 2. Suppose that for the parameter value "0, the system (17.61) has an attracting smooth torus T 2("0) spanned by a stable periodic orbit s, a saddle-type periodic orbit u and its unstable manifold W u(u) (resonance torus). The invariant manifolds of the equilibrium points of the Poincare mapping computed with respect to a surface transversal to the torus in the longitudinal direction, are represented in Fig. 17.36a. The multiplier of the orbit s, which is the nearest to the unit circle, is assumed to be real and simple. Furthermore, let "() : 0 1] ! V be an arbitrary continuous curve in parameter space, for which "(0) = "0 and for which system (17.61) has no invariant resonance torus for " = "(1). Then the following statements are true: a) There exists a value s 2 (0 1) for which Tu 2("(s )) loses its smoothness. Here, either the multiplier (s ) is complex or the unstable manifold W (u) loses its smoothness close to s. b) There exists a further parameter value s 2 (s 1) such that system (17.61) has no resonance torus for s 2 (s 1]. The torus is destroyed in the following way: ) The periodic orbit s loses its stability for " = "(s ). A local bifurcation arises as period doubling or the creation of a torus. ) The periodic orbits u and s coincide for " = "(s ) (saddle-node bifurcation) and so they vanish. ) The stable and unstable manifolds of u intersect each other non-transversally for " = "(s ) (see the bifurcation diagram in Fig. 17.36c). The points of the beak-shaped curve S1 correspond to the fused s and u (saddle-node bifurcation). The tip C1 of the beak-shaped curve is on a curve S0, which corresponds to a splitting of the torus. e2
S4 S1
a)
b)
c)
S P1 4 S3 S1
C1
P3 S2 P2 P0 S0 e1
Figure 17.36 The parameter points where the smoothness is lost, are on the curve S2, while the points on S3 characterize the dissolving of a T 2 torus. The parameter points for which the stable and unstable manifolds of u intersect each other non-transversally, are on the curve S4. Let P0 be an arbitrary point in the beaked shaped tip of the beak such that for this parameter value a resonance torus T 2 arises. The transition from P0 to P1 corresponds to the case ) of the theorem. If the multiplier becomes ;1 on S2, then there is a period doubling. A cascade of further period doublings can lead to a strange attractor. If a pair of complex conjugate multipliers 12 arises on the unit circle passing through S2 , then it can result in the splitting of a further torus, for which the Afraimovich{Shilnikov theorem can be used again. The transition from P0 to P2 represents the case ) of the theorem: The torus loses its smoothness, and
17.3 Bifurcation Theory and Routes to Chaos 843
on passing through on S1, there is a saddle-node bifurcation. The torus is destroyed, and a transition to chaos through intermittence can happen. The transition from P0 to P3, nally, corresponds to the case ): After the loss of smoothness, a non-robust homoclinic curve forms on passing through on S4 . The stable cycle s remains and a hyperbolic set arises which is not attracting for the present. If s vanishes, then a strange attractor arises from this set.
2. Mappings on the Unit Circle and Rotation Number
1. Equivalent and Lifted Mappings The properties of the invariant curves of the Poincare map-
ping play an important role in the loss of smoothness and destruction of a torus. If the Poincare mapping is represented in polar coordinates, then, under certain assumptions, we get decoupled mappings of the angular variables as informative auxiliary mappings on the unit circle. These are invertible in the case of smooth invariant curves (Fig. 17.36a) and in the case of non-smooth curves (Fig. 17.36b) they are not invertible. A mapping F : IR ! IR with F (! + 1) = F (!) + 1 for all ! 2 IR, which generates the dynamical system !n+1 = F (!n) (17.86) is called equivariant . For every such mapping, an associated mapping of the unit circle f : S 1 ! S 1 can be assigned where S 1 = IR n Z = f! mod 1 ! 2 IRg. Here f (x) := F (!) if the relation x = !] holds for the equivalence class !]. F is called a lifted mapping of f . Obviously, this construction is not unique. In contrast to (17.86) xt+1 = f (xt ) (17.87) is a dynamical system on S 1. For two parameters ! and K let the mapping F~ ( ! K ) be de ned by F~ ( ! K ) = + ! ; K sin for all 2 IR. The corresponding dynamical system n+1 = n + ! ; K sin n (17.88) can be transformed by the transformation n = 2!n into the system (17.89) !n+1 = !n + ) ; 2K sin 2!n where ) = 2! . With F (! ) K ) = ! +) ; 2K sin 2! an equivariant mapping arises, which generates the canonical form of the circle mapping. 2. Rotation Number The orbit (!) = fF n(!)g of (17.86) is a q-periodic orbit of (17.87) in S 1 if and only if it is a pq cycle of (17.86), i.e., if there exists an integer p such that !n+q = !n + p (n 2 Z) holds. The mapping f : S 1 ! S 1 is called orientation preserving if there exists a corresponding lifted mapping F , which is monotone increasing. If F from (17.86) is a monotone inreasing homeomorphism, F n(x) for every x 2 IR, and this limit does not depend on x. Hence, then there exists the limit jnlim j!1 n n(x) F 1 1 ~ the expression (F ) := jnlim j!1 n can be de ned. If f : S ! S is a homeomorphism and F and F are two lifted mappings of f , then (F ) = (F~ ) + k , where k is an integer. Based on this last property, the rotation number (f ) of an orientation-preserving homeomorphism f : S 1 ! S 1 can be de ned as (f ) = (F ) mod 1, where F is an arbitrary lifted mapping of f . If f : S 1 ! S 1 in (17.87) is an orientation-preserving homeomorphism, then the rotation number has the following properties (see 17.4]): a) If (17.87) has a q-periodic orbit, then there exists an integer p such that (f ) = pq holds.
b) If (f ) = 0, then (17.87) has an equilibrium point.
844 17. Dynamical Systems and Chaos c) If (f ) = pq , where p 6= 0 is an integer and q is a natural number (p and q are coprimes), then (17.87)
has a q-periodic orbit. d) (f ) is irrational if and only if (17.87) has neither a periodic orbit nor an equilibrium point. Theorem of Denjoy: If f : S 1 ! S 1 is an orientation-preserving C 2-di eomorphism and the rotation number = (f ) is irrational, then f is topologically conjugate to a pure rotation whose lifted mapping is F (x) = x + .
3. Di erential Equations on the Torus T 2 Let
!_ 1 = f1 (!1 !2) !_ 2 = f2(!1 !2) (17.90) be a planar di erential equation, where f1 and f2 are di erentiable and one-periodic functions in both arguments. In this case (17.90) de nes a ow, which can also be interpreted as a ow on the torus T 2 = S 1 S 1 with respect to !1 and !2. If f1(!1 !2) > 0 for all (!1 !2), then (17.90) has no equilibrium points and it is equivalent to the scalar rst-order di erential equation d!2 = f2(!1 !2) : (17.91) d!1 f1(!1 !2) With the relations !1 = t !2 = x and f = f2 , (17.91) can be written as a non-autonomous di erential f1 equation x_ = f (t x) (17.92) whose right-hand side is one-periodic with respect to t and x. Let '( x0) be the solution of (17.92) with initial state x0 at time t = 0. So, a mapping '1() = '(1 ) can be de ned for (17.92), which can be considered as the lifted mapping of a mapping f : S 1 ! S 1. Let !1 !2 2 IR be constants and !_ 1 = !1 !_ 2 = !2 a di erential equation on the torus, which is equivalent to the scalar di erential equation x_ = !!2 for !1 6= 0. Thus, '(t x0) = !!2 t + x0 and 1 1 '1(x) = !!2 + x. 1
4. Canonical Form of a Circle Mapping
1. Canonical Form The mapping F from (17.89) is an orientation-preserving di eomorphism for 0 K < 1, because @F @ = 1 ; K cos 2 > 0 holds. For K = 1, F is nolonger a di eomorphism,
but it is still a homeomorphism, while for K > 1, the mapping is not invertible, and hence nolonger a homeomorphism. In the parameter domain 0 K 1, the rotation number () K ) := (F ( ) K )) is de ned for F ( ) K ). Let K 2 (0 1) be xed. Then ( K ) has the following properties on 0,1]: a) The function ( K ) is not decreasing, it is continuous, but it is not di erentiable. b) For every rational number pq 2 0 1) there exists an interval Ip=q , whose interior is not empty and for which () K ) = p holds for all ) 2 Ip=q . q c) For every irrational number 2 (0 1) there exists exactly one ) with () K ) = . 2. Devil's Staircase and Arnold Tongues For every K 2 (0 1), ( K ) is a Cantor function. The graph of ( K ), which is represented in Fig. 17.37b, is called the devil's staircase. The bifurcation diagram of (17.89) is represented in Fig. 17.37a. At every rational number on the )-axis, a beak-shaped region (Arnold tongue) with a non-empty interior starts, where the rotation number is constant and equal to the rational number. The reason for the formation of the tongue is a synchronization of the frequencies (frequency locking).
17.3 Bifurcation Theory and Routes to Chaos 845
a) For 0 K < 1, these regions are not overlapping. At every irrational number of the )-axis, a
continuous curve starts which always reaches the line K = 1. In the rst Arnold tongue with = 0, the dynamical system (17.89) has equilibrium points. If K is xed and ) increases, then two of these equilibrium points fuse on the boundary of the rst Arnold tongue and vanish at the same time. As a result of such a saddle-node bifurcation, a dense orbit arises on S 1 . Similar phenomena can be observed when leaving other Arnold tongues. b) For K > 1 the theory of the rotation numbers is nolonger applicable. The dynamics become more complicated, and the transition to chaos takes place. Here, similarly to the case of Feigenbaum constants, further constants arise, which are equal for certain classes of mappings to which also the standard circle mapping belongs. One of them is described in the following. K 1 0 1
1 3
1 3 1 1
1 3
a)
1 2
1 2
1 3
W
r(. ,K) 1 0.8 0.6 0.4 0.2 b)
0.2
0.4
0.6
0.8
1W
Figure 17.37
p 3. Golden Mean, Fibonacci Numbers The irrational number 52; 1 is called the golden mean p 1 and it has a simple continued fraction representation 5 ; 1 = = 1 1 1 : : :] (see 1.1.1.4,
1 2 1 + 1+ 1+ 1 3., p. 4). By successive evaluation of the continued fraction we get a sequence frng of rational numbers, p 5 ; 1 which converges to 2 . The numbers rn can be represented in the form rn = FFn , where Fn are n+1 Fibonacci numbers, which are determined by the iteration Fn+1 = Fn + Fn;1 (n = 1 2 ) withpinitial values F0 = 0 and F1 = 1. Now, let )1 be the parameter value of (17.89), for which ()1 1) = 52; 1 and let )n be the closest value to )1, for which ()n 1) = rn holds. Numerical calculation gives the )n ; )n;1 = ;2:8336 : : : . limit nlim !1 ) ;)
n+1
n
846 18. Optimization
18 Optimization
18.1 Linear Programming
18.1.1 Formulation of the Problem and Geometrical Representation 18.1.1.1 The Form of a Linear Programming Problem 1. The Subject
of linear programming is the minimization or maximization of a linear objective function (OF) of nitely many variables subject to a nite number of constraints (CT), which are given as linear equations or inequalities. Many practical problems can be directly formulated as a linear programming problem, or they can be modeled approximately by a linear programming problem.
2. General Form
A linear programming problem has the following general form: OF: f (x) = c1 x1 + + cr xr + cr+1xr+1 + + cnxn = max! a11x1 + + a1r xr + a1r+1 xr+1 + + a1nxn b1 CT: ... ... ... ... ... as1x1 + + asr xr + asr+1xr+1 + + asnxn bs as+11x1 + + as+1r xr + as+1r+1xr+1 + + as+1nxn = bs+1 ... ... ... ... ... am1 x1 + + amr xr + amr+1xr+1 + + amnxn = bm
x1 0 : : : xr 0 xr+1 : : : xn free. In a more compact vector notation this problem becomes:
OF : f (x) = c1 Tx1 + c2 Tx2 = max! Here, we denote: 0 c1 1 0 cr+1 1 B C Bc C c c1 = BB@ ...2 CCA c2 = BB@ r...+2 CCA cr cn 0 a11 a12 a1r 1 B a a a2r C C A11 = BB@ ... 21 22 C A as1 as2 asr
0 as+11 as+12 as+1r 1 B a a as+2r C C C A21 = BB@ ... s+21 s+22 A am1 am2 amr
(18.2a)
(18.1a)
9 > > > > > > > = > > > > > > > "
CT : A11 x1 + A12 x2 b12 A21 x1 + A22 x2 = b x1 0 x2 free.
0 x1 1 0 xr+1 1 B C Bx C x x1 = BB@ ...2 CCA x2 = BB@ r...+2 CCA xr xn 0 a1r+1 a1r+2 a1r+n 1 B a a a2r+n C C A12 = BB@ ... 2r+1 2r+2 C A asr+1 asr+2 asn 0 as+1r+1 as+1r+2 as+1n 1 Ba a as+2n CC A22 = BB@ ... s+2r+1 s+2r+2 C A: amr+1 amr+2 amn
(18.1b)
9 > = > "
(18.2b)
(18.2c)
(18.2d)
(18.2e)
18.1 Linear Programming 847
3. Constraints
with the inequality sign \ " will have the above form if we multiply them by (;1).
4. Minimum Problem
A minimum problem f (x) = min! becomes an equivalent maximum problem by multiplying the objective function by (;1) ;f (x) = max! (18.3)
5. Integer Programming
Sometimes certain variables are restricted to be only integers. We do not discuss this discrete problem here.
6. Formulation with only Non-Negative Variables and Slack Variables
In applying certain solution methods, we have to consider only non-negative variables, and constraints (18.1b), (18.2b) given in equality form. Every free variable xk must be decomposed into the di erence of two non-negative variables xk = x1k ; OF : f (x) = c1 x1 + + cnxn = max! (18.4a) x2k . The inequalities become equalities by adding 9 non-negative variables they are called slack vari- CT : a11. x1 + + a1n. xn = b.1 > .. .. .. > > ables. That is, we can consider the problem in the = form as given in (18.4a,b), where n is the increased am1 x1 + + amn xn = bm > (18.4b) > number of variables. " In vector form we have: x1 0 : : : xn 0: > T OF: f (x) = c x = max! (18.5a) CT: Ax = b x 0 : (18.5b) We can suppose that m n, otherwise the system of equations contains linearly dependent or contradictory equations.
7. Feasible Set
The set of all vectors x satisfying constraints (18.2b) is called the feasible set of the original problem. If we rewrite the free variables as above, and every inequality of the form \" into an equation as in (18.4a) and (18.4b), then the set of all non-negative vectors x 0 satisfying the constraints is called the feasible set M : M = fx 2 IRn : x 0 Ax = bg: (18.6a) A point x 2 M with the property f (x ) f (x) for every x 2 M (18.6b) is called the maximum point or the solution point of the linear programming problem. Obviously, the components of x not belonging to slack variables form the solution of the original problem.
18.1.1.2 Examples and Graphical Solutions 1. Example of the Production of Two Products
Suppose we need primary materials R1 , R2 , and R3 to produce two products E1 and E2 . Scheme 18.1 shows how many units of primary materials are needed to produce each unit of the products E1 and E2 , and there are given also the available amount of the primary materials. Scheme 18.1 Selling one unit of the products E1 or E2 results in 20 or 60 units of R1 / Ei R2 /Ei R3 /Ei pro t, respectively (PR ). Determine a production program E1 12 8 0 which yields maximum pro t, if at E 6 12 10 2 least 10 units must be produced from product E1. Available amount 630 620 350 If we denote by x1 and x2 the num-
848 18. Optimization ber of units produced from E1 and E2 , then we get the following problem: OF: f (x) = 20x1 + 60x2 = max! CT: 12x1 + 6x2 630 8x1 + 12x2 620 10x2 350 x1 10: Introducing the slack variables x3 , x4 , x5 , x6, we get: OF : f (x) = 20x1 + 60x2 + 0 x3 + 0 x4 + 0 x5 + 0 x6 = max! CT : 12x1 + 6x2 + x3 = 630 8x1 + 12x2 + x4 = 620 10x2 + x5 = 350 ;x1 + x6 = ;10:
2. Properties of a Linear Programming Problem
On the basis of this example, we can demonstrate some properties x2 of the linear programming problem by graphical representation. To do this, we do not consider the slack variables only the original two 35 variables are used. a) A line a1 x1 + a2x2 = b divides the x1 x2 plane into two half-planes. 25 M The points (x1 x2 ) satisfying the inequality a1x1 +a2 x2 b are in one of these half-planes. The graphical representation of this set of points in a Cartesian coordinate system can be made by a line, and the half40 x1 plane containing the solutions of the inequalities is denoted by an 0 10 arrow. The set of feasible solutions M , i.e., the set of points satisfying all inequalities is the intersection of these half-planes (Fig. 18.1). Figure 18.1 In this example the points of M form a polygonal domain. It may happen that M is unbounded or empty. If more then two boundary lines go through a vertex of the polygon, we call this vertex a degenerate vertex (Fig. 18.2). x2
x2
x2 P M
0
x1
x1
0
0
x1
Figure 18.2
b) Every point in the x1 x2 plane satisfying the equality f (x) = 20x1 + 60x2 = c0 is on one line, on
the level line associated to the value c0 . With di erent choices of c0 , a family of parallel lines is de ned, on each of which the value of the objective function is constant. Geometrically, those points are the solutions of the programming problem, which belong to the feasible set M and also to the level line 20x1 + 60x2 = c0 with maximal value of c0 . In this example, the solution point is (x1 x2) = (25 35) on the line 20x1 + 60x2 = 2600. The level lines are represented in Fig. 18.3, where the arrows point in the direction of increasing values of the objective function. Obviously, if the feasible set M is bounded, then there is at least one vertex such that the objective function takes its maximum. If the feasible set M is unbounded, it is possible that the objective function
18.1 Linear Programming 849
c0=2600
x3
x2
c0=2400 35
P3=(0,0,1)
P6=(0,2,1) P5=(2,2,2)
25
P1=(1,0,0)
M
P2=(0,1,0)
P7=(0,2,0) x2
c0=0 0
10
25
x1
x1 P4=(4,2,0)
Figure 18.3 is unbounded, as well.
Figure 18.4
18.1.2 Basic Notions of Linear Programming, Normal Form
We consider problem (18.5a,b) with the feasible set M.
18.1.2.1 Extreme Points and Basis 1. Denition of the Extreme Point
A point x 2 M is called an extreme point or vertex of M , if for all x1 x2 2 M with x1 6= x2: x 6= x1 + (1 ; )x2 0 < < 1 i.e., x is not on any line segment connecting two di erent points of M .
(18.7)
2. Theorem about Extreme Points
The point x 2 M is an extreme point of M if the columns of matrix A associated to the positive components of x are linearly independent. If the rank of A is m, then the maximal number of independent columns in A is m. So, an extreme point can have at most m positive components. The other components, at least n ; m, are equal to zero. In the usual case, there are exactly m positive components. If the number of positive components is less then m, we call it a degenerate extreme point.
3. Basis
We can assign m linearly independent column vectors of the matrix A to every extreme point, the columns belonging to the positive components. This system of linearly independent column vectors is called the basis of the extreme point . Usually, exactly one basis belongs to every extreme n point. However possibilities to several bases can be assigned to a degenerate extreme point. There are at most m choose m linearly independent vectors from n columns of A.Consequently, the number of di erent n bases, and therefore the number of di erent extreme points is m . If M is not empty, then M has at least one extreme point.
850 18. Optimization OF: f (x) = 2x1 + 3x2 + 4x3 = max! CT: x1 + x2 + x3 1 x2 2 ;x1 + 2x3 2 2x1 ; 3x2 + 2x3 2:
(18.8)
The feasible set M determined by the constraints is represented in Fig. 18.4. Introduction of slack variables x4 , x5 , x6 , x7 leads to: CT : x1 + x2 + x3 ; x4 =1 x2 + x5 =2 ;x1 + 2x3 + x6 =2 2x1 ; 3x2 + 2x3 + x 7 = 2: The extreme point P2 = (0 1 0) of the polyhedron corresponds to the point x = (x1 x2 x3 x4 x5 x6 x7 ) = (0 1 0 0 1 2 5) of the extended system. The columns 2, 5, 6 and 7 of A form the corresponding basis. The degenerated extreme point P1 corresponds to (1 0 0 0 2 3 0). A basis of this extreme point contains the columns 1 5 6 and one of the columns 2 4 or 7. Remark: Here, the rst inequality was a \ " inequality and we did not add but subtract x4. Frequently these types of additional variables both with a negative sign and a corresponding bi > 0 are called surplus variables, rather than slack variables. As we will see in 18.1.3.3, p. 854, the occurrence of surplus variables requires additional e ort in the solution procedure.
4. Extreme Point with a Maximal Value of the Objective Function T
Theorem: If M is not empty, and the objective function f (x) = c x is bounded from above on M ,
then there is at least one extreme point of M where it has its maximum. A linear programming problem can be solved by determining at least one of the extreme points with maximum value of the objective function. Usually, the number of extreme points of M is very large in practical problems, so we need a method by which we can nd the solution in a reasonable time. Such a method is the simplex method , which is also called the simplex algorithm or simplex procedure.
18.1.2.2 Normal Form of the Linear Programming Problem 1. Normal Form and Basic Solution
The linear programming problem (18.4a,b) can always be transformed to the following form with a suitable renumbering of the variables: (18.9a) OF: f (x) = c1 x1 + + cn;mxn;m + c0 = max! 9 = b1 > CT: a11 x1 + + a1n;mxn;m + xn;m+1 > ... ... ... ... = (18.9b) am1 x1 + + amn;mxn;m + xn = bm > > > " x1 : : : xn;m xn;m+1 : : : xn 0: The last m columns of the coecient matrix are obviously independent, and they form a basis. The basic solution (x1 x2 : : : xn;m xn;m+1 : : : xn ) = (0 : : : 0 b1 : : : bm ) can be determined directly from the system of equations, but if b 0 does not hold, it is not a feasible solution. If b 0, then (18.9a,b) is called a normal form or canonical form of the linear programming problem . In this case, the basic solution is a feasible solution, as well, i.e., x 0, and it is an extreme point of M . The variables x1 : : : xn;m are called non-basic variables and xn;m+1 : : : xn are called basic variables. The objective function has the value c0 at this extreme point, since the non-basic variables are equal to zero.
18.1 Linear Programming 851
2. Determination of the Normal Form
If an extreme point of M is known, then we can get a normal form of the linear programming problem in the following way. We choose a basis from the columns of A corresponding to the extreme point. Suppose the basic variables are collected into the vector xB and the non-basic variables are in xN . The columns associated to the basis form the basis matrix AB , the other columns form the matrix AN . Then, Ax = AN xN + AB xB = b: (18.10) The matrix AB is non-singular and it has an inverse A;B1, the so-called basis inverse. Multiplying (18.10) by A;B1 and changing the objective function according to the non-basic variables results in the canonical form of the linear programming problem: OF : f (x) = cTN xN + c0 (18.11a)
CT : A;B1 AN xN + xB = A;B1b with xN 0 xB 0: (18.11b) Remark: In the original system (18.1b) has only constraints of type \" and simultanously b 0. Then the extended system (18.4b) contains no surplus variables (see 18.1.2.1, p. 849). In this case a normal form is immediatlely known. Selecting all slack variables as basic variables xB we have AB = I and xB = b, xN = 0 is a feasible extreme point. In the above example x = (0 1 0 0 1 2 5) is an extreme point. Consequently: 0 1 0 0 01 0 1 1 ;1 1 0 1 0 0 01 B C B 1 1 0 0 B C B 0 0 0C C AB = @ 0 0 1 0 A A;B1 = BB@ ;10 10 01 00 CCA AN = @ ;1 2 0 A (18.12a) ;3 0 0 1 2 2 0 3 0 0 1 x2 x5 x6 x7 0 1 1 ;1 1 B B ;1 ;1 1 C C A;B1 AN = @ ;1 2 0 A 5 5 ;3 x1 x3 x4
x1 x3 x4
011 AB b = BB@ 1 CCA : ;1
2 5
x1 + x2 + x3 ; x4 =1 9 > ; x3 + x4 + x5 =1 = + 2x3 + x6 =2 > 5x1 + 5x3 ; 3x4 + x 7 = 5: "
;x1 ;x1
From f (x) = 2x1 + 3x2 + 4x3, we get the transformed objective function f (x) = ;x1 + x3 + 3x4 + 3 if we subtract the triple of the rst constraint.
(18.12b)
(18.13)
(18.14)
18.1.3 Simplex Method 18.1.3.1 Simplex Tableau
The simplex method is used to produce a sequence of extreme points of the feasible set with increasing values of the objective function. The transition to the new extreme point is performed starting from the normal form corresponding to the given extreme point, and arriving at the normal form corresponding to the new extreme point. In order to get a clear arrangement, and easier numerical performance, we put the normal form (18.9a,b) in the simplex tableau (Scheme 18.2a, 18.2b):
852 18. Optimization Scheme 18.2a x1 xn;m xn;m+1 a11 a1n;m b1 ... ... ... ... xn am1 amn;m bm c1 cn;m ;c0
Scheme 18.2b or briey
xN xB AN b c ;c0
The k-th row of the tableau corresponds to the constraint xn;m+k + ak1x1 + + akn;mxn;m = bk : (18.15a) We have for the objective function (18.15b) c1x1 + + cn;mxn;m = f (x) ; c0 : From this simplex tableau, we can nd the extreme point (xN xB ) = (0 b). We also get the value of the objective function at this point f (x) = c0. We can always nd exactly one of the following three cases in every tableau: a) cj 0, j = 1 : : : n ; m: The tableau is optimal. The point (xN xB ) = (0 b) is the maximal point. b) There exists at least one j such that cj > 0 and aij 0, i = 1 : : : m: The linear programming problem has no solution, since the objective function is not bounded on the feasible set for increasing values of xj it increases without a bound. c) For every j with cj > 0 there exists at least one i with aij > 0: We can move from the extreme point x to a neighboring extreme point x~ with f (~x) f (x). In the case of a non-degenerate extreme point x, the \>" sign always holds.
18.1.3.2 Transition to the New Simplex Tableau 1. Non-Degenerate Case
If a tableau is not in nal form (case c)), then we determine a new tableau (Scheme 18.3). We interchange a basic variable xp and a non-basic variable xq by the following calculations: a) a~pq = a1 : (18.16a) pq ~bp = bp a~pq : b) a~pj = apj a~pq j 6= q (18.16b) c) a~iq = ;aiq a~pq i 6= p c~q = ;cq a~pq : (18.16c) d) a~ij = aij + apj a~iq i 6= p j 6= q ~bi = bi + bp a~iq i 6= p c~j = cj + apj c~q j 6= q c~0 = c0 + bp c~q : (18.16d) The element apq is called the pivot element , the p-th row is the pivot row, and the q-th column is the pivot column. We must consider the following requirements for the choice of a pivot element: a) c~0 c0 should hold b) the new tableau must also correspond to a feasible solution, i.e., b~ 0 must hold. Then, (~xN x~ B ) = (0 b~ ) is a new extreme point at which the value of the objective function f (~x) = c~0 is not smaller than it was previously. These conditions are satis ed if we choose the pivot element in the following way: a) To increase the value of the objective function, a column with cq > 0 can be chosen for a pivot column b) to get a feasible (solution, the pivot row must be chosen as ) bp = min bi : (18.17) apq 1 i m aiq aiq >0
18.1 Linear Programming 853
If the extreme points of the feasible set are not degenerate, then the simplex method terminates in a nite number of steps (case a) or case b)). The normal form in 18.1.2 can be written in a simplex tableau (Scheme 18.4a). Scheme 18.4a Scheme 18.3 Scheme 18.4b x~ N x1 x3 x4 x1 x3 x5 x 1 1 ; 1 1 x 0 0 1 2 ~ ~ 2 2 x~ B AN b x ; 1 ; 1 1 1 1 : 1 x ; 1 ;1 1 1 5 4 c~ ;c~0 x6 ;1 2 0 2 x6 ;1 2 0 2 2:2 x7 5 5 ;3 5 x7 2 2 3 8 8:2 ; 1 1 3 ;3 2 4 ;3 ;6 This tableau is not optimal, since the objective function has a positive coecient in the third column. The third column is assigned as the pivot column (the second column could also be taken under consideration). We calculate the quotients bi =aiq with every positive element of the pivot column (there is only one of them). The quotients are denoted behind the last column. The smallest quotient determines the pivot row. If it is not unique, then the extreme point corresponding to the new tableau is degenerate. After performing the steps of (18.16a){(18.16d) we get the tableau in Scheme 18.4b. This tableau determines the extreme point (0 2 0 1 0 2 8), which corresponds to the point P7 in Fig. 18.4. Since this new tableau is still not optimal, we interchange x6 and x3 (Scheme 18.4c). The extreme point of the third tableau corresponds to the point P6 in Fig. 18.4. After an additional change we get an optimal tableau (Scheme 18.4d) with the maximal point x = (2 2 2 5 0 0 0), which corresponds to the point P5, and the objective function has a maximal value here: f (x ) = 18. Scheme 18.4c Scheme 18.4d Scheme 18.5 x1 x6 x5 x7 x6 x5 x1 xn x2 0 0 1 2 x2 0 0 1 2 y1 a11 a1n b1 ... ... ... ... 3 1 1 5 x4 ; 2 2 1 2 x4 2 0 2 5 ym am1 amn bm OF c1 cn 0 x3 ; 12 12 0 1 x3 16 13 12 2 m m m P P P OF j=1 aj1 j=1 ajn j=1 bj = ;g(0 b) x7 3 ;1 3 6 6:3 x1 13 ; 13 1 2 4 ;2 ;3 ;10 ; 43 ; 23 ;7 ;18
2. Degenerate Case
If the next pivot element cannot be chosen uniquely in a simplex tableau, then the new tableau represents a degenerate extreme point. A degenerate extreme point can be interpreted geometrically as the coincident vertices of the convex polyhedron of the feasible solutions. There are several bases for such a vertex. In this case, it can therefore happen that we perform some steps without reaching a new extreme point. It is also possible that we get a tableau that we had before, so an in nite cycle may occur. In the case of a degenerate extreme point, one possibility is to perturb the constants bi by adding "i (with a suitable "i > 0) such that the resulting extreme points are nolonger degenerate. We get the solution from the solution of the perturbed problem, if we substitute " = 0. If the pivot column is chosen \randomly" in the non-uniquely determined case, then the occurrence of an in nite cycle is unlikely in practical cases.
854 18. Optimization
18.1.3.3 Determination of an Initial Simplex Tableau 1. Secondary Program, Articial Variables
If there are equalities among the original constraints (18.1b) or inequalities with negative bi , then it is not easy to nd a feasible solution to start the simplex method. In this case, we start with a secondary program to produce a feasible solution, which can be a starting point for a simplex procedure for the original problem. We add an articial variable yk 0 (k = 1 2 : : : m) to every left-hand side of Ax = b with b 0, and we consider the secondary program: (18.18a) OF : g(x y) = ;y1 ; ; ym = max! 9 = b1 > CT : a11 x1 + + a1nxn + y1 ... ... ... > ... = (18.18b) am1 x1 + + amnxn + ym = bm > > > " x1 : : : xn 0 y1 : : : ym 0: For this problem, the variables y1 : : : ym are basic variables, and we can start the rst simplex tableau (Scheme 18.5). The last row of the tableau contains the sums of the coecients of the non-basic variables , and these sums are the coecients of the new secondary objective function OF . Obviously, g(x y) 0 always. If g(x y ) = 0 for a maximal point (x y ) of the secondary problem, then obviously y = 0, and consequently x is a solution of Ax = b. If g(x y ) < 0, then Ax = b does not have any solution.
2. Solution of the Secondary Program
Our goal is to eliminate the arti cial variables from the basis. We do not prepare a scheme only for the secondary program separately. We add the columns of the arti cial variables and the row of the secondary objective function to the original tableau. The secondary objective function now contains the sums of the corresponding coecients from the rows corresponding to the equalities, as shown below. If an arti cial variable becomes a non-basic variable, we can omit its column, since we will never choose it again as a basis variable. If we determined a maximal point (x y ), then we distinguish between two cases: 1. g(x y ) < 0: The system Ax = b has no solution, the linear programing problem does not have any feasible solution. 2. g(x y ) = 0: If there are no arti cial variables among the basic variables, this tableau is an initial tableau for the original problem. Otherwise we remove all arti cial variables among the basic variables by additional steps of the simplex method. By introducing the arti cial variables, the size of the problem can be increased considerably. As we see, it is not necessary to introduce arti cial variables for every equation. If the system of constraints before introducing the slack and surplus variables (see Remark on p. 850) has the form A1x b1 , A2x = b2 , A3x b3 with b1 b2 b3 0, then we have to introduce arti cial variables only for the rst two systems. For the third system the slack variables can be chosen as basic variables. In the example of 18.1.2, p. 850, only the rst equation requires an arti cial variable: OF : g(x y) = ; y1 = max! CT : x1 + x2 + x3 ; x4 + y1 =1 x2 + x5 =2 ;x1 + 2x3 + x6 =2 2x1 ; 3x2 + 2x3 + x7 = 2: The tableau (Scheme 18.6b) is optimal with g(x y ) = 0. After omitting the second column we get the rst tableau of the original problem.
18.1 Linear Programming 855
Scheme 18.6a x1 x 2 x 3 x 4 y1 1 1 1 ;1 0 1 0 0 x5 x6 ;1 0 2 0 x7 2 ;3 2 0 OF 2 3 4 0 OF 1 1 1 ;1
1 2 2 2 0 1
1:1 2:1
18.1.3.4 Revised Simplex Method 1. Revised Simplex Tableau
Scheme 18.6b x1 y1 x3 x4 1 x2 1 1 1 ;1 1 x5 ;1 ;1 ;1 1 1 x6 ;1 0 2 0 2 x7 5 3 5 ;3 5 OF ; 1 ;3 1 3 ;3 OF 0 ;1 0 0 0
Suppose the linear programming problem is given in normal form: (18.19a) OF: f (x) = c1 x1 + + cn;mxn;m + c0 = max! 9 = 1 > CT: 1.1 x1 + + 1n;m. xn;m + xn;m+1 . ... > > .. .. .. = (18.19b) m1x1 + + mn;mxn;m + xn = m > > > " x1 0 : : : xn 0: Obviously, the coecient vectors n;m+i (i = 1 : : : n) are the i-th unit vectors. In order to change into another normal form and therefore to reach another extreme point, it is sucient to multiply the system of equations (18.19b) by the corresponding basis inverse. (We refer to the fact that if AB denotes a new basis, the coordinates of a vector x can be expressed in this new basis as A;B1x. If we know the inverse of the new basis, we can get any column as well as the objective function from the very rst tableau by simple multiplication.) The simplex method can be modi ed so that we determine only the basis inverse in every step instead of a new tableau. From every tableau, we determine only those elements which are required to nd the new pivot element. If the number of variables is considerably larger than the number of constraints (n > 3m), then the revised simplex method requires considerably less computing cost and therefore has better accuracy. The general form of a revised simplex tableau is shown in Scheme 18.7. Scheme 18.7 x1 xn;m xn;m+1 xn xq B x1 a1n;m+1 a1n b1 r1 ... ... ... ... ... xBm amn;m+1 amn bm rm c1 cn;m cn;m+1 cn ;c0 cq The quantities of the scheme have the following meaning: xB1 : : : xBm : Actual basic variables (in the rst step the same as xn;m+1 xn ). c1 : : : cn : Coecients of the objective function (the coecients associated to the basic variables are zeros). b1 : : : bm : Right-hand side of the actual normal form. c0 0 : Value of the objective function at the extreme point (xB1 : : : xBm ) = (b1 : : : bm). a1n;m+1 a1n 1 basis inverse, where the columns of A are the columns of ... C A = B@ ... A: Actual xn;m+1 : : : xn corresponding to the actual normal form amn;m+1 amn r = (r1 : : : rm)T: Actual pivot column.
856 18. Optimization
2. Revised Simplex Step
a) The tableau is not optimal when at least one of the coecients cj (j = 1 2 : : : n) is positiv. We choose a pivot column q for a cq > 0. b) We calculate the pivot column r by multiplying the q-th column of the original coecient matrix (18.19b) by A and we introduce the new vector as the last vector of the tableau. We determine the pivot row k in the same way as in the simplex algorithm (18.17). c) We calculate the new tableau by the pivoting step (18.16a{d), where aiq is formally replaced by ri and the indices are restricted for n ; m + 1 j n. The column ~r is omitted. xq becomes a basic variable. For j = 1 : : : n ; m, we get c~j = cj + Tj c~, where ~c = (~cn;m+1 : : : c~n)T, and j is the j -th
column of the coecient matrix of (18.19b). Consider the normal form of the example in 18.1.2, p. 850. We want to bring x4 into the basis. The corresponding pivot column r = 4 is placed into the last column of the tableau (Scheme 18.8a) (initially A is the unit matrix). Scheme 18.8a Scheme 18.8b x1 x3 x4 x2 x5 x6 x7 x4 x1 x3 x4 x2 x5 x6 x7 x3 x2 1 0 0 0 1 ;1 x2 1 1 0 0 2 0 x5 0 1 0 0 1 1 1:1 x4 0 1 0 0 1 ;1 x6 0 0 1 0 2 0 x6 0 0 1 0 2 2 2:2 x7 0 0 0 1 5 ;3 0 3 0 1 8 2 8:2 x7 ; 1 1 3 0 0 0 0 ;3 3 2 4 ;3 0 ;3 0 0 ;6 4 For j = 1 3 4 we have: c~j = cj ; 32j : (c1 c3 c4) = (2 4 0). The determined extreme point x = (0 2 0 1 0 2 8) corresponds to the point P7 in Fig. 18.4. The next pivot column can be chosen for j = 3 = q. The vector r is determined by 01 1 0 01 0 11 0 01 r = (r1 : : : rm) = A 3 = BB@ 00 10 01 00 CCA BB@ ;21 CCA = BB@ ;21 CCA 0 3 0 1 5 2 and we place it into the very last column of the second tableau (Scheme 18.8b). We proceed as above analogously to the method shown in 18.1.3.2, p. 853. If we want to return to the original method, then we have to multiply the matrix of the original columns of the non-basic variables by A and we keep only these columns.
18.1.3.5 Duality in Linear Programming 1. Correspondence
To any linear programming problem (primal problem) we can assign another unique linear programming problem (dual problem):
Primal problem OF: f (x) = cT1x1 + cT2x2 = max! (18.20a) CT: A11x1 + A12x2 b1 A21x1 + A22x2 = b2 x1 0 x2 free: (18.20b)
Dual problem OF : g(u) = bT1u1 + bT2u2 = min! (18.21a) CT : AT11u1 + AT21u2 c1 AT12u1 + AT22u2 = c2 u1 0 u2 free: (18.21b)
The coecients of the objective function of one of the problems form the right-hand side vector of the constraints of the other problem. Every free variable corresponds to an equation, and every variable with restricted sign corresponds to an inequality of the other problem.
18.1 Linear Programming 857
2. Duality Theorems
a) If both problems have feasible solutions, i.e., M 6= , M 6= (where M and M denote the feasible
sets of the primal and dual problems respectively), then (18.22a) f (x) g(u) for all x 2 M u 2 M and both problems have optimal solutions. b) The points x 2 M and u 2 M are optimal solutions for the corresponding problem, if and only if f (x) = g(u): (18.22b) c) If f (x) has no upper bound on M or g(u) has no lower bound on M , then M = or M = , i.e., the dual problem has no feasible solution. d) The points x 2 M and u 2 M are optimal points of the corresponding problems if and only if: uT1(A11x1 + A12 x2 ; b1 ) = 0 and xT1(AT11u1 + AT21u2 ; c1 ) = 0: (18.22c) Using these last equations, we can nd a solution x of the primal problem from a non-degenerate optimal solution u of the dual problem by solving the following linear system of equations: A21x1 + A22 x2 ; b2 = 0 (18.23a) (A11 x1 + A12x2 ; b1)i = 0 for ui > 0 (18.23b) xi = 0 for (AT11u1 + AT21u2 ; c1 )i 6= 0: (18.23c) We can also solve the dual problem by the simplex method.
3. Application of the Dual Problem
Working with the dual problem may have some advantages in the following cases:
a) If it is simple to nd a normal form for the dual problem, we switch from the primal problem to the dual. b) If the primal problem has a large number of constraints compared to the number of variables, then we can use the revised simplex method for the dual problem. Consider the original problem of the example of 18.1.2, p. 850.
Primal problem
OF: f (x) = 2x1 + 3x2 + 4x3 = max! CT: ;x1 ; x2 ; x3 ;1 x2 2 ;x1 + 2x3 2 2x1 ; 3x2 + 2x3 2 x1 x2 x3 0:
Dual problem OF : g(u) = ;u1 + 2u2 + 2u3 + 2u4 = min! CT : ;u1 ; u3 + 2u4 2 ;u1 + u2 ; 3u4 3 ;u1 + 2u3 + 2u4 4 u1 u2 u3 u4 0:
If the dual problem is solved by the simplex method after introducing the slack variables, then we get the optimal solution u = (u1 u2 u3 u4) = (0 7 2=3 4=3) with g(u) = 18. We can obtain a solution x of the primal problem by solving the system (Ax ; b)i = 0 for ui > 0, i.e., x2 = 2, ;x1 + 2x3 = 2, 2x1 ; 3x2 + 2x3 = 2, therefore: x = (2 2 2) with f (x) = 18.
18.1.4 Special Linear Programming Problems 18.1.4.1 Transportation Problem 1. Modeling
A certain product, produced by m producers E1 E2 : : : Em in quantities a1 a2 : : : am, is to be transported to n consumers V1 V2 : : : Vn with demands b1 b2 : : : bn. Transportation cost of a unit product of producer Ei to consumer Vj is cij . The amount of the product transported from Ei to Vj is xij units.
858 18. Optimization We are looking for an optimal transportation plan with minimum total transportation cost. We suppose the system is balanced, i.e., supply equals demand: m X i=1
ai =
n X
j =1
bj :
(18.24)
We construct the matrix of costs C and the distribution matrix X: P 0 c c 1 EE : 0 x x 1 a : 1n 1 11 1n 1 C = B@ 1..1 ... C ... C X = B@ ... A ... (18.25a) A ... : (18.25b) . cm1 cmn Em P : xmb1 xb mn am V : V1 V n 1 n If condition (18.24) is not ful lled, then we distinguish between two cases: P P a) If ai > bj , then we introduce a ctitious consumer Vn+1 with demand bn+1 = P ai ; P bj and with transportation costs cin+1 = 0. b) If P ai < P bj , then we introduce a ctitious producer Em+1 with capacity am+1 = P bj ; P ai and with transportation costs cm+1j = 0. In order to determine an optimal program, we have to solve the following programming problem:
OF : f (X) = CT :
n X j =1
m X n X
i=1 j =1
cij xij = min!
xij = ai (i = 1 : : : m)
(18.26a) m X i=1
xij = bj (j = 1 : : : n) xij 0):
(18.26b)
The minimum of the problem occurs at a vertex of the feasible set. There are m + n ; 1 linearly independent constraints among the m + n original constraints, so, in the non-degenerate case, the solution contains m + n ; 1 positive components xij . To determine an optimal solution the following algorithm is used, which is called the transportation algorithm.
2. Determination of a Basic Feasible Solution
With the Northwest corner rule we can determine an initial basic feasible solution: a) Choose x11 = min(a1 b1): (18.27a) b) If a1 < b1 we omit the rst row of X and proceed to the next source. (18.27b) If a1 > b1 we omit the rst column of X and proceed to the next destination. (18.27c) If a1 = b1 we omit either the rst row or the rst column of X. (18.27d) If there are only one row but several columns, we cancel one column. The same applies for the rows. c) We replace a1 by a1 ; x11 and b1 by b1 ; x11 and we repeat the procedure with the reduced scheme. The variables obtained in step a) are the basic variables, all the others are non-basic variables with zero values.
05 3 2 71 C= @ 8 2 1 1 A 9 2 6 3 V : V1 V2 V3 V4
E: E1 E2 E3
0 x 1 11 x12 x13 x14 @ x21 x22 x23 x24 A X= 32 x33 x34 P : b x=31 4 b x= 1 2 6 b3 = 5 b4 = 7
Determination of an initial extreme point with the Northwest corner rule:
P: a1 = 9 a2 = 10 : a3 = 3
18.1 Linear Programming 859
rst step
04 1 =9 5 @ A 10 X = 3 =4 6 5 7 0
second step
0 4 5 1 =5 0 A 10 X = @ 3 0 =6 5 7 =1
further steps
04 5 1 0 X = @ 1 5 4 A 10= =9 =4 0 : j j3 3 0 =5 =7 =1 0 3
0 There are alternative methods to nd an initial basic solution which also takes the transportation costs into consideration (see, e.g., the Vogel approximation method in 18.10]) and they usually result in a better initial solution.
3. Solution of the Transportation Problem with the Simplex Method
If we prepare the usual simplex tableau for this problem, we get a huge tableau ((m + n) (m n)) with a large number of zeros: In each column, only two elements are equal to 1. So, we will work with a reduced tableau, and the following steps correspond to the simplex steps working only with the nonzero elements of the theoretical simplex tableau. The matrix of the cost data contains the coecients of the objective function. The basic variables are exchanged for non-basic variables iteratively, while the corresponding elements of the cost matrix are modi ed in each step. The procedure is explained by an example. a) Determination of the modi ed cost matrix C~ from C by c~ij = cij + pi + qj (i = 1 : : : m j = 1 : : : n) (18.28a) with the conditions c~ij = 0 for (i j ) if xij is an actual basic variable. (18.28b) We mark the elements of C belonging to basic variables and we substitute p1 = 0. The other quantities pi and qj , also called potentials or simplex multiplicators, are determined so that the sum of pi, qj and the marked costs cij should be 0: 0 (5) 0 1 (3) 2 7 1 p1 = 0 0 0 05 B C B C B 8 (2) (1) (1) p = 1 ~ C=@ =) C = @ 4 0 0 0 C A 2 A : (18.28c) 3 -2 3 0 9 2 6 (3) p3 = ;1 q1 = ;5 q2 = ;3 q3 = ;2 q4 = ;2 b) We determine: c~pq = min fc~ g: (18.28d) ij ij If c~pq 0, then the given distribution X is optimal otherwise xpq is chosen as a new basic variable. In our example: c~pq = c~32 = ;2. c) In C~ , we mark c~pq and the costs associated to the basic variables. If C~ contains rows or columns with at most one marked element, then these rows or columns will be omitted. We repeat this procedure with the remaining matrix, until no further cancellation is possible. 0 (0) (0) 0 5 1 (18.28e) C~ = B@ 4 (0) (0) (0) CA : 3 (;2) 3 (0) d) The elements xij associated to the remaining marked elements c~ij form a cycle. The new basic variable x~pq is to be set to a positive value . The other variables x~ij associated to the marked elements c~ij are determined by the constraints. In practice, we subtract and add from or to every second element
860 18. Optimization of the cycle. To keep the variables non-negative, the amount must be chosen as = xrs = minfxij : x~ij = xij ; g where xrs will be the non-basic variable. In the example = minf1 3g = 1.
P 04 5 1 9 B C ; B C B C 1 ; 5 4 + B C 10 B " CA @ ;! 3 ; 3
X~ =
04 5
=) X~ = @
1 5 5A
1 2
f (x) = 53:
(18.28f)
(18.28g)
P 4 6
5 7 Then, we repeat this procedure with X = X~ but in the calculation of the new pi qj values we always start with the original C. 0 (5) 0 (0) (0) (;2) 3 1 (3) 2 7 1 p1 = 0 B C B 6 2 (0) (0) C 2 (1) (1) C C = B@ 8 C A p2 = 3 =) C~ = B @ A (18.28h) 9 (2) 6 (3) p3 = 1 5 (0) 3 (0) q1 = ;5 q2 = ;3 q3 = ;4 q4 = ;4
0 1 4 5; B C " B C 5; 5+ C X~ = BBB C @ " CA 1+ ;! 2 ;
0 1 = 2 X~ = @ 4 3 23 7 A =) 3
f (X) = 49:
(18.28i)
The next matrix C~ does not contain any negative element. So, X~ is an optimal solution.
18.1.4.2 Assignment Problem
The representation is made by an example. n shipping contracts should be given to n shipping companies so that each company receives exactly one contract. We want to determine the assignment which minimizes the total costs, if the i-th company charges cij for the j -th contract. An assignment problem is a special transportation problem with m = n and ai = bj = 1 for all i j :
OF : f (x) = CT :
n X j =1
n X n X
i=1 j =1
cij xij = min!
xij = 1 (i = 1 : : : n)
(18.29a) n X i=1
xij = 1 (j = 1 : : : n)
xij 2 f0 1g:
(18.29b)
Every feasible distribution matrix contains exactly one 1 in every row and every column, all other elements are equal to zero. In a general transportation problem of this dimension, however, a nondegenerate basic solution would have 2n ; 1 positive variables. Thus, basic feasible solutions to the assignment problem are highly degenerate, with n ; 1 basic variables equal to zero. Starting with a feasible distribution matrix X, we can solve the assignment problem by the general transportation algorithm. It is time consuming to do so. However, because of the highly degenerate nature of the basic feasible solutions, the assignment problem can be solved with the highly ecient Hungarian method (see 18.6]).
18.1.4.3 Distribution Problem
The problem is represented by an example.
18.2 Non-linear Optimization 861
m products E1 E2 : : : Em should be produced in quantities a1 a2 : : : am . Every product can be produced on any of n machines M1 M2 : : : Mn. The production of a unit of product Ei on machine Mj needs processing time bij and cost cij . The time capacity of machine Mj is bj . Denote the quantity produced by machine Mj from product Ei by xij . We want to minimize the total production costs. We get the following general model for this distribution problem:
OF : f (x) = CT :
m X j =1
m X n X
i=1 j =1
cij xij = min!
xij = ai (i = 1 : : : m)
(18.30a) n X i=1
bij xij bj (j = 1 : : : n) xij 0 for all i j:
(18.30b)
The distribution problem is a generalization of the transportation problem and it can be solved by the simplex method. If all bij = 1, then we can use the more e ective transportation algorithm (see 18.1.4.1, p. 858) after introducing a ctitious product Em+1 (see 18.1.4.1, p. 858).
18.1.4.4 Travelling Salesman
Suppose there are n places O1 O2 : : : On. The travelling time from Oi to Oj is cij . Here, cij 6= cji is possible. We want to determine the shortest route such that the traveller passes through every place exactly once, and returns to the starting point. Similarly to the assignment problem, we have to choose exactly one element in every row and column of the time matrix C so that the sum of the chosen elements is minimal. The diculty of the numerical solution of this problem is the restriction that the marked elements cij should be arranged in order of the following form: ci1i2 ci2i3 : : : cinin+1 with ik 6= il for k 6= l and in+1 = i1 . (18.31) The travelling salesman problem can be solved by the branch and bound method.
18.1.4.5 Scheduling Problem
n di erent products are processed on m di erent machines in a product-dependent order. At any time only one product can be processed on a machine. The processing time of each product on each machine is assumed to be known. Waiting times, when a given product is not in process, and machine idle times are also possible. An optimal scheduling of the processing jobs is determined where the objective function is selected as the time when all jobs are nished, or the total waiting time of jobs, or total machine idle time. Sometimes the sum of the nishing times for all jobs is chosen as the objective function when no waiting time or idle time is allowed.
18.2 Non-linear Optimization
18.2.1 Formulation of the Problem, Theoretical Basis 18.2.1.1 Formulation of the Problem 1. Non-linear Optimization Problem
A non-linear optimization problem has the general form f (x) = min! subject to x 2 IRn with (18.32a) i 2 I = f1 : : : mg hj (x) = 0 j 2 J = f1 : : : rg (18.32b) gi(x) 0 where at least one of the functions f , gi, hj is non-linear. The set of feasible solutions is denoted by M = fx 2 IRn : gi(x) 0 i 2 I hj (x) = 0 j 2 J g: (18.33) The problem is to determine the minimum points.
862 18. Optimization
2. Minimum Points
A point x 2 M is called the global minimum point if f (x ) f (x) holds for every x 2 M . If this relation holds for only the points x of a neighborhood U of x , then x is called a local minimum point. Since the equality constraints hj (x) = 0 can be expressed by two inequalities, ;hj (x) 0 hj (x) 0 (18.34) the set can be supposed to be empty, J = .
18.2.1.2 Optimality Conditions 1. Special Directions
a) The Cone of the Feasible Directions at x 2 M is de ned by Z (x) = fd 2 IRn : 9" > 0 : x + d 2 M 0 "g x 2 M (18.35) where the directions are denoted by d. If d 2 Z (x), then every point of the ray x + d belongs to M
for sucient small values of . b) A Descent Direction at a point x is a vector d 2 IRn for which there exists an " > 0 such that f (x + d) < f (x) 8 2 (0 "): (18.36) There exists no feasible descent direction at a minimum point. If f is di erentiable, then d is a descent direction rf (x)Td < 0. Here, r denotes the nabla operator, so rf (x) represents the gradient of the scalar-valued function f at x.
2. Necessary Optimality Conditions
If f is di erentiable and x is a local minimum point, then rf (x )Td 0 for every d 2 Z (x ): In particular, if x is an interior point of M , then rf (x ) = 0:
3. Lagrange Function and Saddle Point
(18.37a) (18.37b)
Optimality conditions (18.37a,b) should be transformed into a more practical form including the constraints. We construct the so-called Lagrange function or Lagrangian:
L(x u) = f (x) +
m X i=1
uigi(x) = f (x) + uTg(x)
x 2 IR u 2 IRm+
(18.38)
according to the Lagrange multiplier method (see 6.2.5.6, p. 403) for problems with equality constraints. A point (x u ) 2 IRn IRm+ is called a saddle point of L, if L(x u) L(x u ) L(x u ) for every x 2 IRn u 2 IRm+ : (18.39)
4. Global Kuhn{Tucker Conditions n
A point x 2 IR satis es the global Kuhn{Tucker conditions if there is an u 2 IRm+ , i.e., u 0 such that (x u ) is a saddle point of L. For the proof of the Kuhn{Tucker conditions see 12.5.6, p. 625.
5. Su cient Optimality Condition
If (x u ) 2 IRn IRm+ is a saddle point of L, then x is a global minimum point of (18.32a,b). If the functions f and gi are di erentiable, then we can also deduce local optimality conditions.
6. Local Kuhn{Tucker Conditions
A point x 2 M satis es the local Kuhn{Tucker conditions if there are numbers ui 0, i 2 I0(x ) such that X (18.40a) uirgi(x ) where ;rf (x ) = i2I0 (x )
I0(x) = fi 2 f1 : : : mg : gi(x) = 0g (18.40b) is the index set of the active constraints at x. The point x is also called a Kuhn{Tucker stationary point.
18.2 Non-linear Optimization 863
uirgi(x ) = 0:
g1(x)=0
f(x*)
D
i=1
g1(x*) D-
rf (x ) +
m X
D
This means geometrically that a point x 2 M satis es the local Kuhn{Tucker conditions, if the negative gradient ;rf (x ) lies in the cone spanned by the gradients rgi(x ) i 2 I0(x ) of the constraints active at x (Fig. 18.5). The following equivalent formulation for (18.40a,b) is also often used: x 2 IRn satis es the local mKuhn{Tucker conditions, if there is a u 2 IR+ such that g(x ) 0 (18.41a) uigi(x ) = 0 i = 1 : : : m (18.41b)
g2(x*)
x* M g2(x)=0
(18.41c)
level lines f(x) = const
Figure 18.5
7. Necessary Optimality Conditions and Kuhn{Tucker Conditions
If x 2 M is an local minimum point of (18.32a,b) and the feasible set satis es the regularity condition at x : 9 d 2 IR such that rgi(x )Td < 0 for every i 2 I0(x ), then x satis es the local Kuhn{Tucker conditions.
18.2.1.3 Duality in Optimization 1. Dual Problem
With the associated Lagrangian (18.38) we form the maximum problem, the so-called dual of (18.32a,b):
L(x u) = max! subject to (x u) 2 M with M = f(x u) 2 IRn IRm+ : L(x u) = zmin L(z u)g: 2IRn
(18.42a) (18.42b)
2. Duality Theorems
If x1 2 M and (x2 u2) 2 M , then a) L(x2 u2) f (x1): b) If L(x2 u2) = f (x1 ), then x1 is a minimum point of (18.32a,b) and (x2 u2 ) is a maximum point of (18.42a,b).
18.2.2 Special Non-linear Optimization Problems 18.2.2.1 Convex Optimization 1. Convex Problem
The optimization problem f (x) = min! subject to gi(x) 0 (i = 1 : : : m) (18.43) is called a convex problem if the functions f and gi are convex. In particular, f and gi can be linear functions. The following statements are valid for convex problems: a) Every local minimum of f over M is also a global minimum. b) If M is not empty and bounded, then there exists at least one solution of (18.43). c) If f is strictly convex, then there is at most one solution of (18.43).
1. Optimality Conditions a) If f has continuous partial derivatives, then x 2 M is a solution of (18.43), if (x ; x )Trf (x ) 0 for every x 2 M: (18.44) b) The Slater condition is a regularity condition for the feasible set M . It is satis ed if there exists an x 2 M such that gi(x) < 0 for every non-ane linear functions gi.
864 18. Optimization c) If the Slater condition is satis ed, then x is a minimum point of (18.43) if and only if there exists a u 0 such that (x u ) is a saddle point of the Lagrangian. Moreover, if functions f and gi are differentiable, then x is a solution of (18.43) if and only if x satis es the local Kuhn{Tucker conditions. d) The dual problem (18.42a,b) can be formulated easily for a convex optimization problem with dif-
ferentiable functions f and gi: L(x u) = max! subject to (x u) 2 M with (18.45a) M = f(x u) 2 IRn IRm+ : rxL(x u) = 0g: (18.45b) The gradient of L is calculated here only with respect to x. e) For convex optimization problems, the strong duality theorem also holds: If M satis es the Slater condition and if x 2 M is a solution of (18.43), then there exists a u 2 IRm+ , such that (x u ) is a solution of the dual problem (18.45a,b), and f (x ) = xmin f (x) = (xmax L(x u) = L(x u ): (18.46) 2M u)2M
18.2.2.2 Quadratic Optimization 1. Formulation of the Problem
Quadratic optimization problems have the form f (x) = xTCx + pTx = min! subject to x 2 M IRn with M = MI : M = fx 2 IRn : Ax b x 0g: Here, C is a symmetric (n n) matrix, p 2 IRn, A is an (m n) matrix, and b 2 IRm. The feasible set M can be written alternatively in the following way: M = MII : M = fx : Ax = b x 0g M = MIII : M = fx : Ax bg:
2. Lagrangian and Kuhn{Tucker Conditions
The Lagrangian to the problem (18.47a,b) is L(x u) = xTCx + pTx + uT(Ax ; b): By introducing the notation @L T v = @L @ x = p + 2Cx + A u and y = ; @ u = ;Ax + b the Kuhn{Tucker conditions are as follows: a) b) c) d)
Case I: Ax + y = b, 2Cx ; v + ATu = ;p, x 0 v 0 y 0 u 0, xTv + yTu = 0.
3. Convexity
a) b) c) d)
Case II: Ax = b, 2Cx ; v + ATu = ;p, x 0 v 0, xTv = 0.
(18.47a) (18.47b) (18.48a) (18.48b) (18.49) (18.50)
a) b) c) d)
Case III: Ax + y = b, 2Cx + ATu = ;p, u 0 y 0, yTu = 0.
(18.51a) (18.51b) (18.51c) (18.51d)
The function f (x) is convex (strictly convex) if and only if the matrix C is positive semide nite (positive de nite). Every result on convex optimization problems can be used for quadratic problems with a positive semide nite matrix C in particular, the Slater condition always holds, so it is necessary and sucient for the optimality of a point x that there exists a point (x y u v), which satis es the corresponding system of local Kuhn{Tucker conditions.
18.2 Non-linear Optimization 865
4. Dual Problem
If C is positive de nite, then the dual problem (18.45a) of (18.47a) can be expressed explicitly: L(x u) = max! subject to (x u) 2 M where (18.52a) 1 n m ;1 T M = f(x u) 2 IR IR+ : x = ; 2 C (A u + p)g: (18.52b) 1 If we substitute the expression x = ; C;1(AT u + p) into the dual objective function L(x u), then 2 we get the equivalent problem T '(u) = ; 14 uTAC;1 ATu ; 12 AC;1p + b u ; 14 pTC;1p = max! u 0: (18.53) Hence: If x 2 M is a solution of (18.47a,b), then (18.53) has a solution u 0, and (18.54) f (x ) = '(u ): Problem (18.53) can be replaced by an equivalent formulation: (u) = uTEu + hTu = min! subject to u 0 where (18.55a) 1 1 E = 4 AC;1AT and h = 2 AC;1p + b: (18.55b)
18.2.3 Solution Methods for Quadratic Optimization Problems 18.2.3.1 Wolfe's Method
1. Formulation of the Problem and Solution Principle
The method of Wolfe is to solve quadratic problems of the special form: f (x) = xTCx + pTx = min! subject to Ax = b x 0: (18.56) We suppose that C is positive de nite. The basic idea is the determination of a solution (x u v ) of the corresponding system of Kuhn{Tucker conditions, associated to problem (18.56): Ax = b (18.57a) 2Cx ; v + ATu = ;p (18.57b) x 0 v 0 (18.57c) T x v = 0: (18.58) Relations (18.57a,b,c) represent a linear equation system with m + n equations and 2n + m non-negative variables. Because of relation (18.58), either xi = 0 or vi = 0 (i = 1 2 : : : n) must hold. Therefore, every solution of (18.57a,b,c), (18.58) contains at most m + n non-zero components. Hence, it must be a basic solution of (18.57a,b,c).
2. Solution Process
First, we determine a feasible basic solution (vertex) x" of the system Ax = b. The indices belonging to the basis variables of x" form the set IB . In order to nd a solution of system (18.57a,b,c), which also satis es (18.58), we formulate the problem ; = min! ( 2 IR) (18.59) Ax = b (18.60a) 2Cx ; v + ATu ; q = ;p with q = 2Cx" + p (18.60b) (18.60c) x 0 v 0 0
xTv = 0:
(18.61)
866 18. Optimization If (x v u ) is a solution of this problem also satisfying (18.57a,b,c) and (18.58), then = 0. The vector (x v u ) = ("x 0 0 1) is a known feasible solution of the system (18.60a,b,c), and it satis es the relation (18.61), too. We form a basis associated to this basic solution from the columns of the 0coecient matrix1 @ A 0 0T 0 A I denotes the unit matrix, 0 the zero matrix and 0 (18.62) is the zero vector of the corresponding dimension, 2C ;I A ;q in the following way: a) m columns belonging to xi with i 2 IB , b) n ; m columns belonging to vi with i 2= IB , c) all m columns belonging to ui, d) the last column, but then a suitable column determined in b) or c) will be dropped. If q = 0, then the interchange according to d) is not possible. Then x" is already a solution. Now, we can construct a rst simplex tableau. The minimization of the objective function is performed by the simplex method with an additional rule that guarantees that the relation xTv = 0 is satis ed: The variables xi and vi (i = 1 2 : : : n) must not be simultaneously basic variables. The simplex method provides a solution of problem (18.59), (18.60a,b,c), (18.61) with = 0 for positive de nite C considering this additional rule. For a positive semide nite matrix C, based on the restricted pivot choice, it may happen that although > 0, no more exchange-step can be made without violating the additional rules. We can show that in this case cannot be reduced any further. f (x) = x21 + 4x22 ; 10x1 ; 32x2 = min! with x1 + 2x2 + x3 = 7 2x1 + x2 + x4 = 8. 01 0 0 01 0 ;10 1 1 2 1 0 7 B C A= 2 1 0 1 b= 8 C = B@ 00 40 00 00 CA p = BB@ ;032 CCA : 0 0 0 0 0 In this case C is positive semide nite. A feasible basic solution of Ax = b is x" = (0 0 7 8)T , q = 2Cx" + p = (;10 ;32 0 0)T. We choose the basis vectors: a) columns 3 and 4 of 2AC , b) columns 1 and 2 of ;0I , c) the columns of A0T and d)
!
column ;0q instead of the rst column of ;0I . The basis matrix is formed from these columns, and the basis inverse is calculated (see 18.1, p. 846).! Multiplying matrix (18.62) and the vectors ;bp by the basis inverse, we get the rst simplex tableau (Scheme 18.9). Only x1 can be interchanged with v2 in this tableau according to the complementary constraints. After a few steps, we get the solution x = (2 5=2 0 3=2)T. The last two equations of 2Cx ; v + ATu ; q = ;p are: v3 = u1 v4 = u2. Therefore, by eleminating u1 and u2 the dimension of the problem can be reduced.
x3 x4 v2 u1 u2
x1 1 2 64 10 0 0 2 10 ; 102
Scheme 18.9 x2 v1 v3 v4 2 0 0 0 7 1 0 0 0 8 32 12 54 ; 8 ; 10 10 10 0 0 0 ;1 0 0 0 0 0 ;1 0 1 1 2 0 ; 10 10 10 1 1 ; 1 ; 2 ;1 0 10 10 10
18.2 Non-linear Optimization 867
18.2.3.2 Hildreth{d'Esopo Method 1. Principle
The strictly convex optimization problem f (x) = xTCx + pTx = min! Ax b (18.63) has the dual problem (see 18.2.2.2,4., p. 865) (u) = uTEu + hTu = min! u 0 with (18.64a) 1 1 E = 4 AC;1AT h = 2 AC;1p + b: (18.64b) Matrix E is positive de nite and it has positive diagonal elements eii > 0, (i = 1 2 : : : m). The variables x and u satisfy the following relation: x = ; 12 C;1 (ATu + p): (18.65)
2. Solution by Iteration
The dual problem (18.64a), which contains only the condition u 0, can be solved by the following simple iteration method: a) Substitute u1 0, (e.g., u1 = 0), k = 1. b) Calculate uki +1 for i = 1 2 : : : m according to
0i;1 1 m X X wik+1 = ; e1 @ eij ukj +1 + h2i + eij ukj A (18.66a)
n o uki +1 = max 0 wik+1 : (18.66b) ii j =1 j =i+1 c) Repeat step b) with k + 1 instead of k until a stopping rule is satis ed, e.g., (uk+1) ; (uk ) < ", " > 0. Under the assumption that there is an x such that Ax < b, the sequence f(uk )g converges to the minimum value min and sequence fxk g given by (18.65) converges to the solution x of the original problem. The sequence fuk g is not always convergent.
18.2.4 Numerical Search Procedures
By using non-linear optimization procedures we can nd acceptable approximate solutions with reasonable computing costs for several types of optimization problems. They are based on the principle of comparison of function values.
18.2.4.1 One-Dimensional Search
Several optimization methods contain the subproblem of nding the minimum of a real function f (x) for x 2 a b]. It is often sucient to nd an approximation x of the minimum point x .
1. Formulation of the Problem
A function f (x), x 2 IR, is called unimodal in a b] if it has exactly one local minimum point on every closed subinterval J a b]. Let f be a unimodal function on a b] and x the global minimum point. Then we have to nd an interval c d] a b] with x 2 c d] such that d ; c < ", " > 0.
2. Uniform Search
We choose a positive integer n such that = nb ;+ a1 < 2" , and we calculate the values f (xk ) for xk = a + k (k = 1 : : : n). If f (x) is the smallest value among these function values, then the minimum point x is in the interval x ; x + ]. The number of required function values for the given accuracy can be estimated by (18.67) n > 2(b "; a) ; 1 :
868 18. Optimization
3. Golden Section Method, Fibonacci Method
The interval a b] = a1 b1] will be reduced step by step so that the new subinterval always contains the minimum point x . We determine the points 1 1 in the interval a1 b1 ] as 1 = a1 + (1 ; )(b1 ; a1) 1 = a1 + (b1 ; a1) with (18.68a) p 1 = 2 ( 5 ; 1) 0:618: (18.68b) This corresponds to the golden section. We distinguish between two cases: a) f (1) < f ( 1): We substitute a2 = a1 b2 = 1 and 2 = 1: (18.69a) b) f (1) f ( 1): We substitute a2 = 1 b2 = b1 and 2 = 1: (18.69b) If b2 ; a2 ", then we repeat the procedure with the interval a2 b2 ], where one value is already known, f (2) in case a) and f ( 2) in case b), from the rst step. To determine an interval an bn], which contains the minimum point x , we must calculate n function values altogether. From the requirement " > bn ; an = n;1 (b1 ; a1) (18.70) we can estimate the necessary number of steps n. By using the golden section method, at most one more function value should be determined compared to the Fibonacci method. Instead of subdividing the interval according to the golden section, we subdivide the interval according to the Fibonacci numbers (see 5.4.1.5, p. 325, and 17.3.2.4, 4., p. 845).
18.2.4.2 Minimum Search in n-Dimensional Euclidean Vector Space
The search for an approximation of the minimum point x of the problem f (x) = min!, x 2 IRn, can be reduced to the solution of a sequence of one-dimensional optimization problems. We take a) x = x1 k = 1 where x1 is an appropriate initial approximation of x . (18.71a) b) We solve the one-dimensional problems '(r ) = f (xk1+1 : : : xkr;+11 xkr + r xkr+1 : : : xkn) = min! with r 2 IR (18.71b) for r = 1 2 : : : n. If "r is an exact or approximating minimum point of the r-th problem, then we substitute xkr +1 = xkr + "r . c) If two consecutive approximations are close enough to each other, i.e., with some vector norm, jjxk+1 ; xk jj < "1 or jf (xk+1) ; f (xk )j < "2 (18.71c) then xk+1 is an approximation of x . Otherwise we repeat step b) with k + 1 instead of k. The onedimensional problem in b) can be solved, by using the methods given in 18.2.4.1, p. 867.
18.2.5 Methods for Unconstrained Problems
The general optimization problem f (x) = min! for x 2 IRn (18.72) is considered with a continuously di erentiable function f . nEach method described in this section constructs, in general, an in nite sequence of points fxk g 2 IR , whose accumulation point is a stationary point. The sequence of points will be determined starting with a point x1 2 IRn and according to the formula xk+1 = xk + k dk (k = 1 2 : : : ) (18.73) k n k i.e., we rst determine a direction d 2 IR at x and by the step size k 2 IR we indicate how far xk+1 is from xk in the direction dk . Such a method is called a descent method, if (18.74) f (xk+1) < f (xk ) (k = 1 2 : : :):
18.2 Non-linear Optimization 869
The equality rf (x) = 0, where r is the nabla operator (see 13.2.6.1, p. 656), characterizes a stationary point and can be used as a stopping rule for the iteration method.
18.2.5.1 Method of Steepest Descent (Gradient Method)
Starting from an actual point xk , the direction dk in which the function has its steepest descent is and consequently xk+1 = xk ; k rf (xk ): (18.75b) dk = ;rf (xk ) (18.75a) A schematic representation of the steepest descent method with level lines f (x) = f (xi ) is shown in Fig. 18.6. The step size k is determined by a line search, i.e., k is the solution of the onedimensional problem level lines 1 2 f (xk + dk ) = min! 0: (18.76) f(x)=f(x ) x This problem can be solved by the methods 3 given in 18.2.4, p. 867. x* x 1 The steepest descent method (18.75b) conx verges relatively slowly. For every accumulation point x of the sequence fxk g, 1 f(x ) rf (x ) = 0. In the case of a quadratic objective function, i.e., f (x) = xTCx + pTx, Figure 18.6 the method has the special form: D
xk+1 = xk + k dk
kT k (18.77a) with dk = ;(2Cxk + p) and k = dk T d k : 2d Cd
(18.77b)
18.2.5.2 Application of the Newton Method
Suppose we approximate the function f at the actual approximation point xk by a quadratic function: (18.78) q(x) = f (xk ) + (x ; xk )Trf (xk ) + 21 (x ; xk )TH(xk )(x ; xk ): k Here H(x ) is the Hessian matrix, i.e., the matrix of second partial derivatives of f at the point xk . If H(xk ) is positive de nite, then q(x) has an absolute minimum at xk+1 with rq(xk+1) = 0, therefore we get the Newton method: (18.79a) xk+1 = xk ; H;1 (xk )rf (xk ) (k = 1 2 : : :) i.e., dk = ;H;1 (xk )rf (xk ) and k in (18.73): (18.79b) The Newton method converges quickly but it has the following disadvantages: a) The matrix H(xk ) must be positive de nite. b) The method converges only for suciently good initial points. c) We cannot inuence the step size. d) The method is not a descent method. e) The computational cost of computing the inverse of H;1(xk ) is fairly high. Some of these disadvantages can be reduced by the following version of the damped Newton method: xk+1 = xk ; k H;1(xk )rf (xk ) (k = 1 2 : : :) : (18.80) The relaxation factor k can be determined, for example, by the principle given earlier (see 18.2.5.1, p. 869).
18.2.5.3 Conjugate Gradient Methods
Two vectors d1 d2 2 IRn are called conjugate vectors with respect to a symmetric, positive de nite matrix C, if (18.81) d1 TCd2 = 0:
870 18. Optimization If d1 d2 : : : dn are pairwise conjugate nvectors with respect to a matrix C, then the convex quadratic problem q(x) = xTCx + pTx x 2 IR , can be solved in n steps if we construct a sequence xk+1 = xk +k dk starting from x1, where k is the optimal step size. Under the assumption that f (x) is approximately quadratic in the neighborhood of x , i.e., C 1 H(x ), the method developed for quadratic 2 objective functions can also be applied for more general functions f (x), without the explicit use of the matrix H(x ). The conjugate gradient method has the following steps: (18.82) a) x1 2 IRn d1 = ;rf (x1 ) where x1 is an appropriate initial approximation for x . b) xk+1 = xk + k dk (k = 1 : : : n) with k 0 so that f (xk + dk ) will be minimized. (18.83a) dk+1 = ;rf (xk+1) + k dk (k = 1 : : : n ; 1) with (18.83b) T k +1 k +1 k = rf (x k )Trf (xk ) and dn+1 = ;rf (xn+1): (18.83c) rf (x ) rf (x ) c) Repeating steps b) with xn+1 and dn+1 instead of x1 and d1.
18.2.5.4 Method of Davidon, Fletcher and Powell (DFP)
With the DFP method, we determine a sequence of points starting from x1 2 IRn according to the formula xk+1 = xk ; k Mk rf (xk ) (k = 1 2 : : :): (18.84) Here, Mk is a symmetric, positive de nite matrix. The idea of the method is a stepwise approximation of the inverse Hessian matrix by matrices Mk in the case when f (x) is a quadratic function. Starting with a symmetric, positive de nite matrix M1, e.g., M1 = I (I is the unit matrix), the matrix Mk is determined from Mk;1 by adding a correction matrix of rank two k kT k k T M = M + v v ; (Mk;1w )(Mk;1w ) (18.85) k
k;1
wk TMk wk = rf (xk ) ; rf (xk;1) (k = 2 3 : : :). We get the step size k from k k f (x ; Mk rf (x )) = min! 0: (18.86) If f (x) is a quadratic function, then the DFP method becomes the conjugate gradient method with M1 = I. with vk
v k T vk k k ;1 = x ; x and wk
18.2.6 Evolution Strategies
Evolution strategies are stochastic optimization methods that imitate the process of natural evolution. Evolutionary algorithms are based on the principles of mutation, recombination and selection. For a comprehensive representation see 18.1, 18.4]. 1. Mutation A parent point x P is modi ed by adding a random variation d, resulting in an o spring point x O = x P + d: Using normally distributed variations d, smaller steps are more likely than big jumps. 2. Recombination New individuals of the o spring generation can be derived by merging the information of two ore more parents, which are randomly selected from a population of m parents. For example, an o spring can be represented as the weighted mean of n parents corresponding to
xO =
n X i=1
i x Pi
n X i=1
i = 1 n = 2 : : : m :
(18.87)
3. Selection The objective function f (x) provides the measure to compare the quality of the individuals. The best individuals survive to become the parents of the next generation.
18.2 Non-linear Optimization 871
Evolution strategies are classi ed with respect to the number of parents and o springs, the number of parents involved in recombination as well as the rules for mutation and selection.
18.2.6.1 Mutation{Selection{Strategy
This strategy is similar to the gradient method discribed in 18.2.5.1, p. 869. But now the direction xk is a normally distributed random vector. 1. Mutation Step A single parent point of the generation k generates one o spring point by adding a normal distributed variation corresponding to x kO = x kp + d k : (18.88) The factor is a parameter to control the mutation step size and to improve the convergence rate of the algorithm. 2. Selection Step The point with the better objective function value is selected as the parent point of the next generation k + 1 ( k f (x kO ) < f (x kP ) (18.89) xkP+1 = xx Ok ifotherwise . P The method terminates if no improved descendants are determined over a certain number of generations. The step size can be enlarged, if the mutation frequently results in better descendants whereas shoud be reduced in the case of rare improvements.
18.2.6.2 Recombination
Starting with a population of m parents of the kth generation X Pk = fx kP1 : : : x kPm g, we form a set of n o springs X Ok = fx kO1 : : : x kOn g by applying mutation and recombination n times. Each o spring is determined combining 2 or more randomly chosen parents. The best m points are selected out of the m + n points of the union of X Pk and X Ok to form the subsequent parent generation k+1 X Pk+1 = fx kP+1 1 : : : x Pm g : To limit the lifetime of all individuals to one generation the selection can be restricted to the set of descendants X Ok : Thus, the objective function values of the o springs can exceed those of the parents, which makes it possible to leave local minima.
18.2.7 Gradient Method for Problems with Inequality Type Constraints
If the problem f (x) = min! subject to the constraints gi(x) 0 (i = 1 : : : m) (18.90) has to be solved by an iteration method of the type (18.91) xk+1 = xk + k dk (k = 1 2 : : :) then we have to consider two additional rules because of the bounded feasible set: 1. The direction dk must be a feasible descent direction at xk . 2. The step size k must be determined so that xk+1 is in M . The di erent methods based on the formula (18.91) di er from each other only in the construction of the direction dk . To ensure the feasibility of the sequence fxk g M , we determine k0 and k00 in the following way: k0 from f (xk + dk ) = min! 0 k00 = maxf 2 IR : xk + dk 2 M g: (18.92) Then k = minfk0 k00 g: (18.93)
872 18. Optimization If there is no feasible descent direction dk in a certain step k, then xk is a stationary point.
18.2.7.1 Method of Feasible Directions 1. Direction Search Program
A feasible descent direction dk at point xk can be determined by the solution of the following optimization problem: = min! (18.94) T T k k k rgi(x ) d i 2 I0(x ) (18.95a) rf (x ) d (18.95b) jjdjj 1: (18.95c) If < 0 for the result d = dk of this direction search program, then (18.95a) ensures feasibility and (18.95b) ensures the descending property of dk . The feasible set for the direction search program is bounded by the normalizing condition (18.95c). If = 0, then xk is a stationary point, since there is no feasible descent direction at xk . A direction search program, de ned by (18.95a,b,c), can result in a zig-zag behavior of the sequence xk , which can be avoided if the index set I0(xk ) is replaced by the index set I"k (xk ) = fi 2 f1 : : : mg : ;"k gi(xk ) 0g "k 0 (18.96) which are the so-called "k active constraints in xk . Thus, we exclude local directions of descent which are going from xk and lead closer to the boundaries of M consisting of the "k active constraints (Fig. 18.7). g1(x)=0 g1(x)=−e
g1(x)=0 x
1
g2(x)=0
x
1
g2(x)=−e g2(x)=0
Figure 18.7 If = 0 is a solution of (18.95a,b,c) after these modi cations, then xk is a stationary point only if I0(xk ) = I"k (xk ). Otherwise "k must be decreased and the direction search program must be repeated.
2. Special Case of Linear Constraints k
D-
x
k
D-
f(x )
k
k
d
a)
M
x
k
f(x ) d
b)
M
Figure 18.8
k
If the functions gi(x) are linear, i.e., gi(x) = aiTx ; bi , then we can establish a simpler direction search method: = rf (xk )Td = min! with (18.97) T k k ai d 0 i 2 I0(x ) or i 2 I"k (x ) (18.98a) jjdjj 1: (18.98b) The e ect of the choice of q di erent norms jjdjj = maxfjdijg 1 or jjdjj = dTd 1 is shown in Fig. 18.8a,b.
q
In a certain sense, the best choice is the norm jjdjj = jjdjj2 = dT d, since by the direction search program we get the direction dk , which forms the smallest angle with ;rf (xk ). In this case the direction search program is not linear and requires higher computational costs. With the choice jjdjj = jjdjj1 = maxfjdijg 1 we get a system of linear constraints ;1 di 1 (i = 1 : : : n), so the direction search program can be solved, e.g., by the simplex method. In order to ensure that the method of feasible directions for a quadratic optimization problem f (x) = xTCx + pTx = min! with Ax b results in a solution in nitely many steps, the direction search
18.2 Non-linear Optimization 873
program is completed by the following conjugate requirements: If k;1 = k0 ;1 holds in a step, i.e., xk is an \interior" point, then we add the condition dk;1TCd = 0 (18.99) to the direction search program. Furthermore we keep the corresponding conditions from the previous steps. If in a later step k = k00 we remove the condition (18.99). f (x) = x21 + 4x22 ; 10x1 ; 32x2 = min! g1(x) = ;x1 0 g2(x) = ;x2 0 g3(x) = x1 + 2x2 ; 7 0 g4(x) = 2x1 + x2 ; 8 0: Step 1: Starting with x1 =((3 0)T rf (x1) = (;4) ;32)T I0(x1 ) = f2g: ;4d1 ; 32d2 = min! T 1 Direction search program: ;d2 0 jjdjj1 1 =) d = (1 1) : kT k Minimizing constant: k0 = ; d krTf (xk ) with C = 10 04 . 2d Cd ( ) k Maximal feasible step size: k00 = min ;giT(xk ) : for i such that aiTdk > 0 10 = 18 100 = 2 =) 5 3 a d 18 2 2 11 2 T i 2 1 = min 5 3 = 3 x = 3 3 . T Step 2: rf (x2 ) = ; 38 ; 80 I0 (x2) = f4g. 83 8 9 T < ; d1 ; 80 d2 = min! = 2 = ;1 1 20 = 152 Direction search program: : 3 = ) d 200 = 43 =) 3 " 2 51 2d1 + d2 0 jjdjj1 1 x2 4 T 3 2 = 3 x = (3 2) . 4 3 x 3 Step 3: rf (x3) = (;4 ;16)T I0 = (x3 ) = f3 4g. d 3 Direction program: x ( ;4d ; search ) 2 16 d = min! 1 2 2 =) d3 = M d 1 d1 + 2d2 0 2d1 + d2 0 jjdjj1 1 1 d 1 T T 2 1x ;1 2 , 30 = 1 300 = 3 =) 3 = 1 x4 = 2 52 . x 4 x1 1 0 3 2 The next direction search program results in = 0. Here the minimum point is x = x4 (Fig. 18.9). Figure 18.9
18.2.7.2 Gradient Projection Method
1. Formulation of the Problem and Solution Principle
Suppose the convex optimization problem f (x) = min! with aiTx bi (18.100) k k for i = 1 : : : m is given. A feasible descent direction d at the point x 2 M is determined in the following way: If ;rf (xk ) is a feasible direction, then dk = ;rf (xk ) is selected. Otherwise xk is on the boundary of M and ;rf (xk ) points outward from M . The vector ;rf (xk ) is projected by a linear mapping Pk into a linear submanifold of the boundary of M de ned by a subset of active constraints of xk . Fig. 18.10a shows a projection into an edge, Fig. 18.10b shows a projection into a face. Supposing non-degeneracy, i.e., if the vectors ai, i 2 I0 (x) are linearly independent for every x 2 IRn, such a
874 18. Optimization projection is given by (18.101) dk = ;Pk rf (xk ) = ; I ; Ak T(Ak Ak T );1Ak rf (xk ): T Here, Ak consists of all vectors ai , whose corresponding constraints form the submanifold, into which ;rf (xk ) should be projected. k
f(x )
D-
k
d
a)
k
b)
x
k
d
D-
x
k
k
f(x )
Figure 18.10
2. Algorithm
The gradient projection method consists of the following steps, starting with x1 2 M and substituting k = 1 and proceeding in accordance to the following scheme: I: If ;rf (xk ) is a feasible direction, then we Tsubstitute dk =k ;rf (xk ), and we continue with III. Otherwise we construct Ak from the vectors ai with we continue with II. i 2 Ik0 (x ) and ;1 k k T T II: We substitute d = ; I ; Ak (Ak Ak ) Ak rf (x ). If d 6= 0, then we continue with III. If dk = 0 and u = ;(Ak AkPT);1 Ak rf (xk ) 0, then xk is a minimum point. The local Kuhn{Tucker conditions ;rf (xk ) = uiai = Ak Tu are obviously satis ed. k i2I0 (x )
If u = 0, then we choose an i with ui < 0, delete the i-th row from Ak and repeat II. III: Calculation of k and xk+1 = xk + k dk and returning to I with k = k + 1.
3. Remarks on the Algorithm
If ;rf (xk ) is not feasible, then this vector is mapped onto the submanifold of the smallest dimension which contains xk . If dk = 0, then ;rf (xk ) is perpendicular to this submanifold. If u 0 does not hold, then the dimension of the submanifold is increased by one by omitting one of the active constraints, so maybe dk 6= 0 can occur (Fig. 18.10b) (with projection onto a (lateral) face). Since Ak is often obtained from Ak;1 by adding or erasing a row, the calculation of (Ak Ak T);1 can be simpli ed by the use of (Ak;1Ak;1T);1. Solution of the problem of the previous example on p. 873. Step 1: x1 = (3 0)T , I: rf (x1 ) = (;4 ;32)T ;rf (x1) is feasible, d1 = (4 32)T. T III: The step size is determined as in the previous example: 1 = 1 x2 = 16 8 : 20 5 5 Step 2: 96 T (not feasible), I (x2) = f4g A = (2 1). I: rf (x2 ) = ; 18 ; 0 2 1 5;2 5 8 16 T 1 II: P2 = ;2 4 d2 = ; 25 25 6= 0. 5 III: 2 = 58 x3 = (3 2)T. Step 3:
18.2 Non-linear Optimization 875
I: rf (x3 ) = (;4 ;16)T (not feasible), I0 (x3) = f3 4g A3 = 12 21 : T II: P3 = 00 00 d3 = (0 0)T u = 283 ; 83 u2 < 0 : A3 = (1 2). 8 T. II: P3 = 15 ;42 ;12 d3 = ; 16 5 5 5 T 5 4 III: 3 = 16 x = 2 2 . Step 4: I: rf (x4 ) = (;6 ;12)T (not feasible), I0 (x4) = f3g A4 = A3 . II: P4 = P3 d4 = (0 0)T u = 6 0. It follows that x4 is a minimum point.
18.2.8 Penalty Function and Barrier Methods
The basic principle of these methods is that a constrained optimization problem is transformed into a sequence of optimization problems without constraints by modifying the objective function. The modi ed problem can be solved, e.g., by the methods given in Section 18.2.5. With an appropriate construction of the modi ed objective functions, every accumulation point of the sequence of the solution points of this modi ed problem is a solution of the original problem.
18.2.8.1 Penalty Function Method
The problem f (x) = min! subject to gi(x) 0 (i = 1 2 : : : m) (18.102) is replaced by the sequence of unconstrained problems H (x pk ) = f (x) + pk S (x) = min! with x 2 IRn pk > 0 (k = 1 2 : : :): (18.103) Here, pk is a positive parameter, and for S (x) 0 x 2 M, S (x) = = (18.104) > 0 x 2= M , holds, i.e., leaving the feasible set M is punished with a \penalty" term pk S (x). The problem (18.103) is solved with a sequence of penalty parameters pk tending to 1. Then lim H (x pk ) = f (x) x 2 M: (18.105) k!1 k If x is the solution of the k-th penalty problem, then: H (xk pk ) H (xk;1 pk;1) f (xk ) f (xk;1) (18.106) and every accumulation point x of the sequence fxk g is a solution of (18.102). If xk 2 M , then xk is a solution of the original problem. For instance, the following functions are appropriate realizations of S (x): S (x) = maxr f0 g1(x) : : : gm(x)g (r = 1 2 : : :) or (18.107a)
S (x) =
m X i=1
maxr f0 gi(x)g (r = 1 2 : : :):
(18.107b)
If functions f (x) and gi(x) are di erentiable, then in the case r > 1, the penalty function H (x pk ) is also di erentiable on the boundary of M , so analytic solutions can be used to solve the auxiliary problem (18.103). Fig. 18.11 shows a representation of the penalty function method. f (x) = x21 + x22 = min! for x1 + x2 1 H (x pk ) = x21 + x22 + pk max2f0 1 ; x1 ; x2 g: The necessary optimality condition is:
876 18. Optimization
H(x,p2)
M
2
H(x,p1)
1
H(x,q1) H(x,q2) M
x* x x
1
x
Figure 18.11
2
x* x
x
f(x)
x
f(x)
Figure 18.12
f0 1 ; x1 ; x2 g 0 rH (x pk ) = 22xx12 ;; 22ppkk max maxf0 1 ; x1 ; x2 g = 0 . The gradient of H is evaluated here with respect to x. By subtracting the equations we have x1 = x2 . The equation 2x1 ; 2pk maxf0 1 ; 2x1g = 0 has a unique solution xk1 = xk2 = pk . We get the 1 + 2pk p 1 k solution x1 = x2 = klim = 2 by letting k ! 1. !1 1 + 2pk
18.2.8.2 Barrier Method
We consider a sequence of modi ed problems in the form H (x qk ) = f (x) + qk B (x) = min! qk > 0 : (18.108) The term qk B (x) prevents the solution leaving the feasible set M , since the objective function increases unboundedly on approaching the boundary of M . The regularity condition M 0 = fx 2 M : gi(x) < 0 (i = 1 2 : : : m)g 6= and M 0 = M (18.109) must be satis ed, i.e., the interior of M must be non-empty and it is possible to get to any boundary point by approaching it from the interior, i.e., the closure of M 0 is M . The function B (x) is de ned to be continuous on M 0 . It increases to 1 at the boundary of M . The modi ed problem (18.108) is solved by a sequence of barrier parameters qk tending to zero. For the solution xk of the k-th problem (18.108) holds (18.110) f (xk ) f (xk;1) and every accumulation point x of the sequence fxk g is a solution of (18.102). Fig. 18.12 shows a representation of the barrier method. The functions, e.g.,
B ( x) = ;
m X i=1
m X
; ln(;gi (x)) x 2 M 0 or
(18.111a)
1 (r = 1 2 : : :) x 2 M 0 (18.111b) ; g (x)]r i i=1 are appropriate realizations of B (x). f (x) = x21 + x22 = min! subject to x1 + x2 1, H (x qk ) = x21 + x22 + qk (; ln(x1 + x2 ; 1)),
B ( x) =
18.2 Non-linear Optimization 877
0 1 1 2x1 ; qk 0 B x 1 ; x2 ; 1 C C x1 + x2 > 1, rH (x qk ) = B B C @ A= 0 1 2x2 ; qk x + x ; 1 1
x1 + x2 > 1. The gradient of H is given
2
with respect to x. Subtracting the equations results in x1 = x2 , 2x1 ; qk
1 = 0 x > 1 =) x2 ; x1 ; qk = 1 1 2 x ;1 2 2 4 1 s 1 1 1 1 1 0 x1 > 2 , xk1 = xk2 = 4 + 16 + 4 qk k ! 1 qk ! 0: x1 = x2 = 2 . The solutions of problems (18.103) and (18.108) at the k-th step do not depend on the solutions of the previous steps. The application of higher penalty or smaller barrier parameters often leads to convergence problems with numerical solution of (18.103) and (18.108), e.g., in the method of (18.2.4), in particular, if we do not have any good initial approximation. Using the result of the k-th problem as the initial solution for the (k + 1)-th problem we can improve the convergence behavior.
18.2.9 Cutting Plane Methods
1. Formulation of the Problem and Principle of Solution
We consider the problem f (x) = cTx = min! c 2 IRn (18.112) over the bounded region M IRn given by convex functions gi(x) (i = 1 2 : : : m) in the form gi(x) 0. A problem with a non-linear but convex objective function f (x) is transformed into this form, if f (x) ; xn+1 0 xn+1 2 IR (18.113) is considered as a further constraint and f (x) = xn+1 = min! for all x = (x xn+1) 2 IRn+1 (18.114) is solved with gi (x) = gi(x) 0. The basic idea of the method is the iterative linear approximation of M by a convex polyhedron in the neighborhood of the minimum point x , and therefore the original program is reduced to a sequence of linear programming problems. First, we determine a polyhedron P1 = fx 2 IRn : aiTx bi i = 1 : : : sg: (18.115) From the linear program (18.116) f (x) = min! with x 2 P1 an optimal extreme point x1 of P1 is determined with respect to f (x). If x1 2 M holds, then the optimal solution of the original problem is found. Otherwise, we determine a hyperplane H1 = fx : as+1Tx = bs+1 as+1Tx1 > bs+1g, which separates the point x1 from M , so the new polyhedron contains 1
3
M
∆-
2
x
H2
f(x)
x
x* x
f(x)=const. H1
Figure 18.13
(18.117) P2 = fx 2 P1 : as+1Tx bs+1 g: Fig. 18.13 shows a schematic representation of the cutting plane method.
2. Kelley Method
The di erent methods di er from each other in the choice of the separating planes Hk . In the case of the Kelley method Hk is chosen in the following way: A jk is chosen so that gjk (xk ) = maxfgi(xk ) (i = 1 : : : m)g: (18.118)
878 18. Optimization At the point x = xk , the function gjk(x) has the tangent plane T (x) = gjk (xk ) + (x ; xk )Trgjk (xk ): (18.119) The hyperplane Hk = fx 2 IRn : T (x) = 0g separates the point xk from all points x with gjk (x) 0. So, for the (k + 1)-th linear program, T (x) 0 is added as a further constraint. Every accumulation point x of the sequence fxk g is a minimum point of the original problem. In practical applications this method shows a low speed of convergence. Furthermore, the number of constraints is always increasing.
18.3 Discrete Dynamic Programming
18.3.1 Discrete Dynamic Decision Models
A wide class of optimization problems can be solved by the methods of dynamic programming. The problem is considered as a process proceeding naturally or formally in time, and it is controlled by time-dependent decisions. If the process can be decomposed into a nite or countably in nite number of steps, then we talk about discrete dynamic programming, otherwise about continuous dynamic programming (see 18.3]). In this section, we discuss only n-stage discrete processes.
18.3.1.1 n-Stage Decision Processes
An n-stage process P starts at stage 0 with an initial state xa = x0 and proceeds through the intermediate states x1 x2 :m: : xn;1 into a nal state xn = xe 2 Xe IRm . The state vectors xj are in the state space Xj IR . To drive a state xj;1 into the state xj , a decision uj is required. All possible decision vectors uj in the state xj ;1 form the decision space Uj (xj ;1 ) IRs . From xj ;1 we get the consecutive state xj by the transformation (Fig. 18.14) xj = gj (xj;1 uj ) j = 1(1)n: (18.120) xa = x 0
g1(x0, u1)
u1Î U1(x0)
x1
g2(x1, u2)
x2
u2Î U2(x1)
xn-1
gn (xn-1, un)
xn = x e
unÎ Un(xn-1)
Figure 18.14
18.3.1.2 Dynamic Programming Problem
Our goal is to determine a policy (u1 : : : un) which drives the process from the initial state xa into the state xe considering all constraints so that it minimizes an objective function or cost function f (f1(x0 u1) : : : fn(xn;1 un)). The functions fj (xj;1 uj ) are called stage costs. The standard form of the dynamic programming problem is (18.121a) OF: f (f1(x0 u1) : : : fn(xn;1 un)) ;! min! 9 j = 1(1) n CT: xj = gj (xj;1 uj ) = x0 = xa xn = xe 2mXe xj 2 Xj IRm j = 1(1)n " (18.121b) uj 2 Uj (xj;1) IR j = 1(1)n: The rst type of constraints xj are called dynamic and the others x0 uj are called static. Similarly to (18.121a), a maximum problem can also be considered. A policy (u1 : : : un) satisfying all constraints is called feasible. The methods of dynamic programming can be applied if the objective function satises certain additional requirements (see 18.3.3, p. 879).
18.3 Discrete Dynamic Programming 879
18.3.2 Examples of Discrete Decision Models 18.3.2.1 Purchasing Problem
In the j -th period of a time interval which can be divided into n periods, a workshop needs vj units of a certain primary material. The available amount of this material at the beginning of period j is denoted by xj;1, in particular, x0 = xa is given. We have to determine the amounts uj , for a unit price cj , which should be purchased by the workshop at the end of each period. The given storage capacity K must not be exceeded, i.e., xj;1 + uj K . We have to determine a purchase policy (u1 : : : un), which minimizes the total cost. This problem leads to the following dynamic programming problem n n X X (18.122a) OF: f (u1 : : : un) = fj (uj ) = cj uj ;! min!
j =1 j =1 xj = xj;1 + uj ; vj j = 1(1)n 9 = x0 = xa 0 xj K j = 1(1)n " (18.122b) Uj (xj;1) = fuj : maxf0 vj ; xj;1 g uj K ; xj;1g j = 1(1)n: In (18.122b) we ensure that demands are satis ed and the storage capacity is not exceeded. If there is also a storage cost l per unit per period, then intermediate cost in the j -th period is (xj;1 + uj ; vj =2)l, and the modi ed cost function is n X f (x0 u1 : : : xn;1 un) = (cj uj + (xj;1 + uj ; vj =2) l): (18.123)
CT:
j =1
18.3.2.2 Knapsack Problem
We have to select some of the items A1 : : : An with weights w1 : : : wn and with values c1 : : : cn so that the total weight does not exceed a given bound W , and the total value is maximal. This problem does not depend on time. It will be reformulated in the following way: At every stage we make a decision uj about the selection of item Aj . Here, uj = 1 holds if Aj is selected, otherwise uj = 0. The capacity still available at the beginning of a stage is denoted by xj;1, so we get the following dynamic problem: n X (18.124a) OF: f (u1 : : : un) = cj uj ;! max!
CT:
j =1
xj = xj;1 ; wj uj x0 = W 0 xj W uj 2 f0 1g falls xj;1 wj uj = 0 falls xj;1 < wj
9 j = 1(1)n > > = j = 1(1)n > j = 1(1)n: > "
(18.124b)
18.3.3 Bellman Functional Equations 18.3.3.1 Properties of the Cost Function
In order to state the Bellman functional equations, the cost function must satisfy two requirements:
1. Separability
The function f (f1(x0 u1) : : : fn(xn;1 un)) is called separable, if it can be given by binary functions H1 : : : Hn;1 and by functions F1 : : : Fn in the following way: f (f1(x0 u1 ) : : : fn(xn;1 un)) = F1 (f1 (x0 u1) : : : fn(xn;1 un)) F1(f1 (x0 u1 ) : : : fn(xn;1 un)) = H1 (f1 (x0 u1 ) F2(f2(x1 u2) : : : fn(xn;1 un))) ................................................................................. (18.125) Fn;1(fn;1(xn;2 un;1) fn(xn;1 un)) = Hn;1 (fn;1(xn;2 un;1) Fn(fn(xn;1 un))) Fn(fn(xn;1 un)) = fn(xn;1 un):
880 18. Optimization
2. Minimum Interchangeability
A function H (f~(a) F~ (b)) is called minimum interchangeable, if: ~ ~ ~(a) min F~ (b) : ) F ( b ) = min min H f ( a H f a2A b2B (ab)2A B
(18.126)
This property is satis ed, for example, if H is monotone increasing with respect to its second argument for every a 2 A, i.e., if for every a 2 A, H f~(a) F~ (b1 ) H f~(a) F~ (b2 ) for F~ (b1) F~ (b2): (18.127) Now, for the cost function of the dynamic programming problem, the separability of f and the minimum interchangeability of all functions Hj j = 1(1)n ; 1 are required. The following often occurring type of cost function satis es both requirements:
f sum =
n X
j =1
fj (xj;1 uj ) or f max = j=1(1) maxn fj (xj;1 uj ):
(18.128)
The functions Hj are
n X fk (xk;1 uk ) and Hjsum = fj (xj;1 uj ) + k=j +1 ( ) Hjmax = max fj (xj;1 uj ) k=max f ( x k k ;1 uk ) : j +1(1)n
(18.129) (18.130)
18.3.3.2 Formulation of the Functional Equations We de ne the following functions: &j (xj;1) = min Fj (fj (xj;1 uj ) : : : fn(xn;1 un)) uk 2Uk (xk;1 )
j = 1(1)n
(18.131)
k=j (1)n
&n+1(xn) = 0: (18.132) If there is no policy (u1 : : : un) driving the state xj;1 into a nal state xe 2 Xe, then we substitute &j (xj;1) = 1. Using the separability and minimum interchangeability conditions and the dynamic constraints for j = 1(1)n we get: &j (xj;1) = u 2Umin H (f (x u ) min Fj+1(fj+1(xj uj+1) : : : fn(xn;1 un))) (x ) j j j ;1 j j
j j ;1
uk 2Uk (xk;1 ) k=j +1(1)n
= u 2Umin H f ( x j j j ;1 uj ) &j +1 (xj ) j j (xj ;1 ) &j (xj;1) = u 2Umin H f ( x j j j ;1 uj ) &j +1 (gj (xj ;1 uj )) : (x ) j
j j ;1
(18.133)
Equations (18.133) and (18.132) are called the Bellman functional equations. &1(x0 ) is the optimal value of the cost function f .
18.3.4 Bellman Optimality Principle
The evaluation of the functional equation &j (xj;1) = u 2Umin H f (x u ) & (x ) (x ) j j j ;1 j j +1 j j
j j ;1
(18.134)
corresponds to the determination of an optimal policy (uj : : : un) minimizing the cost function, i.e., Fj (fj (xj;1 uj ) : : : fn(xn;1 un)) ;! min! (18.135)
18.3 Discrete Dynamic Programming 881
based on the subprocess Pj starting at state xj;1 and consisting of the last n ; j + 1 stages of the total process P . The optimal policy of the process Pj with the initial state xj;1 is independent of the decisions u1 : : : uj;1 of the rst j ; 1 stages of P which have driven P to the state xj;1. To determine &j (xj;1), we need to know the value &j+1(xj ). Now, if (uj : : : un) is an optimal policy for Pj , then, obviously, (uj+1 : : : un) is an optimal policy for the subprocess Pj+1 starting at xj = gj (xj;1 uj ). This statement is generalized in the Bellman optimality principle. Bellman Principle: If (u1 : : : un) is an optimal policy of the process P and (x0 : : : xn) is the corresponding sequence of states, then for every subprocess Pj j = 1(1)n with initial state xj;1 the policy (uj : : : un) is also optimal (see 18.2]).
18.3.5 Bellman Functional Equation Method 18.3.5.1 Determination of Minimal Costs
With the functional equations (18.132), (18.133) and starting with &n+1(xn) = 0 we determine every value &j (xj;1) with xj;1 2 Xj;1 in decreasing order of j . It requires the solution of an optimum problem over the decision space Uj (xj;1) for every xj;1 2 Xj;1. For every xj;1 there is a minimum point uj 2 Uj as an optimal decision for the rst stage of a subprocess Pj starting at xj;1. If the sets Xj are not nite or they are too large, then the values &j can be calculated for a set of selected nodes xj;1 2 Xj;1. The intermediate values can be calculated by a certain interpolation method. &1(x0 ) is the optimal value of the cost function of process P . The optimal policy (u1 : : : un) and the corresponding states (x0 : : : xn) can be determined by one of the following two methods.
18.3.5.2 Determination of the Optimal Policy
1. Variant 1: During the evaluation of the functional equations, the computed uj is also saved for every xj;1 2 Xj;1. After the calculation of &1(x0 ), we get an optimal policy if we determine x1 = g1(x0 u1) from x0 = x0 and the saved u1 = u1 , then from x1 and the saved decision u2 we get x2 , etc. 2. Variant 2: We save only &j (xj;1) for every xj;1 2 Xj;1. After every &j (xj;1) is known, we make a forward calculation. Starting with j = 1 and x0 = x0 we determine uj in increasing order of j by the evaluation of the functional equation &j (xj;1) = u 2Umin H f (x u ) & (g (x u )) : (18.136) (x ) j j j ;1 j j +1 j j ;1 j j
j j ;1
We obtain xj = gj (xj;1 uj ). During the forward calculation, we again have to solve an optimization problem at every stage. 3. Comparison of the two Variants: The computation costs of variant 1 are less than variant 2 requires because of the forward calculations. However decision uj is saved for every state xj;1, which may require very large memory in the case of a higher dimensional decision space Uj (xj;1), while in the case of variant 2, we have to save only the values &j (xj;1). Therefore, sometimes variant 2 is used on computers.
18.3.6 Examples for Applications of the Functional Equation Method
18.3.6.1 Optimal Purchasing Policy 1. Formulation of the Problem
The problem from 18.3.2.1, p. 879, to determine an optimal purchasing policy
OF f (u1 : : : un) =
n X
j =1
cj uj ;! min!
(18.137a)
882 18. Optimization xj = xj;1 + uj ; vj j = 1(1)n x0 = xa 0 xj K j = 1(1)n Uj (xj;1) = fuj : maxf0 vj ; xj;1g uj K ; xj;1g j = 1(1)n leads to the functional equations &n+1(xn) = 0 (c u + &j+1(xj;1 + uj ; vj )) j = 1(1)n: &j (xj;1) = u 2Umin (x ) j j
CT
j
j j ;1
(18.137b) (18.138) (18.139)
2. Numerical Example
4, c2 = 3, c3 = 5, c4 = 3, c5 = 4, c6 = 2, xa = 2 : cv11 = = 6, v2 = 7, v3 = 4, v4 = 2, v5 = 4, v6 = 3. Backward Calculation: The function values &j (xj;1 ) will be determined for the states xj;1 = 0 1 : : : 10. Now, it is enough to make the minimum search only for integer values of uj . j = 6: &6(x5 ) = u 2min c u = c6 maxf0 v6 ; x5 g = 2 maxf0 3 ; x5 g: U (x ) 6 6
n=6
K = 10
6
6 5
According to variant 2 of the Bellman functional equation method, only the values of &6(x5 ) are written in the last row. For example, we determine &4(0). &4(0) = 2 min (3u + &5(u4 ; 2)) u4 10 4 = min(28 27 26 25 24 25 26 27 30) = 24: xj =0 1 2 3 4 5 6 7 8 9 10 j=1 75 2 59 56 53 50 47 44 41 38 35 32 29 3 44 39 34 29 24 21 18 15 12 9 6 4 24 21 18 15 12 9 6 4 2 0 0 5 22 18 14 10 6 4 2 0 0 0 0 6 6 4 2 0 0 0 0 0 0 0 0
Forward Calculation:
&1(2) = 75 = 4 min (4u + &2(u1 ; 4)): u1 8 1 We get u1 = 4 as the minimum point, therefore x1 = x0 + u1 ; v1 = 0. This method is repeated for &2(0) and for all later stages. The optimal policy is: (u1 u2 u3 u4 u5 u6) = (4 10 1 6 0 3):
18.3.6.2 Knapsack Problem
1. Formulation of the Problem
Consider the problem given in 18.3.2.2, p. 879
OF :f (u1 : : : un) = CT:
n X
j =1
cj uj ;! max!
9 xj = xj;1 ; wj uj j = 1(1)n > = x0 = W 0 xj W j = 1(1)n > uj 2 f0 1g if xj;1 wj j = 1(1)n: > > " uj = 0 if xj;1 < wj
Since we have a maximum problem, the Bellman functional equations are now &n+1(xn) = 0 &j (xj;1) = u 2max (c u + &j+1(xj;1 ; wj uj )) j = 1(1)n: U (x ) j j j
j j ;1
(18.140a) (18.140b)
18.3 Discrete Dynamic Programming 883
The decisions can be only 0 and 1, so it is practical to apply variant 1 of the functional equation method. For j = n n ; 1 : : : 1 we get: + & (x ; w ) if x w and c + & (x ; w ) > & (x ) j +1 j ;1 j j ;1 j j j +1 j ;1 j j +1 j ;1 &j (xj;1) = c&jj+1 (xj;1) otherwise
if x w and c + & (x ; w ) > & (x ) j ;1 j j j +1 j ;1 j j +1 j ;1 uj (xj;1) = 01 otherwise :
2. Numerical Example
1, c2 = 2, c3 = 3, c4 = 1, c5 = 5, c6 = 4, W = 10 n = 6: wc11 = = 2, w2 = 4, w3 = 6, w4 = 3, w5 = 7, w6 = 6. Since the weights wj are integers, the possible values for xj are xj 2 f0 1 : : : 10g j = 1(1)n and x0 = 10. The table contains the function values &j (xj;1) and the actual decision uj (xj;1) for every stage and for every state xj;1. For example, the values of &6 (x5) &3(2) &3(6), and &3 (8) are calculated: x5 < w6 = 4 u (x ) = 0 if x5 < 4 &6(x5 ) = 0c6 = 6 ifotherwise, 6 5 0 otherwise. &3(2): x2 = 2 < w3 = 3: &3(2) = &4(2) = 3 u3(2) = 0: &3(6): x2 > w3 and c3 + &3(x2 ; w3) = 6 + 3 < &4(x2 ) = 10: &3(6) = 10 u3(6) = 0: &3(8): x2 > w3 and c3 + &3(x2 ; w3) = 6 + 9 > &4(x2 ) = 10: &3(8) = 15 u3(8) = 1: The optimal policy is (u1 u2 u3 u4 u5 u6) = (0 1 1 1 0 1) &1 (10) = 19: xj = 0 1 2 3 4 5 6 7 8 9 10 j=1 19 0 2 0 0 3 0 4 1 7 1 9 0 10 1 13 1 13 1 15 0 16 0 19 1 3 0 0 3 0 3 0 6 1 9 1 9 0 10 0 12 1 15 1 16 1 16 0 4 0 0 3 1 3 1 3 1 6 0 9 1 10 1 10 1 10 1 13 0 16 1 5 0 0 0 0 0 0 0 0 6 0 7 1 7 1 7 1 7 1 13 1 13 1 6 0 0 0 0 0 0 0 0 6 1 6 1 6 1 6 1 6 1 6 1 6 1
884 19. Numerical Analysis
19 NumericalAnalysis
The most important principles of numerical analysis will be the subject of this chapter. The solution of practical problems usually requires the application of a professional numerical library of numerical methods, developed for computers. Some of them will be introduced at the end of Section 19.8.3. Special computer algebra systems such as Mathematica and Maple will be discussed with their numerical programs in Chapter 20, p. 953 and in Section 19.8.4, p. 946. Error propagation and computation errors will be examined in Section 19.8.2, p. 939.
19.1 Numerical Solution of Non-Linear Equations in a Single Unknown
Every equation with one unknown can be transformed into one of the normal forms: Zero form: f (x) = 0: (19.1) Fixed point form: x = '(x): (19.2) Suppose equations (19.1) and (19.2) can be solved. The solutions are denoted by x . To get a rst approximation of x , we can try to transfom the equation into the form f1 (x) = f2(x), where the curves of the functions y = f1 (x) and y = f2 (x) are more or less simple to sketch. f (x) = x2 ; sin x = 0. We can see from the shapes of the curves y = x2 and y = sin x that x1 = 0 and x2 0:87 are roots (Fig. 19.1). y 4 3
y y=x
y=(x)
2
2 1
y=sin x
0 1 2 −2 −1 x*1=0 x*2 ~ ~ 0.87
y=j(x)
3 x
Figure 19.1
19.1.1 Iteration Method
0
x*
x
Figure 19.2
The general idea of iterative methods is that starting with known initial approximations xk (k = 0 1 : : : n) we form a sequence of further and better approximations, step by step, hence we approach the solution of the given equation by iteration, by a convergent sequence. We try to create a sequence with convergence as fast as possible.
19.1.1.1 Ordinary Iteration Method
To solve an equation given in or transformed into the xed point form x = '(x), we use the iteration rule xn+1 = '(xn) (n = 0 1 2 : : : x0 given) (19.3) which is called the ordinary iteration method. It converges to a solution x if there is a neighborhood of x (Fig. 19.2) such that '(x) ; '(x ) K < 1 (K const) (19.4) x;x
19.1 Numerical Solution of Non-Linear Equations in a Single Unknown 885
holds, and the initial approximation x0 is in this neighborhood. If '(x) is di erentiable, then the corresponding condition is j'0(x)j K < 1: (19.5) The convergence of the ordinary iteration method becomes faster with smaller values of K . n x2 = sin x, i.e., 0 1 2 3 4 5 p xn 0:87 0:8742 0:8758 0:8764 0:8766 0:8767 xn+1 = sin xn. sin xn 0.7643 0.7670 0.7681 0.7684 0.7686 0.7686 Remark 1: In the case of complex solutions, we substitute x = u + iv. Separating the real and the imaginary part, we get an equation system of two equations for real unknowns u and v. Remark 2: The iterative solution of non-linear equation systems can be found in 19.2.2, p. 896.
19.1.1.2 Newton's Method
1. Formula of the Newton Method
To solve an equation given in the form f (x) = 0, we mostly use the Newton method which has the formula xn+1 = xn ; ff0((xxn)) (n = 0 1 2 : : : x0 is given) (19.6) n i.e., to get a new approximation xn+1, we need the value of the function f (x) and its rst derivative f 0(x) at xn .
2. Convergence of the Newton Method
The condition f 0(x) 6= 0 (19.7a) is necessary for convergence of the Newton method, and the condition f (x)f 00 (x) K < 1 (K const) (19.7b) 0 2 f (x) is sucient. The conditions (19.7a,b) must be ful lled in a neighborhood of x such that it contains all the points xn and x itself. If the Newton method is convergent, it converges very quickly. It has quadratic convergence, which means that the error of the (n+1)-st approximation is less than a constant multiple of the square of the error of the n-th approximation. In the decimal system, this means that after a while the number of exact digits will double step by step. p The solution of the equation f (x) = x2 ; a = 0, i.e., the calculation of x = a (a > 0 is given), with the Newton method results in the iteration formula xn+1 = 12 xn + xa : (19.8) n
We get for a = 2:
n 0 1 2 3 xn 1:5 1:416 666 6 1:414 215 7 1:414 213 6
3. Geometric Interpretation
The geometric interpretation of the Newton method is represented in Fig. 19.3. The basic idea of the Newton method is the local approximation of the curve y = f (x) by its tangent line.
4. Modied Newton Method 0
If the values of f (xn ) barely change during the iteration, we can keep it constant, and we use the so-called modi ed Newton method (19.9) xn+1 = xn ; ff0((xxn)) (m xed m < n): m
886 19. Numerical Analysis y
y=f( x)
y
xn 0 0
x*
xn+1 xn
x
xn+1
)
x*
f(x y=
xm
x
Figure 19.3 Figure 19.4 The goodness of the convergence is hardly modi ed by this simpli cation.
5. Di erentiable Functions with Complex Argument
The Newton method also works for di erentiable functions with complex arguments.
19.1.1.3 Regula Falsi
1. Formula for Regula Falsi
To solve the equation f (x) = 0, the regula falsi method has the rule: xm f (x ) (n = 1 2 : : : m < n x x are given): xn+1 = xn ; f (xxn) ; (19.10) n 0 1 n ; f (xm ) We need to compute only the function values. The method follows from the Newton method (19.6) by approximating the derivative f 0(xn ) by the nite di erence of f (x) between xn and a previous approximation xm (m < n).
2. Geometric Interpretation
The geometric interpretation of the regula falsi method is represented in Fig. 19.4. The basic idea of the regula falsi method is the local approximation of the curve y = f (x) by a secant line.
3. Convergence
The method (19.10) is convergent if we choose m so that f (xm ) and f (xn) always have di erent signs. If the convergence already seems to be quick enough during the process, it will speed up if we ignore the change of sign, and we just substitute xm = xn;1 . f (x) = x2 ; sin x = 0: xn n %xn = xn ; xn;1 xn f (xn ) %yn = f (xn) ; f (xn;1 ) % %yn 0 0:9 0.0267 1 ;0:3 0:87 ;0.0074 ;0.0341 0.8798 2 0.0065 0:8765 ;0.000252 0.007148 0.9093 3 0.000229 0:876729 0.000003 0.000255 0.8980 4 ;0.000003 0:876726 If during the process the value of %xn =%yn only barely changes, we do not need to recalculate it again and again.
4. Ste ensen Method
Applying the regula falsi method with xm = xn;1 for the equation f (x) = x ; '(x) = 0 we can often speed up the convergence, especially in the case '0(x) < ;1. This algorithm is known as the Steensen method. 2 p To solve the equation x = sin x with the Ste ensen method, we should use the form f (x) = x ; sin x = 0.
19.1 Numerical Solution of Non-Linear Equations in a Single Unknown 887
n %xn = xn ; xn;1 0 1 2 3
;0:03
0.006654
xn
f (xn)
0:9 0.014942 0:87 ;0.004259 0:876654 ;0.000046 0:876727 0.000001
%y = f (xn) ; f (xn;1)
;0.019201 0.004213
%xn %yn 1.562419 1.579397
19.1.2 Solution of Polynomial Equations
Polynomial equations of n-th degree have the form f (x) = pn(x) = anxn + an;1 xn;1 + + a1x + a0 = 0: (19.11) For their e ective solution we need ecient methods to calculate the function values and the derivative values of the function pn(x) and an initial estimate of the positions of the roots.
19.1.2.1 Horner's Scheme 1. Real Arguments
To determine the value of a polynomial pn(x) of n-th degree at the point x = x0 from its coecients, rst we consider the decomposition pn(x) = anxn + an;1xn;1 + + a2 x2 + a1 x + a0 = (x ; x0 )pn;1(x) + pn(x0) (19.12) where pn;1(x) is a polynomial of (n ; 1)-st degree: pn;1(x) = a0n;1 xn;1 + a0n;2xn;2 + + a01 x + a00 : (19.13) We get the recursion formula a0k;1 = x0 a0k + ak (k = n n ; 1 : : : 0 a0n = 0 a0;1 = pn(x0 )) (19.14) by coecient comparison in (19.12) with respect to xk . (Note that a0n;1 = an.) This way, we determine the coecients a0k of pn;1(x) and the value pn(x0 ) from the coecients ak of pn(x). Furthermore fewer multiplications are required than by the \traditional" way. By repeating this procedure, we get a decomposition of the polynomial pn;1(x) with the polynomial pn;2(x), pn;1(x) = (x ; x0)pn;2(x) + pn;1(x0 ) (19.15) etc., and we get a sequence of polynomials pn(x), pn;1(x) : : : p1(x) p0(x). The calculations of the coecients and the values of the polynomial is represented in (19.16). an an;1 an;2 : : : a3 a2 a1 a0 x0 x0 a0n;1 x0 a0n;2 : : : x0 a03 x0 a02 x0 a01 x0a00 0 an;1 a0n;2 a0n;3 : : : a02 a01 a00 pn(x0) x0 x0 a00n;2 x0 a00n;3 : : : x0 a002 x0 a001 x0 a000 00 an;2 a00n;3 a00n;4 : : : a001 a000 pn;1(x0 ) (19.16) ............................. ( n ;1) x0 x0 a0 a(1n;1) p1(x0 ) x0 a(0n) = p0 (x0 ) We get from scheme (19.16) the value pn(x0 ), and derivatives p(nk) (x0) as: p0n(x0 ) = 1!pn;1(x0) p00n(x0 ) = 2!pn;2(x0 ) : : : p(nn)(x0 ) = n!p0 (x0 ): (19.17)
888 19. Numerical Analysis p4(x) = x4 + 2x3 ; 3x2 ; 7. The substitution value and derivatives of p4 (x) are calculated at x0 = 2 according to (19.16).
1 2 ;3 0 ;7 2 8 10 20 1 4 5 10 13 2 2 12 34 1 6 17 44 2 2 16 1 8 33 2 2 1 10 2 1 2
We see: p4 (2) = 13 p04 (2) = 44 p004 (2) = 66 p0004 (2) = 60 p(4) 4 (2) = 24:
Remarks: 1. We can rearrange the polynomial pn(x) with respect to the powers of x ; x0 , e.g., in the example above we get p4 (x) = (x ; 2)4 + 10(x ; 2)3 + 33(x ; 2)2 + 44(x ; 2) + 13. 2. The Horner scheme can also be used for complex coecients ak . In this case for every coecient we have to compute a real and an imaginary column according to (19.16).
2. Complex Arguments
If the coecients ak in (19.11) are real, then the calculation of pn(x0 ) for complex values x0 = u0 + iv0 can be made real. In order to show this, we decompose pn(x) as follows: pn(x) = an xn + an;1xn;1 + + a1x + a0 = (x2 ; px ; q)(a0n;2xn;2 + + a00 ) + r1x + r0 with (19.18a) 2 x ; px ; q = (x ; x0 )(x ; x0 ) i.e., p = 2u0 q = ;(u02 + v0 2): (19.18b) Then, we get pn(x0 ) = r1x + r0 = (r1 u0 + r0) + ir1v0: (19.18c) To nd (19.18a) we can construct the so-called two-row Horner scheme introduced by Collatz: an an;1 an;2 : : : a3 a2 a1 a0 q qa0n;2 : : : qa03 qa02 qa01 qa00 p pa0n;2 pa0n;3 : : : pa02 pa01 pa00 (19.18d) a0n;2 a0n;3 a0n;4 : : : a01 a00 r1 r0 = an p4(x) = x4 + 2x3 ; 3x2 ; 7. Calculate the value of p4 at x0 = 2 ; i, i.e., for p = 4 and q = ;5. 1 2 ;3 0 ;7 We see: ;5 ;5 ;30 ;80 p4(x0 ) = 34x0 ; 87 = ;19 ; 34i. 4 4 24 64 1 6 16 34 ;87
19.1.2.2 Positions of the Roots 1. Real Roots, Sturm Sequence
With the Cartesian rule of signs we can get a rst idea of whether the polynomial equation (19.11) has a real root, or not. a) The number of positive roots is equal to the number of sign changes in the sequence of the cooecients an an;1 : : : a1 a0 (19.19a) or it is less by an even number.
19.1 Numerical Solution of Non-Linear Equations in a Single Unknown 889
b) The number of negative roots is equal to the number of sign changes in the coecient sequence a0 ;a1 a2 : : : (;1)nan (19.19b)
or it is less by an even number. p5(x) = x5 ; 6x4 + 10x3 + 13x2 ; 15x ; 16 has 1 or 3 positive roots and 0 or 2 negative roots. To determine the number of real roots in any given interval (a b), Sturm sequences are used (see 1.6.3.2, 2., p. 44). After computing the function values y = pn(x ) at a uniformly distributed set of nodes x = x0 + h (h constant) (which can be easily performed by using the Horner scheme) a good guess of the graph of the function and the locations of roots are obtained. If pn(c) and pn(d) have di erent signs, there is at least one real root between c and d.
2. Complex Roots
In order to localize the real or complex roots into a bounded region of the complex plane consider the following polynomial equation which is a simple consequence of (19.11): f (x) = jan;1jrn;1 + jan;2jrn;2 + + ja1 jr + ja0j = janjrn (19.20) and we determine, e.g., by systematic repeated trial and error, an upper bound r0 for the positive roots of (19.20). Then, for all roots xk (k = 1 2 : : : n) of (19.11), jxk j r0 : (19.21) f (x) = p4(x) = x4 +4:4x3 ;20:01x2 ;50:12x+29:45 = 0 f (x) = 4:4r3 +20:01r2 +50:12r +29:45 = r4. We get for r = 6: f (6) = 2000:93 > 1296 = r4 r = 7: f (7) = 2869:98 > 2401 = r4 r = 8: f (8) = 3963:85 < 4096 = r4: From this it follows that jxk j < 8 (k = 1 2 3 4). Actually, for the root x1 with maximal absolute value we have: ;7 < x1 < ;6. Remark: A special method has been developed in electrotechnics in the so-called root locus theory for the determination of the number of complex roots with negative real parts. It is used to examine stability (see 19.11], 19.30]).
19.1.2.3 Numerical Methods 1. General Methods
The methods discussed in Section 19.1.1, p. 884, can be used to nd real roots of polynomial equations. The Newton method is well suited for polynomial equations because of its fast convergence, and the fact that the values of f (xn) and f 0(xn ) can be easily computed by using Horner's rule. By assuming that an approximation xn of the root x of a polynomial equation f (x) = 0 is suciently good, then the correction term = x ; xn can be iteratively improved by using the xed-point equation
= ; f 0(1x ) f (xn) + 2!1 f 00(xn ) 2 + = '( ): (19.22) n
2. Special Methods
The Bairstow method is well applicable to nd root pairs, especially complex conjugate pairs of roots. The Horner scheme (19.18a{d) is used to nd quadratic factors of the given polynomial having the root pair to determine the coecients p and q which make the coecients of the linear remainder r0 and r1 equal to zero (see 19.29], 19.11], 19.30]). If the computation of the root with largest or smallest absolute value is required, then the Bernoulli method is the choice (see 19.19]). The Graee method has some historical importance. It gives all roots simultaneously including complex conjugate roots however the computation costs are tremendous (see 19.11], 19.30]).
890 19. Numerical Analysis
19.2 Numerical Solution of Equation Systems
In several practical problems, we have m conditions for the n unknown quantities xi (i = 1 2 : : : n) in the form of equations: F1 (x1 x2 : : : xn) = 0 F2 (x1 x2 : : : xn) = 0 (19.23) ... ... Fm (x1 x2 : : : xn) = 0: We have to determine the unknowns xi so that they form a solution of the equation system (19.23). Mostly m = n holds, i.e., the number of unknowns and the number of equations are equal to each other. In the case of m > n, we call (19.23) an overdetermined system in the case of m < n it is an underdetermined system. Overdetermined systems usually have no solutions. Then, we are looking for the \best" solution of (19.23), in the Euclidean metric with the least squares method m X i=1
Fi2(x1 x2 : : : xn) = min!
(19.24)
or in other metrics as another extreme value problem. Usually, the values of n ; m variables of an underdetermined problem can be chosen freely, so the solution of (19.23) depends on n ; m parameters. We call it an (n ; m)-dimensional manifold of solutions. We distinguish between linear and non-linear equation systems, depending on whether the equations are only linear or non-linear in the unknowns.
19.2.1 Systems of Linear Equations
Consider the linear equation system a11 x1 + a12 x2 + + a1nxn = b1 a21 x1 + a22 x2 + + a2nxn = b2 (19.25) ... ... an1 x1 + an2x2 + + annxn = bn: The system (19.25) can be written in matrix form Ax = b (19.26a) with 0 a11 a12 a1n 1 0 b1 1 0 x1 1 B C B C Bx C a a a2n C b A = BB@ ... 21 22 C b = BB@ 2... CCA x = BB@ ...2 CCA : (19.26b) A an1 an2 ann bn xn Suppose the quadratic matrix A = (aik ) (i k = 1 2 : : : n) is regular, so system (19.25) has a unique solution (see 4.4.2.1, 2., p. 273). In the practical solution of (19.25), we distinguish between two types of solution methods: 1. DirectMethods are based on elementary transformations, from which the solution can be obtained immediately. These are the pivoting techniques (see 4.4.1.2, p. 271) and the methods given in 19.2.1.1{ 19.2.1.3. 2. Iteration methods start with a known initial approximation of the solution, and forming a sequence of approximations that converges to the solution of (19.25) (see 19.2.1.4, p. 895).
19.2.1.1 Triangular Decomposition of a Matrix 1. Principle of the Gauss Elimination Method By elementary transformations
19.2 Numerical Solution of Equation Systems 891
1. interchanging rows, 2. multiplying a row by a non-zero number and 3. adding a multiple of a row to another row, the system A x = b is transformed into the so-called row echelon form 0 r11 r12 r13 B r22 r23 B r33 R x = c with R = BBBB @ 0
1 : : : r1n : : : r2n C C : : : r3n C C: C . . . ... C A rnn
(19.27)
Since only equivalent transformations were made, the equation system R x = c has the same solutions as A x = b. We get from (19.27):
0
1
n X xi = r1 @ci ; rik xk A i = n ; 1 n ; 2 : : : 1 xn = rcn : (19.28) ii nn k=i+1 The rule given in (19.28) is called backward substitution, since the equations of (19.27) are used in the opposite order as they follow each other. The transition from A to R is made by n ; 1 so-called elimination steps, whose procedure is shown by the rst step. This transforms matrix A into matrix A1: 0 a11 B a21 B A = BBBB a.31 @ .. an1
1 a12 : : : a1n a22 : : : a2n C C a32 : : : a3n C C C C A an2 : : : ann
0 (1) (1) (1) a11 a12 : : : a1n B B (1) B 0 a(1) 22 : : : a2n B B (1) A1 = BB 0 a32 : : : a(1) 3n .. B . B @ .. (1) . (1) 0 an2 : : : ann
1 C C C C C C : C C C A
(19.29)
We proceed as follows: 1. We choose an ar1 6= 0 (according to (19.33)). If there is none, stop: A is singular. Otherwise ar1 is called the pivot. 2. We interchange the rst and the r-th row of A. The result is A. 3. We subtract li1 (i = 2 3 : : : n)times the rst row from the i-th row of the matrix A. As a result we get the matrix A1 and analogously the new right-hand side b1 with the elements ai1 a(1) ik = aik ; li1 a1k li1 = a 11 b(1) (19.30) i = bi ; li1 b1 (i k = 2 3 : : : n): The framed submatrix in A1 (see (19.29)) is of type (n ; 1 n ; 1) and it will be handled analogously to A, etc. This method is called the Gaussian elimination method or the Gauss algorithm (see 4.4.2.4, p. 276).
2. Triangular Decomposition
The result of the Gauss elimination method can be formulated as follows: To every regular matrix A there exists a so-called triangular decomposition or LU factorization of the form PA = LR (19.31)
892 19. Numerical Analysis with
0 r11 r12 r13 B r22 r23 B r33 R = BBBB @ 0
1 : : : r1n : : : r2n C C : : : r3n C C C . . . ... C A rnn
0 1 B l B 21 L = BBBB l31. @ .. ln1
1
C 1 0 C C l32 1 C : (19.32) C ... C ... A ln2 : : : lnn;1 1 Here R is called an upper triangular matrix, L is a lower triangular matrix and P is a so-called permutation matrix. A permutation matrix is a quadratic matrix which has exactly one 1 in every row and every column, and the other elements are zeros. The multiplication P A results in row interchanges in A, which comes from the choices of the pivot elements during the elimination 0 3 1 6procedure. 10x 1 021 1 @ The Gauss elimination method should be used for the system 2 1 3 A @ x2 A = @ 7 A. In 1 1 1 x3 4 schematic form, where the coecient matrix and the vector from the right-hand side are written next to each other (into the so-called extended coecient matrix), we get: 0 1 0 3 0 3 1 6 1 6 2 1 2 1 3 1 6 2C B B C B C (A b) = B @ 2 1 3 7C A)B @ 2=3 1=3 ;1 17=3 C A ) @ 1=3 2=3 ;1 10=3 A, i.e., 4 2=3 1=2 ;1/2 4 1 1 1 1=3 2/3 ;1 10=3 01 0 01
03 1 61
0 1
0
01
03
1
6 1
P = @ 0 0 1 A ) P A = @ 1 1 1 A L = @ 1=3 1 0 A R = @ 0 2=3 ;1 A. 0 1 0 2 1 3 2=3 1=2 1 0 0 ;1=2 In the extended coecient matrices, the matrices A, A1 and A2 , and also the pivots are shown in boxes.
3. Application of Triangular Decomposition
With the help of triangular decomposition, we can describe the solution of linear equation systems A x = b in three steps: 1. P A = L R: Determination of the triangular decomposition and substitution R x = c. 2. L c = P b: Determination of the auxiliary vector c by forward substitution. 3. R x = c: Determination of the solution x by backward substitution. If the solution of a system of linear equations is handled by the expanded coecient matrix (A b), as in the above example, by the Gauss elimination method, then the lower triangular matrix L is not needed explicitly. This can be especially useful if several systems of linear equations are to be solved after each other with the same coecient matrix, with di erent right-hand sides.
4. Choice of the Pivot Elements
Theoretically, every non-zero element a(i1k;1) of the rst column of the matrix Ak;1 could be used as a pivot element at the k-th elimination step. In order to improve the accuracy of solution (to decrease the accumulated rounding errors of the operations), the following strategies are recommended. 1. Diagonal Strategy The successive diagonal elements are chosen as pivot elements i.e., there is no row interchange. This kind of choice of the pivot element makes sense if the absolute value of the elements of the main diagonal are fairly large compared to the others in the same row. 2. Column Pivoting To perform the k-th elimination step, we choose the row index r for which:
ja(rkk;1)j = max ja(k;1) j: i k ik
(19.33)
19.2 Numerical Solution of Equation Systems 893
If r 6= k, then the r-th and the k-th rows will be interchanged. It can be proven that this strategy makes the accumulated rounding errors smaller.
19.2.1.2 Cholesky's Method for a Symmetric Coecient Matrix
In several cases, the coecient matrix A in (19.26a) is not only symmetric, but also positive denite, i.e., for the corresponding quadratic form Q(x) holds
Q(x) = xT A x =
n X n X
i=1 k=1
for every x 2 IRn, x 6= 0. triangular decomposition A = L LT with 0 l11 B l21 l22 B B L = BBB l31. l32 l33 @ .. ln1 ln2 ln3
q lkk = a(kkk;1)
aik xi xk > 0
(19.34)
Since for every symmetric positive de nite matrix A there exists a unique (19.35) 0 ... : : : lnn (k;1)
lik = aikl
kk
1 C C C C C C A (i = k k + 1 : : : n )
(19.36a)
(19.36b)
a(ijk) = a(ijk;1) ; lik ljk (i j = k + 1 k + 2 : : : n) (19.36c) the solution of the corresponding linear equation system A x = b can be determined by the Cholesky method by the following steps: 1. A = L LT : Determination of the so-called Cholesky decomposition and substitution LTx = c. 2. L c = b: Determination of the auxiliary vector c by forward substitution. 3. LT x = c: Determination of the solution x by backward substitution. For large values of n the computation cost of the Cholesky method is approximately half of that of the LU decomposition given in (19.31), p. 891.
19.2.1.3 Orthogonalization Method 1. Linear Fitting Problem
Suppose an overdetermined linear equation system n X
k=1
aik xk = bi (i = 1 2 : : : m m > n)
(19.37)
is given in matrix form Ax = b: (19.38) Suppose the coecient matrix A = (aik ) with size (m n) has full rank n, i.e., its columns are linearly independent. Since an overdetermined linear equation system usually has no solution, instead of (19.37) we consider the so-called error equations
ri =
n X
k=1
aik xk ; bi (i = 1 2 : : : m m > n)
(19.39)
with residues ri, and we require that the sum of their squares should be minimal: m X i=1
ri2 =
n m "X X
i=1 k=1
2 aik xk ; bi = F (x1 x2 : : : xn ) = min!
(19.40)
894 19. Numerical Analysis The problem (19.40) is called a linear tting problem or a linear least squares problem (see also 6.2.5.5, p. 403). The necessary condition for the relative minimum of the sum of residual squares F (x1 x2 : : : xn) is @F (19.41) @xk = 0 (k = 1 2 : : : n) and it leads to the linear equation system ATAx = ATb: (19.42) The transition from (19.38) to (19.42) is called a Gauss transformation, since the system (19.42) arises by applying the Gaussian least squares method (see 6.2.5.5, p. 403) for (19.38). Since we suppose full rank for A, ATA is a positive de nite matrix of size (n n), and the so-called normal equations (19.42) can be solved numerically by the Cholesky method (see 19.2.1.2, p. 893). We can have numerical diculties with the solution of the normal equation system (19.42) if the condition number (see 19.24]) of the matrix AT A is too large. The solution x can then have a large relative error. Because of this problem, it is better to use the orthogonalization method for solving numerically linear tting problems.
2. Orthogonalization Method
The following facts are the basis of the following orthogonalization method for solving a linear least squares problem (19.40): 1. The length of a vector does not change during an orthogonal transformation, i.e., the vectors x and x~ = Q0 x with QT0Q0 = E (19.43) have the same length. 2. For every matrix A of size (m n) with maximal rank n (n < m) there exists an orthogonal matrix Q of size (m m) such that 0r r ::: r 1 B 11 r1222 : : : r12nn C C ! B B . . . ... C B C R T ^ B C ^ : (19.45) A = QR (19.44) with Q Q = E and R = O = BB C C r nn B C
@
O
A
Here R is an upper triangular matrix of size (n n), and O is a zero matrix of size (m ; n n). The factored form (19.44) of matrix A is called the QR decomposition. So, the error equations (19.39) can be transformed into the equivalent system r11 x1 + r12 x2 + : : : + r1n xn ;^b1 = r^1 r22 x2 + : : : + r2n xn ;^b2 = r^2 ... ... ... = ... ^ (19.46) rnnxn ;bn = r^n ;^bn+1 = r^n+1 ... ... ;^bm = r^m without changing the sum of the squares of the residuals. From (19.46) it follows that the sum of the squares is minimal for r^1 = r^2 = = r^n = 0 and the minimum value is equal to the sum of the squares of r^n+1 to r^m. We get the required solution x by backward substitution Rx = b^ 0 (19.47)
19.2 Numerical Solution of Equation Systems 895
where b^ 0 is the vector with components ^b1 ^b2 : : : ^bn obtained from (19.46). There are two methods most often used for a stepwise transition of (19.39) into (19.46): 1. Givens transformation, 2. Householder transformation. The rst one results in the QR decomposition of matrix A by rotations, the other one by reections. The numerical implementations can be found in 19.23]. Practical problems in linear mean square approximations are solved mostly by the Householder transformation, where the frequently occurring special band structure of the coecient matrix A can be used.
19.2.1.4 Iteration Methods 1. Jacobi Method
Suppose in the coecient matrix of the linear equation system (19.25) every diagonal element aii (i = 1 2 : : : n) is di erent from zero. Then the i-th row can be solved for the unknown xi , and we get immediately the following iteration rule, where is the iteration index: n a X ik ( ) x(i +1) = abi ; xk (i = 1 2 : : : n) (19.48) ii k=1 aii (k6=i)
(0) (0) ( = 0 1 2 : : : x(0) 1 x2 : : : xn are given initial values). Formula (19.48) is called the Jacobi method. Every component of the new vector x( +1) is calculated from the components of x( ) . If at least one of the conditions n a X ik < 1 column sum criterion max (19.49) k i=1 aii (i6=k)
or max i
n a X ik < 1 k=1 aii
row sum criterion
(19.50)
(k6=i)
holds, then the Jacobi method is convergent for any initial vector x(0) .
2. Gauss{Seidel Method ( +1)
If the rst component x1 is calculated by the Jacobi method, then this value can be used in the calculation of x(2 +1) . While we proceed similarly in the calculation of the further components, we get the iteration formula i;1 n a X X ik ( ) xk (19.51) x(i +1) = abi ; aaik x(k +1) ; ii k=1 ii k=i+1 aii (0) (0) (i = 1 2 : : : n x(0) 1 x2 : : : xn given initial value = 0 1 2 : : :) : Formula (19.51) is called the Gauss{Seidel method. The Gauss{Seidel method usually converges more quickly than the Jacobi method, but its convergence criterion is more complicated. 10x1 ; 3x2 ; 4x3 + 2x4 = 14 ;3x1 + 26x2 + 5x3 ; x4 = 22 ;4x1 + 5x2 + 16x3 + 5x4 = 17 2x1 + 3x2 ; 4x3 ; 12x4 = ;20: The corresponding iteration formula according to (19.51) is:
896 19. Numerical Analysis 1 14 + 3x( ) + 4x( ) ; 2x( ) x(1 +1) = 10 2 3 4 1 22 + 3x( +1) ; 5x( ) + x( ) x(2 +1) = 26 1 3 4 1 17 + 4x( +1) ; 5x( +1) ; 5x( ) x(3 +1) = 16 1 2 4 1 ;20 + 2x( +1) + 3x( +1) ; 4x( +1) : x(4 +1) = 12 1 2 3
Some approximations and the solution are given here:
x(0) 0 0 0 0
x(1)
1.4 1.0077 1.0976 1.7861
x(4)
1.5053 0.9946 0.5059 1.9976
x(5)
1.5012 0.9989 0.5014 1.9995
x
1.5 1 0.5 2
3. Relaxation Method
The iteration formula of the Gauss{Seidel method (19.51) can be written in the so-called correction form ! i;1 n X X x(i +1) = x(i ) + abi ; aaik x(k +1) ; aaik x(k ) i.e., ii k=1 ii k=i ii ( +1) ( ) ( ) xi = xi + di (i = 1 2 : : : n = 0 1 2 : : :): (19.52) By an appropriate choice of a relaxation parameter ! and rewriting (19.52) in the form x(i +1) = x(i ) + !d(i ) (i = 1 2 : : : n = 0 1 2 : : :) (19.53) we can try to improve the speed of convergence. It can be shown that convergence is possible only for 0 < ! < 2: (19.54) For ! = 1 we retrieve the Gauss{Seidel method. In the case of ! > 1, which is called overrelaxation, the corresponding iteration method is called the SOR method (successive overrelaxation). The determination of an optimal relaxation parameter is possible only for some special types of matrices. We apply iterative methods to solve linear equation systems in the rst place when the main diagonal elements aii of the coecient matrix have an absolute value much larger than the other elements aik (i 6= k) (in the same row or column), or when the rows of the equation system can be rearranged in a certain way to get such a form.
19.2.2 Non-Linear Equation Systems
Suppose the system of n non-linear equations Fi(x1 x2 : : : xn) = 0 (i = 1 2 : : : n) (19.55) for the n unknowns x1 , x2, . . . , xn has a solution. Usually, a numerical solution can be given only by an iteration method.
19.2.2.1 Ordinary Iteration Method
We can use the ordinary iteration method if the equations (19.55) can be transformed into a xed-point form xi = fi(x1 x2 : : : xn) (i = 1 2 : : : n): (19.56) (0) (0) , we get the improved values either by Then, starting from estimated approximations x(0) , x ,. . . , x 1 2 n
1. iteration with simultaneous steps or by
x(i +1) = fi x(1 ) x(2 ) : : : x(n ) (i = 1 2 : : : n = 0 1 2 : : :)
2. iteration with sequential steps
( ) ( ) ( ) x(i +1) = fi x(1 +1) : : : x(i; +1) (i = 1 2 : : : n = 0 1 2 : : :): 1 xi xi+1 : : : xn
(19.57) (19.58)
19.2 Numerical Solution of Equation Systems 897
It is of crucial importance for the convergence of this method that in the neighborhood of the solution the functions fi should depend only weakly on the unknowns, i.e., if fi are di erentiable, the absolute values of the partial derivatives must be rather small. We get as a convergence condition ! n X @fi : (19.59) K < 1 with K = max max @xk i k=1 With this quantity K , the error estimation is the following: ( +1) ( +1) ( ) max xi ; xi 1 ;KK max xi ; xi : (19.60) i i Here, xi is the component of the required solution, x(i ) and x(i +1) are the corresponding -th and ( + 1)-th approximations.
19.2.2.2 Newton's Method
The Newton method is used for the problem given in the form (19.55). After nding the initial ap(0) (0) proximation values x(0) 1 , x2 : : : xn , the functions Fi are expanded in Taylor form as functions of n independent variables x1 , x2 : : : xn (see p. 417). Terminating the expansion after the linear terms, from (19.55) we get a linear equation system, and with this we can get iterative improvements by the following formula: n @F X i ( ) Fi x(1 ) x(2 ) : : : x(n ) + @x x1 : : : x(n ) x(k +1) ; x(k ) = 0 (19.61) k=1 k (i = 1 2 : : : n = 0 1 2 : : :): The coecient matrix of the linear equation system (19.61), which should be solved in every iteration step, is ! @Fi x( ) x( ) : : : x( ) (i k = 1 2 : : : n) F0 (x( ) ) = @x (19.62) 1 2 n k and it is called the Jacobian matrix. If the Jacobian matrix is invertible in the neighborhood of the solution, the Newton method is locally quadratically convergent, i.e., its convergence essentially depends on how good the initial approximations are. If we substitute x(k +1) ; x(k ) = d(k ) in (19.61), then the Newton method can be written in the correction form x(k +1) = x(k ) + d(k ) (i = 1 2 : : : n = 0 1 2 : : :): (19.63) To reduce the sensitivity to the initial values, analogously to the relaxation method, we can introduce a so-called damping or step length parameter : x(k +1) = x(k ) + d(k ) (i = 1 2 : : : n = 0 1 2 : : : > 0): (19.64) Methods to determine can be found in 19.24].
19.2.2.3 Derivative-Free Gauss{Newton Method
To solve the least squares problem (19.24), we proceed iteratively in the non-linear case as follows: (0) (0) 1. Starting from a suitable initial approximation x(0) 1 , x2 , . . . , xn , we approximate the non-linear functions Fi(x1 x2 : : : xn) (i = 1 2 : : : m) as in the Newton method (see (19.61)) by linear approximations F~i (x1 x2 : : : xn), which are calculated in every iteration step according to n @F X i ( ) x1 : : : x(n ) xk ; x(k ) F~i(x1 : : : xn) = Fi x(1 ) x(2 ) : : : x(n ) + @x k k=1 (i = 1 2 : : : n = 0 1 2 : : :): (19.65)
898 19. Numerical Analysis 2. We substitute d(k ) = xk ; x(k ) in (19.65) and we determine the corrections d(k ) by using the Gaussian least squares method, i.e., by the solution of the linear least squares problem m X F~i2(x1 : : : xn) = min (19.66) i=1
e.g., with the help of the normal equations (see (19.42)), or the Householder method (see 19.6.2.2, p. 921). 3. We get approximations for the required solution by x(k +1) = x(k ) + d(k ) or (19.67a) x(k +1) = x(k ) + d(k ) (k = 1 2 : : : n) (19.67b) where ( > 0) is a step length parameter similar to the Newton method. By repeating steps 2 and 3 with x(k +1) instead of x(k ) we get the Gauss{Newton method. It results in a sequence of approximation values, whose convergence strongly depends on the accuracy of the initial approximation. We can reduce the sum of the error squares by introducing a length parameter . If the evaluation of the partial derivatives @Fi x(1 ) : : : x(n ) (i = 1 2 : : : m k = 1 2 : : : n) requires @xk too much work, we can approximate the partial derivatives by di erence quotients : @Fi x( ) : : : x( ) : : : x( ) 1 F x( ) : : : x( ) x( ) + h( ) x( ) : : : x( ) i 1 n n k k;1 k k k+1 @xk 1 h(k ) ;Fi x(1 ) : : : x(k ) : : : x(n ) (i = 1 2 : : : m k = 1 2 : : : n = 0 1 2 : : :): (19.68) The so-called discretization step sizes h(k ) may depend on the iteration steps and the values of the variables. If we use approximations (19.68), then we have to calculate only function values Fi while performing the Gauss{Newton method, i.e., the method is derivative free.
19.3 Numerical Integration
19.3.1 General Quadrature Formulas
The numerical evaluation of the de nite integral
Zb I (f ) = f (x) dx
(19.69)
a
must be done only approximately if the integrand f (x) cannot be integrated by elementary calculus, or it is too complicated, or when the function is known only at certain points x , the so-called interpolation nodes from the integration interval a b]. We use the so-called quadrature formulas for the approximate calculation of (19.69). They have the general form
Q(f ) =
n X
c0 y +
=0 f ( ) (x
n X
=0
c1 y0 + +
n X
=0
cp y(p)
(19.70)
with y( ) = ) ( = 1 2 : : : p = 1 2 : : : n), y = f (x ), and constant values of c . Obviously, I (f ) = Q(f ) + R (19.71) where R is the error of the quadrature formula. We suppose in the application of quadrature formulas that the required values of the integrand f (x) and its derivatives at the interpolation nodes are known
19.3 Numerical Integration 899
as numerical values. Formulas using only the values of the function are called mean value formulas formulas using also the derivatives are called Hermite quadrature formulas.
19.3.2 Interpolation Quadratures
The following formulas represent so-called interpolation quadratures. Here, the integrand f (x) is interpolated at certain interpolation nodes (possibly the least number of them) by a polynomial p(x) of corresponding degree, and the integral of f (x) is replaced by that of p(x). The formula for the integral over the entire interval is given by summation. In the following, we give the formulas for the most practical cases. The interpolation nodes are equidistant: (19.72) x = x0 + h ( = 0 1 2 : : : n) x0 = a xn = b h = b ;n a : We give an upper bound for the magnitude of the error jRj for every quadrature formula. Here, M means an upper bound of jf ( ) (x)j on the entire domain.
19.3.2.1 Rectangular Formula
In the interval x0 x0 + h], f (x) is replaced by the constant function y = y0 = f (x0 ), which interpolates f (x) at the interpolation node x0, which is the left endpoint of the integration interval. We get in this way the simple rectangular formula xZ0 +h x0
f (x) dx h y0
jRj h2 M1 : 2
(19.73a)
We get the left-sided rectangular formula by summation:
Zb a
f (x) dx h(y0 + y1 + y2 + + yn;1) jRj (b ;2a)h M1:
(19.73b)
M1 denotes an upper bound of jf 0(x)j on the entire domain of integration. We get analogously the right-sided rectangular sum, if we replace y0 by y1 in (19.73a). The formula is: Zb f (x) dx h(y1 + y2 + + yn) jRj (b ;2a)h M1 : (19.74) a
19.3.2.2 Trapezoidal Formula
f (x) is replaced by a polynomial of rst degree in the interval x0 x0 + h], which interpolates f (x) at the interpolation nodes x0 and x1 = x0 + h. We get: xZ0 +h h3 M : f (x) dx h2 (y0 + y1) jRj 12 (19.75) 2 x0 We get the so-called trapezoidal formula by summation: Zb 2 jRj (b ;12a)h M2 : (19.76) f (x) dx h y20 + y1 + y2 + + yn;1 + y2n a M2 denotes an upper bound of jf 00(x)j, on the entire integration domain. The error of the trapezoidal formula is proportional to h2, i.e., the trapezoidal sum has an error of order 2. It follows that it converges
900 19. Numerical Analysis to the de nite integral for h ! 0 (hence, n ! 1), if we do not consider rounding errors.
19.3.2.3 Simpson's Formula
f (x) is replaced by a polynomial of second degree in the interval x0 x0 + 2h], which interpolates f (x) at the interpolation nodes x0 , x1 = x0 + h and x2 = x0 + 2h: x0Z+2h h5 M : f (x) dx h3 (y0 + 4y1 + y2) jRj 90 (19.77) 4 x0 n must be an even number for a complete Simpson formula. We get: Zb f (x) dx h3 (y0 + 4y1 + 2y2 + 4y3 + + 2yn;2 + 4yn;1 + yn) (19.78) a 4 jRj (b ;180a)h M4 : M4 is an upper bound for jf (4)(x)j on the entire integration domain. The Simpson formula has an error of order 4 and it is exact for polynomials up to third degree.
19.3.2.4 Hermite's Trapezoidal Formula
f (x) is replaced by a polynomial of third degree in the interval x0 x0 + h], which interpolates f (x) and f 0(x) at the interplation nodes x0 and x1 = x0 + h: xZ0 +h h2 (y0 ; y0 ) jRj h5 M : f (x) dx h2 (y0 + y1) + 12 (19.79) 0 1 720 4 x0 We get the Hermite trapezoidal formula by summation: h2 Zb 4 f (x) dx h y20 + y1 + y2 + + yn;1 + y2n + 12 (y00 ; yn0 ) jRj (b ; a)h M4 : (19.80) 720 a (4) M4 denotes an upper bound for jf (x)j on the entire integration domain. The Hermite trapezoidal formula has an error of order 4 and it is exact for polynomials up to third degree.
19.3.3 Quadrature Formulas of Gauss Quadrature formulas of Gauss have the general form
Zb a
f (x) dx
n X
=0
c y with y = f (x )
(19.81)
where not only the coecients c are considered as parameters but also the interpolation nodes x . These parameters are determined in order to make the formula (19.81) exact for polynomials of the highest possible degree. The quadrature formulas of Gauss result in very accurate approximations, but the interpolation nodes must be chosen in a very special way.
19.3.3.1 Gauss Quadrature Formulas
If the integration interval in (19.81) is chosen as a b] = ;1 1], and we choose the interpolation nodes as the roots of the Legendre polynomials (see 9.1.2.6, 3., p. 511, 21.12, p. 1066), then the coecients c can be determined so that the formula (19.81) gives the exact value for polynomials up to degree 2n + 1. The roots of the Legendre polynomials are symmetric with respect to the origin. For the cases n = 1 2 and 3 we get:
19.3 Numerical Integration 901
n = 1: x0 = ;x1 x1 = p1 = 0:577 350 269 : : : 3 n = 2: x0 = ;x2
c0 = 1 c1 = 1: c0 = 95 c1 = 89
x1 = 0 (19.82) s 3 x2 = 5 = 0:774 596 669 : : : c2 = c0: n = 3: x0 = ;x3 c0 = 0:347 854 854 : : : x1 = ;x2 c1 = 0:652 145 154 : : : x2 = 0:339 981 043 : : : c2 = c1 x3 = 0:861 136 311 : : : c3 = c0: Remark: A general integration interval a b] can be transformed into ;1 1] by the transformation t = b ;2 a x + a +2 b (t 2 a b] x 2 ;1 1]). Then ! Zb n X f (t) dt b ;2 a c f b ;2 a x + a +2 b (19.83) =0 a with the values x and c given above for the interval ;1 1].
19.3.3.2 Lobatto's Quadrature Formulas
In some cases it is reasonable also to choose the endpoints of the integration interval as interpolation nodes. Then, we have 2n more free parameters in (19.81). These values can be determined so that polynomials up to degree 2n ; 1 can be integrated exactly. We get for the cases n = 2 and n = 3: n = 2: n = 3: 1 x0 = ;1 c0 = 3 x0 = ;1 c0 = 16 x1 = 0 c1 = 43 (19.84a) x1 = ;x2 c1 = 56 (19.84b) x2 = 1 c2 = c0: x2 = p1 = 0:447 213 595 : : : c2 = c1 5 x3 = 1 c3 = c0 : The case n = 2 represents the Simpson formula.
19.3.4 Method of Romberg
To increase the accuracy of numerical integration the method of Romberg can be recommended, where we start with a sequence of trapezoid sums, which is obtained by repeated halving of the integration step size.
19.3.4.1 Algorithm of the Romberg Method The method consists of the following steps:
1. Trapezoid sums determination
We determine the trapezoid sum T (hi) according to (19.76) as approximations of the integral with the step sizes a hi = b ; 2i (i = 0 1 2 : : : m):
Zb a
f (x) dx (19.85)
902 19. Numerical Analysis Here, we consider the recursive relation ! " ! T (hi) = T hi2;1 = hi2;1 12 f (a) + f a + hi2;1 + f (a + hi;1) + f a + 23 hi;1
+f (a + 2hi;1) + + f a + 2n ; 1 hi;1 + 1 f (b) (19.86) 2 2 ! nX ;1 = 12 T (hi;1) + hi2;1 f a + hi2;1 + jhi;1 (i = 1 2 : : : m n = 2i;1): j =0 Recursion formula (19.86) tells that for the calculation of T (hi) from T (hi;1) we need to compute the function values only at the new interpolation nodes.
2. Triangular Scheme
We substitute T0i = T (hi) (i = 0 1 2 : : :) and we calculate recursively the values Tk;1i;1 (k = 1 2 : : : m i = k k + 1 : : :): Tki = Tk;1i + Tk;1i4; (19.87) k;1 The arrangement of the values calculated according to (19.87) is most practical in a triangular scheme, whose elements are calculated in a column-wise manner: T (h0 ) = T00 T (h1 ) = T01 T11 T (h2 ) = T02 T12 T22 (19.88) T (h3 ) = T03 T13 T23 T33 ......................... The scheme will be continued downwards (with a xed number of columns) until the lower values at the right are almost the same. The values T1i (i = 1 2 : : :) of the second column correspond to those calculated by the Simpson formula.
19.3.4.2 Extrapolation Principle
The Romberg method represents an application of the so-called extrapolation principle. This will be demonstrated by deriving the formula (19.87) for the case k = 1. We denote by I the required integral, by T (h) the corresponding trapezoid sum (19.76). If the integrand of I is (2m + 2) times continuously di erentiable in the integration interval, then it can be shown that an asymptotical expansion with respect to h is valid for the error R of the quadrature formula, and it has the form R(h) = I ; T (h) = a1h2 + a2 h4 + + am h2m + O(h2m+2) (19.89a) or T (h) = I ; a1 h2 ; a2 h4 ; ; am h2m + O(h2m+2 ): (19.89b) The coecients a1, a2 ,. . !. ,am are constants and independent of h. We form T (h) and T h according to (19.89b) and consider the linear combination 2 ! T1(h) = 1T (h) + 2 T h2 = (1 + 2 )I ; a1 1 + 42 h2 ; a2 1 + 162 h4 ; : (19.90) If we substitute 1 + 2 = 1 and 1 + 2 = 0, then T1(h) has an error of order 4, while T (h) and T (h=2) 4 both have errors of order only 2. We have ! ! ! T h ; T (h) : (19.91) T1(h) = ; 31 T (h) + 43 T h2 = T h2 + 2 3
19.3 Numerical Integration 903
This is the formula (19.87) for k = 1. Repeated application of the above procedure results in the approximation Tki according to (19.87) and Tki = I + O(h2i k+2): (19.92) Z 1 sin x dx (integral sine, see 8.2.5, 1., p. 460) cannot be obtained in an The de nite integral I = 0 x elementary way. Calculate the approximate values of this integral (calculating for 8 digits). k=0 k=1 k=2 k=3 0.92073549 1. Romberg method: 0.93979328 0.94614588 0.94451352 0.94608693 0.94608300 0.94569086 0.94608331 0.94608307 0:94608307: The Romberg method results in the approximation value 0:94608307. The value calculated for 10 digits is 0:9460830704. The order O (1=8)8 6 10;8 of the error according to (19.92) is proved. 2. Trapezoidal and Simpson Formulas: We can read directly from the scheme of the Romberg method that for h3 = 1=8 the trapezoid formula has the approximation value 0:94569086 and the Simpson formula gives the value 0:94608331. The correction of the trapezoidal formula by Hermite according to (19.79) results in the value I
0:94569086 + 0:30116868 64 12 = 0:94608301. 3. Gauss Formula:
By the formula (19.83) we get for = 0:94604113 n = 1: I 21 c0f 12 x0 + 12 + c1f 12 x1 + 12
n = 2: I 21 c0f 12 x0 + 12 + c1f 12 x1 + 12 + c2f 12 x2 + 12 = 0:94608313
n = 3: I 21 c0f 12 x0 + 12 + + c3f 12 x3 + 12 = 0:94608307: We see that the Gauss formula results in an 8-digit exact approximation value for n = 3, i.e., with only four function values. With the trapezoidal rule this accuracy would need a very large number (> 1000) of function values.
Remarks: 1. Fourier analysis has an important role in integrating periodic functions (see 7.4.1.1, 1., p. 420). The
details of numerical realizations can be found under the title of harmonic analysis (see 19.6.4, p. 927). The actual computations are based on the so-called Fast Fourier Transformation FFT (see 19.6.4.2, p. 928). 2. In many applications it is useful to take the special properties of the integrands under consideration. Further integration routines can be developed for such special cases. A large variety of convergence properties, error analysis, and optimal integration formulas is discussed in the literature (see, e.g., 19.4]). 3. Numerical methods to nd the values of multiple integrals are discussed in the literature (see, e.g., 19.26]).
904 19. Numerical Analysis
19.4 Approximate Integration of Ordinary Dierential Equations
In many cases, the solution of an ordinary di erential equation cannot be given in closed form as an expression of known elementary functions. The solution, which still exists under rather general circumstances (see 9.1.1.1, p. 488), must be determined by numerical methods. These result only in particular solutions, but it is possible to reach high accuracy. Since di erential equations of higher order than one can be either initial value problems or boundary value problems, numerical methods were developed for both types of problems.
19.4.1 Initial Value Problems
The principle of the methods presented in the following discussion to solve initial value problems y0 = f (x y) y(x0) = y0 (19.93) is to give approximate values yi for the unknown function y(x) at a chosen set of interpolation points xi . Usually, we consider equidistant interpolation nodes with a previously given step size h: xi = x0 + ih (i = 0 1 2 : : :): (19.94)
19.4.1.1 Euler Polygonal Method
We get an integral representation of the initial value problem (19.93) by integration
Zx y(x) = y0 + f (x y(x)) dx:
(19.95)
x0
This is the starting point for the approximation
y (x 1 ) = y 0 +
xZ0 +h x0
f (x y(x)) dx y0 + hf (x0 y0) = y1
(19.96)
which is generalized as the Euler broken line method or Euler polygonal method : yi+1 = yi + hf (xi yi) (i = 0 1 2 : : : y(x0) = y0): (19.97) For a geometric interpretation see Fig. 19.5. If we compare (19.96) with the Taylor expansion y
0
y(x1) = y(x0 + h)
y(x)
y0 x0
y1 x1 h
y2 x2 h
y3 x3
h
Figure 19.5
00 = y0 + f (x0 y0)h + y 2( ) h2 (19.98)
x
with x0 < < x0 + h, then we see that the approximation y1 has an error of order h2 . The accuracy can be improved by reducing the step size h. Practical calculations show that halving the step size h results in halving the error of the approximations yi. We can get a quick overview of the approximate shape of the solution curve by using the Euler method.
19.4.1.2 Runge{Kutta Methods 1. Calculation Scheme 0
The equation y (x) = f (x y) determines at every point (x0 y0) a direction, the direction of the tangent line of the solution curve passing through the point (x0 y0). The Euler method follows this direction until the next interpolation node. The Runge{Kutta methods consider more points \between" (x0 y0)
19.4 Approximate Integration of Ordinary Dierential Equations 905
and the possible next point (x0 + h y1) of the curve, and depending on the appropriate choice of these additional points we get more accurate values for y1. We have Runge{Kutta methods of di erent orders depending on the number and the arrangements of these "auxiliary" points. Here we show a fourthorder method (see 19.4.1.5, 1., p. 907). (The Euler method is a rst-order Runge{Kutta method.) The calculation scheme of fourth order for the step from x0 to x1 = x0 + h x y k = h f (x y ) to get an approximate value for y1 of x0 y0 k1 (19.93) is given in (19.99). The further x0 + h=2 y0 + k1=2 k2 steps follow the same scheme. (19.99) x0 + h=2 y0 + k2=2 k3 The error of this Runge{Kutta method x0 + h y0 + k3 k4 has order h5 (at every step) according to (19.99), so with an appropriate x1 = x0 + h y1 = y0 + 16 (k1 + 2k2 + 2k3 + k4) choice of the step size we can have high accuracy. y0 = 14 (x2 + y2) with y(0) = 0. We determine y(0:5) x y k = 81 (x2 + y2) in one step, i.e. h = 0:5 (see the table on the right). The 0 0 0 exact value for 8 digits is 0:01041860. 0.25 0 0.00781250 2. Remarks 0.25 0.00390625 0.00781441 0 1. For the special di erential equation y = f (x), this 0.5 0.00781441 0.03125763 Runge{Kutta method becomes the Simpson formula (see 0.5 0:01041858 19.3.2.3, p. 900). 2. For a large number of integration steps, a change of step size is possible or sometimes necessary. The change of step size can be decided by checking the accuracy so that we repeat the step with a double step size 2h. If we have, e.g., the approximate value y2(h) for y(x0 + 2h) (calculated by the single step size) and y2(2h) (calculated by the doubled step size), then we have the estimation for the error R2 (h) = y(x0 + 2h) ; y2(h): 1 y (h) ; y (2h)]: R2 (h) 15 (19.100) 2 2 Information about the implementation of the step size changes can be found in the literature (see 19.24]). 3. Runge{Kutta methods can easily be used also for higher-order di erential equations, see 19.24]. Higher-order di erential equations can be rewritten in a rst-order di erential equation system (see p. 497). Then, the approximation methods are performed as parallel calculations according to (19.99), as the di erential equations are connected to each other.
19.4.1.3 Multi-Step Methods
The Euler method (19.97) and the Runge{Kutta method (19.99) are so-called single-step methods, since we start only from yi in the calculation of yi+1. In general, linear multi-step methods have the form yi+k + k;1yi+k;1 + k;2yi+k;2 + + 1yi+1 + 0 yi = h(k fi+k + k;1fi+k;1 + + 1fi+1 + 0fi ) (19.101) with appropriately chosen constants j and j (j = 0 1 : : : k k = 1). The formula (19.101) is called a k-step method if j0j + j0j 6= 0. It is called explicit, if k = 0, since in this case the values fi+j = f (xi+j yi+j ) on the right-hand side of (19.101) only contain the already known approximation values yi yi+1 : : : yi+k;1. If k 6= 0 holds, the method is called implicit, since then the required new value yi+k occurs on both sides of (19.101). We have to know the k initial values y0 y1 : : : yk;1 in the application of a k-step method. We can get these initial values, e.g., by one-step methods. We can derive a special multi-step method to solve the initial value problem (19.93) if we replace the
906 19. Numerical Analysis derivative y0(xi ) in (19.93) by a dierence formula (see 9.1.1.5, 1., p. 496) or if we approximate the integral in (19.95) by a quadrature formula (see 19.3.1, p. 898). Examples of special multi-step methods are: 1. Midpoint Rule The derivative y0(xi+1 ) in (19.93) is replaced by the slope of the secant line between the interpolation nodes xi and xi+2 . We get: yi+2 ; yi = 2hfi+1 : (19.102) 2. Rule of Milne The integral in (19.95) is approximated by the Simpson formula: yi+2 ; yi = h3 (fi + 4fi+1 + fi+2 ): (19.103) 3. Rule of Adams and Bashforth The integrand in (19.95) is replaced by the interpolation polynomial of Lagrange (see 19.6.1.2, p. 918) based on the k interpolation nodes xi, xi+1 ,. . . , xi+k;1. We integrate between xi+k;1 and xi+k and get:
yi+k ; yi+k;1 =
kX ;1
2 xi+k 3 ;1 64 Z Lj (x) dx75 f (xi+j yi+j ) = h kX j f (xi+j yi+j ):
j =0 xi+k;1
j =0
(19.104)
The method (19.104) is explicit for yi+k . For the calculation of the coecients j see 19.2].
19.4.1.4 Predictor{Corrector Method
In practice, implicit multi-step methods have a great advantage compared to explicit ones in that they allow much larger step sizes with the same accuracy. But, an implicit multi-step method usually requires the solution of a non-linear equation to get the approximation value yi+k . This follows from (19.101) and has the form
yi+k = h
k X
j =0
j fi+j ;
kX ;1 j =0
j yi+j = F (yi+k ):
(19.105)
The solution of (19.105) is an iterative one. We proceed as follows: An initial value yi(0) +k is determined by an explicit formula, the so-called predictor. Then it will be corrected by an iteration rule ( ) yi(+ +1) (19.106) k = F (yi+k ) ( = 0 1 2 : : :) which is called the corrector coming from the implicit method. Special predictor{corrector formulas are: 1: yi(0)+1 = yi + 12h (5fi;2 ; 16fi;1 + 23fi) (19.107a) +1) = y + h (;f + 8f + 5f ( ) ) ( = 0 1 : : :) yi(+1 (19.107b) i i i+1 12 i;1 2: yi(0)+1 = yi;2 + 9yi;1 ; 9yi + 6h(fi;1 + fi) (19.108a) h ( +1) ( ) yi+1 = yi;1 + 3 (fi;1 + 4fi + fi+1) ( = 0 1 : : :): (19.108b) The Simpson formula as the corrector in (19.108b) is numerically unstable and it can be replaced, e.g., by +1) = 0:9y + 0:1y + h (0:1f + 6:7f + 30:7f + 8:1f ( ) ): yi(+1 (19.109) i;1 i i;2 i;1 i i+1 24
19.4 Approximate Integration of Ordinary Dierential Equations 907
19.4.1.5 Convergence, Consistency, Stability 1. Global Discretization Error and Convergence
Single-step methods can be written generally in the form: yi+1 = yi + hF (xi yi h) (i = 0 1 2 : : : y0 given): (19.110) Here F (x y h) is called the increment function or progressive direction of the single-step method. The approximating solution obtained by (19.110) depends on the step size h and it should be denoted by y(x h). Its di erence from the exact solution y(x) of the initial value problem (19.93) is called the global discretization error g (x h) (see (19.111)), and we say: The single-step method (19.110) is convergent with order p if p is the largest natural number such that (19.111) g(x h) = y(x h) ; y(x) = O(hp) holds. Formula (19.111) says that the approximation y(x h) determined with the step size h = x ; x0 n converges to the exact solution y(x) for every x from the domain of the initial value problem if h ! 0. The Euler method (19.97) has order of convergence p = 1. For the Runge{Kutta method (19.99) p = 4 holds.
2. Local Discretization Error and Consistency
The order of convergence according to (19.111) shows how well the approximating solution y(x h) approximates the exact solution y(x). Beside this, it is an interesting question of how well the increment function F (x y h) approximates the derivative y0 = f (x y). For this purpose we introduce the so-called local discretization error l(x h) (see (19.112)) and we say: The single-step method (19.110) is consistent with order p, if p is the largest natural number with l(x h) = y(x + hh) ; y(x) ; F (x y h) = O(hp): (19.112) It follows directly from (19.112) that for a consistent single-step method lim F (x y h) = f (x y): (19.113) h!0 The Euler method has order of consistency p = 1, the Runge {Kutta method has order of consistency p = 4.
3. Stability with Respect to Perturbation of the Initial Values
In the practical performance of a single-step method, a rounding error O(1=h) adds to the global discretization error O(hp). Consequently, we have to select a not too small, nite step size h > 0. It is also an important question of how the numerical solution yi behaves under perturbations of the initial values or in the case xi ! 1. In the theory of ordinary di erential equations, an initial value problem (19.93) is called stable with respect to perturbations of its initial values if: jy~(x) ; y(x)j jy~0 ; y0j: (19.114) Here y~(x) is the solution of (19.93) with the perturbed initial value y~(x0 ) = y~0 instead of y0. Estimation (19.114) tells that the absolute value of the di erence of the solutions is not larger than the perturbation of the initial values. In general, it is hard to check (19.114). Therefore we consider the linear test problem y0 = y with y(x0) = y0 ( constant 0) (19.115) which is stable, and a single-step method is applied to this special initial value problem. A consistent method is called absolutely stable with step size h > 0 with respect to perturbed initial values if the approximating solution yi of the above linear test problem (19.115) obtained by using the method satis es the condition jyij jy0j: (19.116)
908 19. Numerical Analysis Applying the Euler polygon method for equation (19.115) results in the solution yi+1 = (1 + h)yi (i = 0 1 : : :). Obviously, (19.116) holds if j1 + hj 1, and so the step size must satisfy ;2 h 0.
4. Sti Di erential Equations
Many application problems, including those in chemical kinetics, can be modeled by di erential equations whose solutions consist of terms converging to zero exponentially but in a high di erent kind of exponential decreasing. These equations are called sti dierential equations. For example: y(x) = C1 e 1 x + C2e 2 x (C1 C2 1 2 const) (19.117) with 1 < 0, 2 < 0 and j1j * j2j, e.g., 1 = ;1, 2 = ;1000. The term with 2 does not have a signi cant a ect on the solution function, but it does in selecting the step size h for a numerical method. In such cases the choice of the most appropriate numerical method has special importance (see 19.23]).
19.4.2 Boundary Value Problems
The most important methods for solving boundary value problems of ordinary di erential equations will be demonstrated on the following simple linear boundary value problem for a di erential equation of the second order: y00(x) + p(x)y0(x) + q(x)y(x) = f (x) (a x b) with y(a) = y(b) = : (19.118) The functions p(x), q(x) and f (x) and also the constants and are given. The given method can also be adapted for boundary value problems of higher-order di erential equations.
19.4.2.1 Di erence Method
We divide the interval a b] by equidistant interpolation points x = x0 + h ( = 0 1 2 : : : n x0 = a, xn = b) and we substitute the values of the derivatives in the di erential equation at the interior interpolation points y00(x ) + p(x )y0(x ) + q(x )y(x ) = f (x ) ( = 1 2 : : : n ; 1) (19.119) by so-called nite divided dierences, e.g.: y0(x ) y0 = y+1 2;h y;1 (19.120a) y00(x ) y00 = y+1 ; 2hy2 + y;1 : (19.120b) This way, we get n ; 1 linear equations for the n ; 1 approximation values y y(x ) in the interior of the integration interval a b], considering the conditions y0 = and yn = . If the boundary conditions also contain derivatives, they must also be replaced by nite expressions. Eigenvalue problems of di erential equations (see 9.1.3.2, p. 515) are handled analogously. The application of the dierence method, described by (19.119) and (19.120a,b), leads to a matrix eigenvalue problem (see 4.5, p. 278). The solution of the homogeneous di erential equation y00 + 2y = 0 with boundary conditions y(0) = y(1) = 0 leads to a matrix eigenvalue problem. The di erence method transforms the differential equation into the di erence equation y+1 ; 2y + y;1 + h2 2y = 0. If we choose three interior points, hence h = 1=4, then considering y0 = y(0) = 0, y4 = y(1) = 0 we get the discretized system ! 2 y + ;2 + 16 y2 =0 1 ! 2 y1 + ;2 + 16 y2 + y3 = 0 ! 2 y = 0: y2 + ;2 + 16 3
19.4 Approximate Integration of Ordinary Dierential Equations 909
This homogeneous equation system has a non-trivial solution only when the coecient determinant is zero. This condition results in the eigenvalues 12 = 9:37, 22 = 32 and 32 = 54:63. Among them only the smallest one is close to its corresponding true value 9:87. Remark: The accuracy of the di erence method can be improved by 1. decreasing the step size h, 2. application of a2 derivative approximation of higher order (approximations as (19.120a,b) have an error of order O(h )), 3. application of multi-step methods (see 19.4.1.3, p. 905). If we have a non-linear boundary value problem, the di erence method leads to a system of non-linear equations of the unknown approximation values y (see 19.2.2, p. 896).
19.4.2.2 Approximation by Using Given Functions
For the approximate solution of the boundary value problem (19.118) we apply a linear combination of suitably chosen functions gi(x), which are linearly independent and each one satis es the boundary value conditions: n X y(x) g(x) = ai gi(x): (19.121) i=1
If we substitute g(x) into the di erential equation (19.118), then we get an error, the so-called defect "(x a1 a2 : : : an) = g00(x) + p(x)g0(x) + q(x)g(x) ; f (x): (19.122) To determine the coecients ai , we can use the following principles (see also p. 913): 1. Collocation Method The defect is to be zero at n given points x , the so-called collocation points. The conditions "(x a1 a2 : : : an) = 0 ( = 1 2 : : : n) a < x1 < x2 < : : : < xn < b (19.123) result in a linear equation system for the unknown coecients. 2. Least Squares Method We require that the integral
Zb F (a1 a2 : : : an) = "2(x a1 a2 : : : an) dx a
(19.124)
depending on the coecients, should be minimal. The necessary conditions @F (19.125) @ai = 0 (i = 1 2 : : : n) give a linear equation system for the coecients ai. 3. Galerkin Method We require that the so-called error orthogonality is satis ed, i.e.,
Zb a
"(x a1 a2 : : : an)gi(x) dx = 0 (i = 1 2 : : : n)
(19.126)
and we get in this way a linear equation system for the unknown coecients. 4. Ritz Method The solution y(x) often has the property that it minimizes the variational integral,
Zb I y] = H (x y y0) dx a
(19.127)
(see (10.4), p. 552). If we know the function H (x y y0), then we replace y(x) by the approximation g(x) as in (19.121) and we minimize I y] = I (a1 a2 : : : an). The necessary conditions @I (19.128) @ai = 0 (i = 1 2 : : : n) result in n equation for the coecients ai.
910 19. Numerical Analysis Under certain conditions on the functions p, q, f and y, the boundary value problem ; p(x)y0(x)]0 + q(x)y(x) = f (x) with y(a) = y(b) = (19.129) and the variational problem Zb I y] = p(x)y02(x) + q(x)y2(x) ; 2f (x)y(x)] dx = min! with y(a) = y(b) = (19.130) a are equivalent, so we can get H (x y y0) immediately from (19.130) for the boundary value problem of the form (19.129). Instead of the approximation (19.121), we often consider
g(x) = g0(x) +
n X i=1
aigi(x)
(19.131)
where g0(x) satis es the boundary values and the functions gi(x) satisfy the conditions gi(a) = gi(b) = 0 (i = 1 2 : : : n): (19.132) For the problem (19.118), we can choose, e.g., ; (x ; a): g0(x) = + b ; (19.133) a Remark: In a linear boundary value problem, the forms (19.121) and (19.131) result in a linear equation systems for the coecients. In the case of non-linear boundary value problems we get non-linear equation systems, which can be solved by the methods given in Section 19.2.2, p. 896.
19.4.2.3 Shooting Method
With the shooting method, we reduce the solution of a boundary value problem to the solution of an initial value problem. The basic idea of the method is described below as the single-target method.
1. Single-Target Method
The initial value problem y00 + p(x)y0 + q(x)y = f (x) with y(a) = y0(a) = s (19.134) is associated to the boundary value problem (19.118). Here s is a parameter, which the solution y of the initial-value problem (19.134) depends on, i.e., y = y(x s) holds. The function y(x s) satis es the rst boundary condition y(a s) = according to (19.134). We have to determine the parameter s so that y(x s) satis es the second boundary condition y(b s) = . Therefore, we have to solve the equation F (s) = y(b s) ; (19.135) and the regula falsi (or secant) method is an appropriate method to do this. It needs only the values of the function F (s), but the computation of every function value requires the solution of an initial value problem (19.134) until x = b for the special parameter value s with one of the methods given in 19.4.1.
2. Multiple-Target Method
In a so-called multiple-target method, the integration interval a b] is divided into subintervals, and we use the single-target method on every subinterval. Then, the required solution is composed from the solutions of the subintervals, where the continuous transition at the endpoints of the subintervals must be ensured. This requirement results in further conditions. For the numerical implementation of the multiple-target method, which is used mostly for non-linear boundary value problems, see. 19.24].
19.5 Approximate Integration of Partial Dierential Equations 911
19.5 Approximate Integration of Partial Dierential Equations
In the following, we discuss only the principle of numerical solutions of partial di erential equations using the example of linear second-order partial di erential equations with two independent variables with the corresponding boundary or/and initial conditions.
19.5.1 Dierence Method
We consider a regular grid on the integration domain by the chosen points (x y ). Usually, this grid is chosen to be rectangular and equally spaced: x = x0 + h y = y0 + l ( = 1 2 : : :): (19.136) We get squares for l = h. If we denote the required solution by u(x y), then we replace the partial derivatives occurring in the di erential equation and in the boundary or initial conditions by nite divided dierences in the following way, where u denotes an approximate value for the function value u(x y ):
Partial Derivative
Finite Divided Di erence 1 (u 1 h +1 ; u ) or 2h (u +1 ; u ;1 )
Order of Error 9>>
> > @u (x y ) 2) > O ( h ) or O ( h > > @x > > @u (x y ) 1 (u 1 2 > O ( l ) or O ( l ) +1 ; u ) or (u +1 ; u ;1 ) > @y l 2l = 2 (19.137) @ u (x y ) 1 (u > +1 +1 ; u +1 ;1 ; u ;1 +1 + u ;1 ;1 ) O(hl) > @x@y 4hl > > > @ 2 u (x y ) 1 (u 2) > O ( h +1 ; 2u + u ;1 ) > 2 2 @x h > > 2 > @ u (x y ) 1 (u 2 > " ; 2 u + u ) O ( l ) +1 ;1 2 2 @y l The error order in (19.137) is given by using the Landau symbol O. In some cases, it is more practical to apply the approximation @ 2 u (x y ) u +1+1 ; 2u +1 + u ;1+1 + (1 ; ) u +1 ; 2u + u ;1 (19.138) @x2 h2 h2 with a xed parameter (0 1). Formula (19.138) represents a convex linear combination of two nite expressions obtained from the corresponding formula (19.137) for the values y = y and y = y+1. A partial di erential equation can be rewritten as a dierence equation at every interior point of the grid by the formulas (19.137), where the boundary and initial conditions are considered, as well. This equation system for the approximation values u has a large dimension for small step sizes h and l, so usually, we solve it by an iteration method (see 19.2.1.4, p. 895). A: The function u(x y) should be the solution of the di erential y u0, 2=0 equation %u = uxx+uyy = ;1 for the points (x y) with jxj < 1, jyj < 2 , 2 i.e., in the interior of a rectangle, and it should satisfy the boundary u−1, 1=0 u0, 1 u1, 1=0 conditions u = 0 for jxj = 1 and jyj = 2. The di erence equation correu−1, 0=0 u0, 0 u1, 0=0 sponding to the di erential equation for a square grid with step size h −1 0 1 x is: 4u = u +1 + u +1 + u ;1 + u ;1 + h2. The step size h = 1 (Fig. 19.6) results in a rst rough approximation for the function valu0,−1 ues at the three interior points: 4u01 = 0 + 0 + 0 + u00 + 1 4u00 = −2 0 + u01 + 0 + u0;1 + 1 4u0;1 = 0 + u00 + 0 + 0 + 1. 5 0:357. We get: u00 = 37 0:429, u01 = u0;1 = 14 Figure 19.6
912 19. Numerical Analysis B: The equation system arising in the application of the di erence method for partial di erential equations has a very special structure. We demonstrate it by the following example which is a more general boundary value problem. The integration domain is the square G: 0 x 1, 0 y 1. We are looking for a function u(x y) with %u = uxx + uyy = f (x y) in the interior of G, u(x y) = g(x y) on the boundary of G. The functions f and g are given. The di erence equation associated to this di erential equation is, for h = l = 1=n: u +1 + u +1 + u ;1 + u ;1 ; 4u = n12 f (x y ) ( = 1 2 : : : n ; 1). In the case of n = 5, the left-hand side of this di erence equation system for the approximation values u in the 4 4 interior points has the form (19.139): 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @
1
;4 1 0 0 1 ;4 1 0 0 1 ;4 1 0 0 1 ;4
1 0 0 0 C 0u 1 C 0 1 0 0 11 C O C B 0 0 1 0 u21 C C B C C B 0 0 0 1 u31 C C B C C B C B C u41 C C B C 1 0 0 0 ;4 1 0 0 1 0 0 0 C B u 12 C C B C 0 1 0 0 1 ;4 1 0 0 1 0 0 C B u22 C C B C 0 0 1 0 0 1 ;4 1 0 0 1 0 C B u32 C C B C C B 0 0 0 1 0 0 1 ;4 0 0 0 1 u42 C C B C C B (19.139) B C u13 C 1 0 0 0 ;4 1 0 0 1 0 0 0 C C B C C B C u23 C B 0 1 0 0 1 ;4 1 0 0 1 0 0 C B u33 C C B C 0 0 1 0 0 1 ;4 1 0 0 1 0 C C B C B C 0 0 0 1 0 0 1 ;4 0 0 0 1 C B u43 C u14 C C B C C B u24 C B C 1 0 0 0 ;4 1 0 0 C C B C C @ u 0 1 0 0 1 ; 4 1 0 34 A C O 0 0 1 0 0 1 ;4 1 C A u44 0 0 0 1 0 0 1 ;4 if we consider the grid row-wise from left to right, and considering that the values of the function are given on the boundary. We see that the coecient matrix is symmetric and is a sparse matrix. This form is called block-tridiagonal. We can also see that the form of the matrix depends on how we select the grid-points. For di erent classes of partial di erential equations of second order, such as elliptic, parabolic and hyperbolic di erential equations, more e ective methods have been developed, and also the convergence and stability conditions have been investigated. There is a huge number of books about this topic (see, i.e., 19.22], 19.24]).
19.5.2 Approximation by Given Functions We approximate the solution u(x y) by a function in the form
u(x y) v(x y) = v0 (x y) +
n X i=1
aivi (x y):
(19.140)
Here, we distinguish between two cases: 1. v0 (x y) satis es the given inhomogeneous di erential equation, and the further functions vi(x y) (i = 1 2 : : : n) satisfy the corresponding homogeneous di erential equation (then we are looking for the linear combination approximating the given boundary conditions as well as possible). 2. v0(x y) satis es the inhomogeneous boundary conditions and the other functions vi(x y) (i = 1 2 : : : n) satisfy the homogeneous boundary conditions (then we are looking for the linear combination approximating the solution of the di erential equation on the considered domain as well as
19.5 Approximate Integration of Partial Dierential Equations 913
possible). If we substitute the approximating function v(x y) from (19.140) in the rst case into the boundary conditions, in the second case into the di erential equation, then in both cases we get an error term, the so-called defect " = "(x y a1 a2 : : : an): (19.141) To determine the unknown coecients ai, we can apply one of the following methods:
1. Collocation Method
The defect " should be zero in n reasonably distributed points, at the collocation points (x y ) ( = 1 2 : : : n): "(x y a1 a2 : : : an) = 0 ( = 1 2 : : : n): (19.142) The collocation points in the rst case are boundary points (we talk about boundary collocation), in the second case they are interior points of the integration domain (we talk about domain collocation). From (19.142) we get n equations for the coecients. Boundary collocation is usually preferred to domain collocation. We apply this method to the example solved in 19.5.1 by the di erence method, with the functions satisfying the di erential equation: v(x y a1 a2 a3) = ; 14 (x2 + y2)+ a1 + a2 (x2 ; y2)+ a3(x4 ; 6x2 y2 + y4). The coecients are determined to satisfy the boundary conditions at the points (x1 y1) = (1 0:5), (x2 y2) = (1 1:5) and (x3 y3) = (0:5 2) (boundary collocation). We get the linear equation system ;0:3125 + a1 + 0:75a2 ; 0:4375a3 = 0 ;0:8125 + a1 ; 1:25a2 ; 7:4375a3 = 0 ;1:0625 + a1 ; 3:75a2 + 10:0625a3 = 0 with the solution a1 = 0:4562, a2 = ;0:2000, a3 = ;0:0143. We can calculate the approximate values of the solution at arbitrary points with the approximating function. To compare the values with those obtained by the di erence method: v(0 1) = 0:3919 and v(0 0) = 0:4562.
2. Least Squares Method
Depending on whether the approximation function (19.140) satis es the di erential equation or the boundary conditions, we require 1. either the line integral over the boundary C Z I = "2(x(t) y(t) a1 : : : an) dt = min (19.143a) (C )
where the boundary curve C is given by a parametric representation x = x(t), y = y(t), 2. or the double integral over the domain G ZZ I = "2(x y a1 : : : an) dx dy = min: (G)
(19.143b)
From the necessary conditions, @I = 0 (i = 1 2 : : : n), we get n equations for computing of the @ai parameters a1, a2 ,. . . , an.
19.5.3 Finite Element Method (FEM)
After the appearance of modern computers the nite element methods became the most important technique for solving partial di erential equations. These powerful methods give results which are easy to interpret. Depending on the types of various applications, the FEM is implemented in very di erent ways, so here we give only the basic idea. It is similar to those used in the Ritz method (see 19.4.2.2, p. 909)
914 19. Numerical Analysis for numerical solution of boundary value problems for ordinary di erential equations and is related to spline approximations (see 19.7, p. 931). The nite element method has the following steps: 1. De ning a Variational Problem We formulate a variational problem to the given boundary value problem. The process is demonstrated on the following boundary value problem: %u = uxx + uyy = f in the interior of G, u = 0 on the boundary of G: (19.144) We multiply the di erential equation in (19.144) by an appropriate smooth function v(x y) vanishing on the boundary of G, and we integrate over the entire G to get ZZ @ 2 u @ 2 u ! ZZ fv dx dy: (19.145) 2 + @y 2 v dx dy = @x (G) (G) Applying the Gauss integral formula (see 13.3.3.1, 1., p. 665), where we substitute P (x y) = ;vuy and Q(x y) = vux in (13.119), we get the variational equation from (19.145) a(u v) = b(v) (19.146a) with ZZ @u @v @u @v ! ZZ a(u v) = ; + dx dy b ( v ) = fv dx dy: (19.146b) @x @x @y @y (G) (G) y
y 1
G
yn
5 6 1
0
x
0
4
Gmn
3
2
xm
1
x
Figure 19.7 Figure 19.8 2. Triangularization The domain of integration G is decomposed into simple subdomains. Usually, we use a triangularization, where we cover G by triangles so that the neighboring triangles have a complete side or only a single vertex in common. Every domain bounded by curves can be approximated quite well by a union of triangles (Fig. 19.7). Remark: To avoid numerical diculties, the triangularization should not contain obtuse-angled triangles. A triangularization of the unit square could be performed as shown in Fig. 19.8. Here we start from the grid points with coordinates x = h y = h ( = 0 1 2 : : : N h = 1=N ). We get (N ; 1)2 interior points. Considering the choice of the solution functions, it is always useful to consider the surface elements G composed of the six triangles having the common point (x y ). (In other cases, the number of triangles may di er from six. These surface elements are obviously not mutually exclusive.) 3. Solution We de ne a supposed approximating solution for the required function u(x y) in every triangle. A triangle with the corresponding supposed solution is called a nite element. Polynomials in x and y are the most suitable choices. In many cases, the linear approximation u~(x y) = a1 + a2x + a3y (19.147)
19.5 Approximate Integration of Partial Dierential Equations 915
is sucient. The supposed approximating function must be continuous under the transition from one triangle to neighboring ones, so we get a continuous nal solution. The coecients a1 , a2 and a3 in (19.147) are uniquely de ned by the values of the functions u1, u2 and u3 at the three vertices of the triangle. The continuous transition to the neighboring triangles is ensured by this at the same time. The supposed solution (19.147) contains the approximating values ui of the required function as unknown parameters. For the supposed solution, which is applied as an approximation in the entire domain G for the required solution u(x y), we choose
u~(x y) =
NX ;1 N ;1 X =1 =1
u (x y):
(19.148)
We have to determine the appropriate coecients . The following must be valid for the functions u (x y): They represent a linear function over every triangle of G according to (19.147) with the following conditions: k= l=
1. u (xk yl) = 10 for (19.149a) at any other grid point of G :
2. u (x y) 0 for (x y) 62 G :
1 yn Gmn xm
Figure 19.9 Analogously, we have: 8 x y > >1; h ; + h ;
> > > > 1; x ; > > < h u (x y) = > 1 ; hy ;
> > > 1 + xh ; + hy ;
> > > > : 1 + xh ;
(19.149b) The representation of u (x y) over G is shown in Fig. 19.9. The calculation of u over G , i.e., over all triangles 1 to 6 in Fig. 19.8 is shown here only for triangle 1: u (x y) = a1 + a2 x + a3 with (19.150) 8 1 for x = x y = y < u (x y) = : 0 for x = x ;1 y = y;1 (19.151) 0 for x = x y = y;1: From (19.151) we have a1 = 1 ; a2 = 0 a3 = 1=h, and we get for triangle 1: (19.152) u (x y) = 1 + hy ; : for triangle 2 for triangle 3 for triangle 4
(19.153)
for triangle 5 for triangle 6:
4. Calculation of the Solution Coecients We determine the solution coecients by the
requirements that the solution (19.148) satis es the variational problem (19.146a) for every solution function u , i.e., we substitute u~(x y) for u(x y) and u (x y) for v(x y) in (19.146a). This way, we get a linear equation system NX ;1 N ;1 X =1 =1
a(u ukl) = b(ukl ) (k l = 1 2 : : : N ; 1)
(19.154)
916 19. Numerical Analysis for the unknown coecients, where ZZ @u @ukl @u @ukl ! a(u ukl) = @x @x + @y @y dx dy G
b(ukl) =
kl
ZZ Gkl
fukl dx dy:
(19.155)
In the calculation of a(u ukl) we must pay attention to the fact that we have to integrate only in the cases of domains G and Gkl with non-empty intersection. These domains are denoted by shadowing in Table 19.1. Table 19.1 Auxiliary table for FEM
Surface region =k 1: = l
Triangle of @ukl @u P @ukl @u Graphical representation Gkl G @x @x @x @x 1 1 0 0 2 2 ;1=h ;1=h 4 4 3 3 ;1=h ;1=h 5 3 6
=k 2: = l;1
1
1
=k+1 3: = l
4 4 5 5 6 6
0 1=h 1=h
0 1=h 1=h
h2
1 5 2 4
;1=h
0
1=h 0
0
3
2 6 3 5
;1=h ;1=h
1=h 1=h
; h22
3
3 1 4 6
;1=h
0
0 1=h
0
4 2 5 1
;1=h
0
1=h 0
0
; h22
2
2
2
=k+1 4: = l+1
4
=k 5: = l+1
5
4
=k;1 6: = l
6
5 3 6 2
1=h 1=h
;1=h ;1=h
=k;1 7: = l;1
6
6 4 1 3
1=h 0
;1=h
5
1
0
0
The integration is always performed over a triangle with an area h2 =2, so for the partial derivatives with respect to x we get: 1 h2 (19.156a) h2 (4kl ; 2k+1l ; 2k;1l) 2 :
19.6 Approximation, Computation of Adjustment, Harmonic Analysis 917
Analogously, for the partial derivatives with respect to y we have: 1 h2 h2 (4kl ; 2kl+1 ; 2kl;1) 2 : The calculation of the right-hand side b(ukl ) of (19.154) gives: ZZ b(ukl) = f (x y)ukl(x y) dx dy fkl VP Gkl
(19.156b) (19.157a)
where VP is the volume of the pyramid over Gkl with height 1, determined by ukl(x y) (Fig. 19.9). Since VP = 31 6 12 h2 we have b(ukl) fkl h2: (19.157b) So, the variational equations (19.154) result in the linear equation system 4kl ; k+1l ; k;1l ; kl+1 ; kl;1 = h2fkl (k l = 1 2 : : : N ; 1) (19.158) for the determination of the solution coecients.
Remarks: 1. If the solution coecients are determined by (19.158), then u~(x y) from (19.148) represents an
explicit approximating solution, whose values can be calculated for an arbitrary point (x y) from G. 2. If the integration domain must be covered by an irregular triangular grid then it is useful to introduce triangular coordinates (also called barycentric coordinates). In this way, the position of a point can be easily determined with respect to the triangular grid, and the calculation of the multidimensional integral is made easier as in (19.155), because every triangle can be easily transformed into the unit triangle with vertices (0 0), (0 1), (1 0). 3. If accuracy must be improved or also the di erentiability of the solution is required, we have to apply piecewise quadratic or cubic functions to obtain the supposed approximation (see, e.g., 19.22]). 4. In practical applications, we usually obtain equation systems of huge dimensions. This is the reason why so many special methods have been developed, e.g., for automatic triangularization and for practical enumeration of the elements (the structure of the equation system depends on it). For detailed discussion of FEM see 19.13], 19.7], 19.22].
19.6 Approximation, Computation of Adjustment, Harmonic Analysis 19.6.1 Polynomial Interpolation
The basic problem of interpolation is to t a curve through a sequence of points (x y ) ( = 0 1 : : : n). This can happen graphically by any curve- tting gadget, or numerically by a function g(x), which takes given values y at the points x , at the so-called interpolation points. That is g(x) satis es the interpolation conditions g(x ) = y ( = 0 1 2 : : : n): (19.159) In the rst place, we use polynomials as interpolation functions, or for periodic functions so-called trigonometric polynomials. In this last case we talk about trigonometric interpolation (see 19.6.4.1, 2., p. 927). If we have n + 1 interpolation points, the order of the interpolation is n, and the highest degree of the interpolation polynomial is at most n. Since with increasing degree of the polynomials, strong oscillation may occur, which is usually not required, we decompose the interpolation interval into subintervals and we perform a spline interpolation (see 19.7, p. 931).
19.6.1.1 Newton's Interpolation Formula
To solve the interpolation problem (19.159) we consider a polynomial of degree n in the following form: g(x) = pn(x) = a0 + a1 (x ; x0 ) + a2(x ; x0 )(x ; x1 ) +
918 19. Numerical Analysis +an(x ; x0 )(x ; x1 ) : : : (x ; xn;1 ): (19.160) This is called the Newton interpolation formula, and it gives an easy calculation of the coecients ai (i = 0 1 : : : n), since the interpolation conditions (19.159) result in a linear equation system with a triangular matrix. For n = 2 we get the annexed equation a0 = y0 system from (19.159). The interpolation pp22 ((xx01 )) = = a + a ( x ; x ) y1 0 1 1 0 polynomial pn(x) is uniquely determined p2 (x2 ) = a0 + a1(x2 ; x0 ) + a2(x2 ; x0 )(x2 ; x1 ) = = y2 by the interpolation conditions (19.159). The calculation of the function values can be simpli ed by the Horner schema (see 19.1.2.1, p. 887).
19.6.1.2 Lagrange's Interpolation Formula
We can t a polynomial of n-th degree through n+1 points (x y ) ( = 0 1 : : : n), with the Lagrange formula: n X g(x) = pn(x) = y L (x): (19.161) =0
Here L (x) ( = 0 1 : : : n) are the Lagrange interpolation polynomials. Equation (19.161) satis es the interpolation conditions (19.159), since =
L (x ) = = 10 for (19.162) for 6= : Here is the Kronecker symbol. The Lagrange interpolation polynomials are de ned by the formula n Y L = (x (x;;x x)(0 )(x x;;xx1))((xx ;;xx ;1 )()(xx;;x x+1 ) )(x(x; x;n)x ) = xx ;;xx : (19.163) 0 1 ;1 +1 n =0 6=
We t a polynomial through the points given by the table yx 01 13 32 . We use the Lagrange interpolation formula (19.161) and we get: ; 1)(x ; 3) = 1 (x ; 1)(x ; 3) L0 (x) = ((0x ; 1)(0 ; 3) 3 0)(x ; 3) 1 L1 (x) = ((1x ; ; 0)(1 ; 3) = ; 2 x(x ; 3) 0)(x ; 1) 1 L2 (x) = ((3x ; ; 0)(3 ; 1) = 6 x(x ; 1) p2(x) = 1 L0 (x) + 3 L1 (x) + 2 L2 (x) = ; 56 x2 + 17 x + 1: 6 The Lagrange interpolation formula depends explicitly and linearly on the given values y of the function. This is its theoretical importance (see, e.g., the rule of Adams{Bashforth, 19.4.1.3, 3., p. 906). For practical calculation the Lagrange interpolation formula is rarely reasonable.
19.6.1.3 Aitken{Neville Interpolation
In several practical cases, we do not need to know the explicit form of the polynomial pn(x), but only its value at a given location x of the interpolation domain. We can get this function value in a recursive way due to Aitken and Neville. We apply the useful notation pn(x) = p01:::n(x) (19.164) in which the interpolation points x0 x1 : : : xn and the degree n of the polynomial are denoted. Notice that (19.165) p01:::n(x) = (x ; x0)p12:::n(x)x;;(xx; xn)p012:::n;1(x) n
0
19.6 Approximation, Computation of Adjustment, Harmonic Analysis 919
i.e., the function value p01:::n(x) can be obtained by linear interpolation of the function values of p12:::n(x) and p012:::n;1(x), two interpolation polynomials of degree n ; 1. Application of (19.165) leads to a scheme which is given here for the case of n = 4: x0 y0 = p0 x1 y1 = p1 p01 x2 y2 = p2 p12 p012 (19.166) x3 y3 = p3 p23 p123 p0123 x4 y4 = p4 p34 p234 p1234 p01234 = p4(x): The elements of (19.166) are calculated column-wise. A new value in the scheme is obtained from its west and north-west neighbors ; (x ; x3)p2 = p + x ; x3 (p ; p ) p23 = (x ; x2 )xp3 ; (19.167a) 3 x3 ; x2 3 2 3 x2 ; (x ; x3 )p12 = p + x ; x3 (p ; p ) p123 = (x ; x1 )px23 ; (19.167b) 23 x3 ; x1 23 12 3 x1 ; (x ; x4 )p123 = p + x ; x4 (p ; p ): p1234 = (x ; x1 )p234 (19.167c) 234 x4 ; x1 x4 ; x1 234 123 For performing the Aitken{Neville algorithm on a computer we need to introduce only a vector p with n + 1 components (see 19.4]), which takes the values of the columns in (19.166) after each other according to the rule that the value pi;ki;k+1:::i (i = k k + 1 : : : n) of the k-th column will be the i-th component pi of p. The columns of (19.166) must be calculated from the top down, so we will have all necessary values. The algorithm has the following two steps: 1: For i = 0 1 : : : n set pi = yi: (19.168a) x ; x i 2: For k = 1 2 : : : n and for i = n n ; 1 : : : k compute pi = pi + x ; x (pi ; pi;1): (19.168b) i i;k After nishing (19.168b) we have the required function value pn(x) at x in element pn.
19.6.2 Approximation in Mean
The principle of approximation in mean is known as the Gauss least squares method. In calculations we distinguish between continuous and discrete cases.
19.6.2.1 Continuous Problems, Normal Equations
The function f (x) is approximated by a function g(x) on the interval a b] so that the expression
Zb F = !(x) f (x) ; g(x)]2 dx a
(19.169)
depending on the parameters contained by g(x), should be minimal. !(x) denotes a given weight function, such that !(x) > 0 in the integration interval. If we are looking for the best approximation g(x) in the form
g(x) =
n X i=0
ai gi(x)
(19.170)
with suitable linearly independent functions g0(x), g1(x) : : : gn(x), then the necessary conditions @F (19.171) @ai = 0 (i = 0 1 : : : n)
920 19. Numerical Analysis for an extreme value of (19.169) result in the so-called normal equation system n X i=0
ai (gi gk ) = (f gk ) (k = 0 1 : : : n)
(19.172)
to determine the unknown coecients ai. Here we used the brief notation
Zb
(gi gk ) = !(x)gi(x)gk (x) dx
(19.173a)
a
Zb
(f gk ) = !(x)f (x)gk (x) dx (i k = 0 1 : : : n)
(19.173b)
a
which are considered as the scalar products of the two indicated functions. The system of normal equations can be solved uniquely, since the functions g0 (x), g1(x) : : : , gn(x) are linearly independent. The coecient matrix of the system (19.172) is symmetric, so we could apply the Cholesky method (see 19.2.1.2, p. 893). The coecients ai can be determined directly, without solving the equation system, if the system of functions gi(x) is orthogonal, that is, if (gi gk ) = 0 for i 6= k: We call it an orthonormal system, if
(19.174)
i 6= k, (gi gk ) = 01 for for i = k (i k = 0 1 : : : n): With (19.175), the normal equations (19.172) are reduced to
(19.175)
ai = (f gi) (i = 0 1 : : : n): (19.176) Linearly independent function systems can be orthogonalized. From the power functions gi(x) = xi (i = 0 1 : : : n), depending on the weight function and on the interval, we may obtain the orthogonal polynomials in Table 19.2. Table 19.2 Orthogonal polynomials
a b] !(x) ;1 1] 1 ;1 1] p1 1; x2 0 1) e;x ;x2 =2 (;1 1) e
Name of the polynomials see p. Legendre polynomial Pn(x) Chebysev polynomial Tn(x) Laguerre polynomial Ln(x) Hermite polynomial Hn(x)
511 924 513 514
With these polynomial systems we can work on arbitrary intervals: 1. Finite approximation interval. 2. Approximation interval in nite at one end, e.g., in time-dependent problems. 3. Approximation interval in nite at both ends, e.g., in stream problems. Every nite interval a b] can be transformed by the substitution
x = b +2 a + b ;2 a t (x 2 a b] t 2 ;1 1])
(19.177)
19.6 Approximation, Computation of Adjustment, Harmonic Analysis 921
into the interval ;1 1].
19.6.2.2 Discrete Problems, Normal Equations, Householder's Method Let N pairs of values (x y ) be given, e.g., by measured values. We are looking for a function g(x), whose values g(x ) di er from the given values y in such a way that the quadratic expression
F=
N X =1
y ; g(x )]2
(19.178)
is minimal. The value of F depends on the parameters contained in the function g(x). Formula (19.178) represents the classical sum of residual squares. The minimizationof the sum of residual squares is called @F = 0 (i = the least squares method. From the assumption (19.170) and the necessary conditions @a i 0 1 : : : n) for a relative minimum of (19.178) we obtain a linear equation system for the coecients, which is called the normal equations: n X i=0
ai gigk ] = ygk ] (k = 0 1 : : : n):
(19.179)
Here we used the Gaussean sum symbols in the following notation:
gigk ] =
N X =1
gi(x )gk (x ) (19.180a)
ygk] =
N X =1
y gk (x ) (i k = 0 1 : : : n):
(19.180b)
Usually, n * N . For the polynomial g(x) = a0 + a1x + + anxn , the normal equations are a0 xk ] + a1 xk+1] + + aPn xk+n] = xk y] (k = 0 1 : : : n) with xk ] = PN=1 x k x0 ] = N xk y] = PN=1 x k y y] = N y . The coecient matrix of the normal equation system (19.179) is symmetric, so for the nu =1 merical solution we may apply the Cholesky method. The normal equations (19.179) and the residue sum square (19.178) have the following compact form:
GTGa = GTy F = (y ; Ga)T(y ; Ga) with 0 g (x ) 0 1 B g 0 (x2 ) B G = BBBB g0(.x3 ) @ .. g0 (xN )
g1(x1) g2(x1 ) : : : gn(x1 ) 1 g1(x2) g2(x2 ) : : : gn(x2 ) C C g1(x3) g2(x3 ) : : : gn(x3 ) C C C C A g1(xN ) g2(xN ) : : : gn(xN )
(19.181a)
0y 1 1 B C y B 2C y = BBBB y.3 CCCC @ .. A yN
0 1 a0 B a1 C B C a = BBBB a.2 CCCC : @ .. A an
(19.181b)
If, instead of the minimalization of the sum of residual squares, we want to solve the interpolation problem for the N points (x y ), then we have to solve the following system of equations:
Ga = y: (19.182) This equation system is overdetermined in the case of n < N ; 1, and usually it does not have any solution. We get (19.179) or (19.181a) if we multiply (19.182) by GT.
922 19. Numerical Analysis From a numerical viewpoint, the Householder method (see 4.4.3.2, 2., p. 278) is recommended to solve equation (19.182), and this solution results in the minimal sum of residual squares (19.178).
19.6.2.3 Multidimensional Problems 1. Computation of Adjustments
(2) (N ) Suppose that there is a function f (x1 x2 : : : xn) of n index1 x(1) 1 x1 : : : x1 pendent variables x1 x2 : : : xn. We do not know its explicit (1) (2) x2 x2 x2 : : : x(2N ) form only N substitution values f are given, which are, in gen... ... ... ... (19.183) eral, measured values. We can write these data in a table (see (1) (2) ( (19.183)). xn xn xn : : : xnN ) The formulation of the adjustment problem is clearer if we inf f1 f2 : : : fN troduce the following vectors: T : Vector of n independent variables, x = (x1 x2 : : : xn) x() = (x(1) x(2) : : : x(n) )T : Vector of the -th interpolation node ( = 1 : : : N ) f = (f1 f2 : : : fN )T : Vector of the N function values at the N interpolation nodes. We approximate f (x1 x2 : : : xn) = f (x) by a function of the form
g(x1 x2 : : : xn) =
m X
aigi(x1 x2 i=0 Here, the m + 1 functions gi(x1 x2 : : :
: : : xn):
(19.184)
xn) = gi(x) are suitable, selected functions. A: Linear approximation by n variables: g(x1 x2 : : : xn) = a0 + a1 x1 + a2 x2 + + anxn . B: Complete quadratic approximation with three 2variables:2 2
g(x1 x2 x3) = a0 + a1 x1 + a2 x2 + a3x3 + a4 x1 + a5 x2 + a6x3 + a7x1 x2 + a8 x1 x3 + a9x2 x3 : h i2 The coecients are chosen to minimize PN=1 f ; g x(1) x(2) : : : x(n) .
2. Normal Equation System
Analogously to (19.181b) we form the matrix G, in which we replace the interpolation nodes x by vectorial interpolation nodes x() ( = 1 2 : : : N ). To determine the coecients, we can use the normal equation system GTGa = GTf (19.185) or the overdetermined equation system Ga = f : (19.186) For an example of multidimensional regression see 16.3.4.3, 3., p. 782.
19.6.2.4 Non-Linear Least Squares Problems
We show the main idea for a one-dimensional discrete case. The approximation function g(x) depends non-linearly on certain parameters. A: g(x) = a0 ea1 x + a2 ea3 x. This expression does not depend linearly on the parameters a1 and a3 . B: g(x) = a0 ea1 x cos a2x. This function does not depend linearly on the parameters a1 and a2 . We indicate the fact that the approximation function g(x) depends on a parameter vector a = (a0 a1 : : : an)T by the notation g = g(x a) = g(x a0 a1 : : : an): (19.187) Suppose, N pairs of values (x y ) ( = 1 2 : : : N ) are given. To minimize the sum of residual squares N X
=1
y ; g(x a0 a1 : : : an)]2 = F (a0 a1 : : : an)
(19.188)
19.6 Approximation, Computation of Adjustment, Harmonic Analysis 923
the necessary conditions @F = 0 (i = 0 1 : : : n) lead to a non-linear normal equation system which @ai must be solved by an iterative method, e.g., by the Newton method (see 19.2.2.2, p. 897). Another way to solve the problem, which is usually used in practical problems, is the application of the Gauss{Newton method (see 19.2.2.3, p. 897) given for the solution of the non-linear least squares problem (19.24). We need the following steps to apply it for this non-linear approximation problem (19.188): 1. Linearization of the approximating function g(x a) with the help of the Taylor formula with respect to ai . To do this, we need the approximation values a(0) i (i = 0 1 : : : n): n @g X (0) (0) g(x a) g~(x a) = g(x a ) + @a (x a )(ai ; a(0) (19.189) i ): i=0 i 2. Solution of the linear minimum problem N X
=1
y ; g~(x a)]2 = min!
(19.190)
with the help of the normal equation system ~ = G~ Ty G~ TGa (19.191) or by the Householder method. In (19.191) the components of the vectors a and y are given as %ai = ai ; a(0) (i = 0 1 2 : : : n) and (19.192a) i %y = y ; g(x a(0) ) ( = 1 2 : : : N ): (19.192b) The matrix G~ can be determined analogously to G in (19.181b), where we replace gi(x ) by @g (0) @ai (x a1 ) (i = 0 1 : : : n = 1 2 : : : N ). 3. Calculation of a new approximation (0) (1) (0) a(1) (19.193) i = ai + %ai or ai = ai + %ai (i = 0 1 2 : : : n) where > 0 is a step length parameter. (0) By repeating steps 2 and 3 with a(1) i instead of ai , etc. we obtain a sequence of approximation values for the required parameters, whose convergence strongly depends on the accuracy of the initial approximations. We can reduce the value of the sum of residual squares with the introduction of the multiplyer .
19.6.3 Chebyshev Approximation
19.6.3.1 Problem Denition and the Alternating Point Theorem 1. Principle of Chebyshev Approximation
Chebyshev approximation or uniform approximation in the continuous case is the following: The function f (x) is to be approximated in an interval a x b by the approximation function g(x) = g(x a0 a1 : : : an) so that the error de ned by max jf (x) ; g(x a0 a1 : : : an)j = (a0 a1 : : : an) (19.194) a x b
should be as small as possible for the appropriate choice of the unknown parameters ai (i = 0 1 : : : n). If there exists such an approximating function for f (x), then the maximum of the absolute error value will be taken at least at n +2 points x of the interval, at the so-called alternating points, with changing signs (Fig. 19.10). This is actually the meaning of the alternating point theorem for the characterization of the solution of a Chebyshev approximation problem.
924 19. Numerical Analysis y
y f(x)
f(x)-g(x)
g(x) 0
0 b
a
a
b
x
x
a)
b)
Figure 19.10
;1 1] by a polynomial of degree n ; 1 in the Chebyshev sense, then we get the Chebyshev polynomial Tn(x) as an error function whose maximum is normed to one. The alternating points, being at the endpoints and at exactly n ; 1 points in the interior of the interval, correspond to the extreme points of Tn(x) (Fig. 19.11a{f). If we approximate the function f (x) = xn on the interval
T1(x)
T2(x) 1
1
T3(x)
x
x -1
0
1
-1
-1
a)
b)
T4(x) 1
0
d)
0
1
-1
-1
-1 T5(x)
x
1
0
1
-1
c)
T6(x) 1
1
x
x -1
1
-1
e)
0
x
1
-1
-1
f)
0
1
-1
Figure 19.11
19.6.3.2 Properties of the Chebyshev Polynomials 1. Representation
Tn(x) = cos(n arccos x) h p n p ni Tn(x) = 12 x + x2 ; 1 + x ; x2 ; 1 nt x = cos t for jxj < 1, Tn(x) = cos cosh nt x = cosh t for jxj > 1 (n = 1 2 : : :):
(19.195a) (19.195b) (19.195c)
19.6 Approximation, Computation of Adjustment, Harmonic Analysis 925
2. Roots of Tn (x) x = cos (2 2;n 1) ( = 1 2 : : : n): 3. Position of the extreme values of Tn (x) for x 2 ;1 1] x = cos n ( = 0 1 2 : : : n):
4. Recursion Formula Tn+1 = 2xTn (x) ; Tn;1(x) (n = 1 2 : : : T0 (x) = 1 T1(x) = x): From this recursion we have for example T2 (x) = 2x2 ; 1 T3(x) = 4x3 ; 3x T4 (x) = 8x4 ; 8x2 + 1 T5(x) = 16x5 ; 20x3 + 5x T6 (x) = 32x6 ; 48x4 + 18x2 ; 1 T7 (x) = 64x7 ; 112x5 + 56x3 ; 7x T8 (x) = 128x8 ; 256x6 + 160x4 ; 32x2 + 1 T9 (x) = 256x9 ; 576x7 + 432x5 ; 120x3 + 9x T10 (x) = 512x10 ; 1280x8 + 1120x6 ; 400x4 + 50x2 ; 1:
(19.196) (19.197) (19.198) (19.199a) (19.199b) (19.199c) (19.199d) (19.199e) (19.199f) (19.199g)
19.6.3.3 Remes Algorithm
1. Consequences of the Alternating Point Theorem
The numerical solution of the continuous Chebyshev approximation problem originates from the alternating point theorem. We choose the approximating function
g(x) =
n X i=0
ai gi(x)
(19.200)
with n+1 linearly independent known functions, and we denote by ai (i = 0 1 : : : n) the coecients of the solution of the Chebyshev problem and by % = (a0 a1 : : : an ) the minimal deviation according to (19.194). In the case when the functions f and gi (i = 0 1 : : : n) are di erentiable, from the alternating point theorem we have n X i=0
ai gi(x ) + (;1) % = f (x )
n X i=0
ai gi0 (x ) = f 0(x ) ( = 1 2 : : : n + 2):
(19.201)
The nodes x are the alternating points with a x1 < x2 < : : : < xn+2 b: (19.202) The equations (19.201) give 2n + 4 conditions for the 2n + 4 unknown quantities of the Chebyshev approximation problem: n + 1 coecients, n + 2 alternating points and the minimal deviation %. If the endpoints of the interval belong to the alternating points, then the conditions for the derivatives are not necessarily valid there.
2. Determination of the Minimal Solution according to Remes
According to Remes, we proceed with the numerical determination of the minimal solution as follows: 1. We determine an approximation of the alternating points x (0) ( = 1 2 : : : n + 2) according to (19.202), e.g., equidistant or as the positions of the extrema of Tn+1 (x) (see 19.6.3.2, p. 923). 2. We solve the linear equation system n X i=0
ai gi(x (0) ) + (;1) % = f (x (0) ) ( = 1 2 : : : n + 2)
926 19. Numerical Analysis and as a solution, we get the approximations ai(0) (i = 0 1 : : : n) and %0 . 3. We determine a new approximation of the alternating points x (1) ( = 1 2 : : : n + 2), e.g., as n positions of the extrema of the error function f (x) ; P ai(0) gi(x). Now, it is sucient to apply only i=0 approximations of these points. (1) (1) By repeating steps 2 and 3 with x and ai instead of x (0) and ai(0) , etc. we obtain a sequence of approximations for the coecients and the alternating points, whose convergence is guaranteed under certain conditions, which can be given (see 19.25]). We stop calculations if, e.g., from a certain iteration index
n X
f (x) ; ai( ) gi(x) j% j = amax
x b
(19.203)
i=0
holds with a sucient accuracy.
19.6.3.4 Discrete Chebyshev Approximation and Optimization From the continuous Chebyshev approximation problem
max f (x) ; a x b
n X i=0
aigi(x) = min!
(19.204)
we get the corresponding discrete problem, if we choose N nodes x ( = 1 2 : : : N N n + 2) with the property a x1 < x2 < xN b and require
max f (x ) ; =12:::N We substitute
n X i=0
aigi(x ) = min! :
(19.205)
n f (x ) ; X = =1max a g ( x ) i i 2:::N
and obviously we have
(19.206)
i=0
n f (x ) ; X a g ( x ) i i ( = 1 2 : : : N ): i=0
(19.207)
Eliminating the absolute values from (19.207) we obtain a linear inequality system for the coecients ai and , so the problem (19.205) becomes a linear programming problem (see 18.1.1.1, p. 846):
8 Pn > < + =0 aigi(x ) f (x ) = min! subject to > iP n : ; aigi(x ) ;f (x ) i=0
( = 1 2 : : : N ):
(19.208)
Equation (19.208) has a minimal solution with > 0. For a suciently large number N of nodes and with some further conditions the solution of the discrete problem can be considered as the solution of the continuous problem. n If we use instead of the linear approximation function g(x) = P aigi(x) a non-linear approximation i=0 function g(x) = g(x a0 a1 : : : an), which does not depend linearly on the parameters a0 a1 : : : an, then we obtain analogously a non-linear optimization problem. It is usually non-convex even in the cases of simple function forms. This essentially reduces the number of numerical solution methods for
19.6 Approximation, Computation of Adjustment, Harmonic Analysis 927
non-linear optimization problems (see 18.2.2.1, p. 863).
19.6.4 Harmonic Analysis
We want to approximate a periodic function f (x) with period 2, which is given formally or empirically, by a trigonometric polynomial or a Fourier sum of the form n X (19.209) g(x) = a20 + (ak cos kx + bk sin kx) k=1 where the coecients a0 , ak and bk are unknown real numbers. The determination of the coecients is the topic of harmonic analysis.
19.6.4.1 Formulas for Trigonometric Interpolation 1. Formulas for the Fourier Coe cients
Since the function system 1 cos kx sin kx (k = 1 2 : : : n) is orthogonal in the interval 0 2] with respect to the weight function ! 1, we get the formulas for the coecients Z2 Z2 ak = 1 f (x) cos kx dx bk = 1 f (x) sin kx dx (k = 0 1 2 : : : n) (19.210) 0 0 by applying the continuous least squares method according to (19.172). The coecients ak and bk calculated by formulas (19.210) are called Fourier coecients of the periodic function f (x) (see 7.4, p. 420). If the integrals in (19.210) are complicated or the function f (x) is known only at discrete points, then the Fourier coecients can be determined only approximately by numerical integration. Using the trapezoidal formula (see 19.3.2.2, p. 899) with N + 1 equidistant nodes x = h ( = 0 1 : : : N ) h = 2N (19.211) we get the approximation formula N N X X ak a~k = N2 f (x ) cos kx bk ~bk = N2 f (x ) sin kx (k = 0 1 2 : : : n): (19.212) =1 =1 The trapezoidal formula becomes the very simple rectangular formula in the case of periodic functions. It has higher accuracy here as a consequence of the following fact: If f (x) is periodic and (2m +2) times di erentiable, then the trapezoidal formula has an error of order O(h2m+2).
2. Trigonometric Interpolation
Some special trigonometric polynomials formed with the approximation coecients a~k and ~bk have important properties. Two of them are mentioned here: 1. Interpolation Suppose N = 2n holds. The special trigonometric polynomial nX ;1 g~1(x) = 21 a~0 + (~ak cos kx + ~bk sin kx) + 12 a~n cos nx (19.213) k=1 with coecients (19.212) satis es the interpolation conditions g~1(x ) = f (x ) ( = 1 2 : : : N ) (19.214) at the interpolation nodes x (19.211). Because of the perodicity of f (x) we have f (x0) = f (xN ). 2. Approximation in Mean Suppose N = 2n. The special trigonometric polynomial m X g~2(x) = 12 a~0 + (~ak cos kx + ~bk sin kx) (19.215) k=1
928 19. Numerical Analysis for m < n and with the coecients (19.212) approximates the function f (x) in discrete quadratic mean with respect to the N nodes x (19.211), that is, the residual sum of squares
F=
N X
=1
f (x ) ; g~2(x )]2
(19.216)
is minimal. The formulas (19.212) are the originating point for the di erent ways of e ective calculation of Fourier coecients.
19.6.4.2 Fast Fourier Transformation (FFT)
1. Computation costs of computing Fourier coe cients
The sums in the formulas (19.212) also occur in connection with discrete Fourier transformation, e.g., in electrotechnics, in impulse and picture processing. Here N can be very large, so the occurring sums must be calculated in a rational way, since the calculation of the N approximating values (19.212) of the Fourier coecients requires about N 2 additions and multiplications. For the special case of N = 2p, the number of multiplications can be largely reduced from N 2 (= 22p) to pN (= p2p) with the help of p N 2 pN the so-called fast Fourier transformation FFT. The magnitude of 10 ! 106 ! 104 (19.217) this reduction is demonstrated on the example on the right-hand 20 ! 1012 ! 107 side. By this method, the computation costs and computation time are reduced so e ectively that in some important application elds even a smaller computer is sucient. The FFT uses the properties of the N -th unit roots, i.e., the solutions of equation zN = 1 to a successive sum up in (19.212).
2. Complex Representation of the Fourier Sum
The principle of FFT can be described fairly easily if we rewrite the Fourier sum (19.209) with the formulas cos kx = 1 eikx + e;ikx sin kx = i e;ikx ; eikx (19.218) 2 2 into the complex form ! n n a ; ib X X k k ikx ak + ibk ;ikx e + e : (19.219) g(x) = 21 a0 + (ak cos kx + bk sin kx) = 12 a0 + 2 2 k=1 k=1 If we substitute Z 2 (19.220b) ck = ak ;2 ibk (19.220a) then because of (19.210) ck = 21 0 f (x)e;ikx dx and (19.219) becomes the complex representation of the Fourier sum:
g(x) =
n X
k =;n
ck eikx with c;k = c"k :
(19.221)
If the complex coecients ck are known, then we get the required real Fourier coecients in the following simple way: a0 = 2c0 ak = 2Re(ck ) bk = ;2Im(ck ) (k = 1 2 : : : n): (19.222)
3. Numerical Calculation of the Complex Fourier Coe cients
For the numerical determination of ck we apply the trapezoidal formula for (19.220b) analogously to (19.211) and (19.212), and get the discrete complex Fourier coecients c~k : NX ;1 NX ;1 c~k = N1 f (x )e;ikx = f !Nk (k = 0 1 2 : : : n) with (19.223a) =0 =0
19.6 Approximation, Computation of Adjustment, Harmonic Analysis 929 2 i f = N1 f (x ) x = 2N ( = 0 1 2 : : : N ; 1) !N = e; N :
(19.223b)
Relation (19.223a) with the quantities (19.223b) is called the discrete complex Fourier transformation of length N of the values f ( = 0 1 2 : : : N ; 1). The powers !N = z ( = 0 1 2 : : : N ; 1) satisfy equation zN = 1. So, they are called the N -th unit roots. Since e;2i = 1, !NN = 1 !NN +1 = !N1 !NN +2 = !N2 : : : : (19.224) The e ective calculation of the sum (19.223a) uses the fact that a discrete complex Fourier transformation of length N = 2n can be reduced to two transformations with length N = n in the following 2 way: a) For every coecient c~k with an even index, i.e., k = 2l, we have
c~2l =
2X n;1
=0
f !N2l =
nX ;1 h
;1 i nX f !N2l + fn+ !N2l(n+) = f + fn+ ] !N2l :
=0 Here we use the equality !N2l(n+)
= !N2ln!N2l = !N2l .
=0
(19.225)
If we substitute y = f + fn+ ( = 0 1 2 : : : n ; 1) and consider that !N2 = !n, then
(19.226)
y !nl ( = 0 1 2 : : : n ; 1)
(19.227)
c~2l =
nX ;1 =0
is the discrete complex Fourier transformation of the values y ( = 0 1 2 : : : n ; 1) with length n = N2 .
b) For every coecient c~k with an odd index, i.e., with k = 2l + 1, we get analogously: c~2l+1 =
2X n;1
=0
f !N(2l+1) =
nX ;1 =0
(f ; fn+ )!N ] !N2l :
If we substitute yn+ = (f ; fn+ )!N ( = 0 1 2 : : : n ; 1) and we consider that !N2 = !n, then we have
c~2l+1 =
nX ;1 =0
yn+ !nl ( = 0 1 2 : : : n ; 1)
(19.228) (19.229) (19.230)
which is the discrete complex Fourier transformation of the values yn+ ( = 0 1 2 : : : n ; 1) with length n = N . 2 The reduction according to a) and b), i.e., the reduction of a discrete complex Fourier transformation to two discrete complex Fourier transformations of half the length, can be continued if N is a power of 2, i.e., if N = 2p (p is a natural number). The application of the reduction after p times is called the FFT. Since every reduction step requires N complex multiplications because of (19.229), the computation 2 cost of the FFT method is N p = N log N: (19.231) 2 2 2
930 19. Numerical Analysis
4. Scheme for FFT
For the special case N = 8 = 23, the three corresponding reduction steps of the FFT according to (19.226) and (19.229) are demonstrated in the following Scheme 1:
Scheme 1:
Step 1 Step 2 Step 3 y0 = f0 + f4 y0 := y0 + y2 y0 := y0 + y1 = c~0 y1 = f1 + f5 y1 := y1 + y3 y1 := (y0 ; y1)!20 = c~4 y2 = f2 + f6 y2 := (y0 ; y2)!40 y2 := y2 + y3 = c~2 y3 = f3 + f7 y3 := (y1 ; y3)!41 y3 := (y2 ; y3)!20 = c~6 0 y4 = (f0 ; f4 )!8 y4 := y4 + y6 y4 := y4 + y5 = c~1 y5 = (f1 ; f5 )!81 y5 := y5 + y7 y5 := (y4 ; y5)!20 = c~5 2 0 y6 = (f2 ; f6 )!8 y6 := (y4 ; y6)!4 y6 := y6 + y7 = c~3 y7 = (f3 ; f7 )!83 y7 := (y5 ; y7)!41 y7 := (y6 ; y7)!20 = c~7 N = 8 n := 4 !8 = e; 28 i N := 4 n := 2 !4 = !82 N := 2 n := 1 !2 = !42 We can observe how terms with even and odd indices appear. In Scheme 2 (19.232) the structure of the method is illustrated.
f0 f1 f2 f3 f4 f5 f6 f7
Scheme 2:
8 8 > > > < c~4k > > c ~ ) 2k > > > : c~4k+2 < c~k ) 8 > > > < c~4k+1 > > c ~ ) 2k+1 > > > : : c~4k+3 (k = 0 1 : : : 7) (k = 0 1 2 3) (k = 0
) cc~~88kk+4 ) cc~~88kk+2 +6 ) cc~~88kk+1 +5 ) cc~~88kk+3 +7
(19.232)
(k = 0): Scheme 3: Index Step 1 Step 2 Step 3 Index If the coecients c~k are substic~0 000 c~0 c~0 c~0 000 tuted into Scheme 1 and we c~1 00L c~2 c~4 c~4 L00 consider the binary forms of the indices before step 1 and afc~2 0L0 c~4 c~2 c~2 0L0 ter step 3, then we recognize that the order of the required c~3 0LL c~6 c~6 c~6 LL0 coecients can be obtained by c ~ L00 c ~ c ~ c ~ 00L 4 1 1 1 simply reversing the order of the bits of the binary form of c~5 L0L c~3 c~5 c~5 L0L their indices. This is shown in c~6 LL0 c~5 c~3 c~3 0LL Scheme 3. c~7 LLL c~7 c~7 c~7 LLL 2 x=0 In the case of the function f (x) = 2x2 for for 0 < x < 2 with period 2, the FFT is used for the discrete Fourier transformation. We choose N = 8. With x = 28 f = 18 f (x ) ( = 0 1 2 : : : 7) !8 = 2i e; 8 = 0:707107(1 ; i) !82 = ;i !83 = ;0:707107(1 + i) we get Scheme 4: 1)
19.7 Representation of Curves and Surfaces with Splines 931
Scheme 4:
f0 = 2:467401 f1 = 0:077106 f2 = 0:308425 f3 = 0:693957 f4 = 1:233701 f5 = 1:927657 f6 = 2:775826 f7 = 3:778208
Step 1 y0 = 3:701102 y1 = 2:004763 y2 = 3:084251 y3 = 4:472165 y4 = 1:233700
Step 2 y0 = 6:785353 y1 = 6:476928 y2 = 0:616851 y3 = 2:467402 i y4 = 1:233700 +2:467401 i y5 = ;1:308537(1 ; i) y5 = 0:872358 +3:489432 i y6 = 2:467401 i y6 = 1:233700 ;2:467401 i y7 = 2:180895(1 + i) y7 = ;0:872358 +3:489432 i
Step 3 y0 = 13:262281 = c~0 y1 = 0:308425 = c~4 y2 = 0:616851 + 2:467402 i = c~2 y3 = 0:616851 ; 2:467402 i = c~6 y4 = 2:106058 + 5:956833 i = c~1
y5 = 0:361342 ; 1:022031 i = c~5 y6 = 0:361342 + 1:022031 i = c~3 y7 = 2:106058 ; 5:956833 i = c~7
a0 = 26:524 562 From the third (last) reduction step we get the required a1 = 4:212 116 b1 = ;11:913 666 real Fourier coecients according to (19.222). (See the a2 = 1:233 702 b2 = ; 4:934 804 right-hand side.) a3 = 0:722 684 b3 = ; 2:044 062 In this example, the general property a4 = 0:616 850 b4 = 0 c~N ;k = "c~k (19.233) of the discrete complex Fourier coecients can be observed. For k = 1 2 3, we see that c~7 = "c~1 c~6 = "c~2 c~5 = "c~3 .
19.7 Representation of Curves and Surfaces with Splines 19.7.1 Cubic Splines
Since interpolation and approximation polynomials of higher degree usually have unwanted oscillations, it is useful to divide the approximation interval into subintervals by the so-called nodes and to consider a relatively simple approximation function on every subinterval. In practice, cubic polynomials are mostly used. We require a smooth transition at the nodes of this piecewise approximation.
19.7.1.1 Interpolation Splines
1. Denition of the Cubic Interpolation Splines, Properties
Suppose there are given N interpolation points (xi fi) (i = 1 2 : : : N x1 < x2 < : : : xN ). The cubic interpolation spline S (x) is determined uniquely by the following properties: 1. S (x) satis es the interpolation conditions S (xi ) = fi (i = 1 2 : : : N ). 2. S (x) is a polynomial of degree 3 in any subinterval xi xi+1] (i = 1 2 : : : N ; 1). 3. S (x) is twice continuously di erentiable in the entire approximation interval x1 xN ]. 4. S (x) satis es the special boundary conditions: a) S 00(x1 ) = S 00(xN ) = 0 (we call them natural splines) or b) S 0 (x1) = f10, S 0(xN ) = fN 0 (f10 and fN 0 are given values) or c) S (x1 ) = S (xN ), in the case of f1 = fN , S 0(x1 ) = S 0(xN ) and S 00(x1 ) = S 00(xN ) (we call them
932 19. Numerical Analysis periodic splines). It follows from these properties that for all twice continuously di erentiable functions g(x) satisfying the interpolation conditions g(xi) = fi (i = 1 2 : : : N )
ZxN
x1
S 00 (x)]2 dx
ZxN
x1
g00(x)]2 dx
(19.234)
is valid (Holladay's Theorem). Based on (19.234) we can say that S (x) has minimal total curvature, since for the curvature of a given curve, in a rst approximation, S 00 (see 3.6.1.2, 4., p. 228). It can be shown that if we lead a thin elastic ruler (its name is spline) through the points (xi fi) (i = 1 2 : : : N ), its bending line follows the cubic spline S (x).
2. Determination of the Spline Coe cients
We suppose that the cubic interpolation spline S (x) for x 2 xi xi+1 ] has the form: S (x) = Si(x) = ai + bi (x ; xi ) + ci(x ; xi )2 + di(x ; xi )3 (i = 1 2 : : : N ; 1): (19.235) The length of the subinterval is denoted by hi = xi+1 ; xi . We can determine the coecients of the natural spline on the following way: 1. From the interpolation conditions we get ai = fi (i = 1 2 : : : N ; 1): (19.236) It is reasonable to introduce the additional coecient aN = fN , which does not occur in the polynomials. 2. The continuity of S 00(x) at the interior nodes requires that di;1 = ci3;h ci;1 (i = 2 3 : : : N ; 1): (19.237) i;1 The natural conditions result in c1 = 0, and (19.237) still holds for i = N , if we introduce cN = 0. 3. The continuity of S (x) at the interior nodes results in the relation bi;1 = ai h; ai;1 ; 2ci;13 + ci hi;1 (i = 2 3 : : : N ): (19.238) i;1 0 4. The continuity of S (x) at the interior nodes requires that ! ci;1hi;1 + 2(hi;1 + hi )ci + ci+1hi = 3 ai+1h; ai ; ai h; ai;1 (i = 2 3 : : : N ; 1): (19.239) i i;1 Because of (19.236), the right-hand side of the linear equation system (19.239) to determine the coefcients ci (i = 2 3 : : : N ; 1 c1 = cN = 0) is known. The left hand-side has the following form: 0 2(h + h ) h 10 c 1 1 2 2 2 B C B h2 2(h2 + h3) h3 O c3 C B C B C B C B h3 2(h3 + h4 ) h4 c4 C B C B C B C B : (19.240) ... C ... ... ... B C B C B C B C B C B C @ A@ A O hN ;2 hN ;2 2(hN ;2 + hN ;1) cN ;1 The coecient matrix is tridiagonal, so the equation system (19.239) can be solved numerically very easily by an LR decomposition (see 19.2.1.1, 2., p. 891). We can then determine all other coecients in (19.238) and (19.237) with these values ci.
19.7.1.2 Smoothing Splines
The given function values fi are usually measured values in practical applications so they have some error. In this case, the interpolation requirement is not reasonable. This is the reason why cubic smoothing
19.7 Representation of Curves and Surfaces with Splines 933 splines are introduced. We get this spline if in the cubic interpolation splines we replace the interpolation requirements by ZxN N " f ; S (x ) 2 X i i + S 00(x)]2 dx = min!: (19.241)
i x1 We keep the requirements of continuity of S , S 0 and S 00 , so the determination of the coecients is a constrained optimization problem with conditions given in equation form. The solution can be obtained by using a Lagrange function (see 6.2.5.6, p. 403). For details see 19.26]. In (19.241) ( 0) represents a smoothing parameter , which must be given previously. For = 0 we get the cubic interpolation spline, as a special case, for \large" we get a smooth approximation curve, but it returns the measured values inaccurately, and for = 1 we get the approximating regression line as another special case. A suitable choice of can be made, e.g., on computer by screen dialog. The parameter i (i > 0) in (19.241) represents the standard deviation (see 16.4.1.3, 2., p. 790) of the measurement errors, of the values fi (i = 1 2 : : : N ). Until now, the abscissae of the interpolation points and the measurement points were the same as the nodes of the spline function. For large N this results in a spline containing a large number of cubic functions (19.235). A possible solution is to choose the number and the position of the nodes freely, because in many practical applications only a few spline segments are satisfactory. It is reasonable also from a numerical viewpoint to replace (19.235) by a spline of the form rX +2 S (x) = aiNi4(x): (19.242) i=1
i=1
Here r is the number of freely chosen nodes, and the functions Ni4(x) are the so-called normalized B splines (basis splines ) of order 4, i.e., polynomials of degree three, with respect to the i-th node. For details see 19.5].
19.7.2 Bicubic Splines
19.7.2.1 Use of Bicubic Splines
Bicubic splines are used for the following problem: A rectangle R of the x y plane, given by a x b, c y d, is decomposed by the grid points (xi yj ) (i = 0 1 : : : n j = 0 1 : : : m) with a = x0 < x1 < < xn = b c = y0 < y1 < < ym = d (19.243) into subdomains Rij , where the subdomain Rij contains the points (x y) with xi x xi+1 , yj y yj+1 (i = 0 1 : : : n ; 1 j = 0 1 : : : m ; 1). The values of the function f (x y) are given at the grid points f (xi yj ) = fij (i = 0 1 : : : n j = 0 1 : : : m): (19.244) A possible simple, smooth surface over R is required which approximates the points (19.244).
19.7.2.2 Bicubic Interpolation Splines 1. Properties
The bicubic interpolation spline S (x y) is de ned uniquely by the following properties: 1. S (x y) satis es the interpolation conditions S (xi yj ) = fij (i = 0 1 : : : n j = 0 1 : : : m): 2. S (x y) is identical to a bicubic polynomial on every Rij of the rectangle R, that is,
S (x y) = Sij (x y) =
3 X 3 X
k=0 l=0
aijkl(x ; xi )k (y ; yj )l
(19.245) (19.246)
on Rij . So, Sij (x y) is determined by 16 coecients, and for the determination of S (x y) we need 16 m n coecients.
934 19. Numerical Analysis 3. The derivatives
@S @S @ 2 S @x @y @x@y are continuous on R. So, a certain smoothness is ensured for the entire surface. 4. S (x y) satis es the special boundary conditions: @S (x y ) = p for i = 0 n j = 0 1 : : : m ij @x i j @S (x y ) = q for i = 0 1 : : : n j = 0 m ij @y i j
(19.247)
(19.248)
@2S
@x@y (xi yj ) = rij for i = 0 n j = 0 m: Here pij , qij and rij are previously given values. We can use the results of one-dimensional cubic spline interpolation for the determination of the coefcients aijkl. We see: 1. There is a very large number (2n + m + s) of linear equation systems but only with tridiagonal coecient matrices. 2. The linear equation systems di er from each other only on their right-hand sides. In general, it can be said that bicubic interpolation splines are useful with respect to computation cost and accuracy, and so they are appropriate procedures for practical applications. For practical methods of computing the coecients see the literature.
2. Tensor Product Approach
The bicubic spline approach (19.246) is an example of the so-called tensor product approach having the form
S (x y) =
n X m X
i=0 j =0
aij gi(x)hj (y)
(19.249)
and which is especially suitable for approximations over a rectangular grid. The functions gi(x) (i = 0 1 : : : n) and hj (y) (j = 0 1 : : : m) form two linearly independent function systems. The tensor product approach has the big advantage, from numerical viewpoint, that, e.g., the solution of a twodimensional interpolation problem (19.245) can be reduced to a one-dimensional one. Furthermore, the two-dimensional interpolation problem (19.245) is uniquely solvable with the approach (19.249) if 1. the one-dimensional interpolation problem with functions gi(x) with respect to the interpolation nodes x0 x1 : : : xn and 2. the one-dimensional interpolation problem with functions hj (y) with respect to the interpolation nodes y0 y1 : : : ym are uniquely solvable. An important tensor product approach is that with the cubic B-splines:
S (x y) =
+2 rX +2 pX
i=1 j =1
aij Ni4(x)Nj4(y):
(19.250)
Here, the functions Ni4 (x) and Nj4(y) are normalized B-splines of order four. Here r denotes the number of nodes with respect to x, p denotes the number of nodes with respect to y. The nodes can be chosen freely but their positions must satisfy certain conditions for the solvability of the interpolation problem. The B-spline approach results in an equation system with a band structured coecient matrix, which is a numerically useful structure.
19.7 Representation of Curves and Surfaces with Splines 935
For solutions of di erent interpolation problems using bicubic B-splines see the literature.
19.7.2.3 Bicubic Smoothing Splines
The one-dimensional cubic approximation spline is mainly characterized by the optimality condition (19.241). For the two-dimensional case we could determine a whole sequence of corresponding optimality conditions, however only a few special cases make the existence of a unique solution possible. For appropriate optimality conditions and algorithms for solution of the approximation problem with bicubic B-splines see the literature.
19.7.3 Bernstein{Bezier Representation of Curves and Surfaces 1. Bernstein Basis Polynomials
The Bernstein{Bezier representation (briey B{B representation) of curves and surfaces applies the Bernstein polynomials Bin(t) = ni ti (1 ; t)n;i (i = 0 1 : : : n) (19.251) and uses the following fundamental properties: 1. 0 Bin(t) 1 for 0 t 1 (19.252) n X 2. Bin(t) = 1: (19.253) i=0
Formula (19.253) follows directly from the binomial theorem (see 1.1.6.4, p. 12). A: B01 (t) = 1 ; t B11(t) = t (Fig. 19.12). B: B03 (t) = (1 ; t)3 B13 (t) = 3t(1 ; t)2 B23 (t) = 3t2(1 ; t) B33(t) = t3 (Fig. 19.13). 3
B i (t) 1
1
B i (t) 1
i=0
0
1
t
i=3
i=0
i=1 4 9
i=1
i=2
0
1 3
2 3
Figure 19.12
2. Vector Representation
1
t
Figure 19.13
In the following, a space curve, whose parametric representation is x = x(t), y = y(t), z = z(t), will be denoted in vector form by ~r = ~r(t) = x(t) ~ex + y(t) ~ey + z(t) ~ez : (19.254) Here t is the parameter of the curve. The corresponding representation of a surface is ~r = ~r(u v) = x(u v) ~ex + y(u v) ~ey + z(u v) ~ez : (19.255) Here, u and v are the surface parameters.
19.7.3.1 Principle of the B{B Curve Representation
Suppose there are given n + 1 vertices Pi (i = 0 1 : : : n) of a three-dimensional polygon with the ~ i. Introducing the vector-valued function position vectors P n X ~i ~r(t) = Bin(t)P (19.256) i=0
936 19. Numerical Analysis we assign a space curve to these points, which is called the B{B curve. Because of (19.253) we can consider (19.256) as a \variable convex combination" of the given points. The three-dimensional curve (19.256) has the following important properties: 1. The points;!P0 and Pn;!are interpolated. P1 P2 2. Vectors P0P1 and Pn;1Pn are tangents to ~r(t) at points P0 and Pn. The relation between a polygon and a B{B curve is shown in P5 P0 Fig. 19.14. P3 The B{B representation is considered as a design of the curve, since we can easily inuence the shape of the curve by changing P4 the polygon vertices. We often use normalized B-splines instead of Bernstein polyFigure 19.14 nomials. The corresponding space curves are called the B-spline curves. Their shape corresponds basically to the B{B curves with the following advantages: 1. The polygon is better approximated. 2. The B-spline curve changes only locally if the polygon vertices are changed. 3. In addition to the local changes of the shape of the curve the di erentiability can also be inuenced. So, it is possible to produce break points and line segments for example.
19.7.3.2 B{B Surface Representation
~ ij , Suppose there are given the points Pij (i = 0 1 : : : n j = 0 1 : : : m) with the position vectors P which can be considered as the nodes of a grid along the parameter curves of a surface. Analogously to the B{B curves (19.256), we assign a surface to the grid points by n X m X ~ ij : ~r(u v) = Bin(u)Bjm(v)P (19.257) i=0 j =0
Representation (19.257) is useful for surface design, since by changing the grid points we may change the surface. Anyway, the inuence of every grid point is global, so we should change from the Bernstein polynomials to the B-splines in (19.257).
19.8 Using the Computer
19.8.1 Internal Symbol Representation
Computer are machines that work with symbols. The interpretation and processing of these symbols is determined and controlled by the software. The external symbols, letters, cyphers and special symbols are internally represented in binary code by a form of bit sequence. A bit (binary digit) is the smallest representable information unit with values 0 and 1. Eight bits form the next unit, the byte. In a byte we can distinguish between 28 bit combinations, so 256 symbols can be assigned to them. Such an assignment is called a code. There are di erent codes one of the most widespread is ASCII (American Standard Code for Information Interchange).
19.8.1.1 Number Systems 1. Law of Representation
Numbers are represented in computers in a sequence of consecutive bytes. The basis for the internal representation is the binary system, which belongs to the polyadic systems, similarly to the decimal system. The law of representation for a polyadic number system is
a=
n X
i=;m
zi B i (m > 0 n 0 m n integer)
(19.258)
19.8 Using the Computer 937
with B as basis and zi (0 zi < B ) as a cypher of the number system. The positions i 0 form the integers, those with i < 0 the fractional part of the number. For the decimal number representation, i.e., B = 10, of the decimal number 139:8125 we get 139:8125 = 1 102 + 3 101 + 9 100 + 8 10;1 + 1 10;2 + 2 10;3 + 5 10;4. The number systems occurring most often in computers are shown in Table 19.3.
Number system
Binary system Octal system Hexadecimal system Decimal system
Table 19.3 Number systems
Basis Corresponding cyphers 2 8 16 10
0 1 (or O L) 01234567 0 1 2 3 4 5 6 7 8 9 A,B,C,D,E,F (The letters A{F are for the values 10{15.) 0123456789
2. Conversion
The transition from one number system to another is called conversion. If we use di erent number systems in the same time, in order to avoid confusion we put the basis as an index. The decimal number 139:8125 is in di erent systems: 139:812510 = 10001011:11012 = 213:648 = 8B:D16 . 1. Conversion of Binary Numbers into Octal or Hexadecimal Numbers The conversion of binary numbers into octal or hexadecimal numbers is simple. We form groups of three or four bits starting at the binary point to the left and to the right, and we determine their values. These values are the cyphers of the octal or hexadecimal systems. 2. Conversion of Decimal Numbers into Binary, Octal or Hexadecimal Numbers For the conversion of a decimal numbers into another system, we adept the following rules for the integer and for the fractional part separately: a) Integer Part: If G is an integer in the decimal system, then for the number system with basis B the law of formation (19.258) is:
G=
n X i=0
ziB i (n 0):
(19.259)
If we divide G by B , then we get an integer part (the sum) and a residue: n G X i;1 z0 (19.260) B = i=1 ziB + B : Here, z0 can have the values 0 1 : : : B ; 1, and it is the lowest valued cypher of the required number. If we repeat this method for the quotients, we get further cyphers. b) Fractional Part: If g is a proper fraction, then the method to convert it into the number system with basis B is
gB = z;1 +
m X i=2
z;iB ;i+1
(19.261)
i.e., we get the next cypher as the integer part of the product gB . The values z;2 z;3 : : : can be obtained in the same way.
938 19. Numerical Analysis A: Conversion of the decimal number 139 into a binary number. 139 : 2 = 69 residue 1 (1 = z0 ) 69 : 2 = 34 residue 1 (1 = z1 ) 34 : 2 = 17 residue 0 (0 = z2 ) 17 : 2 = 8 residue 1 : 8 : 2 = 4 residue 0 : 4 : 2 = 2 residue 0 : 2 : 2 = 1 residue 0 : 1 : 2 = 0 residue 1 (1 = z7 ) 13910 = 100010112
B: Conversion of a decimal fraction 0.8125 into a binary fraction. 0.8125 2 = 1.625 (1 = z;1 ) 0.625 2 = 1.25 (1 = z;2 ) 0.25 2 = 0.5 (0 = z;3 ) 0.5 2 = 1.0 (1 = z;4 ) 0.0 2 = 0.0 0:812510 = 0:11012
3. Conversion of Binary, Octal, and Hexadecimal Numbers into a Decimal Number The
algorithm for the conversion of a value from the binary, octal, or hexadecimal system into the decimal system is the following, where the decimal point is after z0 :
a=
n X
i=;m
zi B i (m > 0 n 0 integer):
(19.262)
The calculation is convenient with the Horner rule. LLLOL = 1 24 + 1 23 + 1 22 + 0 21 + 1 20 = 29.L and O see Table 19.3. The corresponding Horner scheme is shown on the right.
19.8.1.2 Internal Number Representation
2
111 0 1 2 6 14 28 1 3 7 14 29
Binary numbers are represented in computers in one or more bytes. We distinguish between two types of form of representation, the xed-point numbers and the oating-point numbers. In the rst case, the decimal point is at a xed place, in the second case it is \oating" with the change of the exponent.
1. Fixed-Point Numbers
The range for xed-point numbers with the given parameters is 0 j a j 2t ; 1: (19.263) Fixed-point numbers can be represented in the form of Fig. 19.15.
2. Floating-Point Numbers
binary number (t bits)
sign u of the fixed-point number
Basically, two di erent forms are in Figure 19.15 use for the representation of oating-point numbers, where the internal implementation can vary in detail.
1. Normalized Semilogarithmic Form
In the rst form, the signs of the exponent E and the mantissa M of the number a are stored separately a = MB E : (19.264a) Here the exponent E is chosen so that for the mantissa sign u sign u 1=B M < 1 (19.264b) of the exponent of the mantissa holds. We call it the normalized semilogarithFigure 19.16 mic form (Fig. 19.16). The range of the absolute value of the oating-point numbers with the given parameters is: (19.265) 2;2p j a j 1 ; 2;t 2(2p;1) : mantissa M (t bits)
exponent E (p bits)
E
M
19.8 Using the Computer 939
2. IEEE Standard The second (nowadays used) form of oating-point numbers corresponds to the IEEE (Institute of Electrical and Electronics Engineers) standard accepted in 1985. It deals with the
requirements of computer arithmetic, roundo behavior, arithmetical operators, conversion of numbers, comparison operators and handling of exceptional cases such as over- and underow. The oating-point number representations are mantissa M characteristic C shown in Fig. 19.17. We get the characteristic C from the exponent E by addition of a suitable constant K . This is chosen so that we get only positive numbers in the characteristic. The representable number is sign u of the floating-point number a = (;1)v 2E 1:b1b2 : : : bt;1 with E = C ; K: (19.266) Figure 19.17 Here: Cmin = 1 Cmax = 254 since C = 0 and C = 255 are reserved. The standard gives two basic forms of representation (single-precision and double-precision oatingpoint numbers), but other representations are also possible. Table 19.4 contains the parameters for the basic forms. Table 19.4 Parameters for the basic forms
Parameter
Word length in bits Maximal exponent Emax Minimal exponent Emin Constant K Number of bits in exponent Number of bits in mantissa
Single precision Double precision 32 +127 ;126 +127 8 24
64 +1023 ;1022 +1023 11 53
19.8.2 Numerical Problems in Calculations with Computers 19.8.2.1 Introduction, Error Types
The general properties of calculations with a computer are basically the same as those of calculations done by hand, however some of them need special attention, because the accuracy comes from the representation of the numbers, and from the missing judgement with respect to the errors of the computer. Furthermore, computers perform many more calculation steps than human can do manually. So, we have the problem of how to inuence and control the errors, e.g., by choosing the most appropriate numerical method among the mathematically equivalent methods. In further discussions, we will use the following notation, where x denotes the exact value of a quantity, which is mostly unknown, and x~ is an approximation value of x: Absolute error: j%xj = jx ; x~j: (19.267) Relative error: %x = x ; x~ : (19.268) x x The notation (19.269) (x) = x ; x~ and rel (x) = x ;x x~ is also often used.
19.8.2.2 Normalized Decimal Numbers and Round-O
1. Normalized Decimal Numbers
Every real number x 6= 0 can be expressed as a decimal number in the form x = 0:b1 b2 : : : 10E (b1 6= 0):
(19.270)
940 19. Numerical Analysis Here 0 b1b2 : : : is called the mantissa formed with the cyphers bi 2 f0 1 2 : : : 9g. The number E is an integer, the so-called exponent with respect to the base 10. Since b1 6= 0, we call (19.270) a normalized decimal number. Since only nitely many cyphers can be handled by a real computer, we have to restrict ourselves to a xed number t of mantissa cyphers and to a xed range of the exponent E . So, from the number x given in (19.270) we obtain the number ( 0:b1 b2 bt 10E for bt+1 5 (round-down) x~ = (19.271) (0:b1 b2 bt + 10;t)10E for bt+1 > 5 (round-up) by round-o (as it is usual in practical calculations). For the absolute error caused by round-o , j%xj = jx ; x~j 0:5 10;t10E : (19.272)
2. Basic Operations and Numerical Calculations
Every numerical process is a sequence of basic calculation operations. Problems arise especially with the nite number of positions in the oating-point representation. We give here a short overview. We suppose that x and y are normalized error-free oating-point numbers with the same sign and with a non-zero value: x = m1 B E1 y = m2 B E2 with (19.273a)
mi =
t X k=1
a(;i)k B ;k
a(;i)1 6= 0 and
(19.273b)
a(;i)k = 0 or 1 or : : : or B ; 1 for k > 1 (i = 1 2): (19.273c) 1. Addition If E1 > E2, then the common exponent becomes E1 , since normalization allows us to make only a left-shift. The mantissas are then added. If B ;1 j m1 + m2 B ;(E1 ;E2) j< 2 (19.274a) and j m1 + m2 B ;(E1;E2 ) j 1 (19.274b) then shifting the decimal point by one position to the left results in an increase of the exponent by one. 0:9604 103 + 0:5873 102 = 0:9604 103 + 0:05873 103 = 1:01913 103 = 0:1019 104. 2. Subtraction The exponents are equalized as in the case of addition, the mantissas are then subtracted. If j m1 ; m2B ;(E1 ;E2) j< 1 ; B ;t (19.275a) and j m1 ; m2 B ;(E1 ;E2) j< B ;1 (19.275b) shifting the decimal point to the right by a maximum of t positions results in the corresponding decrease of the exponent. 0:1004 103 ; 0:9988 102 = 0:1004 103 ; 0:09988 103 = 0:00052 103 = 0:5200 100. This example shows the critical case of subtractive cancellation. Because of the limited number of positions (here four), zeros are carried in from the left instead of the correct characters. 3. Multiplication The exponents are added and the mantissas are multiplied. If m1 m2 < B ;1 (19.276) then the decimal point is shifted to the right by one position, and the exponent is decreased by one. (0:3176 103) (02504 105) = 0:07952704 108 = 0:7953 107. 4. Division The exponents are subtracted and the mantissas are divided. If m1 ;1 (19.277) m2 B then the decimal point is shifted to the left by one position, and the exponent is increased by one. (0:3176 103)=(0:2504 105) = 1:2683706 : : : 10;2 = 0:1268 10;1.
19.8 Using the Computer 941
5. Error of the Result The error of the result in the four basic operations with terms that are
supposed to be error-free is a consequence of round-o . For the relative error with number of positions t and the base B , the limit is B B ;t : (19.278) 2 6. Subtractive cancellation As we have seen above, the critical operation is the subtraction of nearly equal oating-point numbers. If it is possible, we should avoid this by changing the order of operations, or by using certain identities. p p p p x = 1985 ; 1984 = 0:4455 102 ; 0:4454 102 = 0:1 10;1 or x = 1985 ; 1984 = 1984 = 0:1122 10;1 p1985 ; p 1985 + 1984
19.8.2.3 Accuracy in Numerical Calculations 1. Types of Errors
Numerical methods have errors. There are several types of errors, from which the total error of the nal result is accumulated (Fig. 19.18). Total error
Input error
Error of method
Truncation error
2. Input Error
Round-off error
Discretization error
Figure 19.18
1. Notion of Input Error Input error is the error of the result caused by inaccurate input data. Slight inaccuracies of input data are also called perturbations. The determination of the error of the input data is called the direct problem of error calculus. The inverse problem is the following: How large an error the input data may have such that the nal input error does not exceed an acceptable tolerance value. The estimation of the input error in rather complex problems is very dicult and is usually hardly possible. In general, for a real-valued function y = f (x) with x = (x1 x2 : : : xn)T, for the absolute value of the input error we have j%yj = jf (x1 x2 : : : xn ) ; f (~x1 x~2 : : : x~n)j ! n n X X @f (x)j j%x j = j @f ( 1 2 : : : n)(xi ; x~i)j max j (19.279) i x @xi i=1 @xi i=1 if we use the Taylor formula (see 7.3.3.3, p. 417) for y = f (x) = f (x1 x2 : : : xn) with a linear residue. 1 2 : : : n denote the intermediate values, x~1 x~2 : : : x~n denote the approximating values of x1 x2 : : : xn. The approximating values are the perturbed input data. Here, we also consider the Gauss error propagation law (see 16.4.2.1, p. 794). 2. Input Error of Simple Arithmetic Operations The input error is known for simple arithmetical operations. With the notation of (19.267){(19.269) for the four basic operations we get:
942 19. Numerical Analysis (x y) = (x) (y) (19.280) (xy) = y (x) + x (y) + (x) (y) (19.281) ! xy = y1 (x) ; yx2 (y) + terms of higher order in " (19.282) y rel(y) (19.283) (xy) = (x) + (y) + (x) (y) (19.284) rel (x y) = x rel (xx) rel rel rel rel rel y ! rel xy = rel (x) ; rel (y) + terms of higher order in ": (19.285) The formulas show: Small relative errors of the input data result in small relative errors of the result on multiplication and division. For addition and subtraction, the relative error can be very large if jx yj * jxj + jyj.
3. Error of the Method
1. Notion of the Error of the Method The error of the method comes from the fact that we
should be able to numerically approximate theoretically continuous phenomena in many di erent ways as limits. Hence, we have truncation errors in limiting processes (as, e.g., in iteration methods) and discretization errors in the approximation of continuous phenomena by a nite discrete system (as, e.g., in numerical integration). Errors of methods exist independently of the input and round-o errors consequently, they can be investigated only in connection with the applied solution methodology. 2. Applying Iteration Methods If we use an iteration method, we should consider that both cases may occur: We can get a correct solution or also a false solution of the problem. It is also possible that we get no solution by an iteration method although there exists one. To make an iteration method clearer and safer, we should consider the following advice: a) To avoid \in nite" iterations, we should count the number of steps and stop the process if this number exceeds a previously given value (i.e., we stop without reaching the required accuracy). b) We should keep track of the location of the intermediate result on the screen by a numerical or a graphical representation of the intermediate results. c) We should use all known properties of the solution such as gradient, monotonicity, etc. d) We should investigate the possibilities of scaling the variables and functions. e) We should perform several tests by varying the step size, truncation conditions, initial values, etc.
4. Round-o Errors
Round-o errors occur because the intermediate results should be rounded. So, they have an essential importance in judging mathematical methods with respect to the required accuracy. They determine together with the errors of input and the error of the method, whether a given numerical method is strongly stable, weakly stable or unstable. Strong stability, weak stability, or instability occur if the total error, at an increasing number of steps, decreases, has the same order, or increases, respectively. At the instability we distinguish between the sensitivity with respect to round-o errors and discretization errors (numerical instability) and with respect to the error in the initial data at a theoretically exact calculation (natural instability). A calculation process is appropriate if the numerical instability is not greater than the natural instability. For the local error propagation of round-o errors, i.e., errors at the transition from a calculation step to the next one, the same estimation process can be used as the one we have at the input error.
5. Examples of Numerical Calculations
We illustrate some of the problems mentioned above by numerical examples.
A: Roots of a Quadratic Equation:
ax2 + bx + c = 0 with real coecients a b c and D = b2 ; 4ac 0 (real roots). Critical situations are the cases a) j 4ac j* b2 and b) 4ac b2 . Recommended proceeding:
19.8 Using the Computer 943
p b) D x = c (Vieta root theorem, see 1.6.3.1, 3., p. 44). a) x1 = ; b + sign( 2 2a ax1 b) The vanishing ofpD cannot be avoided by a directpmethod. Subtractive cancellation occurs but the error in (b + sign(b D) is not too large since jbj 0 D holds. B: Volume3 of a3 Thin Conical Shell for h * r V = 4 (r + h3) ; r because of (r + h) r there is a case of subtractive cancellation. However in the
2 2 3 equation V = 4 3r h + 33rh + h there is no such problem. 1 X C: Determining the Sum S = k2 1+ 1 (S = 1:07667 : : :) with an accuracy of three signifk=1 icant digits. Performing the calculations with 8 digits, we should add about 6000 terms. After the identical transformation 2 1 = 12 ; 2 21 k + 1 k k (k + 1) we see that 1 1 1 2 X X X S = k12 ; k2(k21 + 1) and S = 6 ; k2(k21 + 1) . By this transformation we have k=1 k=1 k=1 to consider only eight terms. q 2 2 D: Avoiding the 00 Situation in the function z = (1 ; 1 + x2 + y2) xx2 ;+ yy2 for x = y = 0. p Multiplying the numerator and the denominator by (1 + 1 + x2 + y2) we avoid this situation. E: Example for an Unstable Recursive Process. Algorithms s 2 with the general form yn+1 = a ayn + byn;1 (n = 1 2 : : :) are stable if the condition 2 a4 + b < 1 is satis ed. The special case yn+1 = ;3yn + 4yn;1 (n = 1 2 : : :) is unstable. If y0 and y1 have errors " and ;", then for y2 y3 y4 y5 y6 : : : we get the errors 7", ;25", 103", ;409", 1639", : : :. The process is instable for the parameters a = ;3 and b = 4. F: Numerical Integration of a Di erential Equation. For the rst-order ordinary di erential equation y0 = f (x y) with f (x y) = ay (19.286) and the initial value y(x0) = y0 we will represent the numerical solution. a) Natural Instability. Together with the exact solution y(x) for the exact initial values y(x0) = y0 let u(x) be the solution for a perturbed initial value. Without loss of generality, we may assume that the perturbed solution has the form u(x) = y(x) + " (x) (19.287a) where " is a parameter with 0 < " < 1 and (x) is the so-called perturbation function. Considering that u0(x) = f (x u) we get from the Taylor expansion (see 7.3.3.3, p. 417) u0(x) = f (x y(x) + " (x)) = f (x y) + " (x) fy (x y) + terms of higher order (19.287b) which implies the so-called di erential variation equation 0(x) = fy (x y)(x): (19.287c) The solution of the problem with f (x y) = ay is (x) = 0 ea(x;x0 ) with 0 = (x0): (19.287d) For a > 0 even a small initial perturbation 0 results in an unboundedly increasing perturbation (x). So, there is a natural instability. b) Investigation of the Error of the Method in the Trapezoidal Rule. With a = ;1, the stable
944 19. Numerical Analysis di erential equation y0(x) = ;y(x) has the exact solution y(x) = y0e;(x;x0 ) where y0 = y(x0): (19.288a) The trapezoidal rule is xZi+1 y(x)dx yi +2yi+1 h with h = xi+1 ; xi : (19.288b) x1 By using this formula for the given di erential equation we get xZi+1 h y~i+1 = y~i + (;y)dx = y~i ; y~i +2y~i+1 h or y~i+1 = 22 ; + h y~i or xi ! h i y~ : y~i = 22 ; (19.288c) +h 0 With xi = x0 + ih, i.e., with i = (xi ; x0 =h we get for 0 h < 2 ! h (xi;x0 )=h y~ = y~ ec(h)(xi ;x0) with y~i = 22 ; 0 0 +h ! h ln 22 ; + h = ;1 ; h2 ; h4 ; c(h) = (19.288d) h 12 80 ;(xi ;x0 ) If y~0 = y0, then y~i < yi, and so for h ! 0, y~i also tends to the exact solution y0e . c) Input Error We supposed in b) that the exact and the approximate initial values coincide. Now, we investigate the behavior when y0 6= y~0 with j y~0 ; y0 j< "0. Since ! h (~y ; y ) we have (~y ; y ) 2 ; h i+1 (~y ; y ): (~yi+1 ; yi+1) 22 ; (19.289a) i i i+1 i+1 0 0 +h 2+h So, "i+1 is at most of the same order as "0, and the method is stable with respect to the initial values. We have to mention that in solving the above di erential equation with the Simpson method an arti cial instability is introduced. In this case, for h ! 0, we would get the general solution y~i = C1e;xi + C2 (;1)iexi=3 : (19.289b) The problem is that the numerical solution method uses higher-order di erences than those to which the order of the di erential equation corresponds.
19.8.3 Libraries of Numerical Methods
Over time, libraries of functions and procedures have been developed independently of each other for numerical methods in di erent programming languages. An enormous amount of computer experimentation was considered in their development, so in solutions of practical numerical problems we should use the programs from one of these program libraries. Programs are available for current operating systems like WINDOWS, UNIX and LINUX and mostly for every computation problem type and they keep certain conventions, so it is more or less easy to use them. The application of methods from program libraries does not relieve the user of the necessity of thinking about the expected results. This is a warning that the user should be informed about the advantages and also about the disadvantages and weaknesses of the mathematical method he/she is going to use.
19.8.3.1 NAG Library
The NAG library (Numerical Algorithms Group) is a rich collection of numerical methods in the form of functions and subroutines/procedures in the programming languages FORTRAN 77, FORTRAN 90,
19.8 Using the Computer 945
and C. Here is a contents overview: 1. Complex arithmetic 14. Eigenvalues and eigenvectors 2. Roots of polynomials 15. Determinants 3. Roots of transcendental equations 16. Simultaneous linear equations 4. Series 17. Orthogonalization 5. Integration 18. Linear algebra 6. Ordinary di erential equations 19. Simple calculations with statistical data 7. Partial di erential equations 20. Correlation and regression analysis 8. Numeric di erentiation 21. Random number generators 9. Integral equations 22. Non-parametric statistics 10. Interpolation 23. Time series analysis 11. Approximation of curves and surfaces from data 24. Operations research 12. Minimum/maximum of a function 25. Special functions 13. Matrix operations, inversion 26. Mathematical and computer constants Furthermore the NAG library contains extensive software concerning statistics and nancial mathematics.
19.8.3.2 IMSL Library
The IMSL library (International Mathematical and Statistical Library) consists of three synchronized parts: General mathematical methods, Statistical problems, Special functions. The sublibraries contain functions and subroutines in FORTRAN 77, FORTRAN 90 and C. Here is a contents overview:
General Mathematical Methods 1. Linear systems 6. Transformations 2. Eigenvalues 7. Non-linear equations 3. Interpolation and approximation 8. Optimization 4. Integration and di erentiation 9. Vector and matrix operations 5. Di erential equations 10. Auxiliary functions Statistical Problems 1. Elementary statistics 12. Random sampling 2. Regression 13. Life time distributions and reliability 3. Correlation 14. Multidimensional scaling 4. Variance analysis 15. Estimation of reliability function, 5. Categorization and discrete data analysis hazard rate and risk function 6. Non-parametric statistics 16. Line-printer graphics 7. Test of goodness of t and test of randomness 17. Probability distributions 8. Analysis of time series and forecasting 18. Random number generators 9. Covariance and factor analysis 19. Auxiliary algorithms 10. Discriminance analysis 20. Auxiliary mathematical tools 11. Cluster analysis Special Functions 1. Elementary functions 6. Bessel functions 2. Trigonometric and hyperbolic 7. Kelvin functions functions 8. Bessel functions with fractional orders 3. Exponential and related functions 9. Weierstrass elliptic integrals and 4. Gamma function and relatives related functions 5. Error functions and relatives 10. Di erent functions
946 19. Numerical Analysis
19.8.3.3 Aachen Library
The Aachen library is based on the collection of formulas for numerical mathematics of G. Engeln{ Mullges (Fachhochschule Aachen) and F. Reutter (Rheinisch{Westfalische Technische Hochschule Aachen). It exists in the programming languages BASIC, QUICKBASIC, FORTRAN 77, FORTRAN 90, C, MODULA 2 and TURBO PASCAL. Here is an overview: 1. Numerical methods to solve non-linear and special algebraic equations 2. Direct and iterative methods to solve systems of linear equations 3. Systems of non-linear equations 4. Eigenvalues and eigenvectors of matrices 5. Linear and non-linear approximation 6. Polynomial and rational interpolation, polynomial splines 7. Numerical di erentiation 8. Numerical quadrature 9. Initial value problems of ordinary di erential equations 10. Boundary value problems of ordinary di erential equations The programs of the Aachen library are especially suitable for the investigation of individual algorithms of numerical mathematics.
19.8.4 Application of Computer Algebra Systems 19.8.4.1 Mathematica
1. Tools for the Solution of Numerical Problems
The computer algebra system Mathematica o ers a very e ective tool that can be used to solve a large variety of numerical mathematical problems. The numerical procedures of Mathematica are totally different from symbolic calculations. Mathematica determines a table of values of the considered function according to certain previously given principles, similarly to the case of graphical representations, and it determines the solution using these values. Since the number of points must be nite, this could be a problem with \ badly " behaving functions. Although Mathematica tries to choose more nodes in problematic regions, we have to suppose a certain continuity on the considered domain. This can be the cause of errors in the nal result. It is advised to use as much information as possible about the problem under consideration, and if it is possible, then to perform calculations in symbolic form, even if this is possible only for subproblems. In Table 19.5, we represent the operations for numerical computations: Table 19.5 Numerical operations NIntegrate calculates de nite integrals n NSum calculates sums P f (i) i=1 NProduct calculates products NSolve numerically solves algebraic equations NDSolve numerically solves di erential equations After starting Mathematica the \ Prompt " In 1] := is shown it indicates that the system is ready to except an input. Mathematica denotes the output of the corresponding result by Out 1]. In general: The text in the rows denoted by In n] := is the input. The rows with the sign Out n] are given back by Mathematica as answers. The arrow ;> in the expressions means, e.g., replace x by the value a.
2. Curve Fitting and Interpolation
1. Curve Fitting Mathematica can perform the tting of chosen functions to a set of data using the
least squares method (see 6.2.5, p. 401 .) and the approximation in mean to discrete problems (see 19.6.2.2, p. 921). The general instruction is: Fit fy1 y2 : : :g funkt x]: (19.290)
19.8 Using the Computer 947
Here the values yi form the list of data, funkt is the list of the chosen functions, by which the tting should be performed, and x denotes the corresponding domain of the independent variables. If we choose funkt, e.g., as Table x^i fi 0 ng] , then the tting will be made by a polynomial of degree n. Let the following list of data be given: In 1] := l = f1:70045 1:2523 0:638803 0:423479 0:249091 0:160321 0:0883432 0:0570776 0:0302744 0:0212794g With the input In 2] := f 1 = Fit l f1 x x^ 2 x^ 3 x^ 4g x] we suppose that the elements of l are assignd to the values 1 2 : : : 10 of x. We get the following approximation polynomial of degree four: Out 2] = 2:48918 ; 0:853487x + 0:0998996x2 ; 0:00371393x3 ; 0:0000219224x4 With the command In 3] := Plot ListPlot l fx 10g] f 1 fx 1 10g AxesOrigin;> f0 0g] we get a representation of the data and the approximation curve given in Fig. 19.19a. 1.75
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Figure 19.19 For the given data this is completely satisfactory. The terms are the rst four terms of the series expansion of e1;0:5x. 2. Interpolation Mathematica o ers special algorithms for the determination of interpolation functions. They are represented as so-called interpolating function objects, which are formed similarly to pure functions. The directions for using them are in Table 19.6. Instead of the function values yi we can give a list of function values and values of speci ed derivatives at the given points. With In 3] := Plot Interpolation l] x] fx 1 10g] we get Fig. 19.19b. We can see that Mathematica gives a precise correspondence to the data list. Table 19.6 Commands for interpolation Interpolation fy1 y2 : : :g] gives an approximation function with the values yi for the values xi = i as integers Interpolation ffx1 y1 g fx2 y2 g : : :g] gives an approximation function for the point{sequence (xi yi)
3. Numerical Solution of Polynomial Equations
As shown in 20.4.2.1, p. 984, Mathematica can determine the roots of polynomials numerically. The command is NSolve p x] == 0 x n], where n prescribes the accuracy by which the calculations should be done. If we omit n, then the calculations are made to machine accuracy. We get the complete solution, i.e., m roots, if we have an input polynomial of degree m. In 1] := NSolve x^ 6 + 3x^ 2 ; 5 == 0] Out 1] = fx;> ;1:07432g fx;> ;0:867262 ; 1:15292Ig fx;> ;0:867262 + 1:15292Ig
948 19. Numerical Analysis
fx;> 0:867262 ; 1:15292Ig fx;> 0:867262 + 1:15292Ig fx;> 1:07432gg:
4. Numerical Integration
For numerical integration Mathematica o ers the procedure NIntegrate. Di erently from the symbolical method, here it works with a table of values of the integrand. We consider two improper integrals (see 8.2.3, p. 453) as examples. A: In 1] := NIntegrate Exp ;x^ 2] fx ;Infinity Infinityg] Out 1] = 1:77245. B: In 2] := NIntegrate 1=x^2 fx ;1 1g] Power::infy: In nite expression 01 encountered. NIntegrate::inum: Integrand ComplexIn nity is not numerical atfxg = f0g: Mathematica recognizes the discontinuity of the integrand at x = 0 in example B and gives a warning. Mathematica applies a table of values with a higher number of nodes in the problematic domain, and it recognizes the pole. However, the answer can be still wrong. Mathematica applies certain previously speci ed options for numerical integration, and in some special cases they are not sucient. We can determine the minimal and the maximal number of recursion steps, by which Mathematica works in a problematic domain, with parameters MinRecursion and MaxRecursion. The default options are always 0 and 6. If we increase these values, although Mathematica works slower, it gives a better result. In 3] := NIntegrate Exp ;x^ 2] fx ;1000 1000g] Mathematica cannot nd the peak at x = 0, since the integration domain is too large, and the answers is: NIntegrate::ploss: Numerical integration stopping due to loss of precision. Achieved neither the requested PrecisionGoal nor AccuracyGoal suspect one of the following: highly oscillatory integrand or the true value of the integral is 0: Out 3] = 1:34946 10;26 If we require In 4] := NIntegrate Exp ;x^ 2] fx ;1000 1000g MinRecursion;> 3 MaxRecursion;> 10] then we get Out 4] = 1:77245 Similarly, we get a result closer to the actual value of the integral with the command: NIntegrate fun fx xa x1 x2 : : : xe g]: (19.291) We can give the points of singularities xi between the lower and upper limit of the integral to force Mathematica to evaluate more accurately.
5. Numerical Solution of Di erential Equations
In the numerical solution of ordinary di erential equations and also in the solution of systems of di erential equations Mathematica represents the result by an InterpolatingFunction. It allows us to get the numerical values of the solution at any point of the given interval and also to sketch the graphical representation of the solution function. The most often used commands are represented in Table 19.7. Table 19.7 Commands for numerical solution of di erential equations dgl y fx xa xeg] computes the numerical solution of the di erential equation in the domain between xa and xe InterpolatingFunction liste] x] gives the solution at the point x Plot Evaluate y x]=: l os]] fx xa xe g] scetches the gravical representation NDSolve
19.8 Using the Computer 949
Solution of a di erential equation describing the motion of a heavy object in a medium with friction. The equations of motion in two dimension are q q x = ; x_ 2 + y_ 2 x_ y = ;g ; x_ 2 + y_ 2 y:_ The friction is supposed to be proportional to the velocity. If we substitute g = 10 = 0:1, then the following command can be given to solve the equations of motion with initial values x(0) = y(0) = 0 and x_ (0) = 100 y_ (0) = 200: In 1] := dg = NDSolve fx00 t] == ;0:1Sqrt x0 t]^ 2 + y 0 t]^ 2] x0 t] y 00 t] == ;10 ;0:1Sqrt x0 t]^2 + y0 t]^2] y0 t] x 0] == y 0] == 0 x0 0] == 100 y0 0] == 200g fx yg ft 15g] Mathematica gives the answer by the interpolating function: Out 1] = ffx;> InterpolatingFunction f0: 15:g ] y;> InterpolatingFunction f0: 15:g ]gg We can represent the solution In 2] := ParametricPlot fx t] y t]g=:dg ft 0 2g PlotRange;> All] as a parametric curve (Fig. 19.20a). NDSolve accepts several options which a ect the accuracy of the result. The accuracy of the calculations can be given by the command AccuracyGoal. The command PrecisionGoal works similarly. During calculations, Mathematica works according to the so-called WorkingPre cision, which should be increased by ve units in calculations requiring higher accuracy. The numbers of steps by which Mathematica works in the considered domain is prescribed as 500. In general, Mathematica increases the number of nodes in the neighborhood of the problematic domain. In the neighborhood of singularities it can exhaust the step limit. In such cases, it is possible to increase the number of steps by MaxSteps. It is also possible the prescribe Infinity for MaxSteps. The equations for the Foucault pendulum are: x(t) + !2x(t) = 2)y_ (t) y(t) + !2y(t) = ;2)x_ (t): With ! = 1 ) = 0:025 and the initial conditions x(0) = 0 y(0) = 10 x_ (0) = y_ (0) = 0 we get the solution: In 3] := dg 3 = NDSolve fx00 t] == ;x t] + 0:05y 0 t] y 00 t] == ;y t] ; 0:05x0 t] x 0] == 0 y 0] == 10 x0 0] == y0 0] == 0g fx yg ft 0 40g] Out 3] = ffx;> InterpolatingFunction f0: 40:g ] y;> InterpolatingFunction f0: 40:g ]gg With In 4] := ParametricPlot fx t] y t]g=:dg 3 ft 0 40g AspectRatio;> 1] we get Fig. 19.20b.
19.8.4.2 Maple
The computer algebra system Maple can solve several problems of numerical mathematics with the use of built-in approximation methods. The number of nodes, which is required by the calculations, can be determined by specifying the value of the global variable Digits as an arbitrary n. But we should not forget that selecting a higher n than the prescribed value results in a lower speed of calculation.
1. Numerical Calculation of Expressions and Functions
After starting Maple, the symbol \ Prompt " > is shown, which denotes the readyness for input. Connected in- and outputs are often represented in one row, separated by the arrow operator ;! .
950 19. Numerical Analysis 10
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Figure 19.20 1. Operator evalf Numerical values of expressions containing built-in and user-de ned functions which can be evaluated as a real number, can be calculated with the command evalf(expr n): (19.292) expr is the expression whose value should be determined the argument n is optional, it is for evaluation to n digits accuracy. Default accuracy is set by the global p variable Digits. Prepare a table of values of the function y = f (x) = x + ln x. First, we de ne the function by the arrow operator: p > f := z ;> sqrt(z) + ln(z) ;! f := z ;> x + ln x: Then we get the required values of the function with the command evalf(f (x)) , where a numerical value should be substituted for x. A table of values of the function with steps size 0:2 between 1 and 4 can be obtained by > for x from 1 by 0:2 to 4 do print(f x] = evalf(f (x) 12)) od Here, it is required to work with twelve digits. Maple gives the result in the form of a one-column table with elements in the form f3:2] = 2:95200519181. 2. Operator evalhf(expr): Beside evalf there is the operator evalhf. It can be used in a similar way to evalf. Its argument is also an expression which has a real value. It evaluates the symbolic expression numerically, using the hardware oating-point double-precision calculations available on the computer. A Maple oating-point value is returned. Using evalhf speeds up your calculations in most cases, but you lose the de niable accuracy of using evalf and Digits together. For instance in the problem in 19.8.2, p. 939, it can produce a considerable error.
2. Numerical Solution of Equations
As discussed in Chapter 20, see 20.4.4.2, p. 995, by using Maple we can solve equations or systems of equations numerically in many cases. The command to do this is fsolve. It has the syntax fsolve(eqn var option): (19.293) This command determines real solutions. If eqn is in polynomial form, the result is all the real roots. If eqn is not in polynomial form, it is likely that fsolve will return only one solution. The available options are given in Table 19.8. Table 19.8 Options for the command fsolve complex determines a complex root (or all roots of a polynomial) maxsols = n determines at least the n roots (only for polynomial equations) fulldigits ensures that fsolve does not lower the number of digits used during computations intervall looks for roots in the given interval
19.8 Using the Computer 951
A: Determination of all solutions of a polynomial equation x6 + 3x2 ; 5 = 0. With > eq := x^ 6 + 3 # x^2 ; 5 = 0 :
we get > fsolve(eq x) ;! ;1:074323739 1:074323739 Maple determined only the two real roots. With the option complex, we get also the complex roots: > fsolve(eq x complex) ;1:074323739 ;0:8672620244 ; 1:152922012I ;0:8672620244 + 1:152922012I 0:8672620244 ; 1:152922012I 0:8672620244 + 1:152922012I 1:074323739 B: Determination of both solutions of the transcendental equation e;x3 ; 4x2 = 0. After de ning the equation > eq := exp(;x^ 3) ; 4 # x^2 = 0 we get > fsolve(eq x) ;! 0:4740623572 as the positive solution. With > fsolve(eq x x = ;2::0) ;! ;0:5412548544 Maple also determines the second (negative) root.
3. Numerical Integration
The determination of de nite integrals is often possible only numerically. This is the case when the integrand is too complicated, or if the primitive function cannot be expressed by elementary functions. The command to determine a de nite integral in Maple is evalf: evalf(int(f (x) x = a::b) n): (19.294) Maple calculates the integral by using an approximation formula. Z2 Calculation of the de nite integral e;x3 dx. Since the primitive function is not known, for the ;2 integral command we get the following answer Z2 > int(exp(;x^ 3) x = ;2::2) ;! e;x3 dx: ;2 If we type > evalf(int(exp(;x^ 3) x = ;2::2) 15) then we get 277:745841695583. Maple used the built-in approximation method for numerical integration with 15 digits. In certain cases this method fails, especially if the integration interval is too large. Then, we can try to call another approximation procedure with the call to a library readlib( evalf=int ) : which applies an adaptive Newton method. The input > evalf(int(exp(;x^ 2) x = ;1000::1000)) results in an error message. With > readlib(evalf=int) : > evalf=int(exp(;x^ 2) x = ;1000::1000 10 NCrule) 1:772453851 we get the correct result. The third argument speci es the accuracy and the last one speci es the internal notation of the approximation method.
952 19. Numerical Analysis
4. Numerical Solution of Di erential Equations
We solve ordinary di erential equations with the Maple operation dsolve given in 20.4.4, 5., p. 994. However, in most cases it is not possible to determine the solution in closed form. In these cases, we can try to solve it numerically, where we have to give the corresponding initial conditions. In order to do this, we use the command dsolve in the form dsolve(deqn var numeric) (19.295) with the option numeric as a third argument. Here, the argument deqn contains the actual di erential equation and the initial conditions. The result of this operation is a procedure, and if we denote it, e.g., by f , for using the command f (t), we get the value of the solution function at the value t of the independent variable. Maple applies the Runge{Kutta method to get this result (see 19.4.1.2, p. 904). The default accuracy for the relative and for the absolute error is 10;Digits+3. The user can modify these default error tolerances with the global symbols RELERR and ABSERR. If there are some problems during calculations, then Maple gives di erent error messages. At solving the problem given in the Runge{Kutta methods in 19.4.1.2, p. 905, Maple gives: > r := dsolve(fdiff(y(x) x) = (1=4) # (x^ 2 + y(x)^2) y(0) = 0g y(x) numeric) r := proc dsolve=numeric=result2 (x 1592392 1]) end With > r(0:5) ;! fx(:5) = 0:5000000000 y(x)(:5) = 0:01041860472g we can determine the value of the solution, e.g., at x = 0:5.
953
20 ComputerAlgebraSystems 20.1 Introduction
20.1.1 Brief Characterization of Computer Algebra Systems 1. General Purpose of Computer Algebra Systems
The development of computers has made possible the introduction of computer algebra systems for \doing mathematics". They are software systems able to perform mathematicaloperations formally. These systems, such as Macsyma, Reduce, MuPad, Maple, Mathematica, can also be used on relatively small computers (PC), and with their help, we can transform complicated expressions, calculate derivatives and integrals, solve systems of equations, represent functions of one and of several variables variables graphically, etc. They can manipulate mathematical expressions, i.e., they can transform and simplify mathematical expressions according to mathematical rules if this is possible in closed form. They also provide a wide range of numerical solutions to required accuracy, and they can represent functional dependence between data sets graphically. Most computer algebra systems can import and export data. Besides a basic o er of de nitions and procedures which are activated at every start of the system, most systems provide a large variety of libraries and program packages from special elds of mathematics, which can be loaded and activated on request (see 20.4]). Computer algebra systems allow users to build up their own packages. However, the possibilities of computer algebra systems should not be overestimated. They spare us the trouble of boring, time-demanding, and mechanical computations and transformations, but they do not save us from thinking. For frequent errors see 19.8.2, p. 939.
2. Restriction to Mathematica and Maple
The systems are under perpetual developing. Therefore, every concrete representation reects only a momentary state. In the following, we introduce the basic idea and applications of these systems for the most important elds of mathematics. This introduction will help for the rst steps in working with computer algebra systems. In particular, we discuss the two systems Mathematica (version 4.1) and Maple 8. These two systems seem to be very popular among users, and the basic structure of the other systems is similar.
3. Input and output in Mathematica and Maple
In this book, we do not discuss how computer algebra systems are installed on computers. It is assumed that the computer algebra system has already been started by a command, and it is ready to communicate by command lines or in a Windows-like graphical environment. The input and output is always represented for both Mathematica (see 19.8.4.1, 1., p. 946) and Maple (see 19.8.4.2, 1., p. 949) in rows which are distinguished from other text-parts, e.g., in the form In 1] := Solve 3 x ; 5 == 0 x] in Mathematica, (20.1) > solve(3 # x ; 5 = 0 x) in Maple. System speci c symbols (commands, type notation, etc.) will be represented in typewriter style. In order to save space, we often write the input and the output in the same row in this book, and we separate them by the symbol ;!.
20.1.2 Examples of Basic Application Fields 20.1.2.1 Manipulation of Formulas
Formula manipulation means here the transformation of mathematical expressions in the widest sense, e.g., simpli cation or transformation into a useful form, representation of the solution of equations or equation systems by algebraic expressions, di erentiation of functions or determination of inde nite integrals, solution of di erential equations, formation of in nite series, etc.
954 20. Computer Algebra Systems Solution of the following quadratic equation: x2 + ax + b = 0 with a b 2 IR: (20.2a) In Mathematica, we type (note the blank between a and x): Solve x^ 2 + a x + b == 0 x] : (20.2b) After pressing the corresponding input key/keys (ENTER or SHIFT+RETURN, depending on the operation system), Mathematica replaces this row by In 1] := Solve x^ 2 + a x + b == 0 x] (20.2c) and starts the evaluation process. In a moment, the answer appears in a new row ;a + Sqrt a2 ; 4b] g f x ;> ;a ; Sqrt a2 ; 4b] gg: Out 1] = ffx ;> (20.2d) 2 2 Mathematica has solved the equation and both solutions are represented in the form of a list consisting of two sublists. Here Sqrt means the square root. In Maple the input has the following form: > solve(fx^2 + a # x + b = 0g fxg) (20.3a) The semicolon after the last symbol is very important. After the equation is entered with the ENTER key, Maple evaluates this input and the resulting output is displayed directly below the input f1=2(;a + (a2 ; 4b)1=2 )g f1=2(;a ; (a2 ; 4b)1=2 )g (20.3b) The result is given in the form of a sequence of two sets representing the solutions. Except for some special symbols of the systems, the basic structures of commands are very similar. At the beginning there is a symbol, which is interpreted by the system as an operator, which is applied to the operand given in braces or in brackets. The result is displayed as a list or sequence of solutions or answers. Several operations and formula manipulations are represented similarly.
20.1.2.2 Numerical Calculations
Computer algebra systems provide many procedures to handle numerical problems of mathematics. These are solutions of algebraic equations, linear systems of equations, the solutions of transcendental equations, calculation of de nite integrals, numerical solutions of di erential equations, interpolation problems, etc. Problem: Solution of the equation x6 ; 2x5 ; 30x4 + 36x3 + 190x2 ; 36x ; 150 = 0: (20.4a) Although this equation of degree six cannot be solved in closed form, it has six real roots, which are to be determined numerically. In Mathematica the input is: In 2] := NSolve x^ 6 ; 2x^ 5 ; 30x^ 4 + 36x^ 3 + 190x^ 2 ; 36x ; 150 == 0 x] (20.4b) It results in the answer: Out 2] = ffx;> ;4:42228g fx;> ;2:14285g fx;> ;0:937397g fx;> 0:972291g fx;> 3:35802g fx;> 5:17217gg (20.4c) This is a list of the six solutions with a certain accuracy which will be discussed later. The input in Maple is: > fsolve(fx^6 ; 2 # x^5 ; 30 # x^4 + 36 # x^3 + 190 # x^2 ; 36 # x ; 150 = 0g fxg) (20.4d) Here, the input of \= 0" can be omitted, and the assignment of the variable \fxg" is also not necessary here, since it is the only one. Maple automatically considers the entered expression to be equal to zero.
20.1 Introduction 955
The output is the sequence of the six solutions. The application of the command fsolve tells to Maple that the result is wanted numerically in the form of oating-point decimal numbers.
20.1.2.3 Graphical Representations
Most computer algebra systems allow the graphical representation of built-in and self-de ned functions. Usually, this is the representation of functions of one variable in Cartesian and in polar coordinates, parametric representation, and representation of implicit functions. Functions of two variables can be represented as surfaces in space parametric representation is also possible. Further more curves can be demonstrated in three-dimensional space. In addition, there are further graphical possibilities to demonstrate functional dependence between data sets, e.g., in the form of diagrams. All systems provide a wide selection of options of representation such as line thickness of the applied elements, e.g., vectors, di erent colours, etc. Usually, the represented graphics can be exported in an appropriate format such as eps, gif, jpeg, bmp and they can be built into other programs, or directly printed on a printer or plotter.
20.1.2.4 Programming in Computer Algebra Systems
All systems allow useres to develop their own packages to solve special problems. This means both the use of well-known tools to build up procedures, e.g., DO, IF { THEN, WHILE, FOR, etc. and, on the other hand the application of the built-in methods of the system which allow elegant solutions for many problems. Self-constructed program blocks can be introduced into the libraries and they can be reactivated at any time.
20.1.3 Structure of Computer Algebra Systems 20.1.3.1 Basic Structure Elements 1. Type of Objects
Computer algebra systems work with several di erent types of objects. Objects are the known family of numbers, variables, operators, functions, etc., which are loaded at the start of the system, or which are de ned by the user according to a suitable syntax. Classes of objects, like type of numbers or lists, etc., are called types. Most of the objects are identi ed by their names, which one can think of as associated to an object class and which must satisfy the given grammatical rules. The user gives a sequence of objects in the input row, i.e., their names, corresponding to the given syntax, closes the input with the corresponding special key and/or by a special system command, then the system starts evaluation and returns the result in a new row or rows. (Input can be spread over several lines.) The objects, object types and object classes described below are available in every computer algebra system, and their particular specialities are described in the manuals for the system.
2. Numbers
Computer algebra systems usually use the number types integers, rational numbers, real numbers (oating-point numbers), complex numbers some systems also know algebraic numbers, radical numbers, etc. With di erent type-check operations, the type or certain properties of given numbers can be determined, like non-negative, prime, etc. Floating-point numbers can be determined with arbitrary accuracy. Usually, the systems work with a default precision, which can be changed on request. The systems know the special numbers, which have a fundamental importance in mathematics, such as e, and 1. They use these numbers symbolically, but they also know their numerical approximation to an arbitrary accuracy.
956 20. Computer Algebra Systems
3. Variables and Assignments
Variables have names represented by given symbols which are usually determined by the user. There are names that are prede ned and reserved by the system they cannot be chosen. While no value is assigned to the variable, the symbol itself stands for the variable. Values can be assigned to the variables by special assignment operators. These values can be numbers, other variables, special sequences of objects, or even expressions. In general, there exist some assignment operators which di er, rst of all, in the time of their evaluation, i.e., right after their input or for the rst time at a later call of the variable.
4. Operators
All systems have a basic set of operators. The usual operators of mathematics + ; # = ^ (or # #) > < = belong to this set, for which the usual order of precedence is valid for evaluation. The inx form of an expression means that these operators are written between the operands. The set of operators written in prex form { where the operator is written in front of the operand { is dominant in all systems. This type of operator operates on special classes of objects, e.g., on numbers, polynomials, sets, lists, matrices, or on systems of equations, or they operate as functional operators, such as di erentiation, integration, etc. In general, there are operators for organizing the form of the output, manipulating strings and further systems of objects. Some systems allow us to represent some operators in sux form, i.e., the operator is behind the operand. Operators often use optional arguments.
5. Terms and Functions
The notion of term means an arrangement of objects connected by mathematical operators, usually in in x form, hence, it means certain basic elements often occurring in mathematics. A basic task of computer algebra systems is transforming terms and solving equations. The following sequence x^4 ; 5 # x^3 + 2 # x^ 2 ; 8 (20.5) is, e.g., a term, in which x is a variable. Computer algebra systems know the usual elementary functions such as the exponential function, logarithmic function, trigonometric functions, and their inverses and several other special functions. These functions can be placed into terms instead of variables. In this way, complicated terms or functions can be generated.
6. Lists and Sets
All computer algebra systems know the object class of lists, which is considered as a sequence of objects. The elements of a list can be reached by special operators. In general, the elements of a list can be lists themselves. So, we can get nested lists, which are used in the construction of special types of objects, such as matrices, tensors all systems provides such special object classes. They make it possible to manipulate objects like vectors and tensors symbolically in vector spaces, and to apply linear algebra. The notion of set is also known in computer algebra systems. The operators of set theory are de ned. In the following sections, the basic structure elements and their syntax will be discussed for the two chosen computer algebra systems, Mathematica 4.1 and Maple 8.
20.2 Mathematica
Mathematica is a computer algebra system, developed by Wolfram Research Inc. A detailed description of Mathematica 4.1 can be found in 20.7, 20.11].
20.2.1 Basic Structure Elements
In Mathematica the basic structure elements are called expressions. Their syntax is (we emphasize again, the current objects are given by their corresponding symbol, by their names): obj0 obj1 obj2 : : : objn ] (20.6)
20.2 Mathematica 957 obj0 is called the head of the expression the index 0 is assigned to it. The parts obji (i = 1 : : : n) are the elements or arguments of the expression, and one can refer to them by their indices 1 : : : n. In many cases the head of the expression is an operator or a function, the elements are the operands or variables on which the head acts. Also the head, as an element of an expression, can be an expression, too. Square brackets are reserved in Mathematica for the representation of an expression, and they can be applied only in this relation. The term x^ 2+2 # x +1, which can also be entered in this in x form in Mathematica, has the complete form (FullForm) Plus 1 Times 2 x] Power x 2]] which is also an expression. Plus, Power and Times denote the corresponding arithmetical operations. The example shows that all simple mathematical operators exist in pre x form in the internal representation, and the term notation is only a facility in Mathematica. Parts of expressions can be separated. This can be done with Part expr i], where i is the number of the corresponding element. In particular, i = 0 gives back the head of the expression. If we enter in Mathematica In 1] := x^ 2 + 2 # x + 1 where the sign # can be omitted, then after the ENTER key is pressed, Mathematica answers Out 1] = 1 + 2x + x2 Mathematica analyzed the input and returned it in mathematical standard form. If the input had been terminated by a semicolon, then the output would have been suppressed. If we enter In 2] := FullForm %] then the answer is Out 2] = Plus 1 Times 2 x] Power x 2]] The sign % in the square brackets tells Mathematica that the argument of this input is the last output. From this expression it is possible to get, e.g., the third element In 3] := Part % 3] for instance Out 3] = Power x 2] which is an expression in this case. Symbols in Mathematica are the notation of the basic objects they can be any sequence of letters and numbers but they must not begin with a number. The special sign $ is also allowed. Upper-case and lower-case letters are distinguished. Reserved symbols begin with a capital letter, and in compound words also the second word begins with a capital letter. Users should write their own symbols using only lower-case letters.
20.2.2 Types of Numbers in Mathematica
20.2.2.1 Basic Types of Numbers in Mathematica
Mathematica knows four types of numbers represented in Table 20.1. Table 20.1 Types of numbers in Mathematica
Type of number Head
Integers Rational numbers Real numbers Complex numbers
Integer Rational Real Complex
Characteristic
Input
exact integer, arbitrarily long nnnnn fraction of coprimes in form Integer=Integer pppp=qqqq oating-point number, arbitrary given precision nnnn:mmmm complex number in the form number+number #I
Real numbers, i.e., oating-point numbers, can be arbitrarily long. If an integer nnn is written in the form nnn:, then Mathematica considers it as a oating-point number, that is, of type Real.
958 20. Computer Algebra Systems The type of a number x can be determined with the command Head x]. Hence, In 1] := Head 51] results in Out 1] = Integer, while In 2] := Head 51:] Out 2] = Real. The real and imaginary components of a complex number can belong to any type of numbers. A number such as 5:731 + 0 I is considered as a Real type by Mathematica, while 5:731 + 0. I is of type Complex, since 0: is considered as a oating-point approximation of 0. There are some further operations, which give information about numbers. So, if x is a number In 3] := NumberQ x] Out 3] := True (20.7a) Otherwise, if x is not a number, then the output is Out 3] =False. Here, True and False are the symbols for Boolean constants. IntegerQ x] tests if x is an integer, or not, so In 4] := IntegerQ 2:] Out 4] = False (20.7b) Similar tests can be performed for numbers with operators EvenQ, OddQ and PrimeQ. Their meanings are obvious. So, we get In 5] := PrimeQ 1075643] Out 5] = True (20.7c) while In 6] := PrimeQ 1075641] Out 6] = False (20.7d) These last tests belong to a group of test operators, which all end by Q and always answer True or False in the sense of a logical test (in this case a type check).
20.2.2.2 Special Numbers
In Mathematica, there are some special numbers which are often needed, and they can be called with as the transformation arbitrary accuracy. They include with the symbol Pi, e with the symbol E, 180 factor from degree measure into radian measure with the command Degree, Infinity as the symbol for 1 and the imaginary unit I.
20.2.2.3 Representation and Conversion of Numbers
Numbers can be represented in di erent forms which can be converted into each other. So, every real number x can be represented by a oating-point number N x n] with an n-digit precision. IN 7] := N E 20] yields Out 7] = 2:7182818284590452354 (20.8a) With Rationalize x dx], the number x with an accuracy dx can be converted into a rational number, i.e., into the fraction of two integers. 1457 In 8] := Rationalize % 10^ ; 5] Out 8] = (20.8b) 536 With 0 accuracy, Mathematica gives the possible best approximation of the number x by a rational number. Numbers of di erent number systems can be converted into each other. With BaseForm x b], the number x given in the decimal system is converted into the corresponding number in the number system with base b. If b > 10, then the consecutive letters of the alphabet a b c : : : are used for the further digits having a meaning greater than ten.
A:
In 15] := BaseForm 255 16] Out 15] = ==BaseForm = ff16 In 16] := BaseForm N E 10] 8] Out 16] = ==BaseForm = 2:5576052138
(20.9a) (20.9b) The conversion of a number of base b into the decimal system can be performed by b^^ mmmm.
B:
In 17] := 8 ^^ 735 Out 17] = 477
(20.9c) Numbers can be represented with arbitrary precision (the default here is the hardware precision), and for large numbers so-called scienti c form is used, i.e., the form n:mmmm10^ qq.
20.2 Mathematica 959
20.2.3 Important Operators
Several basic operators can be written in in x form (as in the classical form in mathematics) < symb1 op symb2 > . However, in every case, the complete form of this simpli ed notation is an expression. The most often occurring operators and their complete form are collected in Table 20.2. Table 20.2 Important Operators in Mathematica a+b Plus a b] u == v Equal u v] a b or a # b Times a b] w! = v Unequal w v] a^b Power a b] r > t Greater r t] a=b Times a Power b ;1]] r >= t GreaterEqual r t] u;> v Rule u v ] s < t Less s t] r=s Set r s] s v ] or t=: u;> v means that every element u which occurs in the expression t will be replaced by the expression v. In 5] := x + y 2 =: y ;> a + b Out 5] = x + (a + b)2 It is typical in the case of both operators that the right-hand side is evaluated immediately after the assignment or transformation rule. So, the left-hand side will be replaced by this evaluated right-hand side at every later call. Here, we have to mention two further operators with delayed evaluation. u := v FullForm = SetDelayed u v] and (20.10a) u :> v FullForm = RuleDelayed u v] (20.10b) The assignment or the transformation rule are also valid here until it is changed. Although the lefthand side is always replaced by the right side, the right-hand side is evaluated for the rst time only at the moment when the left one is called. The expression u == v or Equal u v] means that u and v are identical. Equal is used, e.g., in manipulation of equations.
20.2.4 Lists
20.2.4.1 Notions
Lists are important tools in Mathematica for the manipulation of whole groups of quantities, which are important in higher-dimensional algebra and analysis. A list is a collection of several objects into a new object. In the list, each object is distinguished only by its place in the list. The de nition of a list is made by the command List a1 a2 a3 : : :] fa1 a2 a3 : : :g (20.11) To explain the work with lists, a concrete list is used, denoted by l1: In 1] := l1 = List a1 a2 a3 a4 a5 a6] Out 1] = fa1 a2 a3 a4 a5 a6g (20.12) Mathematica applies a short form to the output of the list: It is enclosed in curly braces. Table 20.3 represents commands which choose one or more elements from a list, and the output is a \sublist".
960 20. Computer Algebra Systems Table 20.3 Commands for the choice of list elements First l] selects the rst element Last l] selects the last element Part l n] or l n]] selects the n-th element Part l fn1 n2 : : :g] gives a list of the elements with the given numbers l fn1 n2 : : :g]] equivalent to the previous operation Take l m] gives the list of the rst m elements of l Take l fm ng] gives the list of the elements from m through n Drop l n] gives the list without the rst n elements Drop l fm ng] gives the list without the elements from m through n For the list l1 in (20.11) we get, e.g., In 2] := First l1] Out 2] = a1 In 3] := l1 3]] In 4] := l1 f2 4 6g]] Out 4] = fa2 a4 a6g In
20.2.4.2 Nested Lists, Arrays or Tables
Out 3] = a3 5] := Take l1
2]
Out 5] = fa1
a2g
The elements of lists can again be lists, so we get nested lists. If we enter, e.g., for the elements of the previous list l1 In 6] := a1 = fb11 b12 b13 b14 b15g In 7] := a2 = fb21 b22 b23 b24 b25g In 8] := a3 = fb31 b32 b33 b34 b35g and analogously for a4 a5 and a6, then because of (20.12) we get a nested list (an array) which we do not represent here explicitly. We can refer to the j -th element of the i-th sublist with the command Part l i j ]. The expression l i j ]] has the same result. In the above example, e.g., In 12] := l1 3 4]] yields Out 12] = b34 Furthermore, Part l fi1 i2 : : :g fj 1 j 2 : : :g] or l fi1 i2 : : :g fj 1 j 2 : : :g]] results in a list consisting of the elements numbered with j 1 j 2 : : : from the lists numbered with i1 i2 : : :. For the above example In 13] := l1 f3 5g f2 3 4g]] Out 13] = ffb32 b33 b34g fb52 b53 b54gg The idea of nesting lists is obvious from these examples. It is easy to create lists of three or higher dimensions, and it is easy to refer to the corresponding elements. Table 20.4 Operations with lists Position l a] gives a list of the positions where a occurs in the list MemberQ l a] checks whether a is an element of the list FreeQ l a] checks if a does not occur in the list Prepend l a] changes the list by adding a to the front Append l a] changes the list by appending a to the end Insert l a i] inserts a at position i in the list Delete l fi j : : :g] delete the elements at positions i j : : : from the list ReplacePart l a i] replace the element at position i by a
20.2.4.3 Operations with Lists
Mathematica provides several further operations by which lists can be monitored, enlarged or shortened
(Table 20.4). With Delete, the list l1 can be shortened by the term a6: In 14] := l2 = Delete l1 6] Out 14] = fa1 a2 a3 a4 a5g
20.2 Mathematica 961
where in the output the ai are shown by their values { they are lists themselves.
20.2.4.4 Special Lists
In Mathematica, several operations are available to create special lists. One of them, which often occurs in working with mathematical functions, is the command Table shown in Table 20.5. Table 20.5 Operation Table Table f fimaxg] creates a list with imax values of f :f (1) f (2) : : : f (imax) Table f fi imin imaxg] creates a list with values of f from imin to imax Table f fi imin imax dig] the same as the last one, but by steps di Table of binominal coecients for n = 7: In 15] := Table Binomial 7 i] fi 0 7g]] Out 15] = f1 7 21 35 35 21 7 1g With Table, also higher-dimensional arrays can be created. With the expression Table f fi i1 i2g fj j 1 j 2g : : :] we get a higher-dimensional, multiple nested table, i.e., entering In 16] := Table Binomial i j ] fi 1 7g fj 0 ig] we get the binominal coecients up to degree 7: Out 16] = ff1 1g f1 2 1g f1 3 3 1g f1 4 6 4 1g f1 5 10 10 5 1g f1 6 15 20 15 6 1g f1 7 21 35 35 21 7 1gg The operation Range produces a list of consecutive numbers or equally spaced numbers: Range n] yields the list f1 2 : : : ng Similarly, Range n1 n2] and Range n1 n2 dn] produce lists of numbers from n1 to n2 with step-size 1 or dn respectively.
20.2.5 Vectors and Matrices as Lists 20.2.5.1 Creating Appropriate Lists
Several special (list) commands are available for de ning vectors and matrices. A one-dimensional list of the form v = fv1 v2 : : : vng (20.13) can always be considered as a vector in n-dimensional space with components v1 v2 : : : vn. The special operation Array v n] produces the list (the vector) fv 1] v 2] : : : v n]g. Symbolic vector operations can be performed with vectors de ned in this way. The two-dimensional lists l1 of (20.2.4.2, p. 960) and l2 (20.2.4.3, p. 960) introduced above can be considered as matrices with rows i and columns j . In this case bij would be the element of the matrix in the i-th row and the j -th column. A rectangular matrix of type (6,5) is de ned by l1, and a square matrix of type (5 5) by l2. With the operation Array b fn mg] a matrix of type (n m) is generated, whose elements are denoted by b i j ]. The rows are numbered by i, i changes from 1 to n by j the columns are numbered from 1 to m. In this symbolic form l1 can be represented as l1 = Array b f6 5g] (20.14a) where b i j ] = bij (i = 1 : : : 6 j = 1 : : : 5): (20.14b) The operation IdentityMatrix n] produces the n-dimensional unit matrix. With the operation DiagonalMatrix list] a diagonal matrix is produced with the elements of the list in its main diagonal.
962 20. Computer Algebra Systems The operation Dimension list] gives the size (number of rows, columns, . . . ) of a matrix, whose structure is given by a list. Finally, with the command MatrixForm list], we get a matrix-type representation of the list. A further possibility to de ne matrices is the following: Let f (i j ) be a function of integers i and j . Then, the operation Table f fi ng fj mg] de nes a matrix of type (n m), whose elements are the corresponding f (i j ).
20.2.5.2 Operations with Matrices and Vectors
Mathematica allows formal manipulation of matrices and vectors. The operations given in Table 20.6
can be applied.
Table 20.6 Operations with matrices matrix a is multiplied by the scalar c the product of matrices a and b Det a] the determinant of matrix a Inverse a] the inverse of matrix a Transpose a] the transpose of matrix a MatrixPower a n] the n-th power of matrix a Eigenvalues a] the eigenvalues of matrix a Eigenvectors a] the eigenvectors of matrix a
ca a:b
A:
In 18] := r = Array a
In 19] := Transpose r]
f4 4g]
f f a 1 1] a 1 2] a 1 3] a 1 4] g f a 2 1] a 2 2] a 2 3] a 2 4] g f a 3 1] a 3 2] a 3 3] a 3 4] g f a 4 1] a 4 2] a 4 3] a 4 4] gg 19] = f f a 1 1] a 2 1] a 3 1] a 4 1] g f a 1 2] a 2 2] a 3 2] a 4 2] g f a 1 3] a 2 3] a 3 3] a 4 3] g f a 1 4] a 2 4] a 3 4] a 4 4] gg
Out 18] =
Out
Here, the transpose matrix rT of r is produced. If the general four-dimensional vector v is de ned by In 20] := v = Array u 4] then we get Out 20] = fu 1] u 2] u 3] u 4]g Now, the product of the matrix r and the vector v is again a vector (see Calculations with Matrices, 4.1.4, p. 254). In 21] := r: v Out 21] = f a 1 1] u 1] + a 1 2] u 2] + a 1 3] u 3] + a 1 4] u 4] a 2 1] u 1] + a 2 2] u 2] + a 2 3] u 3] + a 2 4] u 4] a 3 1] u 1] + a 3 2] u 2] + a 3 3] u 3] + a 3 4] u 4] a 4 1] u 1] + a 4 2] u 2] + a 4 3] u 3] + a 4 4] u 4] g: There is no di erence between row and column vectors in Mathematica. In general, matrix multiplication is not commutative (see Calculations with Matrices 4.1.4, p. 254). The expression r: v corresponds to the product in linear algebra when a matrix is multiplied by a column vector from the right, while v: r means a multiplication by a row vector from the left. B: In the section on Cramer's rule (4.4.2.3, p. 275) the linear system of equations pt = b is solved with the matrix In 22] := p = MatrixForm ff2 1 3g f1 ;2 1g f3 2 2gg] Out 22] = ==MatrixForm = 2 1 3 1 ;2 1 3 22
20.2 Mathematica 963
and vectors
3] Out 23] = fx 1] x 2] x 3]g f9 ;2 7g Out 24] = f9 ;2 7g: Since in this case det(p) = 6 0 holds, the system can be solved by t = p;1 b. This can be done by In 25] := Inverse p]: b with the output of the solution vector Out 25] = f;1 2 3g: In 23] := t = Array x In 24] := b =
20.2.6 Functions
20.2.6.1 Standard Functions
Mathematica knows several mathematical standard functions, which are listed in Table 20.7.
Exponential function Logarithmic functions Trigonometric functions Arc functions Hyperbolic functions Area functions
x] x], x],
Table 20.7 Standard functions
Exp Log Log Sin Cos Tan Cot Sec Csc ArcSin ArcCos ArcTan ArcCot ArcSec ArcCsc Sinh Cosh Tanh Coth Sech Csch ArcSinh ArcCosh ArcTanh ArcCoth ArcSech ArcCsch
b,x] x],
x], x], x], x], x], x], x], x], x], x], x], x], x],
x] x], x],
x],
x]
x],
x]
x],
x]
All these functions can also be applied with complex arguments. In every case we have to consider the single-valuedness of the functions. For real functions we have to choose one branch of the function (if it is needed) for functions with complex arguments the principal value (see 14.5, p. 698) should be chosen.
20.2.6.2 Special Functions
Mathematica knows several special functions, which are not elementary functions.
some of these functions.
Table 20.8 lists
Tabelle 20.8 Special functions Bessel functions Jn(z) and Yn(z) BesselJ n,z], BesselY n,z] Modi ed Bessel functions In(z) and Kn(z) BesselI n,z], BesselK n,z] Legendre polynomials Pn(x) LegendrP n,x] Spherical harmonic Ylm ( &) SphericalHarmonicY l m theta phi]
Further functions can be loaded with the corresponding special packages (see also 20.11]).
20.2.6.3 Pure Functions
Mathematica supports the use of so-called pure functions. A pure function is an anonymous function,
an operation with no name assigned to it. They are denoted by Function x body]. The rst argument speci es the formal parameters and the second one is the body of the function, i.e., body is an expression for the function of the variable x. In 1] := Function x x^ 3 + x^ 2] Out 1] = Function x x3 + x2 ] (20.15) and so In 2] := Function x x^ 3 + x^ 2] c] gives Out 2] = c3 + c2 : (20.16) We can use a simpli ed version of this command. It has the form body &, where the variable is denoted by 7 . Instead of the previous two rows we can also write (20.17) In 3] := (7^ 3 + 7^ 2) & c] Out 3] = c3 + c2 It is also possible to de ne pure functions of several variables: Function fx1 x2 : : :g body ] or in short form body &, where the variables in body are denoted by the
964 20. Computer Algebra Systems elements 71 72 : : :. The sign & is very important for closing the expression, since it can be seen from this sign that the previous expression should be considered as a pure function.
20.2.7 Patterns
Mathematica allows users to de ne their own functions and to use them in calculations.
With the command In 1] := f x ] := Polynom(x) (20.18) with Polynom(x) as an arbitrary polynomial of variable x, a special function is de ned by the user. In the de nition of the function f, there is no simple x, but x (pronounced x-blank) with a symbol for the blank. The symbol x means \something with the name x". From here on, every time when the expression f something] occurs, Mathematica replaces it by its de nition given above. This type of de nition is called a pattern. The symbol blank denotes the basic element of a pattern y stands for y as a pattern. It is also possible to apply in the corresponding de nition only a \ ", that is y^ . This pattern stands for an arbitrary power of y with any exponent, thus, for an entire class of expressions with the same structure. The essence of a pattern is that it de nes a structure. When Mathematica checks an expression with respect to a pattern, it compares the structure of the elements of the expression to the elements of the pattern, Mathematica does not check mathematical equality! This is important in the following example: Let l be the list In 2] := l = f1 y y ^a y ^Sqrt x] ff y ^ (r=q )] 2^ y gg (20.19) If we write In 3] := l =: y ^ ;> ja (20.20) then Mathematica returns the list Out 3] = f1 y ja ja ff ja] 2y gg (20.21) Mathematica checked the elements of the list with respect to its structural identity to its pattern y ^ and in every case when it determined coincidence it replaced the corresponding element by ja. The elements 1 and y were not replaced, since they have not the given structure, even though y0 = 1 y1 = y holds. Remark: Pattern comparison always happens in FullForm. If we examine In 4] := b=y =: y ^ ;> ja then we get Out 4] = b ja (20.22) This is a consequence of the fact that FullForm of b=y is Times b Power y ;1] ], and for structure comparison the second argument of Times is identi ed as the structure of the pattern. With the de nition In 5] := f x ] := x^ 3 (20.23a) Mathematica replaces, corresponding to the given pattern, In 6] = f r] by Out 6] = r3 etc. (20.23b) 3 3 In 7] := f a] + f x] yields Out 7] = a + x (20.23c) If In 8] := f x] := x^ 3 so for the same input In 7] := : : : (20.23d) then the output would be Out 7] = f a] + x3 (20.23e) In this case only the \identical" input corresponds to the de nition.
20.2.8 Functional Operations
Functions operate on numbers and expressions. Mathematica can also perform operations with functions, since the names of functions are handled as expressions so they can be manipulated as expressions.
20.2 Mathematica 965
1. Inverse Function The determination of the inverse function of a given function f (x) can be made by the functional operation InverseFunction. A: In 1] := InverseFunction f ] x] Out 1] = f ;1 x] B: In 2] := InverseFunction Exp] Out 2] = Log 2. Di erentiation Mathematica uses the possibility that the di erentiation of functions can be considered as a mapping in the space of functions. In Mathematica, the di erentiation operator is Derivative 1] f ] or in short form f0 . If the function f is de ned, then its derivative can be got by f0 . In 3] := f x ] := Sin x] Cos x] With In 4] := f0 we get Out 4] = Cos 71]2 ; Sin 71]2 & hence f0 is represented as a pure function and it corresponds to In 5] := % x] Out 5] = Cos x]2 ; Sin x]2 3. Nest The command Nest f x n] means that the function f nested n times into itself should be applied on x. The result is f f : : : f x]] : : :]. 4. NestList By NestList f x n] a list fx f x] f f x]] : : :g will be shown, where nally f is nested n times. 5. FixedPoint For FixedPoint f x], the function is applied repeatedly until the result does not change. 6. FixedPointList The functional operation FixedPointList f x] shows the continued list with the results after f is applied, until the value no longer changes. As an example for this type of functional operation the NestList operation will be used for the approximation of a root of an equation f (x) = 0 with Newton's method (see 19.1.1.2, p. 885). We seek a root of the equation x cos x = sin x in the neighborhood of 3=2: In 6] := f x ] := x ; Tan x] In 7] := f0 x] Out 7] = 1 ; Sec x]2 In 8] = g x ] := x ; f x]=f0 x] In 9] := NestList g 4:6 4] == N Out 9] = f4:6 4:54573 4:50615 4:49417 4:49341g In 10] := FixedPoint g 4:6] Out 10] = 4:49341 A higher precision of the result can also be achieved. 7. Apply Let f be a function which is considered in connection with a list fa b c : : :g. Then, we get Apply f fa b c : : :g] f a b c : : :] (20.24) In 1] := Apply Plus fu v wg] Out 1] = u + v + w In 2] := Apply List a + b + c] Out 2] = fa b cg Here, the general scheme of how Mathematica handles expressions of expressions can be easily recognized. We write the last operation in FullForm: In 3] := Apply List Plus a b c]] Out 3] = List a b c] The functional operation Apply obviously replaces the head of the considered expression Plus by the required List. 8. Map With a de ned function f the operation Map gives: Map f fa b c : : :g] ;! ff a] f b] f c] : : :g (20.25) Map generates a list whose elements are the values when f is applied for the original list. Let f be the function f (x) = x2 . It is de ned by In 4] := f x ] := x^ 2
966 20. Computer Algebra Systems With this f we get In 5] := Map f fu v wg] Out 5] = fu2 v 2 w2 g Map can be applied for more general expressions: In 6] := Map f Plus a b c]] Out 6] = a2 + b2 + c2
20.2.9 Programming
Mathematica can handle the loop constructions known from other languages for procedural programming (IF, FOR, WHILE, DO). The two basic commands are Do expr fi i1 i2 dig] and (20.26a) While test expr ] (20.26b) The rst command evaluates the expression expr, where i runs over the values from i1 to i2 in steps di. If di is omitted, the step size is one. If i1 is also missing, then it starts from 1. The second command evaluates the expression while test has the value True. In order to determine an approximate value of e2 , the series expansion of the exponential function is used: In 1] := sum = 1:0 Do sum = sum + (2^ i=i!) fi 1 10g] (20.27) sum Out 1] = 7:38899 The Do loop evaluates its argument a previously given number of times, while the While loop evaluates until a previously given condition becomes false. Among other things, Mathematica provides the possibility of de ning and using local variables. This can be done by the command Module ft1 t2 : : :g procedure] (20.28) The variables or constants enclosed in the list are locally usable in the module their values assigned here are not valid outside of this module. A: We have to de ne a procedure which calculates the sum of the square roots of the integers from 1 to n. In 1] := sumq n ] := Module fsum = 1:g (20.29) Do sum = sum + N Sqrt i]] fi 2 ng] sum ] The call sumq 30] results in 112:083. The real power of the programming capabilities of Mathematica is, rst of all, the use of functional methods in programming, which are made possible by the operations Nest, NestWhile, Apply, Map, ReplaceList and by some further ones. B: Example A can be written in a functional manner for the case when an accuracy of ten digits is required: sumq n ] := N Apply Plus Table Sqrt i] fi 1 ng]] 10] sumq 30] results in 112:0828452. For the details, see 20.6].
20.2.10 Supplement about Syntax, Information, Messages 20.2.10.1 Contexts, Attributes
Mathematica must handle several symbols among them there are those which are used in further program modules loaded on request. To avoid many-valuedness, the names of symbols in Mathematica
20.2 Mathematica 967
consist of two parts, the context and the short name. Short names mean here the names (see 20.2, p. 956) of heads and elements of the expressions. In addition, in order to name a symbol Mathematica needs the determination of the program part to which the symbol belongs. This is given by the context, which holds the name of the corresponding program part. The complete name of a symbol consists of the context and the short name, which are connected by the ' sign. When Mathematica starts there are always two contexts present: System' and Global'. We can get information about other available program modules by the command Contexts ]. All built-in functions of Mathematica belong to the context System', while the functions de ned by the user belong to the context Global'. If a context is actual, thus, the corresponding program part is loaded, then the symbols can be referred to by their short names. For the input of a further Mathematica program module by whattype(obj ) (20.30) The semicolon must be written at the end of the input. The output is the basic type of the object. Maple knows the following basic types of objects, collected in Table 20.9. The more detailed type structure can be determined by the help of commands like type(obj,typname), the values of which are the Boolean functions true or false. Table 20.10 contains some type names known by Maple. Table 20.9 Basic types in Maple # + : :: :: p := x^ 3 ; 4 # x^ 2 + 3 # x + 5 : Here, an expression, namely, a polynomial of degree 3 in x is de ned. With the substitution operator subs a value can be assigned to the variable x in the polynomial (expression) and then the evaluation is performed: > subs(x = 3 p) ;! 5 > subs(x = 3=4 p) ;! 347 64 > subs(x = 1:54 p) ;! 3:785864 The operator op displays the internal structure, the subexpressions from an expression. With op(p) (20.40) we get the sequence (see next section) of subexpressions on the rst level, x3 ;4x2 3x 5 (20.41) In the form op(i p) , the i-th term is displayed, so, e.g., op(2 p) yields the term ;4x2 . op(0 p) gives the basic type of p, here +. The number of terms of the expression is given by nops(p) .
20.3.5 Sequences and Lists
In Maple, a sequence consists of consecutive expressions separated by commas. The order of the elements is important. Sequences with the same elements but in di erent orders are considered as di erent objects. The sequence is a basic type of Maple: exprseq. > f 1 := x^3 ;4 # x^ 2 3 # x 5 : (20.42a) de nes a sequence, then > type(f 1 exprseq) results in true: (20.42b) With the command > seq(f (i) i = 1::n) the sequence f (1) f (2) : : : f (n) (20.43) is shown. With > seq(i2 i = 1::5) we get 1 4 9 16 25. The range operator range de nes the range of integer variables represented in the form i = n::m, and it means that the index variable i takes the consecutive values n n + 1 n + 2 : : : m. The type of this structure is ::. An equivalent form to generate a sequence is provided by the simpli ed form > f (i)$i = n::m (20.44) which also generates f (n) f (n + 1) : : : f (m). Consequently, $n::m results in the sequence n n + 1 : : : m and x$i the sequence with i terms x. Indexed variables can be generated by > a i]$i = 1::4 ;! a1 a2 a3 a4 Sequences can be completed by further terms: sequence a b : : : (20.45) If we put a sequence f into square brackets, then we get a list, which has the type list. > l := i$i = 1::6)] ;! l := 1 2 3 4 5 6]
20.3 Maple 973
With the already known operator op, for the command op(liste) we get the sequence back, which was the base of the list. A list can be completed if rst it is changed by op(liste) into a sequence, this sequence is completed, then it is changed into a new list by square brackets. Lists can have elements which are themselves lists their type is listlist. These types of constructions have an important role when matrices are constructed. The selection of one particular element of a list can be done by op(n liste) . This command gives the n-th element of the list. If the list has a name, like L, then it is easier to type L n] . For a double list we nd the elements on a lower level with op(m op(n L)) or with the equivalent call L n] m] . There are no diculties in building up lists with higher levels. Generating a simple list: > L1 := a b c d e f ] ;! L1 := a b c d e f ] Selecting the fourth element of this list: > op(4 L) or > L 4] ;! d Generating a double nested list: > L2 := a b c] d e f ]] : (output is suppressed!) Selecting the third element of the second sublist: > op(3 op(2 L)) or L 2] 3] ;! f Generating a triple nested list: > L3 := a b c] d e f ]] s t] u v]] x y] w z]]] :
20.3.6 Tables, Arrays, Vectors and Matrices 20.3.6.1 Tables and Arrays
Maple knows the commands table, array and Array to construct tables and arrays. With table
(ifc list)
(20.46)
Maple generates a table-type structure. Here, ifc is an indexing function (see 20.3.6.3, p. 974), list is a list of expressions, whose elements are equations. In this case Maple use the left-hand side of the
equation as the indexing of the table elements and the right-hand side as the current table element. If the list contains only elements, then Maple uses the natural indexing, starting at one. > T := table( a b c]) ;! table( 1 = a 2 = b 3 = c]) > R := table( a = x b = y c = z]) ;! table( a = x b = y c = z]) The repeated call of T or R gives only the symbols T or R. With op(T ) or eval(T ), the output is the table. For the call op(op(T )) we get the components of the table in the form of a list, a list of the equations for the table elements. Here, we can see that the evaluation principle for these structures is di erent from the general one. In general, Maple evaluates an expression until the end, i.e., until no further transformations are possible. However, while the de nition is recognized in the example above, further evaluation is suppressed until it is explicitly required with the special command op. The indices of T form a sequence with the command indices(T ) , a sequence of the elements can be obtained by entries (T ) . A table also can be constructed implicit, taking an indicialed name and assigning values of this name: Tab(index). Tab a]:= x : Tab b]:= y : Tab c]:= z : > eval(Tab) ;! table( a = x b = y c = z])
974 20. Computer Algebra Systems With the command > table() we get an empty table, table( ]). The values of the table can be obtained by > T(index). > Tab b] ;! y For the previous examples > indices(T ) yields 1] 2] 3] > indices(R) yields a] b] c] and correspondingly > entries(R) yields x] y] z] With the command array(ifc ber list) (20.47) special tables can be generated, which can be of several dimensions, but they can have only integer indices in every dimension.
20.3.6.2 One-Dimensional Arrays
With array(1::5) , e.g., a one-dimensional array of length 5 is generated without explicit elements with v := array(1::5 a(1) a(2) a(3) a(4) a(5)]) one gets the same but with given components. These onedimensional arrays are considered by Maple as vectors. With the type check function type(v vector) we get true. If we ask whattype(v) , then the answer is symbol. This is in connection with the special evaluation mentioned above.
20.3.6.3 Two-Dimensional Arrays
Two-dimensional arrays can be de ned similarly with A := array(1::m 1::n a(1 1) : : : a(1 n)] : : : a(m 1) : : : a(m n)]]) (20.48) The structure de ned in this way is considered by Maple as a matrix of size m n. The values of a(i j ) are the corresponding matrix elements. > X := array(1::3 x1 x2 x3]) X := x1 x2 x3] results in a vector. A matrix is obtained, e.g., by > A := array(1::3 1::4 ]) A := array(1::3 1::4 ]) 2? ? ? ? 3 The input 11] 12] 13] 14] > eval(A) yields the output 64 ?21] ?22] ?23] ?24] 75 ?31] ?32] ?33] ?34] Maple characterizes the unde ned values of the matrix by the question mark ?ij ]. If a value is assigned to all or some of these elements, like > A 1 1] := 1 : A 2 2] := 1 : A 3 3] := 0 : then the renewed call for A will be displayed with the given values: 2 1 ? ? ? 3 12] 13] 14] > eval(A) ;! 64 ?21] 1 ?23] ?24] 57 ?31] ?32] 0 ?34] With the command > B := array( b11 b12 b13] b21 b22 b23] b31 b32 b33]]) 2 b11 b12 b13 3 B := 64 b21 b22 b23 75 b31 b32 b33
20.3 Maple 975 Maple displays the generated matrix with its elements, since they are given explicitly in the de nition.
The optional assignment of dimension is not necessary here, since the complete determination of the matrix elements makes the de nition unique. If only a few elements are known, then the dimensions must be given Maple replaces the non-de ned values by their formal values: > C := array(1::3 1::4 c11 c12 c13] c21 c22] ]]) 2 c11 c12 c13 C 3 66 c21 c22 C23] C124]4] 77 C := 66 C C C C 77 4 31] 32] 33] 34] 5 C41] C42] C43] C44] (Since some elements are given, Maple cannot return an empty list as it did in the de nition of A. If we call eval(C), then we get question marks instead of C in the matrix.) Index functions, such as diagonal, identity,symmetric, antisymmetric, sparse can be used as optional arguments. With them we can get the corresponding matrices. 2 0 ? ? 3 12] 13] > array(1::3 1::3 antisymmetric) ;! 64 ;?12] 0 ?23] 75 ;?13] ;?23] 0 Another command Array can be used to construct objects of several dimensions. This command has the same form as array but allows much more options. Internally the rst is a hash-array, the second a hardblock-array.
20.3.6.4 Special Commands for Vectors and Matrices
Problems of linear algebra can be solved in Maple with the help of two special packages. The package linalg makes use of the command-structure array and provides special commands like matrix, vector and others as well as corresponding operators. The package Linear Algebra makes use of the command-structure Array, provides commands like Matrix, Vector and a great number of rules for constructions and manipulations of linear algebra. The product of the matrix B and vector X from the rst example in 20.3.6.3, p. 974 by the help of with(linalg ) is > evalm(B & # x) b11 x1 + b12 x2 + b13 x3 b21 x1 + b22 x2 + b23 x3 b31 x1 + b32 x2 + b33 x3] The multiplication of a matrix by a column vector is a column vector. A multiplication in the opposite order would give an error message.
20.3.7 Procedures, Functions and Operators
20.3.7.1 Procedures
A procedure has the following form: > P := proc(parm1 parm2 : : : parmn) > commands > endproc > f := proc(x) ^ > xn > end proc ;! f := proc(x) xn end proc > f (t) ;! tn
(20.49)
20.3.7.2 Functions
Maple has a huge number of prede ned functions, which are available immediately after the start or
they can be loaded with packages. They belong to type mathfunc. An enumeration can be got by ?inifcns. We give a collection of the standard and special functions in Tables 20.13 and 20.14.
976 20. Computer Algebra Systems Table 20.13 Standard functions Exponential function exp Logarithmic functions log, ln Trigonometric functions sin, cos, tan, cot, sec, csc Arcus functions arcsin, arccos, arctan, arccot, Hyperbolic functions sinh, cosh, tanh, coth, sech, csch Area functions arcsinh, arcsosh, arctanh, arccoth, Table 20.14 Special functions Bessel functions Jn(z) and Yn(z) BesselJ(v z ), BesselY(v,z) Modi ed Bessel functions In(z) and Kn(z) BesselI(v z), BesselK(v z) Gamma function Gamma(x) Integral exponential function Ei(x) The Fresnel functions can also be found among the special functions. The package for orthogonal polynomials contains among others the Hermite, Laguerre, Legendre, Jacobi and Chebyshev polynomials. For more details see 20.6]. In Maple the functions behave as procedures. Slightly simpli ed this means that the name of a function, as de ned in Maple, is considered as a procedure. In other words, type(sin procedure) yields true. If the argument, or several arguments if this is needed, is attached to the procedure in parentheses, then one gets the corresponding function of the given variables. > type(cos procedure) yields true and > type(cos function) yields false: If we replace the argument cos by cos(x), then type checking will give the opposite results. Maple provides the possibility to generate self-de ned functions in procedure form. A function can be de ned by the arrow operator ;>. With > F := x;> mathexpr : (20.50) and with mathexpr as an algebraic expression of the variable x, a new function with name F is de ned in procedure form. The algebraic expression can contain previously de ned and/or built-in functions. If an independent variable is attached in parentheses to the procedure symbol generated in this way, then it becomes a function of this independent variable. > F := x;> sin(x) # cos(x) + x^ 3 # tan(x) + x^2 : > F (y) ;! F (y) := sin(y) cos(y) + y3 tan(y) + y2 If a numerical value (e.g., a oating-point number) is assigned to this argument, say, with the call > F (nn:mmm) Maple gives the corresponding function value. Conversely, it can be generated from a function (e.g., from a polynomial of the variable x) the corresponding procedure with the command unapply(function var). So, we get back from F (y) with > unapply(F (y) y) ;! F the procedure with symbol F .
20.3.7.3 Functional Operators
Functional operators are special forms of procedures. They are used as commands for manipulation and combination of functions (procedures). It is possible to work with operators according to the usual rules. The sum and di erence of two operators is again an operator. For multiplication we have to be careful that the product is again an operator. Maple uses the special multiplication symbol @ for operator multiplication. In general, this multiplication is not commutative. Let F := x;> cos(2 # x) and G := x;> x^ 2. Then > (G @ F )(x) ;! cos2(2x) while
20.3 Maple 977
> (F @ G)(x) ;! cos(2x2) : The product of two functions given in operator representation (F # G)(x) = (G # F )(x), results in F (x) # G(x).
20.3.7.4 Di erential Operators
The di erentiation operator in Maple is D. Its application on functions in procedure form is D(F ) or D i](G). In the rst case the derivative of a function of one variable is de ned in procedure form. The attachment of the variable in parentheses results in the derivative as a function. In another form, it can be written as D(F )(x) = diff(F (x) x). Higher derivatives can be got by repeated application of the operator D, which can be simpli ed by the notation (D @ @ n)(F ) where @ @ n means the n-th \power" of the di erential operator. If G is a function of several variables, then D i](G) generates the partial derivatives of G with respect to the i-th variable. This result is also a procedure. With D i j ](G) we get D i](D j ](G)), i.e., the second partial derivative with respect to the j -th and i-th variables. Higher derivatives can be formed similarly. The rules of di erential calculus (see 6.1.2.2, p. 380) are valid for the di erential operator D, where F and H are di erentiable functions: D(F + H ) = D(F ) + D(H ) (20.51a) D(F # H ) = (D(F ) # H ) + (F # D(H )) (20.51b) D(F @ H ) = D(F ) @ H # D(H ): (20.51c)
20.3.7.5 The Functional Operator map
The operator map can be used in Maple to apply an operator or a procedure to an expression or to its components. Let, e.g., F be an procedure representing a function. Then map(F x + x^ 2 + x # y) yields the expression F (x) + F (x2 ) + F (x y). Similarly, with map(F y # z) the result is F (y) # F (z). map(f a b c d]) ;! f (a) f (b) f (c) f (d)]
20.3.8 Programming in Maple
Maple provides the usual control and loop structures in a special form to build procedures and programs.
Case distinction is made by the if command. Its basic structure is if cond then stat 1 else stat 2 end if (20.52) The else branch can be omitted. Before the else branch, arbitrarily many further branches can be introduced with the structure elif cond i then stat i end if (20.53) Loops are generated with for and while, which require in the command part the form do : : : stat : : : end do In the for loop the running index must be written in the form i from n to m by di where di is the step size. If the initial value and step size are missing, then they are automatically replaced by 1. In the while loop, the rst part is while cond do stat end do Also loops can be multiply nested into each other. In order to write a closed program, the procedure command is needed in Maple. It can have several rows, and if it is stored appropriately, then it can be recalled by name if it is needed. Its basic structure
978 20. Computer Algebra Systems is:
(args)
proc local options
::: ::: commands
(20.54)
end proc
The number of arguments of the procedure is not necessarily equal to number of the variables used by the kernel of the procedure in particular they can be completely missing. All variables de ned by local are known only in the procedure. Write a procedure which calculates the sum of the square roots of the rst n natural numbers: > sumqw := proc(n) > local s i > s 0] := 0 > for i to n > do s i] := s i ; 1] + sqrt(i) end do > evalf(s n]) > end proc Maple displays the procedure de ned in this way. Then the procedure can be called by name with the required argument n: > sumqw(30) Output: 112:0828452
20.3.9 Supplement about Syntax, Information and Help 20.3.9.1 Using the Maple Library
Maple consists of three main parts: the kernel, a Maple-library and a user-panel. The kernel is written
in the programming language C and guarantees the main work of the system including basic operations of mathematics. The library holds the main part of mathematical formalism. Corresponding parts will be loaded automatically on demand. Beyond it to Maple belongs a big library of special packages. A special package can be loaded with the command > with(name) (20.55) Here the name is the name of the current package, hence linalg for the package of linear algebra. After loading, Maple lists all the commands of the package and gives a warning if in the new de nitions there are commands already available which were introduced earlier. If only one particular command is needed from a package, then it can be called by paket command] (20.56)
20.3.9.2 Environment Variable
The output of Maple is controled by several environment variables. We already introduced the variable Digits (see 20.3.2.3, 1., p. 970), by which the number of the displayed digits of oating-point numbers is de ned. The general form of the output of the result is de ned by prettyprint. Default is here > interface(prettyprint = true) (20.57) This provides centered output in mathematical style. If this option is de ned false, then the output starts at the left-hand side with the form of input.
20.3.9.3 Information and Help
Help about the meaning of commands and keywords is available by the input ?notion
(20.58)
20.4 Applications of Computer Algebra Systems 979
Instead of the question mark, help(notion) can also be written. This results in a help screen, which contains the corresponding part of the library handbook for the required notion. If Maple runs under Windows, then the call for HELP opens a menu usually on the right-hand side, and the explanation about the required notion can be obtained by clicking on it with the mouse.
20.4 Applications of Computer Algebra Systems
This section describes how to handle mathematical problems with computer algebra systems. The choice of the considered problems is organized according to their frequency in practice and also according to the possibilities of solving them with a computer algebra system. Examples will be given for functions, commands, operations and supplementary syntax, and hints for current computer algebra systems. When it is important, the corresponding special package is also discussed briey.
20.4.1 Manipulation of Algebraic Expressions
In practice, further operations must usually be performed with the occurring algebraic expressions (see 1.1.5, p. 10) such as di erentiation, integration, series representation, limiting or numerical evaluation, transformations, etc. In general, these expressions are considered over the ring of integers (see 5.3.6, p. 313) or over the eld (see 5.3.6.1, 2., p. 313) of real numbers. Computer algebra systems can handle, e.g., polynomials also over nite elds or over extension elds (see 5.3.6.1, 3., p. 313) of the rational numbers. Interested people should study the special literature. The algebraic operations with polynomials over the eld of rational numbers have special importance.
20.4.1.1 Mathematica
Mathematica provides the functions and operations represented in Table 20.15 for transformation of
algebraic expressions.
1. Multiplication of Expressions
The operation of multiplication of algebraic expressions can always be performed. The coecients can also be unde ned expressions. In 1] : = Expand (x + y ; z )^ 4] gives Out 1] = x4 + 4 x3 y + 6 x2 y 2 + 4 x y 3 + y 4 ; 4 x3 z ; 12 x2 y z ; 12 x y 2 z ; 4 y 3 z +6 x2 z2 + 12 x y z2 + 6 y2 z2 ; 4 x z3 ; 4 y z3 + z4 Similarly, In 2] : = Expand (a x + b y ^2)(c x^ 3 ; d y ^2)] Out 2] = a c x4 ; a d x y 2 + b c x3 y 2 ; b d y 4 Tabelle 20.15 Commands for manipulation of algebraic expressions Expand p] expands the powers and products in a polynomial p by multiplication Expand p r ] multiplies only the parts in p, which contain r PowerExpand a] expands also the powers of products and powers of powers Factor p] factorizes a polynomial completely Collect p x] orders the polynomial with respect the powers of x Collect p fx y : : :g] the same as the previous one, with several variables ExpandNumerator r ] expands only the numerator of a rational expression ExpandDenominator r ] expands only the denominator ExpandAll r ] expands both numerator and denominator completely Together r ] combines the terms in the expression over a common denominator Apart r ] represents the expression in partial fractions Cancel r ] cancels the common factors in the fraction
980 20. Computer Algebra Systems
2. Factorization of Polynomials
Mathematica performs factorization over the integer or rational numbers if it is possible. Otherwise the
original expression is returned. In 2] := p = x^ 6 + 7x^ 5 + 12x^ 4 + 6x^ 3 ; 25x^ 2 ; 30x ; 25 In 3] := Factor p] gives Out 3] = ((5 + x) (1 + x + x2 ) (;5 + x2 + x3 )) Mathematica decomposes the polynomial into three factors which are irreducible over the rational numbers. If a polynomial can be completely decomposed over the complex rational numbers, then this can be obtained by the option GaussianIntegers. In 4] := Factor x2 ; 2x + 5] ;! Out 4] = 5 ; 2x + x2 but In 5] := Factor % GaussianIntegers ;> True] Out 5] = (;1 ; 2I + x)(;1 + 2I + x)
3. Operations with Polynomials
Table 20.16 contains a collection of operations by which polynomials can be algebraically manipulated over the eld of rational numbers. Table 20.16 Algebraic polynomial operations PolynomialGCD p1 p2] determines the greatest common divisor of the two polynomials p1 and p2 PolynomialLCM p1 p2] determines the least common multiple of the polynomials p1 and p2 PolynomialQuotient p1 p2 x] divides p1 (as a function of x) by p2, the residue is omitted PolynomialRemainder p1 p2 x] determines the residue on dividing p1 by p2 Two polynomials are de ned: In 1] := p = x^ 6 + 7x^ 5 + 12x^ 4 + 6x^ 3 ; 25x^ 2 ; 30x ; 25 q = x^4 + x^ 3 ; 6x^2 ; 7x ; 7 With these polynomials the following operations are performed: In 2] := PolynomialGCD p q ] ;! Out 2] = 1 + x + x2 In 3] := PolynomialLCM p q ] Out 3] = 3(;7 + x)(1 + x + x2 )(;25 ; 5x + 5x2 + 6x3 + x4 ) In 4] := PolynomialQuotient p q x] ;! Out 4] = 12 + 6x + x2 In 5] := PolynomialRemainder p q x] ;! Out 5] = 59 + 96x + 96x2 + 37x3 With the two last results we get x6 + 7x5 + 12x4 + 6x3 ; 25x2 ; 30x ; 25 = x2 + 6x + 12 + 37x3 + 96x2 + 96x + 59 x4 + x3 ; 6x2 ; 7x ; 7 x4 + x3 ; 6x2 ; 7x ; 7
4. Partial Fraction Decomposition
Mathematica can decompose a fraction of two polynomials into partial fractions, of course, over the
eld of rational numbers. The degree of the numerator of any part is always less than the degree of the denominator. Using the polynomials p and q from the previous example we get ;6 + ;55 + 11 x + 6 x2 In 6] := Apart q=p] ;! Out 6] = 35 (5 + x) 35 (;5 + x2 + x3 )
20.4 Applications of Computer Algebra Systems 981
5. Manipulation of Non-Polynomial Expressions
Complicated expressions, not necessarily polynomials, can often be simpli ed by the help of the command Simplify. Mathematica will always try to manipulate algebraic expressions, independently of the nature of the symbolic quantities. Here, certain built-in knowledge is used. Mathematica knows the rules of powers (see 1.1.4.1, p. 7): In 1] := Simplify a^ n=a^ m)] ;! Out 1] = a(;m+n) (20.59) With the command FullSimplifyExpr] in Mathematica simpli cations are possible. For the manipulation of trigonometric expressions there are the commands TrigExpand, TrigFactor, TrigFactorList, and TrigReduce. In 1] := TrigExpand Sin 2x] Cos 2x]] Out 1] = 2 Cos x] Cos y ]2 Sin x] ; 2 Cos x] (Cos y ] + Sin y ]) In 2] := TrigFactor %] Out 2] = 2 Cos x] Sin x](Cos y ] ; Sin y ]) (Cos y ] + Sin y ]) In 3] := TrigReduce %] Out 3] = (1=2)(Sin 2x ; 2y ] + Sin 2x + 2y ]) Some trigonometric formulas (see 2.7.2.3, p. 79) can be veri ed with the following input: In 2] := Factor Sin 4x] Trig;> True] ;! Out 2] = 4 cos(x)3 sin(x) ; 4 cos(x) sin(x)3 In 3] := Factor Cos 5x] Trig;> True] ;! Out 3] = cos(x) (1 ; 2 cos(2 x) + 2 cos(4 x)) : Finally, we mention that the command ComplexExpand expr] assumes a real variable expr, while in ComplexExpand expr fx1 x2 : : :g] the variables xi are supposed to be complex. In 1] := ComplexExpand Sin 2 x] fxg] Out 1] = Cosh 2 Im x]] Sin 2 Re x]] + I Cos 2 Re x]] Sinh 2 Im x]]
20.4.1.2 Maple
Maple provides the operations enumerated in Table 20.17 for transformation and simpli cation of
algebraic expressions. Table 20.17 Operations to manipulate algebraic expressions expand(p q 1 q 2 : : :) expands the powers and the products in an algebraic expression p. The optional arguments qi prevent the further expansion of the subexpressions qi. factor(p K ) factorizes the expression p. K is an optional RootOf argument. simplify(p q 1 q 2 : : :) applies built-in simplifying rules on p. In the presence of the optional arguments these rules are applied only for them. radsimp(p) simpli es p containing radicals. normal(p) normalizes the rational expression p. sort(p) sorts the terms of polynomial p in order of decreasing degree. coeff(p x i) determines the coecient of xi . collect(p v ) collects the terms of a polynomial of several variables containing the variable v.
1. Multiplication of Expressions
In the simplest case Maple decomposes the expression into the sum of powers of the variables: > expand((x + y ; z)^ 4) 4 x3 y ; 4 x3 z + 6 x2y2 + 6 x2z2 + 4 xy3 ; 4 xz3 ; 4 y3z + 6 y2z2 ; 4 yz3 + x4 + y4 + z4 ;12 x2 yz ; 12 xy2z + 12 xyz2
982 20. Computer Algebra Systems Here we can demonstrate the e ect of the absence and presence of an optional argument on the Maple procedure. > expand((a # x^ 3 + b # y^4) # sin(3 # x) # cos(2 # x)) 8 ax3 sin(x) cos(x)4 ; 6 ax3 sin(x) cos(x)2 + ax3 sin(x) + 8 by4 sin(x) cos(x)4 ;6 by4 sin(x) cos(x)2 + by4 sin(x) The expression is completely expanded. > expand((a # x^ 2 ; b # y^3) # sin(3 # x) # cos(2 # x) a # x^ 2 ; b # y^3) 8 (ax2 ; by3) sin(x) cos(x)4 ; 6 (ax2 ; by3) sin(x) cos(x)2 + (ax2 ; by3) sin(x) Maple kept the expression of the optional argument unchanged. The following example shows the e ectiveness of Maple: > expand(exp(2 # a # x) # sinh(2 # x) + ln(x3 ) # sin(4 # x)) 2 eax2 sinh(x) cosh(x) + 24 ln(x) sin(x) cos(x)3 ; 12 ln(x) sin(x) cos(x)
2. Factorization of Polynomials
Maple can decompose polynomials over algebraic extension elds (if it is possible anyway).
> p := x^6 + 7 # x^ 5 + 12 # x^ 4 + 6 # x^3 ; 25 # x^ 2 ; 30 # x ; 25: q := x^ 4 + x^ 3 ; 6 # x^ 2 ; 7 # x ; 7: > p1 := factor(p) (x + 5) (x2 + x + 1) (x3 + x2 ; 5) and > q1 := factor(q) (x2 + x + 1) (x2 ; 7) Here, Maple decomposed both polynomials into irreducible factors over the eld of rational numbers. If we want to have a decomposition over an algebraic extension eld, then we can continue: > p2 := factor(p (;3)^(1=2)) p p (x3 + x2 ; 5) 2 x + 1 ; ;3 2 x + 1 + ;3 (x + 5) 4 p Maple decomposed the second factor (in this case after a formal extension of the eld by ;3). In general, we do not know if such an extension is possible. If the degrees of the factors are 4, then it is possible. With the operation RootOf, the roots can be determined as algebraic expressions. > r := RootOf(x3 + x2 ; 5): k := allvalues(r): > k 1] s p p 1 3 133 + 655 3 + s p p ; 1 =3 54 18 9 3 133 + 655 3 54 18 > k 2] s p p 3 133 + 655 3 ; 54 2 18 ; s 1 p p ; 1=3 655 3 18 3 133 54 + 18
20.4 Applications of Computer Algebra Systems 983
+
0 1 s p p B C p;1p3 BB 3 133 + 655 3 ; s 1 C C p p B C 18 @ 54 A 655 3 3 133 9 54 + 18
2 The call for k 3] results in the complex conjugate of k 2]. The procedure given in this example yields a sequence of oating-point numbers in case the polynomial can be decomposed only over the eld of real or complex numbers.
3. Operations with Polynomials
Besides the operations discussed above, the operations gcd and lcm have special importance. They nd the greatest common divisor and the least common multiple of two polynomials. Correspondingly, quo(p q x) yields the integer part of the ratio of polynomials p and q , and rem(p q x) gives the residue. > p := x6 + 7 # x5 + 12 # x4 + 6 # x3 ; 25 # x2 ; 30 # x ; 25: q := x4 + x3 ; 6 # x2 ; 7 # x ; 7: > gcd(p q) x2 + x + 1 > lcm(p q) 210 x + 5 x6 ; 43 x5 ; 109 x4 ; 72 x3 + 150 x2 + 175 + 7 x7 + x8 With the command normal the ratio of two polynomials can be transformed into normal form over the eld of rational numbers, i.e., the quotient of two relatively prime polynomials with integer coecients. With the polynomials from the previous example > normal(p=q) x4 + 6 x3 + 5 x2 ; 5 x ; 25 x2 ; 7 With numer and denom the numerator and the denominator can be represented separately. 2 ^ > factor (xp+ 1)=(x ; 7) (7) (1=2)) p(denom x+ 7 x; 7
4. Partial Fraction Decomposition
The partial fraction decomposition in Maple is performed by the command convert, which is called with the option parfrac. Using the polynomials p and q from the previous examples we get > convert(p=q parfrac x) x2 + 6 x + 12 + 37xx2 ;+ 759 and > convert(q=p parfrac x) + 11 x + 6 x2 ; 35 x +6 175 + 35;55 x3 + 35 x2 ; 175
5. Manipulation of General Expressions
The operations introduced in the following allow the transformation of algebraic and transcendental expressions with rational and algebraic functions containing functions which are self-de ned or built-in in Maple. In general, optional arguments can be given, which modify the transformations under certain conditions. The command simplify is introduced here in an example. In the simple form simplify(expr) Maple
984 20. Computer Algebra Systems performs some built-in simpli cation rules on the expression. > t := sinh(3 # x) + cosh(4 # x): > simplify(t) 4 sinh(x) cosh(x)2 ; sinh(x) + 8 cosh(x)4 ; 8 cosh(x)2 + 1 Similarly, > r := sin(2 # x) # cos(3 # x): > simplify(r) 8 sin(x) cos(x)4 ; 6 sin(x) cos(x)2 There exists a command combine, which is the reverse command of expand in a certain sense. > t := tan(2 # x)2 : > t1 := expand(t) 2 x )2 t1 := 4 sin(x) cos( 2 (2 cos(x) ; 1)2 > combine(t1 trig) cos(2 x);2 ; 1 Here, combine was called with the option trig, which provides the use of the basic rules of trigonometry. Using the command simplify we get x)2 ; 1 > t2 := simplify(t) ;! ; cos(2 cos(2 x)2 Here, Maple reduced the tangent function to the cosine function. Transformations can be performed with the exponential, logarithmic and further functions as Bessel and Gamma Functions.
20.4.2 Solution of Equations and Systems of Equations
Computer algebra systems know procedures to solve equations and systems of equations. If the equation can be solved explicitly in the domain of algebraic numbers, then the solution will be represented with the help of radicals. If it is not possible to give the solution in closed form, then at least numerical solutions can be found with a given accuracy. In the following, we introduce some basic commands. The solution of systems of linear equations (see 20.4.3.2, 1., p. 990) is discussed in a special section.
20.4.2.1 Mathematica 1. Equations
Mathematica allows the manipulation and solution of equations within a wide range. In Mathematica,
an equation is considered as a logical expression. If one writes In 1] := g = x^ 2 + 2x ; 9 == 0 (20.60a) then Mathematica considers it as a de nition of an identity. Giving the input In 2] := %=: x;> 2 we get Out 2] = False (20.60b) since with this value of x the left-hand side and right-hand side are not equal. The command Roots g x] transforms the above identity into a form which contains x explicitly. Mathematica represents the result with the help of the logical OR in the form of a logical statement: In 2] : = Roots g x] yields p p Out 2] = x == ;1 ; 10jjx == ;1 + 10 (20.60c) In this sense, logical operations can be performed with equations.
20.4 Applications of Computer Algebra Systems 985
With the operation ToRules, the last logical type equations can be transformed as follows: In 3] : = fToRules %]g ! p p (20.60d) Out 3] = ffx ;> ;1 ; 10g fx ;> ;1 + 10gg
2. Solution of Equations
Mathematica provides the command Solve to solve equations. In a certain sense, Solve perform the
operations Roots and ToRules after each other. Mathematica solves polynomial equations in symbolic form up to fourth degree, since for these equations solutions can be given in the form of algebraic expressions. However, if equations of higher degree can be transformed into a simpler form by algebraic transformations, such as factorization, then Mathematica provides symbolic solutions. In these cases, Solve tries to apply the built-in operations Expand and Decompose. In Mathematica numerical solutions are also available. The general solution of an equation of third degree: In 4] := Solve x^ 3 + a x^ 2 + b x + c == 0 x] Mathematica gives ;a Out 4] = ffx;> 3 1 2 3 (;a2 + 3 b) ;
3
q
3 ;2 a3 + 9 a b ; 27 c + 3 2 ; (a2 b2 ) + 4 b3 + 4 a3 c ; 18 a b c + 27 c2
3
q
;2 a3 + 9 a b ; 27 c + 3 2 ; (a2 b2 ) + 4 b3 + 4 a3 c ; 18 a b c + 27 c2
13
13
+ g 1 32 3 : : :g The solution list shows only the rst term explicitly because of the length of their terms. If we want to solve an equation with given coecients a b c, then it is better to handle the equation itself with Solve than to substitute a b c into the solution formula. A: For the cubic equation (see 1.6.2.3, p. 40) x3 + 6x + 2 = 0 we get: In 5] : = Solve x^ 3 + 6 x + 2 == 0 x] p p p p 1 ; i 3 ; 1 + i 3 g fx;> ; 1 ; i 3 + 1 + i 3 gg Out 5] = ffx;> 21=3 ; 22=3 g fx;> 21=3 22=3 22=3 21=3 B: Solution of an equation of sixth degree: In 6] : = Solve x^ 6 ; 6x^ 5 + 6x^ 4 ; 4x^ 3 + 65x^ 2 ; 38x ; 120 == 0 x] Out 6] = ffx;> ;1g fx;> 4g fx;> 3g fx;> 2g fx;> ;1 ; 2 ig fx;> ;1 + 2 igg Mathematica succeeded in factorizing the equation in B with internal tools then it is solved without diculty. If numerical solutions are required, then the command NSolve is recommended, since it is faster. The following equation is solved by NSolve: In 7] : = NSolve x^ 6 ; 4x^ 5 + 6x^ 4 ; 5x^ 3 + 3x^ 2 ; 4x + 2 == 0 x] Out 7] = ffx;> ;0:379567 ; 0:76948 ig fx;> ;0:379567 + 0:76948 ig fx;> 0:641445g fx;> 1: ; 1: ig fx;> 1: + 1: ig fx;> 2:11769gg
3. Solution of Transcendental Equations
Mathematica can solve transcendental equations, as well. In general, this is not possible symbolically,
986 20. Computer Algebra Systems and these equations often have in nitely many solutions. In these cases, a domain should be given, where Mathematica has to nd the solutions. This is possible with the command FindRoot g fx xsg], where xs is the initial value for the search of the root. In 8] : = FindRoot x + ArcCoth x] ; 4 == 0 fx 1:1g] Out 8] = fx;> 1:00502g and In 9] : = FindRoot x + ArcCoth x] ; 4 == 0 fx 5g] ;! Out 9] = fx;> 3:72478g
4. Solution of Systems of Equations
Mathematica can solve simultaneous equations. The operations, built-in for this purpose, are repre-
sented in Table 20.18, and they present the symbolical solutions, not the numerical ones. Similarly to the case of one unknown, the command NSolve gives the numerical solution. The solution of systems of linear equations is discussed in 20.4.3, p. 988. Table 20.18 Operations to solve systems of equations Solve fl1 == r1 l2 == r2 : : :g fx y : : :g] solves the given system of equations with respect to the unknowns Eliminate fl1 == r1 : : :g fx : : :g] eliminates the elements x : : : from the system of equations Reduce fl1 == r1 : : :g fx : : :g] simpli es the system of equations and gives the possible solutions
20.4.2.2 Maple
1. Important Operations
The two basic operations in Maple to solve equations symbolically are solve and RootOf or roots. With them, or with their variations with certain optional arguments, it is possible to solve several equations, even transcendental ones. If an equation cannot be solved in closed form, Maple provides numerical solutions. RootOf represents the roots of an equation of one variable: k := RootOf(x^3 ; 5 # x + 7 x) ;! k := RootOf( Z 3 ; 5 Z + 7) (20.61) In Maple k denotes the set of the roots of the equation x3 ; 5x + 7 = 0. Here, the given expression is transformed into a simple form, if it is possible, and the global variable is represented by Z . (Maple returns an unevaluated RootOf.) The command allvalues(k) results in a sequence of the roots. The command solve yields the solution of an equation if any exists. > k := solve(x^ 4 + x^ 3 ; 6 # x^2 ; 7 # x ; 7 x) p p p p k := ; 12 + 12 I 3 ; 12 ; 12 I 3 7 ; 7 If the equation entered is of degree greater then four, then answers are provided in terms of RootOf: > r := solve(x^6 + 4 # x^ 5 ; 3 # x + 2 x) ;! r := RootOf( Z 6 + 4 Z 5 ; 3 Z + 2) This equation has no solution for rational numbers. With allvalues we get the approximate numerical solutions.
2. Solution of Equations with One Variable
1. Polynomial Equations Polynomial equations with one unknown, whose degree is 4, can be solved by Maple symbolically. p p > solve(x^4 ; 5 # x^2 ; 6) ;! I ;I 6 ; 6
20.4 Applications of Computer Algebra Systems 987
2. Equations of Degree Three Maple can solve the general third degree equation with general
coecients. > r := solve(x^ 3 + a # x^ 2 + b # x + c x) : > r 1] s p p c ; a3 + 4 b3 ; b2 a2 ; 18 bac + 27 c2 + 4 ca3 3 3 ba ; 6 2 27 18 b ; a2 a ;s p 3 3 2 29 p ;3 3 2 + 4 ca3 3 c a 4 b ; b a ; 18 bac + 27 c 3 ba ; ; + 6 2 27 18 We get the corresponding expression for the other roots r 2] r 3], too, but we do not give them here because of their length. If the input equation in solve has oating-point numbers for coecients, then Maple solves the equation numerically. > solve(1: # x^3 + 6: # x + 2: x) ;3:27480002 :1637400010 ; 2:46585327 I :1637400010 + 2:46585327 I 3. General Equation of Degree Four The general solution is given by Maple also for polynomial equations of degree four. 4. Other Algebraic Expressions Maple can solve equations containing radical expressions of the unknown. 5. Extra root We must be careful, when taking roots, because occasionally we may get equations whose roots are not roots of the original equation. These roots are called extra roots. Hence, every root o ered by Maple should be substituted p intopthe original expression. The solution of the equation x + 7 + 2x ; 1 ; 1 = 0 is to be determined. The input is > p := sqrt(x + 7) + sqrt(2 # x ; 1) ; 1 : l := solve(p = 0 x) : With > s := allvalues(l) : we obtain p p s 1] := 12 + 12 (2 + 17)2 and s 2] := 21 + 12 (2 ; 17)2 : By > subs(x = s i] p) i = 1 2 we can convince ourselves that only s 2] is a solution.
3. Solution of Transcendental Equations
Equations containing transcendental parts can usually be solved only numerically. Maple provides the command fsolve for numerical solution of any kind of equation. With this command, Maple nds real roots of the equation. Usually, we get only one root. However, transcendental equations often have several roots. The command fsolve has an optional third argument, the domain where the root is to be found. > fsolve(x + arccoth(x) ; 4 = 0 x 2::4) ;! 3:724783628 but (20.62) > fsolve(x + arccoth(x) ; 4 = 0 x 1005) ;! 1:005020099
4. Solution of Nonlinear System of Equations
Systems of equations can be solved by the same commands solve and fsolve. The rst argument contains all of the equations in curly braces, and the second one, also in curly braces, lists the unknowns for which the equations are to be solved: > solve(fgl 1 gl 2 : : :g fx1 x2 : : :g) (20.63)
>
solve
(fx^2 ; y^2 = p p p 2 x^2 + y^2 = 4gpfx yg) fy = 1 x = 3g fy = 1 x = ; 3g fy = ;1 x = 3g fy = ;1 x = 3g
(20.64)
988 20. Computer Algebra Systems
20.4.3 Elements of Linear Algebra 20.4.3.1 Mathematica
In 20.2.4, p. 959, the notion of matrix and several operations with matrices were de ned on the basis of lists. Mathematica applies these notions in the theory of systems of linear equations. In the followings p = Array p fm ng] (20.65) de nes a matrix of type (m n) with elements pij = p i j ]]. Furthermore x = Array x fng] und b = Array b fmg] (20.66) are n- or m-dimensional vectors. With these de nitions the general system of linear homogeneous or inhomogeneous equations can be written in the form (see 4.4.2, p. 272) p : x == b p : x == 0 (20.67)
1. Special Case n = m det p 6= 0
In the special case n = m detp 6= 0, the system of inhomogeneous equations has a unique solution, which can be determined directly by x = Inverse p]: b (20.68) Mathematica can handle such systems of up to ca. 50 unknowns in a reasonable time, depending on the computer system. An equivalent, but much faster solution is obtained by LinearSolve p b].
2. General Case
With the commands LinearSolve and NullSpace, all the possible cases can be handled as discussed in 4.4.2, p. 272, i.e., it can be determined rst if any solution exists, and if it does, then it is calculated. In the following, we discuss some of the examples from Section 4.4.2, p. 272 . A: The example in 4.4.2.1, 2., p. 274, is a system of homogeneous equations x1 ; x2 + 5x3 ; x4 = 0 x1 + x2 ; 2x3 + 3x4 = 0 3x1 ; x2 + 8x3 + x4 = 0 x1 + 3x2 ; 9x3 + 7x4 = 0 which has non-trivial solutions. These solutions are the linear combinations of the basis vectors of the null space of matrix p. It is the subspace of the n-dimensional vector space which is mapped into the zero by the transformation p. A basis for this space can be generated by the command NullSpace p]. With the input In 1] := p = ff1 ;1 5 ;1g f1 1 ;2 3g f3 ;1 8 1g f1 3 ;9 7gg we de ne the matrix whose determinant is actually zero, which can be checked by Det p]. Now we take: 3 7 1 0g f;1 ;2 0 1gg In 2] := NullSpace p] and get Out 2] = ff; 2 2 is displayed, a list of two linearly independent vectors of four-dimensional space, which form a basis for the two-dimensional null-space of matrix p. An arbitrary linear combination of these vectors is also in the null-space, so it is a solution of the system of homogeneous equations. This solution coincides with the solution found in Section 4.4.2.1, 2., p. 274. B: Consider the example A in 4.4.2.1, 2., p. 273, x1 ; 2x2 + 3x3 ; x4 + 2x5 = 2 3x1 ; x2 + 5x3 ; 3x4 ; x5 = 6 2x1 + x2 + 2x3 ; 2x4 ; 3x5 = 8 with matrix m1 of type (3 5), and vector b1 In 3] := m1 = ff1 ;2 3 ;1 2g f3 ;1 5 ;3 ;1g f2 1 2 ;2 ;3gg In 4] := b1 = f2 6 8g
20.4 Applications of Computer Algebra Systems 989
For the command In 4] := LinearSolve m1 b1] the response is LinearSolve : : nosol: Linear equation encountered which has no solution. The input appears as output. C: According to example B from Section 4.4.2.1, 1., p. 273, x1 ; x2 + 2x3 = 1 x1 ; 2x2 ; x3 = 2 3x1 ; x2 + 5x3 = 3 ;2x1 + 2x2 + 3x3 = ;4 the input is In 5] := m2 = ff1 ;1 2g f1 ;2 ;1g f3 ;1 5g f;2 2 3gg In 6] := b2 = f1 2 3 ;4g If we want to know how many equations have independent left-hand sides, then we call In 7] := RowReduce m2] ;! Out 7] = ff1 0 0g f0 1 0g f0 0 1g f0 0 0gg Then the input is 10 ; 1 ; 2 g In 8] := LinearSolve m2 b2] ;! Out 8] = f 7 7 7 The answer is the known solution.
3. Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors of matrices are de ned in 4.5, p. 278. Mathematica provides the possibility of determining eigenvalues and eigenvectors by special commands. So, the command Eigenvalues m] produces a list of eigenvalues of a square matrix m Eigenvectors m] creates a list of the eigenvectors of m. If N m] is substituted instead of m, then we get the numerical eigenvalues. In general, if the order of the matrix is greater than four (n > 4), then no algebraic expression can be obtained, since the characteristic polynomial has degree higher than four. In this case, we should ask for numerical values. In 9] := h = Table 1=(i + j ; 1) fi 5g fj 5g] This generates a ve-dimensional so-called Hilbert matrix. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Out 9] = ff1 2 3 4 5 g f 2 3 4 5 6 g f 3 4 5 6 7 g f 4 5 6 7 8 g f 5 6 7 8 9 gg With the command In 10] := Eigenvalues h] we get the answer Eigenvalues::eival: Unable to nd all roots of the characteristic polynomial. But with the command In 11] := Eigenvalues N h]] we get Out 11] = f1:56705
20.4.3.2 Maple
0:208534 0:0114075 0:000305898 3:28793 10;6g
The Maple library provides the special packages linalg and LinearAlgebra. After the command > with (linalg): (20.69)
990 20. Computer Algebra Systems all the 100 commands and operations of the package linalg are available for the user. For a complete list and description see 20.6]. It is important that matrices and vectors must be generated by the commands matrix and vector while using this package, and not by the general structure array. With matrix(m n s), an m n matrix is generated. If s is missing, then the elements of this matrix are not speci ed, but they can be determined later by the assignments A i j ] := : : :. If s is a function f = f (i j ) of the indices, then Maple generates the matrix with these elements. Finally, s can be a list with elements, e.g., vectors. Analogously, the de nition of vectors can happen by vector(n e). The input of a vector is similar to a 1 n matrix, but a vector is always considered to be a column vector. Table 20.19 gives the most important matrix and vector operations. Addition of vectors and matrices can be performed by the command add(u v k l). This procedure adds the vectors or matrices u and v multiplied by the scalars k and l. The optional arguments k and l can be omitted. The addition is performed only if the corresponding matrices have the same number of rows and columns. Multiplication of matrices can be performed by multiply(u v) or with the short form (see 20.3.6.4, p. 975) &# as an in x operator. Table 20.19 Matrix operations transpose (A) determines the transpose of A det(A) determines the determinant of the square matrix A inverse(A) determines the inverse of the square matrix A adjoint(A) determines the adjoint of the square matrix A, i.e., A & # adjoint(A) = det(A) mulcol(A s expr ) multiplies the s-th column of the matrix A by expr mulrow(A r expr ) multiplies the r -th row by expr
1. Solution of Systems of Linear Equations
To handle systems of linear equations Maple provides special operations contained in the linear algebra package. One of them is the command linsolve(A c). It handles the system of linear equations in the form Ax=c (20.70) where A denotes the coecient matrix and c is the vector on the right-hand side. If there is no solution, then the null-sequence Null is returned. If the system has several linearly independent solutions, they will be given in parametric form. The operation nullspace(A) nds a basis in the null space of matrix A , which is di erent from zero if the matrix is singular. It is also possible to solve a system of linear equations with the operators of multiplication and evaluation of the inverse. A: Consider the example E from 4.4.2.1, 2., p. 274, of the homogeneous system x1 ; x2 + 5x3 ; x4 = 0 x1 + x2 ; 2x3 + 3x4 = 0 3x1 ; x2 + 8x3 + x4 = 0 x1 + 3x2 ; 9x3 + 7x4 = 0 whose matrix is singular. This system has non-trivial solutions. In order to solve it, rst we de ne matrix A: > A := matrix( 1 ;1 5 ;1] 1 1 ;2 3] 3 ;1 8 1] 1 3 ;9 7]]): (With det(A) we can be convinced that the matrix is singular.) With the command
> a := nullspace(A) ;! a := ; 32 72 1 0 ;1 ;2 0 1]
20.4 Applications of Computer Algebra Systems 991
the list of two linearly independent vectors is determined. They form a basis in the two-dimensional null space of matrix A. For the general case, operations to apply the Gaussian algorithm are available in Maple. They are enumerated in Table 20.20. If the number of unknowns is equal to the number of equations and the coecient matrix is non-singular, then the command linsolve is recommended. Table 20.20 Operations of the Gaussian algorithm pivot(A i j ) generates a matrix from A by adding an appropriate multiple of the i-th row to the others, whose j -th column consists of zeros except Aij gausselim (A) generates the Gaussian triangle matrix by row-pivoting the elements of A have to be rational numnbers gaussjord (A) generates a diagonal matrix according to the Gauss{Jordan method augment(A u) generates the augmented matrix consisting of A and the column vector u
B: The system from 19.2.1.4, 2., p. 895, 10x1 ; 3x2 ; 4x3 + 2x4 = 14 ;3x1 + 26x2 + 5x3 ; x4 = 22 ;4x1 + 5x2 + 16x3 + 5x4 = 17 2x1 + 3x2 ; 4x3 ; 12x4 = ;20
is to be solved. Now, the input is > A := matrix( 10 ;3 ;4 2] ;3 26 5 ;1] ;4 5 16 5] 2 3 ;4 ;12]]): > v := vector( 14 22 17 ;20]): With linsolve we get
> linsolve(A v) ;! 32 1 21 2 2 33 The Gaussian algorithm results in 66 1 0 0 0 2 77 7 6 > F := gaussjord(augment(A v)) ;! F := 666 0 1 0 0 11 777 64 0 0 1 0 75 2 0001 2 C: The inhomogeneous system of example B from 4.4.2.1, 2., p. 273, x1 ; x2 + 2x3 = 1 x1 ; 2x2 ; x3 = 2 3x1 ; x2 + 5x3 = 3 ;2x1 + 2x2 + 3x3 = ;4 is to be solved. The input is the corresponding matrix and vector: > A := matrix( 1 ;1 2] 1 ;2 ;1] 3 ;1 5] ;2 2 3]]): > v := vector( 1 2 3 ;4]): The system is overdetermined. The command linsolve cannot be used. We can start with 2 1 ;1 2 1 3 6 2 ;1 2 7 > F := augment(A v) ;! F := 64 13 ; ;1 5 3 75 ;2 2 3 ;4
992 20. Computer Algebra Systems 2 then by gaussjord the matrix F is 10 3 1 0 0 6 transformed into an upper triangular form: 6 7 777 66 17 > F 1 := gaussjord(F ) ;! F 1 := 666 0 1 0 ; 7 777 66 0 0 1 ; 2 77 4 75 The solution can be read from F 1. 2. Eigenvalues and Eigenvectors 0 0 0 0 In Maple, the eigenvalues and eigenvectors of a square matrix can be determined by the special operators eigenvals and eigenvects . Usually, the eigenvalue equation cannot be solved in closed form for matrices of order n > 4. Then Maple gives the results as approximating oating-point numbers. Find the eigenvalues of the ve-dimensional Hilbert matrix (see 20.4.3.1, 3., p. 989). In the package linalg there is a special command to generate n-dimensional Hilbert matrices. It is hilbert(n x). The elements of the matrix are 1=(i + j ; x). If x is not de ned, then Maple substitutes x = 1 automatically. We solve the problem with the input > eigenvals(hilbert(5)) Maple answers RootOf( ;1 + 307505 Z ; 1022881200 Z 2 + 92708406000 Z 3 ;476703360000 Z 4 + 266716800000 Z 5 ) With allvalues, a sequence of the approximated values can be produced.
20.4.4 Dierential and Integral Calculus 20.4.4.1 Mathematica
The notation of the derivative as a functional operator was introduced in 20.2.8, p. 964. Mathematica provides several possibilities to apply the operations of analysis, e.g., determination of the di erential quotient of arbitrarily high order, of partial derivatives, of the complete di erential, determination of inde nite and de nite integrals, series expansion of functions, and also solutions of di erential equations.
1. Calculation of Di erential Quotients
1. Di erentiation Operator The di erentiation operator (see Section 20.2.8, p. 964) has the name
. Its complete form is n1 n2 : : :] (20.71) The arguments say how many times the function is to be di erentiated with respect to the current variables. In this sense, it is an operator of partial di erentiation. Mathematica tries to represent the result as a pure function. 2. Di erentiation of Functions The di erentiation of a given function can be performed in a simpli ed manner with the operator D. With D f x] x], the derivative of the function f (x) will be determined. D belongs to a group of di erential operations, which are enumerated in Table 20.21. Tabelle 20.21 Operations of di erentiation D f x] fx ng] yields the n-th derivative of function f (x) with respect to x D f fx1 n1 g fx2 n2 g ] multiple derivatives, ni -th derivative with respect to xi (i = 1 2 ) Dt f ] the complete di erential of the function f df of the function f Dt f x] the complete derivative dx Dt f x1 x2 : : :] the complete derivative of a function of several variables Derivative
Derivative
20.4 Applications of Computer Algebra Systems 993
For both examples in 6.1.2.2, p. 382, we get A : In 1] : = D Sqrt x^ 3 Exp 4x] Sin x]] x] E4 x x2 (x Cos x] + 3 Sin x] + 4 x Sin x]) Out 1] = 2 Sqrt E4 x x3 Sin x]] ^ B : In 2] := D (2x + 1) (3x) x] ;! Out 2] = 6 x (1 + 2 x);1+3 x + 3 (1 + 2 x)3 x Log 1 + 2 x]
The command Dt results in the complete derivative or complete di erential. C : In 3] := Dt x^ 3 + y^3] ;! Out 3] = 3x2Dt x] + 3y2 Dt y] D : In 4] := Dt x^ 3 + y^3 x] ;! Out 4] = 3x2 + 3y2Dt y x] In this last example, Mathematica supposes y to be a function of x, which is not known, so it writes the second part of the derivative in a symbolic way. If Mathematica nds a symbolic function while calculating a derivative, it leaves it in this general form, and expresses its derivative by f 0. E : In 5] := D x f x]^ 3 x] ;! Out 5] = f x]3 + 3xf x]2 f' x] Mathematica knows the rules for di erentiation of products and quotients, it knows the chain rule, and it can apply these formally: F : In 6] := D f u x]] x] Out 6] = f0 u x]] u0 x] 0 0 G : In 7] := D u x]=v x] x] Out 7] = uv xx]] ; u xv]xv]2 x]
2. Indenite Integrals
Z
With the command Integrate f x], Mathematica tries to determine the inde nite integral f (x) dx. If Mathematica knows the integral, it gives it without the integration constant. Mathematica supposes that every expression not containing the integration variable does not depend on it. In general, Mathematica nds an inde nite integral, if there exists one which can be expressed in closed form by elementary functions, such as rational functions, exponential and logarithmicfunctions, trigonometric and their inverse functions, etc. If Mathematica cannot nd the integral, then it returns the original input. Mathematica knows some special functions which are de ned by non-elementary integrals, such as the elliptic functions, and some others. To demonstrate the possibilities of Mathematica, some examples will be shown, which are discussed in 8.1, p. 427 . 1. Integration of Rational Functions (see 8.1.3.3, p. 432 .) A : In 1] : = Integrate (2x + 3)=(x^3 + x^2 ; 2x) x] 5 Log ;1 + x] ; 3 Log x] ; Log 2 + x] Out 1] = 3 2 6 B : In 2] : = Integrate (x^ 3 + 1)=(x(x ; 1)^3) x] 1 (20.72) Out 2] = ; (;1 + x);2 ; ;1 + x + 2 Log ;1 + x] ; Log x] 2. Integration of Trigonometric Functions (see 8.1.5, p. 438 .) A: ZThe example A in 8.1.5.2, p. 439, with the integral Z Z 2 5 2 sin x cos x dx = sin x (1 ; sin2 x)2 cos x dx = t2(1 ; t2 )2 dt with t = sin x
994 20. Computer Algebra Systems is calculated: In 3] : = Integrate Sin x]^ 2Cos x]^ 5 x] 5 Sin x] ; Sin 3 x] ; 3 Sin 5 x] ; Sin 7 x] Out 3] = 64 192 320 448 B: The example B in 8.1.5.2, p. 439, with the integral Z sin x Z pcos x dx = ; pdtt with t = cos x is calculated: In 4] := Integrate Sin x]=Sqrt Cos x]] x] ;! Out 4] = ;2Sqrt Cos x]] Remark: In the case of non-elementary integrals Mathematica tries to substitute them by special integrals. If that is not possible it does nothing. In 5] := Integrate x^ x x] ;! Out 5] = Integrate xx x]
3. Denite Integrals
With the command Integrate f fx xa xe g], Mathematica can evaluate the de nite integral of the function f (x) with a lower limit xa and upper limit xe . A : In 1] := Integrate Exp ;x2 ] fx ;Infinity Infinityg] we get Out 1] = Sqrt Pi] After Mathematica has loaded a special package for integration, it gives the value (see Table 21.8, p. 1056, Nr. 9 for a = 1). B: It can happen, with earlier releases of Mathematica, that if the input is In 2] := Integrate 1=x^ 2 fx ;1 1g] we get Out 2] = 1 After a slightly longer working time, Mathematica gives 1 because the integrand has a pole at x = 0: In the calculation of de nite integrals, we should be careful. If the properties of the integrand are not known, it is recommended before integration to ask for a graphical representation of the function in the considered domain.
4. Multiple Integrals
De nite double integrals can be called by the command Integrate f x y ] fx xa xe g fy ya ye g] (20.73) The evaluation is performed from right to left, so, rst the integration is evaluated with respect to y. The limits ya and ye can be functions of x, which are substituted into the primitive function. Then the integral is evaluated with respect to x. For the integral A, which calculates the area between a parabola and a line intersecting it twice, in 8.4.1.2, p. 472, we get 32 : In 3] := Integrate x y ^ 2 fx 0 2g fy x^2 2xg] ;! Out 3] = 5 Also in this case, it is important to be careful with the discontinuities of the integrand.
5. Solution of Di erential Equations
Mathematica can handle ordinary di erential equations symbolically if the solution can be given in closed form. In this case, Mathematica gives the solution in general. The commands discussed here are
listed in Table 20.22. The solutions (see 9.1, p. 487) are represented as general solutions with the arbitrary constants C i]. Initial values and boundary conditions can be introduced in the part of the list which contains the equation or equations. In this case we get a special solution. As examples, two di erential equations are solved here from 9.1.1.2, p. 489.
20.4 Applications of Computer Algebra Systems 995
Table 20.22 Commands to solve di erential equations DSolve dgl y x] x] solves the di erential equation for y x] (if it is possible) y x] may be given in implicit form DSolve dgl y x] gives the solution of the di erential equation in the form of a pure function DSolve fdgl1 dgl2 : : :g y x] solves a system of ordinary di erential equations
A: The solution of the di erential equation y0(x) ; y(x) tan x = cos x is to be determined. In 1] := DSolve y 0 x] ; y x] Tan x] == Cos x] y x]
Mathematica solves this equation, and gives the solution as a pure function with the integrations con-
stant C 1]
Out 1] = ffy
!
Function
Sec Slot
1]] (4 C(1) + Sin 2 Slot 1]] + 2 Slot 1]) ]gg 4
The symbol Slot is for 7, it is its FullForm. If it is required to get the solution for y x], then Mathematica gives Sec x] (2 x + 4 C(1) + Sin 2 x]) In 2] := y x]=: %1 ;! Out 2] = f g 4 0 We also could make the substitution for other quantities, e.g., for y x] or y 1]. The advantage of using pure functions is obvious here. B: The solution of the di erential equation y0(x)x(x ; y(x)) + y2(x) = 0 (see 9.1.1.2, 2., p. 489) is to be determined. In 3] := DSolve y 0 x] x(x ; y x]) + y x]^2 == 0 y x] x] Mathematica returns: InverseFunction : : ifun: Inverse Functions are beeng used. Values may be lost : : : (( " C i] )) Out 3] := y x] ;> ;xProductLog e x Here ProductLog(z) is the principal velue of the solution of z = !e! in !: The reason for doing so is that Mathematica cannot solve this di erential equation for y. The solution of this di erential equation was given in implicit form (see 9.1.1.2, 2., p. 489). In such cases, the solutions can be found by numerical solutions (see 19.8.4.1, 5., p. 948). Also in the case of symbolic solutions of di erential equations, like in the evaluation of inde nite integrals, the eciency of Mathematica should not be overestimated. If the result cannot be expressed as an algebraic expression of elementary functions, the only way is to nd a numerical solution. By the help of DSolve gleich y x1 : : : xn] fx1 : : : xng] also partial di erential equations can be solved.
20.4.4.2 Maple
Maple provides many possibilities to handle the problems of analysis. Besides di erentiation of func-
tions, it can also evaluate inde nite and de nite integrals, multiple integrals, and expansion of functions into power series. The basic elements of the theory of analytic functions are also provided. Several differential equations can be solved, as well.
1. Di erentiation
The operator of di erentiation D was introduced in 20.3.7, p. 975 . Its application with di erent optional arguments allows us to di erentiate functions in procedure representation. Its complete syntax is D i](f ) (20.74a)
996 20. Computer Algebra Systems Here, the partial derivative of the (procedure) function f is determined with respect to the i-th variable. The result is a function in procedure representation. D i j ](f ) is equivalent to D i](D j ](f )) and D ](f ) = f: (20.74b) The argument f is here a function expression represented in procedure form. The argument can contain self-de ned functions besides the built-in functions, functions de ned by the arrow operator, etc. Let > f := (x y) ;> exp(x # y) + sin(x + y): : Then one sets > D ](f ) ;! f > D 1](f ) ;! (x y) ;> y exp(x y) + cos(x + y) > D 2](f ) ;! (x y) ;> x exp(x y) + cos(x + y) > D 1 2](f ) ;! (x y) ;> exp(x y) + x y exp(x y) ; sin(x + y) Besides the di erentiation operator there exists the operator diff with the syntax diff(expr x1 x2 : : : xn) (20.75a) Here expr is an algebraic expression of the variables x1 x2 : : : . The result is the partial derivative of the expression with respect to the variables x1 : : : xn. If n > 1 holds, then the same result can be obtained by repeated application of the operation diff: diff(a x1 x2) = diff(diff(a x1) x2) (20.75b) Multiple di erentiation with respect to the same argument can be got by the sequence operator $. > diff(sin(x) x$5) ( diff(sin(x) x x x x x)) ;! cos(x) If the function f (x) is not de ned, then the operation diff gives the derivatives symbolically: d f (x). d f (x) f (x) d g(x) dx A > diff(f (x)=g(x)) ;! dxg(x) ; g(dx x)2 d f (x) B > diff(x # f (x) x) ;! f (x) + x dx
2. Indenite Integrals
If the primitive function F (x) can be represented as an expression of elementary functions for a given function f (x), then Maple can usually nd it with int(f x). The integration constant is not displayed. If the primitive function does not exist or is not known in closed form, then Maple returns the integrand. Maple knows many special functions and will substitute them if possible into the output. Instead of the operator int, the long form integrate can be used.
1. Integration of Rational Functions A : > int((2 # x + 3)=(x^3 + x^2 ; 2 # x) x) ;! ; 23 ln(x) ; 16 ln(x + 2) + 53 ln(x ; 1) B : > int((x^ 3 + 1)=(x # (x ; 1)^3) x) ;! ; ln(x) ; (x ;1 1)2 ; x ;1 1 + 2 ln(x ; 1)
2. Integral of Radicands Maple can determine the inde nite integrals given in the table of inde nite integrals (see 21.7, p. 1023 .). If the input is > X := sqrt(x^2 ; a^2): then we have the output: p p (20.76) > int(X x) ;! 21 x x2 ; a2 ; 12 a2 ln(x + x2 ; a2 ) p2 2 x > int(X=x x) ;! x ; a ; aarcsec a > int(X # x x) ;! 31 (x2 ; a2)3=2
20.4 Applications of Computer Algebra Systems 997
3. Integrals with Trigonometric Functions 3 3 2 2 ) ; 6 sin(ax) + 6ax cos(ax) A : > int(x^ 3 # sin(a # x) x) ;a x cos(ax) + 3a x sin(ax a4 B : > int(1=(sin(a # x))^3 x) ;! ; 1 cos(ax) + 1 ln(csc(ax) ; cot(ax)
2 a sin(ax)2 2 a Remark: In the case of non-elementary integrals, which can not be substituted by known special functions, only a formal transformation is performed. Z > int(x^ x x) the output is xx dx since this integral cannot be given in an elementary way.
3. Denite Integrals
To determine a de nite integral the command int is used with the second argument x = a :: b. Here, x is the integration variable, and a :: b are the lower and upper limits of the integration interval. A : > int(x^ 2 x = a::b) ;! 31 b3 ; 13 a3 > int(x^ 2 x = 1::3) ;! 263 (20.77) B : > int(exp(;x^ 2) x = ;infinity::infinity) ;! p C : > int(1=x^4 x = ;1::1) ;! 1 If Maple cannot solve the integral symbolically, it returns the input. In this case we can try a numerical integration (see 19.3, p. 898 .) with the commands evalf(int(expr,var=a..b)) or evalf (Int(expr,var=a..b)).
4. Multiple Integrals
Maple can even calculate multiple integrals if it can be done explicitly. The operation int can be nested.
A : > int(int(x^ 2 + y^2 # exp(x + y) x) y)
1 3 x+y 2 x+y x+y x+y x+y x+y 2 3 x y + e (x + y) ; 2(x + y)e + 2e ; 2 (x + y)e ; e x + e x B : > int(int(x # y^2 y = x^2::2 # x) x = 0::2) ;! 325
5. Solution of Di erential Equations
Maple provides the possibility of solving ordinary di erential equations and di erential equation sys-
tems with the di erent forms of the operator dsolve. The solution can be a general solution or a particular solution with given initial conditions. The solution is given either explicitly or implicitly as a function of a parameter. The operator dsolve accepts as a last argument the options shown in Table 20.23. Table 20.23 Options of operation dsolve explicit gives the solution in explicit form if it is possible laplace applies the Laplace transformation for the solution series the power series expansion is used for the solution numeric the result is a procedure to calculate numerical values of the solution
1. General Solution
( (y(x) x) ; y(x) # tan(x) = cos(x) y(x)) x) + x + 2 C 1 y(x) = 21 cos(x) sin(cos( x)
>
dsolve diff
(20.78a) (20.78b)
998 20. Computer Algebra Systems Maple gives the general solution in explicit form with a constant. In the following example the solution
is given in implicit form, since y(x) cannot be expressed from the de ning equation. The additional option explicit has no e ect here. > dsolve(diff(y(x) x) # (x ; y(x)) + y(x)^2 y(x)) (20.79a)
! 1 e; y(x) x ; Ei 1 y(1x) = C 1
(20.79b)
2. Solution with Initial Values Consider the di erential equation y0 ; ex ; y2 = 0 with y(0) = 0. Here we give the option series. If we use this option, the initial values should belong to x = 0. The same is valid for the option laplace. > dsolve(fdiff(y(x) x) ; exp(x) ; y(x)^2 y(0) = 0g y(x) series) (20.80a) 7 x4 + 31 x5 + O(x6) y(x) = x + 12 x2 + 12 x3 + 24 120
(20.80b)
The equation and the initial values have to be in curly braces. The same is true for systems of di erential equations.
20.5 Graphics in Computer Algebra Systems
By providing routines for graphical representation of mathematical relations such as the graphs of functions, space curves, and surfaces in three-dimensional space, modern computer algebra systems provide extensive possibilities for combining and manipulating formulas, especially in analysis, vector calculus, and di erential geometry, and they provide immeasurable help in engineering designing.
20.5.1 Graphics with Mathematica 20.5.1.1 Basic Elements of Graphics
Mathematica builds graphical objects from built-in (so-called) graphics primitives. These are objects
such as points (Point), lines (Line) and polygons (Polygon) and properties of these objects such as thickness and colour. Mathematica has several options to specify the environment for graphics and how the graphical objects should be represented. With the command Graphics list], where list is a list of graphics primitives, Mathematica is called to generate a graphic from the listed objects. The object list can follow a list of options about the appearance of the representation. With the following input In 1] := g = Graphics fLine ff0 0g f5 5g f10 3gg] Circle f5 5g 4] (20.81a) \Example",\Helvetica-Bold",25] f5 6g]g AspectRatio;> Automatic] (20.81b) a graphic is built from the following elements: a) Broken line of two line segments starting at the point (0 0) through the point (5 5) to the point (10 3). b) Circle with the center at (5 5) and radius 4. c) Text with the content \Example", written in Helvetica-Bold font (the text appears centered with respect to the reference point (5 6)). Text FontForm
20.5 Graphics in Computer Algebra Systems 999
example
Figure 20.1
With the call Show g], Mathematica displays the picture of the generated graphic (Fig. 20.1). Certain options should be previously speci ed. Here the option AspectRatio is set to Automatic. By default Mathematica makes the ratio of the height to the width of the graph 1 : GoldenRatio. It corresponds to a relation between the extension in the x direction to the one in the y direction of 1 : 1=1:618 = 1 : 0:618. With this option the circle would be deformed into an ellipse. The value of the option Automatic ensures that the representation is not deformed.
20.5.1.2 Graphics Primitives
Mathematica provides the two-dimensional graphic objects enumerated in Table 20.24.
20.5.1.3 Syntax of Graphical Representation 1. Building Graphic Objects
If a graphic object is to be built from primitives, then rst a list of the corresponding objects with their global de nition should be given in the form fobject1 object2 : : :g (20.82a) where the objects themselves can be lists of graphic objects. Table 20.24 Two-dimensional graphic objects Point fx y g] point at position x y Line ffx1 y1 g fx2 y2 g : : :g] broken line through the given points Rectangle fxlu ylu g fxro yro g] shaded rectangle with the given coordinates left-down, right-up Polygon ffx1 y1 g fx2 y2 g : : :g] shaded polygon with the given vertices Circle fx y g r ] circle with radius r around the center x y Circle fx y g r f1 2 g] circular arc with the given angles as limits Circle fx y g fa bg] ellipse with half-axes a and b Circle fx y g fa bg f1 2 g] elliptic arc Disk fx y g r ], Disk fx y g fa bg] shaded circle or ellipse Text text fx y g] writes text centered to the point x y Besides these objects Mathematica provides further primitives to control the appearance of the representation, the graphics commands. They specify how graphic objects should be represented. The commands are listed in Table 20.25. Table 20.25 Graphics commands PointSize a] point is drawn with radius a as a fraction of the total picture AbsolutePointSize b] denotes the absolute radius b of the point (measured in pt (0.3515 mm)) Thickness a] draws lines with relative thickness a AbsoluteThickness b] draws lines with absolute thickness b (also in pt) Dashing fa1 a2 a3 : : :g] draws a line as a sequence of stripes with the given length (in relative measure) AbsoluteDashing fb1 b2 : : :g] the same as the previous one but in absolute measure GrayLevel p] speci es the level of shade (p = 0 is for black, p = 1 is for white)
1000 20. Computer Algebra Systems There is a wide scale of colors to choose from but their de nitions are not discussed here.
20.5.1.4 Graphical Options
Mathematica provides several graphical options which have an inuence on the appearance of the entire
picture. Table 20.26 gives a selection of the most important commands. For a detailed explanation, see 20.5]. Table 20.26 Some graphical options AspectRatio ;> w sets the ratio w of height and width. Automatic determines w from the absolute coordinates the default setting is w = 1 : GoldenRatio Axes ;> True draws coordinate axes Axes ;> False does not draw coordinate axes Axes ;> fTrue Falseg shows only the x-axis Frame ;> True shows frames GridLines ;> Automatic shows grid lines AxesLabel ;> fxsymbol ysymbol g denotes axes with the given symbols Ticks ;> Automatic denotes scaling marks automatically with None they can be suppressed Ticks ;> ffx1 x2 : : :g fy1 y2 : : :gg scaling marks are placed at the given nodes Let object 1 be, e.g., In 1] := o1 = fCircle f5 5g f5 3g] Line ff0 5g f10 5gg]g and corresponding to it In 2] := o2 = fCircle f5 5g 3]g: If a graphic object, e.g., o2, is to be provided with certain graphical commands, then it should be written into one list with the corresponding command In 3] := o3 = fThickness 0:01] o2g: This command is valid for all objects in the corresponding braces, and also for nested ones, but not for the objects outside of the braces of the list. From the generated objects two di erent graphic lists are de ned: In 4] := g 1 = Graphics fo1 o2g] g 2 = Graphics fo1 o3g] which di ers only in the second object by the thickness of the circle. With the call Show g 1] and Show g 2 Axes ;> True] (20.82b) we get the picture represented in Fig. 20.2. In the call of the picture in Fig. 20.2b, the option Axes ;> True was activated. This results in the representation of the axes with marks on them chosen by Mathematica and with the corresponding scaling. 8 7 6 5 4 3 a)
b)
Figure 20.2
2
4
6
8
10
20.5 Graphics in Computer Algebra Systems 1001
1. Graphical Representation of Functions
Mathematica has special commands for the graphical representation of functions. With
Plot f x] fx xmin xmax g] (20.83) the function f (x) is represented graphically in the domain between x = xmin and x = xmax . Mathematica produces a function table by internal algorithms and reproduces the graphic following from this table by graphics primitives. If the function sin 2x is to be graphically represented in the domain between ;2 and 2, then the input is In 5] := Plot Sin 2x] fx ;2Pi 2Pig] Mathematica produces the curve shown in Fig. 20.3. 1 It is obvious that Mathematica uses certain default graph0.5 ical options in the representation. So, the axes are automatically drawn, they are scaled and denoted by the corresponding x and y values. In this example, the inuence 2 4 -6 -4 -2 6 of the default AspectRatio can be seen. The ratio of the -0.5 total width to the total height is 1 : 0:618. With the command InputForm %] the whole representa-1 tion of the graphic objects can be shown. For the previous Figure 20.3 example we get: Graphics ffLine ff;6:283185307179587 4:90059381963448 # 10^ ; 16g List of points from the function table calculated by Mathematica f6:283185307179587 ;(4:90059381963448 # 10^ ; 16)gg]gg fPlotRange;> Automatic AspectRatio;> GoldenRatio^(;1) DisplayFunction :> $DisplayFunction ColorOutput ;> Automatic Axes;> Automatic AxesOrigin ;> Automatic PlotLabel;> None AxesLabel;> None Ticks;> Automatic GridLines;> None Prolog;> fg Epilog;> fg AxesStyle;> Automatic Background ;> Automatic DefaultColor ;> Automatic DefaultFont :> $DefaultFont RotateLabel ;> True Frame;> False FrameStyle;> Automatic FrameTicks;> Automatic FrameLabel;> None PlotRegion;> Automatic g] Consequently, the graphic object consists of two sublists. The rst one contains the graphics primitive Line, with which the internal algorithm connects the calculated points of the curve by lines. The second sublist contains the options needed by the given graphic. These are the default options. If the picture is to be altered at certain positions, then the new settings in the Plot command must be set after the main input. With In 6] := Plot Sin 2x] fx ;2Pi 2Pig AspectRatio;> 1] (20.84) the representation would be done with equal x and y absolute scaling. It is possible to give several options at the same time after each other.With the input Plot ff1 x] f2 x] : : :g fx xmin xmax g] (20.85) several functions are shown in the same graphic. With the command Show plot options] (20.86) an earlier picture can be renewed with other options.With (20.87) Show GraphicsArray list]]
1002 20. Computer Algebra Systems (with list as lists of graphic objects) pictures can be placed next to each other, under each other, or they can be arranged in matrix form.
20.5.1.5 Two-Dimensional Curves
A series of curves from the chapter on functions and their representations (see 2.1, p. 47 .) is shown as examples.
1. Exponential Functions
A family of curves with several exponential functions (see 2.6.1, p. 71) is generated by Mathematica
(Fig. 20.4a) with the following input: In 1] := f x
] := 2^x g x ] := 10^x In 2] := h x ] := (1=2)^ x j x ] := (1=E)^ x k x ] := (1=10)^x These are the de nitions of the considered functions. The function ex need not be de ned, since it is built into Mathematica. In the second step the following graphics are generated: In 3] := p1 = Plot ff x] h x]g fx ;4 4g PlotStyle;> Dashing f0:01 0:02g]] In 4] := p2 = Plot fExp x] j x]g fx ;4 4g] In 5] := p3 = Plot fg x] k x]g fx ;4 4g PlotStyle;> Dashing f0:005 0:02 0:01 0:02g]] The whole picture (Fig. 20.4a) can be obtained by: In 6] := Show fp1 p2 p3g PlotRange;> f0 18g AspectRatio;> 1:2] The question of how to write text on the curves is not discussed here. This is possible with the graphics primitive Text. 10
15 2.5
10
-1 1
-6
6
5 a)
-4
-2
0
2
4
2. Function y = x + Arcoth x
b)
Figure 20.4
Considering the properties of the function Arcoth x discussed in 2.10, p. 91, the function y = x + Arcoth x can be graphically represented in the following way: In 1] := f 1 = Plot x + ArcCoth x] fx 1:000000000005 7g] In 2] := f 2 = Plot x + ArcCoth x] fx ;7 ;1:000000000005g] In ] := 3Show ff 1 f 2g PlotRange;> f;10 10g AspectRatio;> 1:2 Ticks;> fff;6 ;6g f;1 ;1g f1 1g f6 6gg ff2:5 2:5g f10 10ggg The high precision of the x values in the close neighborhood of 1 and ;1 was chosen to get suciently large function values for the required domain of y. The result is shown in Fig. 20.4b.
3. Bessel Functions
With the calls In 1] := bj 0 = Plot fBesselJ 0 z ]
BesselJ
2 z]
BesselJ
4 z]g fz 0 10g
20.5 Graphics in Computer Algebra Systems 1003 PlotLabel;> \J (n z ) n = 0 2 4"] := bj 1 = Plot fBesselJ 1 z] BesselJ 3 z] BesselJ 5 z]g fz 0 10g PlotLabel;> \J (n z ) n = 1 3 5"] (20.88) the graphics of the Bessel function Jn(z) for n = 0 2 4 and n = 1 3 5 are generated, which are then represented by the call In 3] := Show GraphicsArray fbj 0 bj 1g]] next to each other in Fig. 20.5.
In 2]
1 0.8 0.6 0.4 0.2
0.6
J(n,z) n=0,2,4
J(n,z) n=1,3,5
0.4 0.2 2
-0.2 a) -0.4
4
6
8
2
10
b)
4
6
8
10
-0.2
Figure 20.5
20.5.1.6 Parametric Representation of Curves
Mathematica has a special graphics command, with which curves given in parametric form can be graph-
ically represented. This command is: ParametricPlot ffx (t) fy (t)g ft t1 t2 g]: (20.89) It provides the possibility of showing several curves in one graphic. A list of several curves must be given in the command. With the option AspectRatio;> Automatic, Mathematica shows the curves in their natural forms. The parametric curves in Fig. 20.6 are the Archimedean spiral (see 2.14.1, p. 103) and the logarithmic spiral (see 2.14.3, p. 104). They are represented with the input In 1] := ParametricPlot ft Cos t] t Sin t]g ft 0 3Pig AspectRatio;> Automatic] and In 2] := ParametricPlot fExp 0:1t] Cos t] Exp 0:1t] Sin t]g ft 0 3Pig AspectRatio ;> Automatic] With In 3] := ParametricPlot ft ; 2 Sin t] 1 ; 2 Cos t]g ft ;Pi 11Pig AspectRatio;> 0:3] a trochoid (see 2.13.2, p. 100) is generated (Fig. 20.7). 8 6 4 2
2
3 2 1
1
-7.5 -5 -2.5 -2 2.5 5 -4 a)
-2 b)
Figure 20.6
-1
1 -1
-1
5
10
15
20
25
30
Figure 20.7
20.5.1.7 Representation of Surfaces and Space Curves
Mathematica provides the possibility of representing three-dimensional graphics primitives.
35
1004 20. Computer Algebra Systems Similarly to the two-dimensional case, three-dimensional graphics can be generated by applying di erent options. The objects can be represented and observed from di erent viewpoints and from di erent perspectives. Also the representation of curved surfaces in three-dimensional space, i.e., the graphical representation of functions of two variables, is possible. Furthermore it is possible to represent curves in three-dimensional space, e.g., if they are given in parametric form. For a detailed description of three-dimensional graphics primitives see 20.5]. The introduction of these representations is similar to the two-dimensional case.
1. Graphical Representation of Surfaces
The command Plot3D in its basic form requires the de nition of a function of two variables and the domain of these two variables: In ] := Plot3D f x y ] fx xa xe g fy ya yeg] (20.90) All options have the default setting. For the function z = x2 + y2, with the input In 1] := Plot3D x^ 2 + y ^2 fx ;5 5g fy ;5 5g PlotRange;> f0 25g] we get Fig. 20.8a, while Fig. 20.8b is generated by the command In 2] := Plot3D (1 ; Sin x]) (2 ; Cos 2 y ]) fx ;2 2g fy ;2 2g] For the paraboloid, the option PlotRange is given with the required z values, because the solid is cut at z = 25.
2. Options for 3D Graphics
The number of options for 3D graphics is large. In Table 20.27, only a few are enumerated, where options known from 2D graphics are not included. They can be applied in a similar sense. The option ViewPoint has special importance, by which very di erent observational perspectives can be chosen.
20 10 0
a)
4 2 -4
-2
0 0
2
4
6 4 2 0 -2
-2 -4
b)
2 1 -1
0 0
1
-1 2 -2
Figure 20.8
3. Three-Dimensional Objects in Parametric Representation
Similarly to 2D graphics, three-dimensional objects given in parametric representation can also be represented. With ParametricPlot3D ffx t u] fy t u] fz t u]g ft ta te g fu ua ue g] (20.91) a parametrically given surface is represented, with ParametricPlot3D ffx t] fy t] fz t]g ft ta te g] (20.92)
20.5 Graphics in Computer Algebra Systems 1005
a three-dimensional curve is generated parametrically. Boxed HiddenSurface ViewPoint Shading PlotRange
Table 20.27 Options for 3D graphics default setting is True it draws a three-dimensional frame around the surface sets the non-transparency of the surface default setting is True speci es the point (x y z) in space, from where the surface is observed. Default values are f1:3 ;2:4 2g default setting is True the surface is shaded False yields white surfaces fza zeg, ffxa xeg fya yeg fza zegg can be chosen for the values All. Default is Automatic
-0.5 -1 1
0
0.5
1 -1
0.5
4
0 -0.5 -1 -1 -0.5
2
a)
0
0.5
1
b)
0
1-1 0
1
0
Figure 20.9 The objects in Fig. 20.9a and Fig. 20.9b are represented with the commands In 3] := ParametricPlot3D fCos t] Cos u] Sin t] Cos u] Sin u]g ft 0 2Pig fu ;Pi=2 Pi=2g] (20.93) In 4] := ParametricPlot3D fCos t] Sin t] t=4g ft 0 20g] Mathematica provides further commands by which density, and contour diagrams, bar charts and sector diagrams, and also a combination of di erent types of diagrams, can be generated. The representation of the Lorenz attractor (see 17.2.4.3, p. 827) can be generated by Mathematica.
20.5.2 Graphics with Maple
20.5.2.1 Two-Dimensional Graphics
Maple can graphically represent functions with the command plot with several di erent options. The
input functions can be explicit functions of one variable, functions given in parametric form and lists of two-dimensional points. Maple prepares a table of values from the input function by internal algorithms, and its points are connected by a spline method to get a smooth curve. There are several options by which the shape of the graphic can be inuenced. The graphic itself is represented in a special environment, and it can be connected to the work document by the corresponding system commands, or it can be sent to the printer or plotter. The data can be saved in various formats, for example as a Postscript le.
1. Syntax of Two-Dimensional Graphics
The two-dimensional plot command has the basic structure plot(funct hb vb options)
(20.94)
1006 20. Computer Algebra Systems The rst argument funct can have the following meanings: a) a real function of one independent variable, e.g., f (x) b) a procedure of a function, generated by, e.g., the arrow symbol c) the parametric representation of a real function in the form of a list u(t) v(t) t = a::b], where t = a::b is the domain of the parameter d) several functions enclosed in curly braces, which should be represented together e) a list of numbers, which are considered to be the coordinates (x y) of the points to be represented. The second argument hb is the domain of the independent variable it has the form x = a::b. If no argument is given, then Maple automatically takes the domain ;10::10. It is possible to assign to one or to both limits the values ;1 and/or 1. In this case, Maple chooses a representation of the x-axis with arctan. The third argument vb directs the domain of the dependent variable (vertical). It should be given in the form y = a::b. If it is omitted, Maple takes the values determined from the equation of the function for the domain of the independent variable. It can cause problems if in this domain there is, e.g., a pole. Then, if it is necessary, this domain should be limited. One or several options can follow as further arguments. Some of them are represented in Table 20.28. The representation of several functions by Maple in one graphic is made in general by di erent colors or by di erent line structure. Maple provides the possibility of making changes directly on the graphic according to corresponding menus, e.g., the ratio of the horizontal and vertical measure, the frame of the picture, etc. Table 20.28 Options for Plot command coords = polar yields the representation of a parametric input in polar coordinates (the rst variable is the radius, the second one is the angle) numpoints = n sets the minimal number of the generated points (default 49) resolution = m sets the horizontal resolution of the representation in pixels (default m = 200) xtickmarks = p sets the number of scaling marks on the x-axis style = SPLINE generates the connection with cubic spline interpolation (default) style = LINE generates linear interpolation style = POINT shows only the points title = T places a title for the graphic, T must be a string y 20
y 2
-2 a) -4
-2
0
2
4x
b)
0
2x
-2
Figure 20.10
2. Examples for Two-Dimensional Graphics
The following graphics are generated by Maple, then vectorized by Coreltrace and nished by Coreldraw!. This was necessary because the direct conversion of a Maple graphic in EPS data results in very thin lines and so unattractive pictures. 1. Exponential and Hyperbolic Functions With the construction > plot(f2^x 10^x (1=2)^x (1=10)^x exp(x) 1=exp(x)g x = ;4::4 y = 0::20 (20.95a)
20.5 Graphics in Computer Algebra Systems 1007
> xtickmarks = 2 ytickmarks = 2) (20.95b) we get the exponential functions represented in Fig. 20.10a. Similarly, the command > plot(fsinh(x) cosh(x) tanh(x) coth(x)g x = ;2:1::2:1 y = ;2:5::2:5) (20.95c) yields the common representation of the four hyperbolic functions (see 2.9.1, p. 87) in Fig. 20.10b. Additional structures, such as arrow heads on axes, captions, etc., are added subsequently with the help of graphic programs. 2. Bessel Functions With the calls > plot(fBesselJ(0 z) BesselJ(2 z) BesselJ(4 z)g z = 0::10) (20.96a) > plot(fBesselJ(1 z) BesselJ(3 z) BesselJ(5 z)g z = 0::10) (20.96b) we get the rst three Bessel functions J (n z) with even n (Fig. 20.11a) and with odd n (Fig. 20.11b). y 1
y 5
J(n,z) n=0,2,4
0
J(n,z) n=1,3,5
0
10 z
10 z
b)
a)
Figure 20.11 The other special functions built into Maple can be represented in a similar way. 3. Parametric Representation With the call > plot( t # cos(t) t # sin(t) t = 0::3 # Pi]) (20.97a) we get the curve represented in Fig. 20.12a. For the following two commands Maple gives a loop function similar to a trochoid Fig. 20.12b (compare: Curtate trochoid in 2.13.2, p. 100) and the hyperbolic spiral Fig. 20.12c (see 2.14.2, p. 104). > plot( t ; sin(2 # t) 1 ; cos(2 # t) t = ;2 # Pi::2 # Pi]) (20.97b) > plot( 1=t t t = 0::2 # Pi] x = ;:5::2 coords = polar) (20.97c) Because of the introduction the option coords, Maple interprets the parametric representation as polar coordinates at the execution of the command.
0 a)
b) -6
0
6
c) 0
1
2
3
Figure 20.12
3. Special Package plots
The special package plots with additional graphical operations can be found in the Maple library. In the two-dimensional case, the commands conformal and polarplot have special interest. With polarplot(L options) (20.98)
1008 20. Computer Algebra Systems curves given in polar coordinates can be drawn. L denotes a set (enclosed in curly braces) of several functions r('). Maple interprets the variable ' as an angle and it denotes the curve in the domain between ; ' if no other domain is prescribed. The command conformal(F r 1 r 2 options) (20.99) displays the mapping of the standard grid lines of the plane to a set of grid curves with the help of a function F of complex variables. The new grid lines also intersect each other orthogonally. The domain r1 determines the original grid lines. Default values are 0::1 + (;1)1=2 . The domain r2 gives the size of the window in which the range lies. The default range here is calculated to completely enclose the resulting conformal lines.
20.5.2.2 Three-Dimensional Graphics
Maple provides the command plot3d to represent functions of two independent variables as surfaces in space or to represent space curves. Maple represents the objects generated by this command anal-
ogously to two-dimensional ones in one window. The number of options for representation is usually larger, especially by the additional options about the viewpoint (of observation).
1. Syntax of plot3d Commands
This command can be used in four di erent forms: a) plot3d(funct x = a::b y = c::d). In this form, funct is a function of two independent variables, whose domain is given by x = a::b and y = c::d . The result is a space surface. b) plot3d(f a::b c::d). Here f is a procedure with two arguments, e.g., generated by the arrow operator, the domains are associated to these variables. c) plot3d( u(s t) v(s t) w(s t)] s = a::b t = c::d). The three functions u v w depending on the two parameters s and t de ne a parametric representation of a space surface, restricted to the domain of the parameters. d) plot3d( f g h] a::b c::d). This is the equivalent form of the parametric representation, where f g h must be procedures of two arguments. All further arguments of the operator plot3d are interpreted by Maple as options. Some important options are represented in Table 20.29. They should be used in the form option = value. Table 20.29 Options of command plot3d numpoints = n sets the minimal number of generated points (default number is n = 625) grid m n] speci es that an m n grid of equally spaced points is sampled (default 25 25) labels = x y z ] indicates the labels used along the axes (string is required) style = s s is a value from POINT, HIDDEN, PATCH, WIREFIRE. Here it de nes how the surface is represented axes = f f can have the values BOXED, NORMAL, FRAME or NONE. The representation of the axes is speci ed coords = c speci es the required coordinate system. Values can be cartesian, spherical, cylindrical . Default is cartesian projection = p p takes values between 0 and 1 and it de nes the observational perspective. Default value is 1 (orthogonal projection) orientation = theta phi] speci es the angle of the point in space in a spherical coordinate system from which the surface is observed view = z 1::z 2 gives the domain of the z values for which the surface should be represented. Default is the total surface In general, almost all options can be reached and appropriately set in the corresponding menu in the screen. In this way, the picture can subsequently be improved.
20.5 Graphics in Computer Algebra Systems 1009
1
6
0.5 0 -0.5 -1
0 -1 -0.5 0
a)
0.5
11
0.5
0
-1 -0.5
-1
-1 0
0 b)
1
Figure 20.13
2. Additional Operations from Package plots
The library package plots provides further possibilities for representation of space structures. Especially important is the representation of space curves with the command spacecurve. The rst argument is a list of three functions depending on a parameter, the second argument must specify the domain of this parameter. So, the options of the command plot3d are kept until they have any meaning for this case. For further information about this package one should study the literature. With the inputs > plot3d( cos(t) # cos(u) sin(t) # cos(u) sin(u)] t = 0::2 # Pi u = 0::2 # Pi) (20.100a) > spacecurve( cos(t) sin(t) t=4] t = 0::7 # Pi) (20.100b) the graphics of a perspectively represented sphere (Fig. 20.13a) and a perspectively represented space spiral (Fig. 20.13b) are generated.
1010 21. Tables
21 Tables 21.1 Frequently Used Mathematical Constants 1 = (=180) rad 1 / e ln 10 = 1=M
3:141592654 : : : 0:017453293 : : : 0.01 2:718281828 : : : 2:302585093 : : :
C Euler constant 1 rad = 1 1 / lg e = M
0:577215665 : : : 57:29577951 : : : 0:001 0:434294482 : : :
21.2 Natural Constants
The table contains values of the physical constants, recommended in 21.11] and 21.12]. In parenthesis is given the standard uncertainty of the last two digits.
Fundamental constants
velocity of light in vacuum gravitation constant standard acceleration of gravity permeability of free space permittivity of vacuum Planck constant, Planck quantum
c0 c G gn 0 "0 h h"
characteristic impedance of vacuum
= 299 792 458 ms;1 (exact) = 6:6742 (10) 10;11 m3kg;1 s;2 = 9:806 65 ms;2 (exact) = 4 10;7 NA;2 = 12:566 370 614 : : : 10;7 NA;2 (exact) = 1= ec2 = 8:854 187 817 : : : 10;12 Fm;1 (exact) = 6:626 069 3(11) 10;34 Js = 4:135 667 43(35) 10;15 eVs = h=2 = 1:054 571 68(18) 10;34 Js = 6:582 119 15(56) 10;16 eVs
Z0
= 376:730 313 461 : : : ) (exact)
Electromagnetic constants
e e=h speci c fundamental electric charge ;e=me ne structure constant 1= quantum of magnetic ux 0 Josephson{constant KJ v. Klitzing{constant RK quantum of conductance G0 quantum of circulation s fundamental electric charge
Atomic electron shell Bohr magneton
B
Bohr radius
a0 re R1 hcr1 0
classical electron radius Rydberg{constant Rydberg{energy Thomson{cross-section
= 1:602 176 53(14) 10;19 C = 2:417 989 40(21) 1014 AJ;1 = ;1:758 820 12(15) 1011 C kg;1 = ece2 =2h = 7:297 352 568(24) 10;3 = 137:035 999 11(46) = h=2e = 2:067 833 72(18) 10;15 Wb = 2e=h = 483 597:879(41) 109 HzV;1 = h=e2 = 25812:807 ) (exact). = 2e=h = 7:748 091 733(26) 10;5 S = h=2me = 3:636 947 550(24) 10;4 m2s;1 )
= eh" =2me = 9:274 00849 (80) 10;26 JT;1 = 5:788 381 804(39) 10;5 eVT;1 = h" 2 =E0(e)e2 = re=2 = 0:529 177 2108(18) 10;10 m = 2a0 = 2:817 940 325(28) 10;15m = 02 me4 c3=8h3 = 10 973 731:568 525(73) m;1 = 13:605 692 3(12) eV = 8re2 =3 = 0:665 245 854(15) 10;28 m2
21.2 Natural Constants 1011
Atomic nuclei and particles
= 1 u = 1 (kg mol;1 )=NA = 121 m(12 C) = 1:660 538 86(28)
atomic mass unit
mu
magic neutron{ and proton{numbers Rest energy atomic mass unit electron proton neutron Rest mass electron proton
Nm Zm = 2, 8, 20, 28, 50, 82 Nm = 126, may be 184 Zm = may be 114, 126
10;27kg = 931:494 043(80) MeV c;2 = 1822:89me molar mass of carbon-12 M(12 C) = 12 10;3kg/mol nuclear magneton k = eh" =2mp = 5:050 783 43(43) 10;27 JT;1 = 3:152 451 259(21) 10;8 eVT;1 nuclear radius R = r0A1=3 r0 = (1:2 : : : 1:4) fm 1 A 250: r0 R 9 fm
neutron magnetic moment electron proton neutron
E0 (u) E0 (e) E0 ( p ) E0 (n)
= 931:494 013(37) MeV = 0:510 998 902(21) MeV = 938:271 998(38) MeV = 939:565 330(38) MeV
me mp
= 9:109 382 6(16) 10;31 kg = 5:485 799 0945(24) 10;4 u = 1:672 621 71(29) 10;27 kg =1 836:152 672 61(85)me = 1:007 276 466 88(13) u = 1:674 927 28(29) 10;27 kg = 1 838:683 659 8(13)me = 1:008 664 915 60(55) u
mn
= ;1:001 159 652 1859(41) B = ;928:476 412(80) 10;26 JT;1 = +2:792 847 337(28) k = 1:410 606 71(12) 10;26 JT;1 = ;1:913 042 72(45) k = 0:966 236 45(26) 10;26 JT;1
e p n
Physical chemistry, Thermodynamics Avogadro{constant Loschmidt{constant Boltzmann{constant
Universal gas constant Faraday{constant molar volum of inert gas (T = 273:15 K p = 101:325 kPa) 1. Planck radiation constant 2. Planck radiation constant Stefan{Boltzmann{constant Wien{constant b = maxT = c2=4:965 114 231 : : :
NA = 6:022 141 5(10) 1023 mol;1 n0 = NA=Vm = 2:686 777(47) 1025 m;3 k = R0 =NA = 1:380 6505(24) 10;23 J K;1 = 8:617 343(15) 10;5 eVK;1 R0 = NAk = 8:314 472(15) J mol;1K;1 F = NAe = 96 485:3383(83) C mol;1 Vm = R0 T0=p0 = 2:2 413 996(39) 10;2 m3 mol;1 c1 c2
= 2hc20 = 3:741 771 38(64) 10;16 W m2 = hc0 =k = 1:438 775 2(25) 10;2 m K = (2=60)k4=h" 3 c2 = 5:670 400(40) 10;8 Wm;2K;4
b
=2:897 768 5(51) 10;3 m K
1012 21. Tables
21.3 Metric Prexes Pre x yocto zepto atto femto pico nano mikro milli centi deci
Factor 10;24 10;21 10;18 10;15 10;12 10;9 10;6 10;3 10;2 10;1
Abbreviation y z a f p n m c d
Pre x deca hecto kilo mega giga tera peta exa zetta yotta
Factor 101 102 103 106 109 1012 1015 1018 1021 1024
Abbreviation da h k M G T P E Z Y
103 = 1000 . 10;3 = 0 001 . 1 m (1 mikrometer) = 10;6 m . 1 nm (1 nanometer) = 10;9 m . Remark: The metric system is built up by adding pre xes which are the same for every kind of measure. These pre xes should be used in steps of powers with base 10 and exponent 3: milli-, micro-, nano- ruther than in the smaller steps hecto-, deca-, deci-. The British system, unlike the metric one, is not built up in 10's e. g.: 1 lb = 16 oz = 7000 grains.
21.4 International System of Physical Units (SI-Units) Further information about physical units see 21.11], 21.12].
SI Base Units
Base quantity length time mass thermodynamic temperature electric current amount of substance luminous intensity
Symbol m s kg K A mol cd
Name meter second kilogram kelvin ampere mole (1 mol = A particles, A = Avogadro-constant) candela
plain angle solid angle
rad sr
radian = l=r , 1 rad = 1 m/1 m (see p. 130) steradian ) = S=r2 , 1 sr = 1 m2=1 m2 (see p. 151)
Additional SI-units
Examples of SI derived units without special names and symbols Derived quantity area volume speed, velocity acceleration angular velocity angular acceleration wave number momentum angular momentum
Symbol m2 m3 m/s m/s2 rad/s rad/s2 1/m kg m s;1 kg m2s;1
Derived quantity moment of inertia mass density particle number density speci c volume magnetic eld strength amount-of-substance concentration luminance mass fraction
Symbol kg m2 kg/m3 m;3 m3 =kg A/m mol/m3 cd/m2 kg/kg = 1
21.4 International System of Physical Units (SI-Units) 1013
Examples of SI derived units with special names and symbols Derived quantity frequency force pressure, stress energy, work, quantity of heat power, radiation ux electric charge, quantity of electrity electric potential electric capacitance electric resistance electric conductance magnetic ux magnetic ux density inductance Celsius temperature luminous ux illuminance activity (radionuclides) dose equivalent absorbed dose catalytic activity
Symbol Hz N Pa J kWh W C V F ) S Wb T H C lm lx Bq Sv Gy kat
Name hertz newton pascal joule kilowatthour watt coulomb volt farad ohm siemens weber tesla henry degree Celsius lumen lux becquerel sievert gray katal
Expression in terms of SI base units 1Hz = 1/s 1N = 1kg m/s2 1Pa = 1N/m2 1J = 1N m = 1kg m2=s2 1kWh = 3:6 106 J 1W = 1J/s = 1N m/s = 1kg m2=s3 1C = 1A s 1V = 1W/A = 1kg m2 =(A s3) 1F = 1C/V = 1A2 s4 =(m2 kg) 1) = 1V/A = 1kg m2 =(A2s3) 1S = 1A/V = 1A2 s3 =(kg m2) 1Wb = 1V s = 1kg m2 =(A s2) 1T = 1Wb/m2 = 1kg=(A s2 ) 1H = 1Wb/A = 1kg m2 =(A2s2) t =C = T =K ; 273:15 1cd sr = 1m2 m;2cd = 1cd 1lx = 1lm/m2 = 1cd=m2 1Bq = 1=s 1Sv = 1J=kg = 1m2 =s2 1Gy = 1J=kg = 1m2 =s2 1/s mol
Examples of SI derived units whose names and symbols include SI derived units with special names and symbols Quantity dynamic viscosity surface tension action heat capacity, entropy molar entropy, molar heat capacity thermal conductivity electric eld strength electric charge density permittivity radiant intensity exposure (X and rays) catalytic (activity) concentration
SI derived unit Pa s N=m Js J=K J=(mol K) W=(m K) V=m C/m3 F/m W/sr C/kg kat/m3
Quantity moment of force energy density speci c energy speci c heat capacity enthalpy irradiance, heat ux density molar energy magnetic eld strength electric ux density permeability radiance absorbed dose rate
SI derived unit Nm J=m3 J=kg J=(K kg) J W=m2 J=mol A m;1 C/m2 H/m W/(m2 sr) Gy/s
1014 21. Tables Some units outside the SI that are accepted for use with the SI Name astronomical unit parsec light year volume litre time minute hour day year plain angle degree radiant grade minute second mass uni ed atomic mass unit metric ton energy electronvolt focal power diopter neper bel length
Symbol ua ps ly L (l) min h d a 1 = =180 rad rad gon 10 = (1=60) = =10800 rad 100 = (1=60)0 = =648000 rad u t eV dpt Np B (1 dB = 0.1B)
Value in SI units 1ua = 1:49597870 1011 m 1pc = 30:857 1015 m 1ly = 9:46044763 1015 m 1L = 1dm3 = 10;3 m3 1min = 60 s 1h = 3600 s 1d = 86400 s 1a = 365 d = 8760 h 1 = 0:017453293 : : : rad 1rad = 57:29577951 : : : rad 1gon = 200 0 1 = 000290888 : : : rad 100 = 000004848 : : : rad 1u = 1:66054 10;27 kg 1t = 103 kg 1eV = 1:602 18 J 1dpt = 1=m 1Np = 1 1B = 21 ln 10 Np
Some units outside the SI that are currently accepted for use with the SI, subject to further review Name nautical (intern.) mile 9 Angstr:m area are hectare area cross-section volume U.S. barrel petroleum velocity knot
Symbol sm 9 A a ha b (barn) bbl kn
energy calorie pressure bar
cal bar
length
mmHg column standard atmosphere curie roentgen rad rem
mmHg atm Ci R rad rem
Value in SI units 1sm = 1852m 19 A = 0:1 nm = 10;10 m 1a = 1dam2 = 100 m2 1ha = 1hm2 = 104 m2 1b = 100 fm2 = 10;28 m2 1bbl = 0:158988 m3 1kn = 1sm/h = 1:852 km/h = 0:5144 m/s 1cal = 4:1868 J 1bar = 0:1 MPa = 100 kPa = 105 kg=(s2m) = 105 Pa 1mmHg = 133:322 Pa 1atm = 1:01325 105 Pa 1Ci = 3:7 1010 Bq 1R = 2:58 10;4 C/kg 1rad = 1cGy = 10;2 Gy 1rem = 1cSv = 10;2 Sv
21.5 Important Series Expansions 1015
21.5 Important Series Expansions Function
Series Expansion Algebraic Functions Binomial Series
m (a x)m After transforming to the form am 1 xa we get the following series:
Binomial Series with Positive Exponents m ; 2) x3 + 1 mx + m(m2!; 1) x2 m(m ; 1)( 3! (m > 0) + (1)n m(m ; 1) : : : (m ; n + 1) xn +
Convergence Region
jxj a for m > 0 jxj < a for m < 0
(1 x)m (1 x)
1 4
(1 x) 13 (1 x) 12 (1 x) 32 (1 x) 52
n! 1 1 3 1 7 x3 ; 1 3 7 11 x4 2 1 4 x ; 4 8 x 4 8312 4 8 12 16 1 1 2 1 2 5 1 8 x4 2 3 1 3 x ; 3 6 x 3 6 9 x ; 3 629512 1 21 x ; 12 14 x2 12 14 36 x3 ; 12 14 36 58 x4 1 23 x + 32 14 x2 32 14 16 x3 + 32 14 16 38 x4 1 5 x + 5 3 x2 5 3 1 x3 ; 5 3 1 1 x4 2 24 246 2468
jxj 1 jxj 1 jxj 1 jxj 1 jxj 1 jxj 1
Binomial Series with Negative Exponents m + 2) x3 + (1 x);m 1 mx + m(m2!+ 1) x2 m(m + 1)( 3! (m > 0) + (1)n m(m + 1) : : : (m + n ; 1) xn + n! 1 1 5 1 9 x3 + 1 5 9 13 x4 2 ; 41 (1 x) 1 4 x + 4 8 x 4 8512 4 8 12 16 1 1 4 1 4 7 1 2 3 4 ; 31 (1 x) 1 3 x + 3 6 x 3 6 9 x + 3 46 79 10 12 x (1 x); 21 1 1 x + 1 3 x2 1 3 5 x3 + 1 3 5 7 x4 2 24 246 2468 (1 x);1 1 x + x2 x3 + x4 (1 x); 23 1 3 x + 3 5 x2 3 5 7 x3 + 3 5 7 9 x4 2 24 246 2468 (1 x);2 1 2x + 3x2 4x3 + 5x4
jxj < 1 jxj < 1 jxj < 1 jxj < 1 jxj < 1 jxj < 1 jxj < 1
1016 21. Tables Function (1 x); 52 (1 x);3 (1 x);4 (1 x);5
Convergence Series Expansion Region 4 1 52 x + 52 74 x2 25 47 69 x3 + 52749611 jxj < 1 8 x 1 1 (2 3x 3 4x2 + 4 5x3 5 6x4 + ) jxj < 1 12 1 1 1 2 3 (2 3 4x 3 4 5x2 +4 5 6x3 5 6 7x4 + ) jxj < 1 1 1 1 2 3 4 (2 3 4 5x 3 4 5 6x2 +4 5 6 7x3 5 6 7 8x4 + ) jxj < 1 Trigonometric Functions
3 5 2n+1 sin x x ; x3! + x5! ; + (;1)n (2xn + 1)! 3 2 sin(x + a) sin a + x cos a ; x sin a ; x cos a 2! 3! 4 sin a xn sin a + n2 x + 4! + + n! 2 4 6 2 n cos x 1 ; x + x ; x + + (;1)n x 2! 4! 6! (2n)! 2 cos a x3 sin a x + cos(x + a) cos a ; x sin a ; 2! 3! 4 cos a xn cos a + n2 x + 4! ; + n! 1 2 17 62 tan x x + 3 x3 + 15 x5 + 315 x7 + 2835 x9 + 2n 2n + 2 (2 ; 1)Bn x2n;1 + (2n)! " 3 2x5 7 1 x x x cot x x ; 3 + 45 + 945 + 4725 +
2n + 2(2nB)!n x2n;1 + 5 x4 + 61 x6 + 277 x8 + sec x 1 + 21 x2 + 24 720 8064 + (2Enn)! x2n +
jxj < 1 jxj < 1 jxj < 1 jxj < 1 jxj < 2 0 < jxj <
jxj < 2
21.5 Important Series Expansions 1017
Function cosec x
Series Expansion 1 1 7 3 31 5 127 x7 + x + 6 x + 360 x + 15120 x + 604800 2n;1 + 2(2 ; 1) Bnx2n;1 (2n)!
Exponential Functions
ex ax = ex ln a x ex ; 1
ln x
ln x
2 3 n 1 + 1!x + x2! + x3! + + xn! + ln a + (x ln a)2 + (x ln a)3 + + (x ln a)n + 1 + x 1! 2! 3! n! 2 B x4 B x6 x B 1x 2 3 1 ; 2 + 2! ; 4! + 6! ; 2n +(;1)n+1 B(2nnx)!
"
0 < jxj <
jxj < 1 jxj < 1 jxj < 2
Logarithmic Functions
1 (x ; 1)3 (x ; 1)5 2 xx ; + 1 + 3(x + 1)3 + 5(x + 1)5 +
2n+1 + (x ; 1) 2n+1 + (2n + 1)(x + 1) 2 (x ; 1)3 (x ; 1)4 ( x ; 1) (x ; 1) ; 2 + 3 ; 4 + n +(;1)n+1 (x ;n 1)
x ; 1 + (x ; 1)2 + (x ; 1)3 + + (x ; 1)n + x 2x2 3x3 nxn 2 x 3 x4 n x x ln (1 + x) x ; 2 + 3 ; 4 + + (;1)n+1 n "
2 3 4 5 n ln (1 ; x) ; x + x2 + x3 + x4 + x5 + + xn + "
+ x x3 + x5 + x7 + + x2n+1 + ln 11 ; 2 x + x 3 5 7 2n + 1 = 2 Artanh x ln x
Convergence Region
x>0
0<x2
x > 21
;1 < x 1 ;1 x < 1 jxj < 1
1018 21. Tables Function
Series Expansion "
1 ln x + 1 2 1 + 1 3 + 1 5 + 1 7 + + + x;1 x 3x 5x 7x (2n + 1) x2n+1 =2 Arcoth x 2 x4 ; x6 ; ; 22n;1 Bn x2n ; ln jsin xj ln jxj ; x6 ; 180 2835 n (2n) ! 2 x4 x6 8 x 17 x ln cos x ; 2 ; 12 ; 45 ; 2520 ; 2n;1 2n 2n ; 2 (2n(2;n)1)! Bn x ; 7 x4 + 62 x6 + ln jtan xj ln jxj + 13 x2 + 90 2835 2n 2n;1 + 2 (2 ; 1)Bn x2n + n(2n) ! arcsin x
arccos x
arctan x arctan x
arccot x
Inverse Trigonometric Functions 3 5 7 x + 2x 3 + 21 43 x 5 + 21 43 65 x 7 + 2n+1 + 123456(2(2nn;)(21)n x+ 1) + " ; x + x3 + 1 3 x5 + 1 3 5 x7 + 2 23 245 2467
2n+1 + 123456(2(2nn;)(21)n x+ 1) + 3 5 7 2n+1 x ; x3 + x5 ; x7 + + (;1)n 2xn + 1 2 ; x1 + 3 1x3 ; 5 1x5 + 7 1x7 ; 1 +(;1)n+1 (2n + 1) x2n+1 "
; x ; x3 + x5 ; x7 + + (;1)n x2n+1 2 3 5 7 2n + 1
Convergence Region
jxj > 1 0 < jxj <
jxj < 2 0 < jxj < 2
jxj < 1 jxj < 1 jxj < 1 jxj > 1 jxj < 1
21.5 Important Series Expansions 1019
Function
Series Expansion
Convergence Region
sinh x
Hyperbolic Functions 7 2n+1 x + 3 ! + 5 ! + x7 ! + + (2xn + 1) ! +
jxj < 1
cosh x tanh x
coth x sech x cosech x
x3
x5
4 6 2n 2 1 + x2 ! + x4 ! + x6 ! + + (2xn) ! + 2 x5 ; 17 x7 + 62 x9 ; x ; 13 x3 + 15 315 2835 n+1 2n 2n + (;1) (22n)(2! ; 1) Bnx2n;1 1 + x ; x3 + 2x5 ; x7 + x 3 45 945 4725 n+1 2n + (;1)(2n) ! 2 Bnx2n;1 6 1385 8 1 ; 21! x2 + 45! x4 ; 61 6 ! x + n8 ! x ; + (;1) En x2n (2n) ! 1 ; x + 7 x3 ; 31 x5 + x 6 360 15120 n 2n;1 + 2(;1) (2(2n) ! ; 1) Bnx2n;1 +
jxj < 1 jxj < 2 0 < jxj <
jxj < 2 0 < jxj <
Area Functions
Arsinh x x ; 1 x3 + 1 3 x5 ; 1 3 5 x7 + 23 245 2467 n +(;1) 1 3 5 (2n ; 1) x2n+1 2 4 6 2n (2n + 1)
1 ; 1 3 ; 1 3 5 ; 2 2x2 2 4 4x4 2 4 6x6 3 5 7 2n+1 Artanh x x + x3 + x5 + x7 + + 2xn + 1 + 1 Arcoth x x1 + 31x3 + 51x5 + 71x7 + + (2n + 1) x2n+1 + Arcosh x ln (2x) ;
jxj < 1 x>1
jxj < 1 jxj > 1
1020 21. Tables
21.6 Fourier Series 1. y = x for 0< x < 2
y = ; 2 sin1 x + sin22x + sin33x +
y 2π -2π
0
2π
4π
6π
x
2. y = x for 0 x y = 2 ; x for < x 2
y = 2 ; 4 cos x + cos323x + cos525x +
y -π 0 π 2π
π 4π
6π
x
3. y = x for ; < x <
y = 2 sin1 x ; sin22x + sin33x ;
y π -π
0π
x
3π 5π 7π 2π 4π 6π
4. y = x for ; 2 x 2 y = ; x for 2 x 32
y = 4 sin x ; sin323x + sin525x ;
y -π
0
π
π/2 x
2π 3π
5. y = a for 0< x < y = ;a for < x < 2
y = 4a sin x + sin33x + sin55x +
y a -π
0 π 2π 3π 4π 5π 6π 7π
x
21.6 Fourier Series 1021
a
6. y = 0 for 0 x < and for ; < x + and 2 ; < x 2 y = a for < x < ; y = ;a for + < x 2 ; y α π-α y = 4a cos sin x + 13 cos 3 sin 3x π 3π 2 2 0 π 5π x + 1 cos 5 sin 5x + 5 2 2 7. y = ax for ;a x a y = a for x ; y = a( ; x) for ; x + y = ;a for + x 2 ; π-α 3π 2 0 π π 2
α
a
y π 2
y = 4 a sin sin x + 312 sin 3 sin 3x + 512 sin 5 sin 5x +
x
5π 2
p y = 6 23a sin x ; 512 sin 5x + 712 sin 7x ; 111 2 sin 11x +
Especially, for = 3 holds:
8. y = x2 for ; x
2 y = 3 ; 4 cos1 x ; cos222x + cos323x ;
y π
2
-π 0 π 2π 3π 4π 5π 6π 7π
x
9. y = x( ; x) for 0 x
2 y = 6 ; cos122x + cos224x + cos326x +
y 2
-π
0
π
2π
3π
4π
π /4
x
10. y = x( ; x) for 0 x y = ( ; x)(2 ; x) for x 2
y = 8 sin x + 313 sin 3x + 513 sin 5x +
y 2
-π
0
π
2π 3π
4π
3π /4
x
1022 21. Tables 11. y = sin x for 0 x 1
0
-π
2x cos 4x cos 6x y = 2 ; 4 cos + + + 13 35 57
y π
3π
2π
4π x
12. y = cos x for 0 < x <
2x 4 sin 4x 6 sin 6x y = 4 2 sin + + + 13 35 57
y 1 -π
0 -1
π
2π
3π
x
13. y = sin x for 0 x y = 0 for x 2 1 -π
0
y
π
2π
3π
x
14. y = cos ux for ; x
15. y = sin ux for ; < x <
16. y = x cos x for ; < x <
17. y = ; ln 2 sin x for 0 < x 2
18. y = ln 2 cos x2 for 0 x < 19. y = 12 ln cot x2 for 0 < x <
2x cos 4x y = 1 + 12 sin x ; 2 cos 1 3 + 3 5 cos 6 + 5 7x + u 1 ; cos x + cos 2x ; cos 3x + y = 2u sin 2u2 u2 ; 1 u2 ; 4 u2 ; 9 (u arbitrary, but not integer number)
x 2 sin 2x 3 sin 3x ; + + y = 2 sin u 1sin 2 2 2 ;u 4;u 9;u (u arbitrary, but not integer number) 2x 6 sin 3x 8 sin 4x y = ; 12 sin x + 42sin 2 ; 1 ; 32 ; 1 + 42 ; 1 ;
y = cos x + 12 cos 2x + 13 cos 3x + y = cos x ; 12 cos 2x + 13 cos 3x ; y = cos x + 13 cos 3x + 15 cos 5x +
21.7 Indenite Integrals 1023
21.7 Indenite Integrals
(For instructions on using these tables see 8.1.1.2, 2., p. 429).
21.7.1 Integral Rational Functions 21.7.1.1 Integrals with X = ax + b 1:
Z
X n dx = a(n 1+ 1) X n+1
Notation: X = ax + b
(n 6= ;1)
Z dx 1 X = a ln X: Z 3: xX n dx = a2 (n1+ 2) X n+2 ; a2 (nb+ 1) X n+1
(for n = ;1 see No. 2):
2:
4:
Z
xm X n dx =
(n 6= ;1 6= ;2) (for n = ;1 = ;2 see No. 5 und 6): 1 Z (X ; b)m X n dX (n 6= ;1 6= ;2 : : : = 6 ;m): am+1
The integral is used for m < n or for integer m and fractional n in these cases (X ; b)m is expanded by the binomial theorem (see 1.1.6.4, p. 12). Z 5 xXdx = xa ; ab2 ln X: Z 6: xXdx2 = a2bX + a12 ln X:
! Z x dx 1 1+ b : = ; X 3 a2 X 2X 2 ! Z x dx 1 8: X n = a2 (n ; ;2)1X n;2 + (n ; 1)b X n;1 (n 6= 1 6= 2): Z 2 9: x Xdx = a13 12 X 2 ; 2bX + b2 ln X : ! Z 2 1 X ; 2b ln X ; b2 : 10: xXdx = 2 a3 X Z x2 dx 1 2 ! 11: X 3 = a3 ln X + 2Xb ; 2bX 2 : "
Z 2 1 ;1 + 2b b2 12: xXdx = ; n a3 (n ; 3)X n;3 (n ; 2)X n;2 (n ; 1)X n;1 ! Z x3 dx 1 X 3 3bX 2 13: = 4 ; + 3b2 X ; b3 ln X : X a 3 2 Z x3 dx 1 X 2 3! 14: X 2 = a4 2 ; 3bX + 3b2 ln X + Xb : ! Z 3 1 X ; 3b ln X ; 3b2 + b3 : 15: xXdx = 3 a4 X 2X 2
7:
(n 6= 1 6= 2 6= 3):
1024 21. Tables 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29:
30:
! Z x3 dx 1 3b ; 3b2 + b3 : = ln X + X 4 a4 X 2X 2 3X 3
Z x3 dx 1 " ;1 3b 3b2 b3 = + ; + X n a4 (n ; 4)X n;4 (n ; 3)X n;3 (n ; 2)X n;2 (n ; 1)X n;1 (n 6= 1 6= 2 = 6 3 6= 4): Z dx 1 X xX = ; b ln x : Z dx 1 ln X + ax : = ; 2 xX b2 x X ! Z dx 1 X 2ax ; a2 x2 : = ; ln + xX 3 b3 x X 2X 2 " !
Z dx ;1 1 ln X ; nX n ; 1 (;a)ixi = ; (n 1): xX n bn x i=1 i iX i Z dx 1 a X x2 X = ; bx + b2 ln x :
1 Z dx 1 2 X x2 X 2 = ;a b2 X + ab2 x ; b3 ln x :
1 Z dx 2 + 1 ; 3 ln X : = ; a + x2 X 3 2b2 X 2 b3 X ab3 x b4 x " n !
Z dx 1 ;X n (;a)i xi;1 + X ; na ln X = ; (n 2): x2 X n bn+1 i=2 i (i ; 1)X i;1 x x " Z dx 2 = ; 13 a2 ln X ; 2aX + X 2 : 3 xX b x x 2x "
Z dx 1 3a2 ln X + a3 x + X 2 ; 3aX : = ; x3 X 2 b4 x X 2x2 x "
Z dx 3 x a4 x2 2 1 X 4 a = ; 5 6a2 ln + ; + X 2 ; 4aX : 3 3 2 xX b x X 2X 2x x " nX ! Z dx +1 i i ;2 2 1 ; n + 1 (;a) x + a X 2 ; (n + 1)aX = ; 3 n n +2 xX b i (i ; 2)X i;2 2x2 x i=3
2 + n(n +2 1)a ln Xx (n 3): ! Z dx m+ Xn;2 m + n ; 2 X m;i;1(;a)i = ; m+1n;1 : m n x X b i (m ; i ; 1)xm;i;1 i=0
If the denominators of the terms behind the sum vanish, then such terms should be replaced by ! m + n ; 2 (;a)m;1 ln X : x m;1
21.7 Indenite Integrals 1025
31: 32: 33: 34: 35: 36: 37: 38: 39:
Notation: % = bf ; ag Z ax + b ax % fx + g dx = f + f 2 ln(fx + g): Z dx 1 fx + g (% 6= 0): (ax + b)(fx + g) = % ln ax + b "
Z x dx = 1 b ln(ax + b) ; g ln(fx + g) (% 6= 0): (ax + b)(fx + g) % a f ! Z dx 1 1 + f ln fx + g = (% 6= 0): (ax + b)2 (fx + g) % ax + b % ax + b Z x dx b = ; a ln a + x (a 6= b): (a + x)(b + x)2 (a ; b)(b + x) (a ; b)2 b + x Z x2 dx b2 a2 ln(a + x) + b2 ; 2ab ln(b + x) (a 6= b): = + 2 (a + x)(b + x) (b ; a)(b + x) (b ; a)2 (b ; a)2 Z dx ;1 1 + 1 + 2 ln a + x (a 6= b): = 2 2 (a + x) (b + x) (a ; b)2 a + x b + x (a ; b)3 b + x ! Z x dx 1 a + b ; a + b ln a + x (a 6= b): = (a + x)2 (b + x)2 (a ; b)2 a + x b + x (a ; b)3 b + x ! Z x2 dx ;1 a2 + b2 + 2ab ln a + x (a 6= b): = (a + x)2 (b + x)2 (a ; b)2 a + x b + x (a ; b)3 b + x
21.7.1.2 Integrals with X = ax2 + bx + c
Notation: X = ax2 + bx + c % = 4ac ; b2
Z 2ax p2 p+b 40: dx X = % arctan % ax + b = ; p 2 Artanh 2p
;%
;p% = p 1 ln 2ax + b ; p;% ;% 2ax + b + ;%
(for % > 0) (for % < 0) (for % < 0):
Z dx 2ax + b 2a Z dx 2 = %X + % X Z Xdx 2ax 2Z 42: X 3 = %+ b 2X1 2 + %3Xa + 6%a2 dx X Z dx Z dx 2 ax + b (2 n ; 3)2 a 43: X n = (n ; 1)%X n;1 + (n ; 1)% X n;1 :
41:
(see No. 40): (see No. 40):
1026 21. Tables Z
Z
44: xXdx = 21a ln X ; 2ba dx Z ZX 45: xXdx2 = ; bx%+X2c ; %b dx X 46: 47: 48: 49: 50: 51: 52: 53: 54: 55: 56:
(see No. 40): (see No. 40):
Z x dx bx + 2c ; b(2n ; 3) Z dx : = ; n X (n ; 1)%X n;1 (n ; 1)% X n;1 Z x2 dx x b Z 2 = ; 2 ln X + b ; 22 ac dx (see No. 40): X a 2 a 2 a Z x2 dx (b2 ; 2ac)x + bc 2c Z dx X +% X (see No. 40): 2 = a%X Z dx (n ; 2)b Z x dx Z xX2 dx ; x c (see No. 43 and 46): X n = (2n ; 3)aX n;1 + (2n ; 3)a X n ; (2n ; 3)a X n Z xm dx xm;1 (m ; 1)c Z xm;2 dx X n = ; (2n ; m ; 1)aX n;1 + (2n ; m ; 1)a Xn Z m ;1 ; (2n(n;;mm;)b1)a x X ndx (m 6= 2n ; 1) (for m = 2n ; 1 see No. 51): Z x2n;1 dx 1 Z x2n;3 dx c Z x2n;3 dx b Z x2n;2 dx X n = a X n;1 ; a Xn ; a Xn : Z dx 1 x2 b Z dx = 2c ln X ; 2c X (see No. 40): Z xX Z Z dx 1 b dx 1 dx xX n = 2c(n ; 1)X n;1 ; 2c X n + c xX n;1 : ! Z dx b ln X ; 1 + b2 ; a Z dx = (see No. 40): x2 X 2c2 x2 cx 2c2 c X Z dx 1 (2n + m ; 3)a Z dx = ; ; m n m ; 1 n ; 1 x X (m ; 1)cx X (m ; 1)c xm;2 X n Z ; (n(+mm; ;1)c2)b xm;dx1 X n (m > 1): "
Z dx = 1 (fx + g)2 f ln 2 2 (fx + g)X 2(cf ; gbf + g a) X Z dx + 22ga ; bf 2 (see No. 40): 2(cf ; gbf + g a) X
21.7.1.3 Integrals with X = a2 x2
Notation: X = a2 x2 8 > arctan xa for the \+" sign > > < x 1 a+x Y = > Artanh a = 2 ln a ; x for the \;" sign and jxj < a > a > : Arcoth xa = 12 ln xx + ; a for the \;" sign and jxj > a: If there is a double sign in a formula, then the upper one belongs to X = a2 + x2, and the lower one to X = a2 ; x2 , a > 0.
21.7 Indenite Integrals 1027
57: 58: 59: 60: 61: 62: 63: 64: 65: 66: 67: 68: 69: 70: 71: 72: 73: 74: 75: 76:
Z dx 1 X = a Y: Z dx x 1 X 2 = 2a2 X + 2a3 Y: Z dx x 3x 3 X 3 = 4a2 X 2 + 8a4 X + 8a5 Y: Z dx x + 2n ; 1 Z dx : = n +1 X 2na2 X n 2na2 X n Z x dx 1 X = 2 ln X: Z x dx 1 X 2 = 2X : Z x dx 1 X 3 = 4X 2 : Z x dx 1 X n+1 = 2nX n (n 6= 0): Z x2 dx X = x aY: Z x2 dx x 1 X 2 = 2X 2a Y: Z x2 dx x x 1 X 3 = 4X 2 8a2X 8a3 Y: Z x2 dx x 1 Z dx = (n 6= 0): n +1 X 2nX n 2n X n Z x3 dx x2 a2 X = 2 ; 2 ln X: Z x3 dx a2 1 X 2 = 2X + 2 ln X: Z x3 dx 1 a2 X 3 = ; 2X + 4X 2 : Z x3 dx 1 a2 X n+1 = ; 2(n ; 1)X n;1 + 2nX n (n > 1): Z dx 1 x2 xX = 2a2 ln X : Z dx 2 = 12 + 1 4 ln x : 2 xX 2a X 2a X Z dx 1 1 1 x2 xX 3 = 4a2X 2 + 2a4X + 2a6 ln X : Z dx 1 1 x2 X = ; a2 x a3 Y:
1028 21. Tables 77: 78: 79: 80: 81: 82:
Z dx 1 x 3 x2 X 2 = ; a4 x 2a4X 2a5 Y: Z dx 1 x 7x 15 x2 X 3 = ; a6 x 4a4X 2 8a6X 8a7 Y: Z dx 1 1 ln x2 : = ; 3 xX 2a2 x2 2a4 X Z dx 1 1 1 x2 x3 X 2 = ; 2a4 x2 2a4X a6 ln X : Z dx 1 1 1 3 x2 x3 X 3 = ; 2a6 x2 a6X 4a4X 2 2a8 ln X : "
Z dx = 1 c ln X b Y : c ln( b + cx ) ; (b + cx)X a2 c2 b2 2 a
21.7.1.4 Integrals with X = a3 x3 83: 84: 85: 86: 87: 88: 89: 90: 91: 92: 93: 94:
Z Z Z Z Z Z Z Z Z Z Z Z
Notation: a3 x3 = X if there is a double sign in a formula, the upper sign belongs to X = a3 + x3 , the lower one to X = a3 ; x3. dx = 1 ln (a x)2 + 1p arctan 2xp a : X 6a2 a2 ax + x2 a2 3 a 3 dx = x + 2 Z dx X 2 3a3 X 3a3 X x dx = 1 ln a2 ax + x2 p1 arctan 2xp a : X 6a (a x2 ) a 3 a 3 2 x dx = x + 1 Z x dx X 2 3a3 X 3a3 X x2 dx = 1 ln X: X 3 x2 dx = 1 : X2 3X x3 dx = x a3 Z dx X X x3 dx = x 1 Z dx X2 3X 3 X dx = 1 ln x3 : xX 3a3 X dx = 1 + 1 ln x3 : xX 2 3a3X 3a6 X dx 1 1 Z x dx x2 X = ; a3 x a3 X dx = ; 1 x2 4 Z x dx x2 X 2 a6 x 3a6X 3a6 X
(see No. 83):
(see No. 85):
(see No. 83): (see No. 83):
(see No. 85): (see No. 85):
21.7 Indenite Integrals 1029
Z Z dx = ; 13 2 13 dx 3 xX 2a x a X Z dx Z 96: x3 X 2 = ; 2a16 x2 3ax6X 35a6 dx X
95:
21.7.1.5 Integrals with X p= a4 + x4 Z
(see No. 83): (see No. 83):
p
dx 1p x2 + axp2 + a2 1p ax 2 a4 + x4 = 4a3 2 ln x2 ; ax 2 + a2 + 2a3 2 arctan a2 ; x2 : Z 2 98: a4x+dxx4 = 21a2 arctan xa2 : p 2 p Z 2 2 2 p2 + a2 + 1p arctan aax 99: ax4 +dxx4 = ; 4a1p2 ln xx2 +; ax 2 ; x2 : ax 2 + a 2a 2 Z 3 100: ax4 +dxx4 = 14 ln(a4 + x4 ):
97:
21.7.1.6 Integrals with X = a4 ; x4 Z
dx 1 a+x 1 x a4 ; x4 = 4a3 ln a ; x + 2a3 arctan a : Z 2 2 102: a4x;dxx4 = 41a3 ln aa2 ;+ xx2 : Z 2 103: ax4 ;dxx4 = 41a ln aa ;+ xx ; 21a arctan xa : Z 3 104: ax4 ;dxx4 = ; 14 ln(a4 ; x4 ):
101:
21.7.1.7 Some Cases of Partial Fraction!Decomposition 105: (a + bx)(1f + gx) fb ;1 ag a +b bx ; f +g gx :
106: (x + a)(x +1 b)(x + c) x A+ a + x B+ b + x C+ c where it holds A = (b ; a)(1 c ; a) B = (a ; b)(1 c ; b) C = (a ; c)(1 b ; c) : 107: (x + a)(x + b)(1 x + c)(x + d) x A+ a + x B+ b + x C+ c + x D+ d A = (b ; a)(c ;1 a)(d ; a) B = (a ; b)(c ;1 b)(d ; b) usw. ! 108: (a + bx2 )(1 f + gx2) fb ;1 ag a +bbx2 ; f +ggx2 :
where it holds
1030 21. Tables
21.7.2 Integrals of Irrational Functions 21.7.2.1 Integrals with px and a2 b2x Notation:
p 8 > > arctan b x for the sign \+", < X = a2 b2 x Y = > 1 a + abpx > ln p : 2 a ; b x for the sign \;".
109: 110: 111: 112: 113: 114: 115: 116:
Z Z Z Z Z Z Z Z
If there is a double sign in a formula, then the upper one belongs to X = a2 + b2 x, the lower one to X = a2 ; b2 x. px dx 2px 2a X = b2 b3 Y: p3 p p x dx = 2 x3 ; 2a2 x + 2a3 Y: X 3 b2 b4 b5 px dx px 1 X 2 = b2 X ab3 Y: p3 p p x dx = 2 x3 + 3a2 x ; 3a Y: X2 b2 X b4 X b5 dx 2 p X x = ab Y: dx p = ; a2 2px 2ab3 Y: X x3 p dxp = x + 1 Y: X 2 x a2 X a3 b p dx 2p 3b2 x 3b Y: p = ; a2 X x a4X a5 X 2 x3
px 21.7.2.2 Other Integrals with p
p Z px dx 1p ln x + ap2x + a2 + p1 arctan a 2x : = ; 4 2 2 a +x a ;x 2a 2 x ; a 2x + a2 a 2 p p Z 2 118: (a4 +dxx2 )px = 2a31p2 ln xx ;+ aap22xx ++ aa2 + a31p2 arctan aa2 ;2xx : p p Z p 119: a4 x;dxx2 = 21a ln aa ;+ pxx ; a1 arctan ax : p p Z 120: (a4 ;dxx2 )px = 21a3 ln aa ;+ pxx + a13 arctan ax : 117:
21.7 Indenite Integrals 1031
p 21.7.2.3 Integrals with ax + b 121: 122: 123: 124: 125: 126: 127: 128: 129: 130: 131: 132: 133: 134: 135: 136:
Zp
2p
Notation: X = ax + b
X dx = 3a X 3:
p Z p ; 2b ) X 3 : x X dx = 2(3ax 15 a2 p Z p 2 2 abx + 8b2 ) X 3 : x2 X dx = 2(15a x ; 12 3 105a Z dx 2pX p = a : X Z x dx 2(ax ; 2b) p p = 3a2 X: X Z x2 dx 2(3a2x2 ; 4abx + 8b2)pX p = : 15a3 X s 8 p p > 2 X = p1 ln pX ; pb p Z dx > ; Arcoth for b > 0 < b X+ b sb p => b x X > p2 arctan X for b < 0: : ;b ;b p Z X Z dx p dx =2 X +b p x x X p Z dx Z dx X a p = ; bx ; 2b p x2 X x X p p Z X Z X a p dx x2 dx = ; x + 2 x X p Z dx X (2n ; 3)a Z dxp : p = ; ; n ; 1 (n ; 1)bx (2n ; 2)b xn;1 X xn X p 5 Zp X 3 dx = 2 5aX : Z p p p x X 3 dx = 352a2 5 X 7 ; 7b X 5 : p 9 p 7 2p 5 ! Z p x2 X 3 dx = a23 9X ; 2b 7X + b 5X : p 3 p Z pX 3 Z dx dx = 2 X + 2b X + b2 p x 3 x X ! Z x dx 2 p p 3 = a2 X + pb : X X
(see No. 127): (see No. 127): (see No. 127):
(see No. 127):
1032 21. Tables 137: 138: 139: 140: 141: 142: 143: 144: 145:
Z x2 dx 2 pX 3 2 ! p p 3 = a3 3 ; 2b X ; pb : X X Z dx Z dx 1 2 p p = p + x X3 b X b x X Z dx Z p 3 = ; p1 ; 23pa ; 23ba2 pdx 2 bx X b X x X x X Z (2n)=2 X n=2 dx = 2aX(2 n) : Z (4n)=2 (2n)=2 ! xX n=2 dx = a22 X4 n ; bX2 n : Z (6n)=2 (4n)=2 2 (2n)=2 ! x2 X n=2 dx = a23 X6 n ; 2bX4 n + b X2 n : Z X n=2 dx 2X n=2 Z X (n;2)=2 x = n +b x dx: Z dx Z dx 2 = +1 n= 2 ( n ; 2) = 2 xX (n ; 2)bX b xX (n;2)=2 : Z dx 1 na Z dx : = ; ; x2 X n=2 bxX (n;2)=2 2b xX n=2
(see No. 127): (see No. 127):
p 21.7.2.4 Integrals with ax + b and pfx + g Z 146: pdx XY
Notation: X = ax + b 8 s > 2 arctan ; fX > p ; > > > < ;af s aY = > p2 Artanh fX > aYq af > 2 p > > : paf ln aY + fX p XY ; ag + bf Z pdx = af 2af XY
Y = fx + g % = bf ; ag for af < 0 for af > 0 for af > 0 :
Z x dx p XY p Z dx 148: p p 3 = ; 2 pX : % Y X Y p 8 > 2 arctan pf X > p for %f < 0 Z dx < ;%f p = > 1 f pX ; ;p%%ff 149: Y X > p ln p p for %f > 0 : : %f f X + %f Zp 2 Z p 150: XY dx = % +4af2aY XY ; 8%af pdx XY
147:
(see No. 146):
(see No. 146):
21.7 Indenite Integrals 1033
151: 152: 153: 154: 155: 156:
Z sY 1 pXY ; % Z pdx dx = X a 2a XY Z pX dx 2pX % Z dx p Y = f +f Y X Z Y n dx Z n;1 ! p p = (2n +2 1)a XY n ; n% Y p dx : X X (p ) Z dx Z p n = ; (n ;11)% Y nX;1 + n ; 23 a p dxn;1 : XY XY ! Zp Z n p XY n dx = (2n +1 3)f 2 XY n+1 + % Yp dx X p ! Z pX dx Z 1 X a p dx : Y n = (n ; 1)f ; Y n;1 + 2 XY n;1
p 21.7.2.5 Integrals with a2 ; x2 157: 158: 159: 160: 161: 162: 163: 164: 165: 166: 167:
Notation: X = a2 ; x2
p X dx = 12 x X + a2 arcsin xa :
Zp
Z p p x X dx = ; 13 X 3: Z p 2 p p x2 X dx = ; x4 X 3 + a8 x X + a2 arcsin xa : p 5 p 3 Z p x3 X dx = 5X ; a2 3X : p Z pX p a + X x dx = X ; a ln x : p Z pX dx = ; X ; arcsin x : 2 x x a p p Z pX X 1 a + X: dx = ; + ln x3 2x2 2a x Z dx x p = arcsin a : X Z x dx p p = ; X: X Z x2 dx 2 p p = ; x2 X + a2 arcsin xa : X Z x3 dx pX 3 p p = 3 ; a2 X: X
(see No. 146): (see No. 149):
(see No. 153):
1034 21. Tables 168: 169: 170: 171: 172: 173: 174: 175: 176: 177: 178: 179: 180: 181: 182: 183: 184:
p Z dx p = ; a1 ln a +x X : x X p Z dx X p = ; a2 x : x2 X p p Z dx X ; 1 ln a + X : p = ; 2a2 x2 2a3 x x3 X ! Zp 2 4 p p X 3 dx = 14 x X 3 + 3a2 x X + 32a arcsin xa : Z p p x X 3 dx = ; 15 X 5: p 5 2p 3 4p Z p a6 arcsin x : x2 X 3 dx = ; x 6X + a x24X + a x16 X + 16 a p 7 2p 5 Z p X a X x3 X 3 dx = 7 ; 5 : p 3 p p Z pX 3 X + a2 X ; a3 ln a + X : dx = x 3 x p 3 Z pX 3 p X 3 3 2 x x2 dx = ; x ; 2 x X ; 2 a arcsin a : p 3 p p Z pX 3 X ; 3 X + 3a ln a + X : dx = ; x3 2x2 2 2 x Z dx x p 3 = 2p : X a X Z x dx p 3 = p1 : X X Z x2 dx x p 3 = p ; arcsin xa : X X Z x3 dx p 2 p 3 = X + pa : X X p Z dx p 3 = 2 p1 ; a13 ln a +x X : x X a X p ! Z dx 1 ; X + px : p = 4 x X x2 X 3 a p Z dx 1 p + 3p ; 3 ln a + X : p = ; x 2a2 x2 X 2a4 X 2a5 x3 X 3
21.7 Indenite Integrals 1035
p 21.7.2.6 Integrals with x2 + a2 185:
Notation: X = x2 + a2 p X dx = 12 x X + a2 Arsinh xa + C hp p i = 12 x X + a2 ln x + X + C1 :
Zp
Z p p x X dx = 13 X 3: Z p 2 p p 187: x2 X dx = x4 X 3 ; a8 x X + a2 Arsinh xa + C 2h p p i p = x4 X 3 ; a8 x X + a2 ln x + X + C1 : p 5 2p 3 Z p 3 188: x X dx = 5X ; a 3X : p Z p p 189: xX dx = X ; a ln a +x X : p p Z p p 190: xX2 dx = ; xX + Arsinh xa + C = ; xX + ln x + X + C1: p p Z p 191: xX3 dx = ; 2xX2 ; 21a ln a +x X : Z p 192: pdx = Arsinh xa + C = ln x + X + C1: X Z x dx p 193: p = X: X Z x2 dx x p 2 2 p p 194: pX = 2 X ; a2 Arsinh xa + C = x2 X ; a2 ln x + X + C1: p 3 p Z 3 195: xp dx = 3X ; a2 X: X p Z dx 196: xpX = ; a1 ln a +x X : p Z p = ; a2Xx : 197: 2dx x X p p Z dx X 1 a + p 198: 3 = ; 2a2 x2 + 2a3 ln x X : x X ! Zp 2 p 4 p 199: X 3 dx = 14 x X 3 + 3a2 x X + 32a Arsinh xa + C 2 p 4 p p ! = 14 x X 3 + 3a2 x X + 32a ln x + X + C1:
186:
1036 21. Tables Z p p x X 3 dx = 15 X 5: p 5 2p 3 4p Z p x a6 Arsinh x + C 2 3 201: x X dx = 6X ; a x24X ; a x16 X ; 16 a p 5 2p 3 4p a6 ln x + pX + C : = x 6X ; a x24X ; a x16 X ; 16 1 p 7 2p 5 Z p 202: x3 X 3 dx = 7X ; a 5X : p 3 p p Z pX 3 203: dx = X + a2 X ; a3 ln a + X : x 3 x p 3 Z pX 3 p 204: x2 dx = ; xX + 32 x X + 32 a2 Arsinh xa + C p 3 p p = ; X + 3 x X + 3 a2 ln x + X + C1: x 2 2 p 3 p p ! Z p 3 3 X ; 3 a ln a + X : 205: xX3 dx = ; 2X + x2 2 2 x Z dx x 206: p 3 = a2pX : X Z x dx 207: p 3 = ; p1X : X Z x2 dx p 208: p 3 = ; px + Arsinh xa + C = ; px + ln x + X + C1: X X X Z x3 dx p 2 a 209: p 3 = X + pX : X p Z dx 210: p 3 = 2 p1 ; a13 ln a +x X : x X a X p ! Z dx 211: 2 p 3 = ; a14 xX + px : X x X p Z dx 212: 3 p 3 = ; 2a2 x12 pX ; 2a43pX + 23a5 ln a +x X : x X
200:
p 21.7.2.7 Integrals with x2 ; a2
Notation: X = x2 ; a2 Zp p 213: X dx = 12 x X ; a2 Arcosh xa + C p i hp = 12 x X ; a2 ln x + X + C1 : Z p p 214: x X dx = 13 X 3:
21.7 Indenite Integrals 1037
215: 216: 217: 218: 219: 220: 221: 222: 223: 224: 225: 226: 227:
Z Z Z Z Z Z Z Z Z Z Z Z Z
2 p p p x2 X dx = x4 X 3 + a8 x X ; a2 Arcosh xa + C 2h p p i p = x X 3 + a x X ; a2 ln x + X + C1: 4 8 p 5 2p 3 p X a x3 X dx = 5 + 3X : p p X a x dx = X ; a arccos x : p p p X dx = ; X + Arcosh x + C = ; X + ln x + pX + C : 1 x2 x a x p p X X 1 a x3 dx = ; 2x2 + 2a arccos x : p pdx = Arcosh xa + C = ln x + X + C1: X xpdx = pX: X 2 xp dx = x pX + a2 Arcosh x + C = x pX + a2 ln x + pX + C : 1 2 a 2 2 X 2 p xp3 dx = X 3 + a2 pX: 3 X dx 1 p = arccos xa : x X a p dx X: p = x2 X a2x p dx X + 1 arccos a : p = x x3 X 2a2x2 2a3 ! 2 p 4 p 3 p X dx = 14 x X 3 ; 3a2 x X + 32a Arcosh xa + C 2 p 4 p p ! = 1 x X 3 ; 3a x X + 3a ln x + X + C1: 4 2 2
Z p p x X 3 dx = 15 X 5: p 5 2p 3 4p Z p x a6 Arcosh x + C 2 3 229: x X dx = 6X + a x24X ; a x16 X + 16 a p 5 2p 3 4p 6 p = x X + a x X ; a x X + a ln x + X + C :
228:
230:
Z
6 24 p 7 a2 X 5 p X x3 X 3 dx = 7 + 5 :
p
16
16
1
1038 21. Tables 231: 232: 233: 234: 235: 236: 237: 238: 239: 240:
p 3 p Z pX 3 X a 2 3 dx = x 3 ; a X + a arccos x : p 3 Z pX 3 X + 3 xpX ; 3 a2 Arcosh x + C dx = ; x2 2 a p 3 2 p 2 p 3 X 3 2 =; + x X ; a ln x + X + C1 : 2 2 2 p 3 p Z pX 3 X 3 X 3 a dx = ; x3 2x2 + 2 ; 2 a arccos x : Z dx p 3 = ; 2 px : a X X Z x dx p 3 = ; p1 : X X Z x2 dx p p 3 = ; px + Arcosh xa + C = ; px + ln x + X + C1 : X X X Z x3 dx p 2 p 3 = X ; pa : X X Z dx 1 p = ; 2 p ; a13 arccos xa : a X x X3 p ! Z dx 1 X + px : p = ; 4 a x X x2 X 3 Z dx p = 1 p ; 3p ; 3 arccos xa : x3 X 3 2a2 x2 X 2a4 X 2a5
p 21.7.2.8 Integrals with ax2 + bx + c
Notation: X = ax2 + bx + c % = 4ac ; b2 k = 4%a
241:
242: 243: 244:
Z
Z Z Z
8 1 p > p ln 2 aX + 2 ax + b +C for a > 0 > > a > > 2ax + b 1 > for a > 0 % > 0 < pa Arsinh p% + C1 dx p => 1 X > p ln(2ax + b) for a > 0 % = 0 > a > > 1 2 ax + b > : ; p arcsin p for a < 0 % < 0 : ;a ;% dx p = 2(2axp + b) : X X % X dx 2(2 p = axp+ b) X1 + 2k : 2 X X 3% X dx = 2(2ax + b) 2k(n ; 1) Z dx : + (2 n +1) = 2 (2 n ; 1) = 2 X (2n ; 1)%X 2n ; 1 X (2n;1)=2
21.7 Indenite Integrals 1039
245: 246: 247: 248: 249: 250: 251: 252: 253: 254: 255: 256: 257:
258:
259: 260:
p Z X dx = (2ax +4ab) X + 21k pdx X p 3 Z dx p X (2 ax + b ) 3 p X X dx = X + 8a p 2k + 8k2 X 5 Z dx p X (2 ax + b ) 5 X 15 2 2 X X dx = X + 4k + 8k2 + 16k3 p 12a X Z (2n+1)=2 2 n + 1 (2 ax + b ) X + 2k(n + 1) X (2n;1)=2 dx: X (2n+1)=2 dx = 4a(n + 1) p xpdx = X ; b Z pdx a 2a X X
Zp Z Z Z Z
Z x dx p = ; 2(bxp+ 2c) : X X % X Z x dx 1 b Z dx X (2n+1)=2 = ; (2n ; 1)aX (2n;1)=2 ; 2a X (2n+1)=2 !p Z x2 dx Z 2 p = 2xa ; 43ab2 X + 3b 8;a24ac pdx X X Z x2 dx (2b2 ; 4ac)x + 2bc 1 Z dx p = p +a p X X a% X X p p R xpX dx = X X ; b(2ax + b) X ; b Z pdx 3a p 8a2 4ak X Z p Z p 2 X X b xX X dx = 5a ; 2a X X dx Z X (2n+3)=2 ; b Z X (2n+1)=2 dx xX (2n+1)=2 dx = (2 n + 3)a 2a ! p Z p Zp 2 x2 X dx = x ; 65ab X4aX + 5b 16;a24ac X dx p 8 ! > 1 ln 2 cX + 2c + b + C > p for c > 0 ; > c x x > > 1 bx + 2c > Z dx > for c > 0 % > 0 < ; p Arsinh p + C1 p = > 1c bx + 2xc % x X > ; p ln for c > 0 % = 0 > c x > > > 1 bx + 2 c > for c < % < 0 : : p;c arcsin xp;% p Z dx Z p = ; cxX ; 2bc pdx x2 X x X Z Z pX dx p dx b pdx + c Z p = X + x 2 X x X
(see No. 241): (see No. 241): (see No. 241):
(see No. 241).
(see No. 244): (see No. 241): (see No. 241): (see No. 241): (see No. 246): (see No. 248): (see No. 245).
(see No. 258): (see No. 241 and 258):
1040 21. Tables 261: 262: 263: 264: 265: 266: 267:
p Z pX dx dx X Z pdx b Z p = ; x2 x +a X + 2 x X Z X (2n+1)=2 X (2n+1)=2 + b Z X (2n;1)=2 dx + c Z X (2n;1)=2 dx dx = x 2n + 1 2 x Z p p dx = ; bx2 ax2 + bx: x ax2 + bx Z p dx 2 = arcsin x ;a a : 2ax ; x Z x dx p p = ; 2ax ; x2 + a arcsin x ;a a : 2ax ; x2 Zp 2 p 2ax ; x2 dx = x ;2 a 2ax ; x2 + a2 arcsin x ;a a : p Z dxp 1 x pag ; bf p p = arctan p (ax2 + b) fx2 + g b ag ; bf p p b fx2 + g p 2 = p p1 ln pbpfx2 + g + xpbf ; ag 2 b bf ; ag b fx + g ; x bf ; ag
(see No. 241 and 258): (see No. 248 and 260):
(ag ; bf > 0) (ag ; bf < 0) :
21.7.2.9 Integrals with other Irrational Expressions 268: 269: 270: 271: 272:
Z pn p ax + b dx = n((nax+ +1)ba) n ax + b: Z dx pn = n(ax + b) p 1 : ax + b (n ; 1)a n ax + b p Z 2 ln a + pxn + a2 : p dx = ; na xn x xn + a2 Z p dx = 2 arccos paxn : x xn ; a2 na s 3 Z px dx 2 p 3 3 = 3 arcsin xa : a ;x
21.7.2.10 Recursion Formulas for an Integral with Binomial Di erential Z 273:
xm (axn + b)p dx
Z 1 m+1 (axn + b)p + npb xm (axn + b)p;1 dx = x m + np + 1 1 ;xm+1 (axn + b)p+1 + (m + n + np + 1) Z xm (axn + b)p+1 dx = bn(p + 1)
Z = (m +1 1)b xm+1 (axn + b)p+1 ; a(m + n + np + 1) xm+n(axn + b)p dx
21.7 Indenite Integrals 1041
=
Z 1 m;n+1 (axn + b)p+1 ; (m ; n + 1)b xm;n (axn + b)p dx : x a(m + np + 1)
21.7.3 Integrals of Trigonometric Functions
Integrals of functions also containing sin x and cos x together with hyperbolic and exponential functions are in the table of integrals of other transcendental functions (see 21.7.4, p. 1050).
21.7.3.1 Integrals with Sine Function Z
sin ax dx = ; 1 cos ax: a Z 1 2 275: sin ax dx = 2 x ; 41a sin 2ax: Z 276: sin3 ax dx = ; a1 cos ax + 31a cos3 ax: Z 277: sin4 ax dx = 83 x ; 41a sin 2ax + 321a sin 4ax: Z n;1 ax cos ax n ; 1 Z 278: sinn ax dx = ; sin na + n sinn;2 ax dx (n integer numbers, > 0): Z 279: x sin ax dx = sina2ax ; x cosa ax : ! Z 2 280: x2 sin ax dx = 2ax2 sin ax ; xa ; a23 cos ax: ! ! Z 2 3 281: x3 sin ax dx = 3ax2 ; a64 sin ax ; xa ; 6ax3 cos ax: Z Z n 282: xn sin ax dx = ; xa cos ax + na xn;1 cos ax dx (n > 0): Z 3 5 7 283: sinxax dx = ax ; (3ax 3!) + (5ax 5!) ; (7ax 7!) + : Zx The de nite integral sin t dt is called the sine integral (see 8.2.5, 1., p. 460) and it is denoted si(x). t 0 For the calculation of the integral see 14.4.3.2, 2., p. 696. The power series expansion is 3 5 7 si(x) = x ; 3x 3! + 5x 5! ; 7x 7! + see 8.2.5, 1., p. 460.
274:
Z sin ax sin ax + a Z cos ax dx dx = ; 2 x x x Z sin ax 1 sin ax a Z cos ax 285: xn dx = ; n ; 1 xn;1 + n ; 1 xn;1 dx Z Z 286: sindxax = cosec ax dx = a1 ln tan ax2 = a1 ln(cosec ax cot ax): Z 287: sindx2 ax = ; a1 cot ax:
284:
(see No. 322): (see No. 324):
1042 21. Tables Z
dx = ; cos ax + 1 ln tan ax : 2 sin3 ax 2a sin2 ax 2a Z dx Z 1 cos ax n 289: sinn ax = ; a(n ; 1) sinn;1 ax + n ;; 21 sinndx ;2 ax Z x dx 1 3 5 7 ) + 31(ax) 290: sin ax = a2 ax + (3ax 3!) + 37( ax 5 5! 3 7 7!
288:
(n > 1):
!
9 2n;1 + 127(ax) + + 2(2 ; 1) Bn(ax)2n+1 + 3 5 9! (2n + 1)! Bn denote the Bernoulli numbers (see 7.2.4.2, p. 412). Z 291: sinx 2dxax = ; xa cot ax + a12 ln sin ax: Z ax 1 n ; 2 Z x dx 292: sinx ndxax = ; (n ; x1)cos a sinn;1 ax ; (n ; 1)(n ; 2)a2 sinn;2 ax + n ; 1 sinn;2 ax (n > 2): Z 293: 1 + dx = ; 1 tan ; ax : sin ax a 4 2 Z 1 tan + ax : 294: 1 ; dx = sin ax a 4 2 Z x dx x 295: 1 + sin ax = ; a tan 4 ; ax2 + a22 ln cos 4 ; ax2 : Z dx = x cot ; ax + 2 ln sin ; ax : 296: 1 ;xsin ax a 4 2 a2 4 2 Z sin ax dx 1 ax 297: 1 sin ax = x + a tan 4 2 : Z 298: sin ax(1dx sin ax) = a1 tan 4 ax2 + a1 ln tan ax2 : Z dx 299: (1 + sin = ; 1 tan ; ax ; 1 tan3 ; ax : 2 ax) 2a 4 2 6a 4 2 Z dx 1 ax 1 3 ax 300: (1 ; sin ax)2 = 2a cot 4 ; 2 + 6a cot 4 ; 2 : Z ax dx = ; 1 tan ; ax + 1 tan3 ; ax : 301: (1sin + sin ax)2 2a 4 2 6a 4 2 Z sin ax dx 302: (1 ; sin ax)2 = ; 21a cot 4 ; ax2 + 61a cot3 4 ; ax2 : ! Z dx = p1 arcsin 3 sin2 ax ; 1 : 303: 1 + sin 2 ax sin2 ax + 1 2 2a Z Z dx = dx 1 304: 1 ; sin 2 ax cos2 ax = a tan ax:
21.7 Indenite Integrals 1043
305: 306: 307: 308: 309: 310: 311: 312:
Z Z Z Z Z Z Z Z
a ; b)x sin(a + b)x sin ax sin bx dx = sin( 2(a ; b) ; 2(a + b)
(jaj 6= jbj
dx b tan 2+c p2 p ax= for b2 > c2) b + c sin ax = a b2 ; c2 arctan b2 ; c2 p 2 2 = p 21 2 ln b tan ax=2 + c ; pc2 ; b2 for b2 < c2): a c ; b b tan ax=2 + c + c ; b sin ax dx = x ; b Z dx b + c sin ax c c b + c sin ax Z dx dx = 1 ln tan ax ; c sin ax(b + c sin ax) ab 2 b b + c sin ax Z dx dx = 2 c2 cos ax + 2b 2 2 (b + c sin ax) a(b ; c )(b + c sin ax) b ; c b + c sin ax sin ax dx = b cos ax c Z dx + 2 2 2 2 (b + c sin ax) a(c ; b )(b + c sin ax) c ; b2 b + c sin ax p2 2 dx 1 b + c tan ax (b > 0): p = arctan b b2 + c2 sin2 ax ab b2 + c2 p2 2 dx b ; c tan ax p1 (b2 > c2 b > 0) b2 ; c2 sin2 ax = ab b2 ; c2 arctan p2 2 b = p 12 2 ln pc2 ; b2 tan ax + b (c2 > b2 b > 0): 2ab c ; b c ; b tan ax ; b
21.7.3.2 Integrals with Cosine Function Z cos ax dx = 1 sin ax: a Z 2 314: cos ax dx = 21 x + 41a sin 2ax:
313: 315: 316: 317: 318: 319: 320: 321:
Z Z Z Z Z Z Z
for jaj = jbj see No.275):
cos3 ax dx = a1 sin ax ; 31a sin3 ax: cos4 ax dx = 3 x + 1 sin 2ax + 1 sin 4ax: 8 4a 32a n;1 ax sin ax n ; 1 Z cos cosn ax dx = + n cosn;2 ax dx (n 1): na x cos ax dx = cosa2ax + x sina ax : ! 2 x2 cos ax dx = 2ax2 cos ax + xa ; a23 sin ax: ! ! 3 2 x3 cos ax dx = 3ax2 ; a64 cos ax + xa ; 6ax3 sin ax: n ax ; n Z xn;1 sin ax dx: xn cos ax dx = x sin a a
(see No. 306): (see No. 306): (see No. 306): (see No. 306):
1044 21. Tables Z cos ax (ax)2 (ax)4 (ax)6 x dx = ln(ax) ; 2 2! + 4 4! ; 6 6! + Z1 The de nite integral ; cost t dt is called the cosine integral (see 14.4.3.2, p. 696) and it is denoted by x 2 4 6 Ci(x). The power series expansion is Ci(x) = C +ln x ; x + x ; x + see 8.2.5, 2., p. 460
322:
2 2! 4 4! 6 6! C denotes the Euler constant (see 8.2.5, 2., p. 460). Z Z dx 323: cosx2ax dx = ; cosxax ; a sin ax (see No. 283): x Z cos ax Z cos ax a sin ax dx 324: (n 6= 1) (see No. 285): xn dx = ; (n ; 1)xn;1 ; n ; 1 xn;1 Z dx 325: cos ax = a1 Artanh Artanh(sin ax) = a1 ln tan ax2 + 4 = a1 ln(sec ax + tan ax): Z 326: cosdx2 ax = a1 tan ax: Z ax + 1 ln tan + ax : 327: cosdx3 ax = 2asin cos2 ax 2a 4 2 Z dx Z 1 sin ax n ; 328: cosn ax = a(n ; 1) cosn;1 ax + n ; 21 cosndx;2 ax (n > 1): ! Z x dx 1 (ax)2 (ax)4 5(ax)6 61(ax)8 1385(ax)10 En(ax)2n+2 + 329: cos = + + + + + + ax a2 2 4 2! 6 4! 8 6! 10 8! (2n + 2)(2n!) En denote the Euler numbers (see 7.2, p. 413). Z x 1 330: cosx dx 2 ax = a tan ax + a2 ln cos ax: Z x sin ax 1 n ; 2 Z x dx 331: cosx ndxax = (n ; 1) a cosn;1 ax ; (n ; 1)(n ; 2)a2 cosn;2 ax + n ; 1 cosn;2 ax (n > 2): Z 1 ax 332: 1 + dx cos ax = a tan 2 : Z 1 ax 333: 1 ; dx cos ax = ; a cot 2 : Z dx = x tan ax + 2 ln cos ax : 334: 1 +xcos ax a 2 a2 2 Z x dx x ax 2 335: 1 ; cos ax = ; a cot 2 + a2 ln sin ax2 : Z ax dx = x ; 1 tan ax : 336: 1cos + cos ax a 2 Z cos ax dx 1 337: 1 ; cos ax = ;x ; a cot ax2 : Z 338: cos ax(1dx+ cos ax) = a1 ln tan 4 + ax2 ; a1 tan ax2 :
21.7 Indenite Integrals 1045
339: 340: 341: 342: 343: 344: 345: 346: 347: 348: 349: 350: 351: 352: 353:
dx = 1 ln tan + ax ; 1 cot ax : cos ax(1 ; cos ax) a 4 2 a 2 Z dx 1 ax 1 ax 3 (1 + cos ax)2 = 2a tan 2 + 6a tan 2 : Z dx = ; 1 cot ax ; 1 cot3 ax : (1 ; cos ax)2 2a 2 6a 2 Z cos ax dx 1 ax 1 ax 3 (1 + cos ax)2 = 2a tan 2 ; 6a tan 2 : Z cos ax dx 1 ax 1 3 ax (1 ; cos ax)2 = 2a cot 2 ; 6a cot 2 : Z 2 ! dx = p1 arcsin 1 ; 3 cos2 ax : 2 1 + cos ax 2 2a 1 + cos ax Z Z dx dx = ; 1 cot ax: = 1 ; cos2 ax a sin2 ax R cos ax cos bx dx = sin(a ; b)x + sin(a + b)x (jaj 6= jbj) (for jaj = jbj see No. 314): 2(a ; b) 2(a + b) Z dx (b ;pc) tan ax=2 p2 (for b2 > c2 ) b + c cos ax = a b2 ; c2 arctan b2 ; c2 p 2 2 = p 21 2 ln (c ; b) tan ax=2 + pc2 ; b2 (for b2 < c2 ) : a c ; b (c ; b) tan ax=2 ; c ; b Z cos ax dx x b Z dx (see No. 347): b + c cos ax = c ; c b + c cos ax Z Z dx 1 ax c dx (see No. 347): cos ax(b + c cos ax) = ab ln tan 2 + 4 ; b b + c cos ax Z Z dx c sin ax dx = ; b (see No. 347): (b + c cos ax)2 a(c2 ; b2 )(b + c cos ax) c2 ; b2 b + c cos ax Z cos ax dx b sin ax c Z dx (see No. 347): (b + c cos ax)2 = a(b2 ; c2 )(b + c cos ax) ; b2 ; c2 b + c cos ax Z dx p1 pb tan ax b2 + c2 cos2 ax = ab b2 + c2 arctan b2 + c2 (b > 0): Z dx p1 pb tan ax (b2 > c2 b > 0) b2 ; c2 cos2 ax = ab b2 ; c2 arctan b2 ; cp2 2 2 = p 12 2 ln b tan ax ; pc2 ; b2 (c2 > b2 b > 0) : 2ab c ; b b tan ax + c ; b Z
21.7.3.3 Integrals with Sine and Cosine Function 354:
Z
sin ax cos ax dx = 21a sin2 ax:
1046 21. Tables Z
sin2 ax cos2 ax dx = x ; sin 4ax : 8 32a Z 1 n 356: sin ax cos ax dx = a(n + 1) sinn+1 ax (n 6= ;1): Z 357: sin ax cosn ax dx = ; a(n 1+ 1) cosn+1 ax (n 6= ;1):
355:
358:
359: 360: 361: 362: 363: 364: 365: 366: 367: 368: 369:
Z
sinn ax cosm ax dx = ; sin
Z
dx 1 sin ax cos ax = a ln tan ax:
dx 1 : = a1 ln tan 4 + ax ; 2 2 sin ax sin ax cos ax dx 1 ax 1 sin ax cos2 ax = a ln tan 2 + cos ax : dx = a1 ln tan ax ; 12 : 3 sin ax cos ax 2 sin ax dx 1 1 sin ax cos3 ax = a ln tan ax + 2 cos2 ax : dx = ; a2 cot 2ax: sin2 ax cos2 ax
ax dx 1 + 3 ln tan + ax : = a1 2 sin ; 2 2 3 cos ax sin ax 2 4 2 sin ax cos ax 1 1 cos ax 3 ax dx = a cos ax ; + 2 ln tan 2 : 3 2 2 sin ax cos ax 2 sin ax Z dx 1 dx = + (n 6= 1) (see No.361 and 363): sin ax cosn ax a(n ; 1) cosn;1 ax sin ax cosn;2 ax Z dx 1 dx (n 6= 1)(see No. 360 and 362): sinn ax cos ax = ; a(n ; 1) sinn;1 ax + sinn;2 ax cos ax dx 1 1 n+m;2Z dx = ; + n n ;1 m m ; 1 sin ax cos ax a(n ; 1) sin ax cos ax n;1 sinn;2 ax cosm ax (lowering the exponent n m > 0 n > 1) dx ;2Z = a(m1; 1) n;1 1 m;1 + n +n m ; 1 sinn ax cosm;2 ax sin ax cos ax (lowering the exponent m n > 0 m > 1):
Z Z Z Z Z Z Z Z Z Z
Z + n ; 1 sinn;2 ax cosm ax dx a(n + m) n+m (lowering the exponent n m and n > 0) Z n+1 m;1 = sin ax cos ax + m ; 1 sinn ax cosm;2 ax dx a(n + m) n+m (lowering the exponent m m and n > 0): n;1 ax cosm+1 ax
21.7 Indenite Integrals 1047
370: 371: 372: 373: 374: 375: 376: 377: 378: 379: 380:
381: 382: 383: 384: 385:
Z sin ax dx = 1 = 1 sec ax: cos2 ax a cos ax a Z sin ax dx 1 = + C = 1 tan2 ax + C1 : cos3 ax 2a cos2 ax 2a Z sin ax dx 1 cosn ax = a(n ; 1) cosn;1 ax : Z sin2 ax dx 1 sin ax + 1 ln tan + ax : = ; cos ax a a 4 2 Z sin2 ax dx 1 sin ax 1 = ; ln tan + ax : 3 2 cos ax a 2 cos ax 2 4 2 Z sin2 ax dx Z dx sin ax = ; 1 (n 6= 1) cosn ax a(n ; 1) cosn;1 ax n ; 1 cosn;2 ax ! Z sin3 ax dx 1 sin2 ax + ln cos ax : = ; cos ax a 2 Z sin3 ax dx 1 = cos ax + 1 : 2 cos ax a cos ax
Z sin3 ax dx 1 " 1 1 = ; (n 6= 1 cosn ax a (n ; 1) cosn;1 ax (n ; 3) cosn;3 ax Z sinn ax sinn;1 ax + Z sinn;2 ax dx dx = ; (n 6= 1): cos ax a(n ; 1) cos ax Z sinn ax sinn+1 ax n ; m + 2 Z sinn ax dx dx = = ; cosm ax a(m ; 1) cosm;1 ax m ; 1 Z cosm;2 ax n;1 ax sin n ; 1 sinn;2 ax dx =; a(n ; m) cosm;1 ax + n ; mZ cosm ax n;1 ax n;1 ax dx sin n ; 1 sin = a(m ; 1) cosm;1 ax ; m ; 1 cosm;2 ax Z cos ax dx = ; a sin1 ax = ; a1 cosec ax: sin2 ax Z cos ax dx 2 = ; 1 2 + C = ; cot ax + C1: 3 2 a sin ax 2a sin ax Z cos ax dx 1 sinn ax = ; a(n ; 1) sinn;1 ax : Z cos2 ax dx 1 ax : = cos ax + ln tan sin ax a 2 cos ax Z cos2 ax dx 1 = ; 2a ; ln tan ax 3 2 2 : sin ax sin ax
(see No. 325, 326, 328):
n 6= 3):
(m 6= 1) (m 6= n) (m 6= 1):
1048 21. Tables 386: 387: 388: 389: 390: 391:
392: 393: 394: 395: 396: 397: 398: 399: 400: 401: 402:
! Z cos2 ax dx cos ax + Z dx =; 1 (n 6= 1) (see No. 289): n n ;1 n ;2 sin ax (n ; 1) a sin ax sin ax ! Z cos3 ax dx 1 cos2 ax = + ln sin ax : sin ax a 2 Z cos3 ax dx = ; 1 sin ax + 1 : 2 a sin ax sin ax
Z cos3 ax dx 1 " 1 1 = ; (n 6= 1 n 6= 3): n n ;3 n ;1 sin ax a (n ; 3) sin ax (n ; 1) sin ax Z cosn ax Z n;2 n;1 dx = cos ax + cos ax dx (n 6= 1): sin ax a(n ; 1) sin ax Z cosn ax dx n+1 ax cos n ; m + 2 Z cosn ax dx (m 6= 1) = ; ; m m ;1 sin ax m ;Z 1 n;2sinm;2 ax a(m ;n1) sin ax ;1 cos ax n ; 1 cos ax dx = (m 6= n) m;1 ax + m ; 1 sinnm;2ax a(n ; m) sin Z cos n;1 ax cos n ; 1 ax dx =; (m 6= 1): a(m ; 1) sinm;1 ax ; m ; 1 sinm;2 ax Z dx 1 1 ax sin ax(1 cos ax) = 2a(1 cos ax) + 2a ln tan 2 : Z dx 1 = + 1 ln tan + ax : cos ax(1 sin ax) 2a(1 sin ax) 2a 4 2 Z sin ax dx 1 1 cos ax cos ax(1 cos ax) = a ln cos ax : Z cos ax dx 1 1 sin ax sin ax(1 sin ax) = ; a ln sin ax : Z sin ax dx 1 1 ln tan + ax : = cos ax(1 sin ax) 2a(1 sin ax) 2a 4 2 Z cos ax dx 1 =; 1 ln tan ax2 : sin ax(1 cos ax) 2a(1 cos ax) 2a Z sin ax dx x 1 sin ax cos ax = 2 2a ln(sin ax cos ax): Z cos ax dx x 1 sin ax cos ax = 2 + 2a ln(sin ax cos ax): Z dx 1 ln tan ax : p = sin ax cos ax a 2 2 8 Z dx 1 ax : = ln 1 tan 1 + cos ax sin ax a 2 Z ax +' 1 dx p with sin ' = p 2c 2 and tan ' = cb : b sin ax + c cos ax = a b2 + c2 ln tan 2 b +c
21.7 Indenite Integrals 1049
Z sin ax dx 1 b + c cos ax = ; ac ln(b + c cos ax): Z ax dx 1 404: bcos + c sin ax = ac ln(b + c sin ax):
403:
405:
Z
! d x +' dx a p b + c cos ax + f sin ax = b + c2 + f 2 sin(ax + ') Z
with sin ' = p 2 c 2 and tan ' = c f c +f Z = 1 arctan cb tan ax : 406: b2 cos2 axdx + c2 sin2 ax abc Z 1 c tan ax + b 407: b2 cos2 axdx ; c2 sin2 ax = 2abc ln c tan ax ; b : Z a + b)x ; cos(a ; b)x (a2 6= b2 ) for a = b 408: sin ax cos bx dx = ; cos( 2(a + b) 2(a ; b)
(see No. 306):
(see No. 354):
21.7.3.4 Integrals with Tangent Function 409: 410: 411: 412:
Z Z Z Z
tan ax dx = ; a1 ln cos ax: tan2 ax dx = tanaax ; x:
tan3 ax dx = 21a tan2 ax + a1 ln cos ax: tann ax dx =
1
Z
tann;1 ax ; tann;2 ax dx:
a(n ; 1) 3 3 5 a5x7 + 17a7 x9 + + 22n(22n ; 1)Bna2n;1 x2n+1 + 413: x tan ax dx = ax3 + a15x + 2105 2835 (2n + 1)! Bn denote the Bernoulli numbers (see 7.2.4.2, p. 412). Z dx (ax)3 2(ax)5 17(ax)7 22n(22n ; 1)Bn(ax)2n;1 + 414: tan ax x = ax + 9 + 75 + 2205 + + (2n ; 1)(2n!) Z tann ax 415: cos2 ax dx = a(n 1+ 1) tann+1 ax (n 6= ;1): Z dx = x + 1 ln(sin ax cos ax): 416: tan ax 1 2 2a Z tan ax dx x 1 417: tan ax 1 = 2 2a ln(sin ax cos ax):
Z
21.7.3.5 Integrals with Cotangent Function Z 418:
cot ax dx = a1 ln sin ax:
1050 21. Tables 419: 420: 421: 422:
Z Z Z Z
cot2 ax dx = ; cotaax ; x:
cot3 ax dx = ; 21a cot2 ax ; a1 ln sin ax:
Z cotn ax dx = ; a(n 1; 1) cotn;1 ax ; cotn;2 ax dx
(n 6= 1):
3 3 x5 2n 2n;1 2n+1 x cot ax dx = xa ; ax9 ; a225 ; ; 2 B(2nna + 1)!x ; :
Bn denote the Bernoulli numbers (see 7.2.4.2, p. 412). Z dx = ; 1 ; ax ; (ax)3 ; 2(ax)5 ; ; 22nBn(ax)2n;1 ; : 423: cot ax x ax 3 135 4725 (2n ; 1)(2n)! Z cotn ax 1 n +1 424: sin2 ax dx = ; a(n + 1) cot ax (n 6= ;1): Z Z tan ax dx 425: 1 dx = cot ax tan ax 1
(see No. 417):
21.7.4 Integrals of other Transcendental Functions 21.7.4.1 Integrals with Hyperbolic Functions Z 426:
427:
Z Z
sinh ax dx = a1 cosh ax:
cosh ax dx = a1 sinh ax:
sinh2 ax dx = 1 sinh ax cosh ax ; 1 x: 2a 2 Z 1 2 429: cosh ax dx = 2a sinh ax cosh ax + 12 x: Z 430: sinhn ax dx 1 sinhn;1 ax cosh ax ; n ; 1 Z sinhn;2 ax dx = an n Z = 1 sinhn+1 ax cosh ax ; n + 2 sinhn+2 ax dx a(n + 1) n+1
428:
431:
Z
coshn ax dx 1 sinh ax coshn;1 ax + n ; 1 Z coshn;2 ax dx = an n Z = ; 1 sinh ax coshn+1 ax + n + 2 coshn+2 ax dx a(n + 1) n+1 Z dx 432: sinh ax = a1 ln tanh ax2 : Z dx 2 ax 433: cosh ax = a arctan e :
(for n > 0) (for n < 0) (n 6= ;1):
(for n > 0) (for n < 0) (n 6= ;1) :
21.7 Indenite Integrals 1051
434: 435: 436: 437: 438: 439: 440: 441: 442: 443: 444: 445: 446:
Z Z Z Z Z Z Z Z Z Z Z Z Z
x sinh ax dx = a1 x cosh ax ; a12 sinh ax:
x cosh ax dx = a1 x sinh ax ; a12 cosh ax:
tanh ax dx = 1 ln cosh ax: a 1 coth ax dx = a ln sinh ax: tanh2 ax dx = x ; tanh ax : a coth coth2 ax dx = x ; a ax : sinh ax sinh bx dx = 2 1 2 (a sinh bx cosh ax ; b cosh bx sinh ax) a ;b cosh ax cosh bx dx = a2 ;1 b2 (a sinh ax cosh bx ; b sinh bx cosh ax) cosh ax sinh bx dx = a2 ;1 b2 (a sinh bx sinh ax ; b cosh bx cosh ax) sinh ax sin ax dx = 21a (cosh ax sin ax ; sinh ax cos ax): cosh ax cos ax dx = 21a (sinh ax cos ax + cosh ax sin ax): sinh ax cos ax dx = 21a (cosh ax cos ax + sinh ax sin ax): cosh ax sin ax dx = 21a (sinh ax sin ax ; cosh ax cos ax):
21.7.4.2 Integrals with Exponential Functions 447: 448:
Z Z
eax dx = a1 eax:
xeax dx = ea2 (ax ; 1): ax
! 2 x2 eax dx = eax xa ; 2ax2 + a23 : Z Z 450: xn eax dx = a1 xn eax ; na xn;1 eax dx: Z ax (ax)2 (ax)3 451: ex dx = ln x + 1ax 1! + 2 2! + 3 3! +
449:
Z
(a2 6= b2 ): (a2 6= b2 ): (a2 6= b2 ):
1052 21. Tables Zx et t dt is called the exponential function integral (see 8.2.5, 4., p. 461) and it is ;1 denoted by Ei(x). For x > 0 the integrand is divergent at t = 0 in this case we consider the principal value of the improper integral Ei(x) (see 8.2.5, 4., p. 461). Zx et 2 3 n dt = C + ln jxj + 1 x1! + 2x 2! + 3x 3! + + nx n! + : t ;1 The de nite integral
C denotes the Euler constant (see 8.2.5, 2., p. 460). ax Z ax Z ax (n 6= 1): 452: exn dx = n ;1 1 ; xen;1 + a xen;1 dx Z dx ax 453: 1 + eax = a1 ln 1 +e eax : Z 454: b +dxceax = xb ; ab1 ln(b + ceax): Z ax 455: be+ cedxax = ac1 ln(b + ceax ): 0 s 1 Z (bc > 0) 456: beax +dxce;ax = p1 arctan @eax bc A a bc p ax = p1 ln c + eax p;bc (bc < 0): 2a ;bc c ; e ;bc Z xeax dx ax 457: (1 + ax)2 = a2 (1e+ ax) : Z Z ax ax 458: eax ln x dx = e aln x ; a1 ex dx Z ax 459: eax sin bx dx = a2e+ b2 (a sin bx ; b cos bx): Z ax 460: eax cos bx dx = a2e+ b2 (a cos bx + b sin bx): Z ax n;1 461: eax sinn x dx = e a2sin+ n2 x (a sin x ; n cos x) Z + na(2n+;n1)2 eax sinn;2 x dx Z ax n;1 462: eax cosn x dx = e a2cos+ n2 x (a cos x + n sin x) Z + n(2n ; 1)2 eax cosn;2 x dx a +n
463:
Z
(see No. 451):
(see No. 447 and 459):
(see No. 447 and 460):
ax ax xeax sin bx dx = a2xe+ b2 (a sin bx ; b cos bx) ; (a2 e+ b2 )2 (a2 ; b2 ) sin bx ; 2ab cos bx]:
21.7 Indenite Integrals 1053
464:
Z
ax ax xeax cos bx dx = a2xe+ b2 (a cos bx + b sin bx) ; (a2 e+ b2 )2 (a2 ; b2 ) cos bx + 2ab sin bx]:
21.7.4.3 Integrals with Logarithmic Functions Z 465:
Z
ln x dx = x ln x ; x:
466: (ln x)2 dx = x(ln x)2 ; 2x ln x + 2x: Z
467: (ln x)3 dx = x(ln x)3 ; 3x(ln x)2 + 6x ln x ; 6x: Z
Z
468: (ln x)n dx = x(ln x)n ; n (lnx)n;1 dx
(n 6= ;1):
Z dx 2 3 = ln ln x + ln x + (ln x) + (ln x) + : ln x 2 2! 3 3! Zx dt The de nite integral ln t is called the logarithm integral (see 8.2.5, p. 460) and it is denoted by Li(x). 0 For x > 1 the integrand is divergent at t = 1. In this case we consider the principal value of the improper integral Li(x) (see 8.2.5, p. 460). The relation between the logarithm integral and the exponential function integral (see 8.2.5, p. 461) is: Li(x) = Ei(ln x). Z x 1 Z dx 470: (lndxx)n = ; (n ; 1)(ln + (n 6= 1) (see No. 469): n ; 1 x) n ; 1 (ln x)n;1
" Z (m 6= ;1): 471: xm ln x dx = xm+1 mln+x1 ; (m +1 1)2 Z m+1 x)n n Z xm (ln x)n;1 dx (m 6= ;1 n 6= ;1 (see No. 470): 472: xm (ln x)n dx = x m (ln ; +1 m+1 Z (ln x)n n+1 (ln x ) 473: x dx = n + 1 : Z ln x 474: xm dx = ; (m ;ln1)xxm;1 ; (m ; 1)1 2 xm;1 (m 6= 1): Z n x)n + n Z (ln x)n;1 dx 475: (lnxmx) dx = ; (m (ln (m 6= 1) (see No. 474): ; 1)xm;1 m ; 1 xm Z m Z e;y 476: xln dx with y = ;(m + 1) ln x (see No. 451): x = y dy Z xm dx xm+1 m + 1 Z xm dx 477: (ln = ; + (n 6= 1): x)n (n ; 1)(ln x)n;1 n ; 1 (ln x)n;1 Z 478: x dx ln x = ln ln x: Z dx 2 2 3 3 479: xn ln x = ln ln x ; (n ; 1) ln x + (n ; 21) 2!(ln x) ; (n ; 31) 3!(ln x) + :
469:
1054 21. Tables Z
;1 dx x(ln x)n = (n ; 1)(ln x)n;1 (n 6= 1): Z dx ;1 p;1 Z dx 481: xp(ln = ; n p ; 1 n ; 1 x) x (n ; 1)(ln x) n ; 1 xp (ln x)n;1 (n 6= 1): Z x3 ; x5 ; ; 22n;1 Bnx2n+1 ; : 482: ln sin x dx = x ln x ; x ; 18 900 n(2n + 1)! Bn denote the Bernoulli numbers (see 7.2.4.2, p. 412). Z 3 x5 x7 ; ; 22n;1(22n ; 1)Bn x2n+1 ; : 483: ln cos x dx = ; x6 ; 60 ; 315 n(2n + 1)! Z 3 7x5 2n 2n;1 x 484: ln tan x dx = x ln x ; x + 9 + 450 + + 2 (2n(2n +;1)!1)Bn x2n+1 + : Z 485: sin ln x dx = x2 (sin ln x ; cos ln x): Z 486: cos ln x dx = x2 (sin ln x + cos ln x): Z Z ax 487: eax ln x dx = a1 eax ln x ; a1 ex dx 480:
(see No. 451):
21.7.4.4 Integrals with Inverse Trigonometric Functions Z 488: 489: 490: 491: 492: 493: 494:
Z Z Z Z Z Z Z
p arcsin xa dx = x arcsin xa + a2 ; x2 :
2 2! p x arcsin xa dx = x2 ; a4 arcsin xa + x4 a2 ; x2 : 3 p x2 arcsin xa dx = x3 arcsin xa + 19 (x2 + 2a2 ) a2 ; x2 : arcsin xa dx x 3 5 7 = a + 2 31 3 xa3 + 2 14 35 5 xa5 + 2 14 36 57 7 xa7 + : x p arcsin xa dx 1 arcsin x ; 1 ln a + a2 ; x2 : = ; x2 x a a x p2 2 x x arccos a dx = x arccos a ; a ; x : 2 2! p x arccos xa dx = x2 ; a4 arccos xa ; x4 a2 ; x2 :
3 p x2 arccos xa dx = x3 arccos xa ; 19 (x2 + 2a2) a2 ; x2 : Z arccos x dx a = ln x ; x ; 1 x3 ; 1 3 x5 ; 1 3 5 x7 ; : 496: x 2 a 2 3 3 a3 2 4 5 5 a5 2 4 6 7 7 a7 x p Z arccos dx a = ; 1 arccos x + 1 ln a + a2 ; x2 : 497: x2 x a a x
495:
21.7 Indenite Integrals 1055
498: 499: 500: 501: 502: 503: 504: 505: 506: 507: 508: 509: 510: 511:
Z Z Z Z Z Z Z Z Z Z Z Z Z Z
arctan xa dx = x arctan xa ; a2 ln(a2 + x2):
x arctan xa dx = 21 (x2 + a2) arctan xa ; ax 2:
3 2 3 x2 arctan xa dx = x3 arctan xa ; ax6 + a6 ln(a2 + x2 ): Z n+1 n+1 xn arctan xa dx = nx + 1 arctan xa ; n +a 1 xa2 + dx (n 6= ;1): x2 arctan xa dx x x3 5 7 = a ; 32a3 + 5x2a5 ; 7x2a7 + (jxj < jaj): x x arctan a dx 2 2 = ; 1 arctan x ; 1 ln a +2 x : x2 x a 2a x arctan xa dx 1 x+ a Z dx = ; arctan n n ; 1 x (n ; 1)x a n ; 1 xn;1 (a2 + x2 ) (n 6= 1): arccot xa dx = x arccot xa + a2 ln(a2 + x2 ): x arccot xa dx = 12 (x2 + a2 ) arccot xa + ax 2: 3 2 3 x2 arccot xa dx = x3 arccot xa + ax6 ; a6 ln(a2 + x2 ): Z n+1 n+1 xn arccot xa dx = nx + 1 arccot xa + n +a 1 ax2 + dx x2 (n 6= ;1): arccot x dx a = ln x ; x + x3 ; x5 ; x7 ; : x 2 a 32a3 52a5 72a7 x arccot a dx 2 2 = ; 1 arccot x + 1 ln a +2 x : x2 x a 2a x arccot xa dx 1 x a Z dx = ; (n ; 1) xn xn;1 arccot a ; n ; 1 xn;1 (a2 + x2 ) (n 6= 1):
21.7.4.5 Integrals with Inverse Hyperbolic Functions Z p Arsinh xa dx = x Arsinh xa ; x2 + a2 : Z p 513: Arcosh xa dx = x Arcosh xa ; x2 ; a2 : Z 514: Artanh xa dx = x Artanh xa + a2 ln(a2 ; x2 ): Z 515: Arcoth xa dx = x Arcoth xa + a2 ln(x2 ; a2 ): 512:
1056 21. Tables
21.8 Denite Integrals
21.8.1 De nite Integrals of Trigonometric Functions
For natural numbers m n:
1: 4:
Z2 0
Z2 0
sin nx dx = 0: (21.1) sin nx sin mx dx =
2:
Z2 0
cos nx dx = 0: (21.2)
( 0 for m 6= n for m = n:
(21.4) 5:
8 2 4 6 8 n;1 > Z2 < for n odd 6: sinn x dx = > 3 51 73 95 n n; 1 : 0 2 2 4 6 n for n even
7a:
Z=2
sin2
0
+1 x cos2 +1 x dx =
Z2 0
3:
Z2 0
sin nx cos mx dx = 0:
cos nx cos mx dx =
(n 2):
; ( + 1); ( + 1) = 1 B( + 1 + 1): 2; ( + + 2) 2
(21.3)
( 0 for m 6= n (21.5) for m = n: (21.6) (21.7a)
B(x y) = ;;((xx);+(yy)) denotes the beta function or the Euler integral of the rst kind, ; (x) denotes the gamma function or the Euler integral of the second kind (see 8.2.5, 6., p. 461). The formula (21.7a) is valid for arbitrary and we use it, e.g., to determine the integrals
Z=2p
Z=2p3
0
0
sin x dx
sin x dx
For positive integer :
7b:
Z=2 0
sin2
+1 x cos2 +1 x dx =
Z=2 dx p 0
3 cos x
! ! 2( + + 1)! :
etc.
(21.7b)
8 > Z1 sin ax < 2 for a > 0 8: dx = > x : ; for a < 0: 0 2
(21.8)
9:
(21.9)
Z cos ax dx x = 1 ( arbitrary): 0 8 Z1 tan ax dx > for a > 0 < 10: = > 2 x : ; for a < 0: 0 2
11:
Z1 cos ax ; cos bx dx = ln ab : x 0
(21.10) (21.11)
21.8 Denite Integrals 1057
8 > > 2 for jaj < 1 Z1 sin x cos ax < for jaj = 1 dx = 12: > x > : 04 for jaj > 1: 0 r Z1 x Z1 x px dx = cos px dx = 2 : 13: sin 0 0 Z1 x sin bx 14: a2 + x2 dx = 2 e;jabj (the sign is the same as the sign of b) : 0 Z1 cos ax 15: 1 + x2 dx = 2 e;jaj: 0 Z1 sin2 ax 16: x2 dx = 2 jaj: 0 + + r Z1 Z1 17: sin(x2 ) dx = cos(x2 ) dx = 2 : ;1
18: 19: 20:
(21.13) (21.14) (21.15) (21.16) (21.17)
;1
Z=2 0 = Z2 0 = Z2 0
(21.12)
p sin x2 dx 2 = 21k ln 11 ;+ kk for jkj < 1: 1 ; k sin x
(21.18)
p cos x2 dx 2 = k1 arcsin k for jkj < 1: 1 ; k sin x
(21.19)
2
p sin x2 dx 2 = k12 (K ; E) for jkj < 1 : 1 ; k sin x
(21.20)
cos ax dx = ba for integer a 0 jbj < 1: 1 ; 2b cos x + b2 1 ; b2
(21.22)
Here, and following, E and K mean complete elliptic integrals (see 8.1.4.3, 2., p. 437): in the E = E k K = F k (see also the table of elliptic integrals 21.9, p. 1061). 2 2 Z=2 cos2 x dx 21: p 2 2 = k12 E ; (1 ; k2) K]: (21.21) 1 ; k sin x 0
22:
Z 0
21.8.2 De nite Integrals of Exponential Functions
(partially combined with algebraic, trigonometric, and logarithmic functions) Z1 + 1) for a > 0 n > ;1 23: xn e;ax dx = ; (ann+1 0 ! = ann+1 for a > 0 n = 0 1 2 : : : :
(21.23a) (21.23b)
1058 21. Tables ; (n) denotes the gamma function (see 8.2.5, 6., p. 461) see also the table of the gamma function 21.10, p. 1063). Z1 ; n+1 n ;ax2 24: x e dx = ( n2+1 ) for a > 0 n > ;1 (21.24a) 2a 2 0 (2k ; 1)p for n = 2k (k = 1 2 : : :) a > 0 (21.24b) = 1 3 k+1 2 ak+1=2 k ! = 2ak+1 for n = 2k + 1 (k = 0 1 2 : : :) a > 0: (21.24c)
25: 26: 27:
Z1
p
0
e;a2 x2 dx = 2a for a > 0:
(21.25)
0
x2 e;a2 x2 dx = 4a3 for a > 0:
(21.26)
0
e;a2 x2 cos bx dx = 2a e;b2 =4a2 for a > 0:
(21.27)
Z1 Z1
p
p
Z1 x dx 2 ex ; 1 = 6 : 0 Z1 2 29: exx +dx1 = 12 : 0 Z1 e;ax sin x 30: dx = arccot a = arctan a1 for a > 0: x 0 Z1 31: e;x ln x dx = ;C ;0 5772
28:
0
(21.28) (21.29) (21.30) (21.31)
C denotes the Euler constant (see 8.2.5, 2., p. 460).
21.8.3 De nite Integrals of Logarithmic Functions
(combined with algebraic and trigonometric functions)
32:
Z1 0
ln j ln x j dx = ;C = ;0 5772 (reduced to Nr. 21.31):
C is the Euler constant (see 8.2.5, 2., p. 460). Z1 2 33: xln;x1 dx = 6 (reduced to Nr. 21.28): 0
34:
Z1 ln x 2 (reduced to Nr. 21.29): dx = ; x+1 12 0
(21.32)
(21.33) (21.34)
21.8 Denite Integrals 1059
Z1 ln x 2 : dx = (21.35) 2 x ;1 8 0 Z1 + x) 36: ln(1 (21.36) 2 + 1 dx = 8 ln 2: x 0 Z1 1 a 37: (21.37) x dx = ; (a + 1) for (;1 < a < 1): 0 ; (x) denotes the gamma function (see 8.2.5, 6., p. 461 see also the table of the gamma function 21.10,
35:
p. 1063).
38: 39: 40: 41: 42: 43: 44:
Z=2
Z=2
0 Z
0
ln sin x dx =
ln cos x dx = ; 2 ln 2:
2 x ln sin x dx = ; 2ln 2 :
(21.39)
0 Z=2 0 Z 0 Z
sin x ln sin x dx = ln 2 ; 1:
(21.40)
p
2 2 ln(a b cos x) dx = ln a + 2a ; b fora b:
(21.41)
ln(a2 ; 2ab cos x + b2 ) dx =
(21.42)
0 Z=2
( 2 ln a for (a b > 0) 2 ln b for (b a > 0):
ln tan x dx = 0:
(21.43)
ln(1 + tan x) dx = 8 ln 2:
(21.44)
0 Z=4 0
(21.38)
21.8.4 De nite Integrals of Algebraic Functions 45:
Z1 0
Z1 ; ( + 1) xa (1 ; x) dx = 2 x2 +1 (1 ; x2 ) dx = ; (;(+ 1) + + 2) 0
= B ( + 1 + 1)
(reduced to Nr. 21.7a): (21.45) ; ( x ) ; ( y ) B(x y) = ; (x + y) denotes the beta function (see 21.8.1, p. 1056) or the Euler integral of the rst kind, ; (x) denotes the gamma function (see 8.2.5, 6., p. 461) or the Euler integral of the second kind. Z1 46: (1 +dxx)xa = sina for a < 1: (21.46) 0
1060 21. Tables 47:
Z1 0
dx (1 ; x)xa = ; cot a for a < 1:
Z1 xa;1 for 0 < a < b: b dx = 1 + x b sin a 0 b p ; 1 Z1 dx 49: p1 ; xa = 2 +aa : a ; 2a 0
48:
(21.47) (21.48) (21.49)
; (x) denotes the gamma function (see 8.2.5, 6., p. 461 see also the table of the gamma function 21.10, p. 1063). Z1 dx a : 50: 1 + 2x cos = 0 < a < (21.50) a + x2 2 sin a 2 0 Z1 dx a : 51: 1 + 2x cos = 0 < a < (21.51) 2 a + x sin x 2 0
21.9 Elliptic Integrals 1061
21.9 Elliptic Integrals
21.9.1 Elliptic Integral of the First Kind ( F
' = 0 10 20 30 40 50 60 70 80 90
' k
= sin
0
10
20
30
40
50
60
70
80
90
0.0000 0.1745 0.3491 0.5236 0.6981 0.8727 1.0472 1.2217 1.3963 1.5708
0.0000 0.1746 0.3493 0.5243 0.6997 0.8756 1.0519 1.2286 1.4056 1.5828
0.0000 0.1746 0.3499 0.5263 0.7043 0.8842 1.0660 1.2495 1.4344 1.6200
0.0000 0.1748 0.3508 0.5294 0.7116 0.8982 1.0896 1.2853 1.4846 1.6858
0.0000 0.1749 0.3520 0.5334 0.7213 0.9173 1.1226 1.3372 1.5597 1.7868
0.0000 0.1751 0.3533 0.5379 0.7323 0.9401 1.1643 1.4068 1.6660 1.9356
0.0000 0.1752 0.3545 0.5422 0.7436 0.9647 1.2126 1.4944 1.8125 2.1565
0.0000 0.1753 0.3555 0.5459 0.7535 0.9876 1.2619 1.5959 2.0119 2.5046
0.0000 0.1754 0.3561 0.5484 0.7604 1.0044 1.3014 1.6918 2.2653 3.1534
0.0000 0.1754 0.3564 0.5493 0.7629 1.0107 1.3170 1.7354 2.4362
E
0 10 20 30 40 50 60 70 80 90
k
=
21.9.2 Elliptic Integral of the Second Kind ( ' =
)
=
' k
)
k
1
= sin
0
10
20
30
40
50
60
70
80
90
0.0000 0.1745 0.3491 0.5236 0.6981 0.8727 1.0472 1.2217 1.3963 1.5708
0.0000 0.1745 0.3489 0.5229 0.6966 0.8698 1.0426 1.2149 1.3870 1.5589
0.0000 0.1744 0.3483 0.5209 0.6921 0.8614 1.0290 1.1949 1.3597 1.5238
0.0000 0.1743 0.3473 0.5179 0.6851 0.8483 1.0076 1.1632 1.3161 1.4675
0.0000 0.1742 0.3462 0.5141 0.6763 0.8317 0.9801 1.1221 1.2590 1.3931
0.0000 0.1740 0.3450 0.5100 0.6667 0.8134 0.9493 1.0750 1.1926 1.3055
0.0000 0.1739 0.3438 0.5061 0.6575 0.7954 0.9184 1.0266 1.1225 1.2111
0.0000 0.1738 0.3429 0.5029 0.6497 0.7801 0.8914 0.9830 1.0565 1.1184
0.0000 0.1737 0.3422 0.5007 0.6446 0.7697 0.8728 0.9514 1.0054 1.0401
0.0000 0.1736 0.3420 0.5000 0.6428 0.7660 0.8660 0.9397 0.9848 1.0000
1062 21. Tables
21.9.3 Complete Elliptic Integral, = sin
K
E
0
1.5708 1.5709 1.5713 1.5719 1.5727
1.5708 1.5707 1.5703 1.5697 1.5689
30 1.6858 1.4675
60 2.1565 1.2111
5
1.5738 1.5751 1.5767 1.5785 1.5805
1.5678 1.5665 1.5649 1.5632 1.5611
35 1.7312 1.4323
65 2.3088 1.1638
10 1.5828 1.5589
40 1.7868 1.3931
70 2.5046 1.1184
15 1.5981 1.5442
45 1.8541 1.3506
75 2.7681 1.0764
20 1.6200 1.5238
50 1.9356 1.3055
80 3.1534 1.0401
25 1.6490 1.4981
55 2.0347 1.2587
85 3.8317 1.0127
1 2 3 4 6 7 8 9
11 12 13 14 16 17 18 19 21 22 23 24 26 27 28 29
1.5854 1.5882 1.5913 1.5946 1.6020 1.6061 1.6105 1.6151 1.6252 1.6307 1.6365 1.6426 1.6557 1.6627 1.6701 1.6777
1.5564 1.5537 1.5507 1.5476 1.5405 1.5367 1.5326 1.5283 1.5191 1.5141 1.5090 1.5037 1.4924 1.4864 1.4803 1.4740
=
k
=
31 32 33 34 36 37 38 39 41 42 43 44 46 47 48 49 51 52 53 54 56 57 58 59
K
1.6941 1.7028 1.7119 1.7214 1.7415 1.7522 1.7633 1.7748 1.7992 1.8122 1.8256 1.8396 1.8691 1.8848 1.9011 1.9180 1.9539 1.9729 1.9927 2.0133 2.0571 2.0804 2.1047 2.1300
E
1.4608 1.4539 1.4469 1.4397 1.4248 1.4171 1.4092 1.4013 1.3849 1.3765 1.3680 1.3594 1.3418 1.3329 1.3238 1.3147 1.2963 1.2870 1.2776 1.2681 1.2492 1.2397 1.2301 1.2206
= 61 62 63 64 66 67 68 69 71 72 73 74 76 77 78 79 81 82 83 84 86 87 88 89
90
K 2.1842 2.2132 2.2435 2.2754 2.3439 2.3809 2.4198 2.4610 2.5507 2.5998 2.6521 2.7081 2.8327 2.9026 2.9786 3.0617 3.2553 3.3699 3.5004 3.6519 4.0528 4.3387 4.7427 5.4349
1
E 1.2015 1.1920 1.1826 1.1732 1.1545 1.1453 1.1362 1.1272 1.1096 1.1011 1.0927 1.0844 1.0686 1.0611 1.0538 1.0468 1.0338 1.0278 1.0223 1.0172 1.0080 1.0053 1.0026 1.0008 1.0000
21.10 Gamma Function 1063
21.10 Gamma Function x
; (x)
x
; (x)
x
; (x)
x
; (x)
01 02 03 04
0.99433 0.98884 0.98355 0.97844
26 27 28 29
0.90440 0.90250 0.90072 0.89904
51 52 53 54
0.88659 0.88704 0.88757 0.88818
76 77 78 79
0.92137 0.92376 0.92623 0.92877
1.00 1.00000 1.25 0.90640 1.50 0.88623 1.75 0.91906
1.05 0.97350 1.30 0.89747 1.55 0.88887 1.80 0.93138 06 07 08 09
0.96874 0.96415 0.95973 0.95546
31 32 33 34
0.89600 0.89464 0.89338 0.89222
56 57 58 59
0.88964 0.89049 0.89142 0.89243
81 82 83 84
0.93408 0.93685 0.93969 0.94261
1.10 0.95135 1.35 0.89115 1.60 0.89352 1.85 0.94561 11 12 13 14
0.94740 0.94359 0.93993 0.93642
36 37 38 39
0.89018 0.88931 0.88854 0.88785
61 62 63 64
0.89468 0.89592 0.89724 0.89864
86 87 88 89
0.94869 0.95184 0.95507 0.95838
1.15 0.93304 1.40 0.88726 1.65 0.90012 1.90 0.96177 16 17 18 19
0.92980 0.92670 0.92373 0.92089
41 42 43 44
0.88676 0.88636 0.88604 0.88581
66 67 68 69
0.90167 0.90330 0.90500 0.90678
91 92 93 94
0.96523 0.96877 0.97240 0.97610
1.20 0.91817 1.45 0.88566 1.70 0.90864 1.95 0.97988 21 22 23 24
0.91558 0.91311 0.91075 0.90852
46 47 48 49
0.88560 0.88563 0.88575 0.88592
71 72 73 74
0.91057 0.91258 0.91467 0.91683
96 97 98 99
0.98374 0.98768 0.99171 0.99581
1.25 0.90640 1.50 0.88623 1.75 0.91906 2.00 1.00000 The values of the gamma function for x < 1 (x = 6 0 ;1 ;2 : : :) and x > 2 can be calculated by the following formula: ; (x) = ; (xx+ 1) ; (x) = (x ; 1) ; (x ; 1): :7) 0:90864 A: ; (0:7) = ; (1 0:7 = 0:7 = 1:2981: B: ; (3:5) = 2:5 ; (2:5) = 2:5 1:5 ; (1:5) = 2:5 1:5 0:88623 = 3:32336:
1064 21. Tables
21.11 Bessel Functions (Cylindrical Functions)
Di erential equation and formulas to the Bessel Functions see 9.1.2.6,2., p. 509.
x
J0(x)
J1(x)
0.1 0.2 0.3 0.4
0.9975 0.9900 0.9776 0.9604 +0.9385 0.9120 0.8812 0.8463 0.8075 +0.7652 0.7196 0.6711 0.6201 0.5669 +0.5118 0.4554 0.3980 0.3400 0.2818 +0.2239 0.1666 0.1104 0.0555 0.0025 ;0.0484 0.0968 0.1424 0.1850 0.2243 ;0.2601 0.2921 0.3202 0.3443 0.3643 ;0.3801 0.3918 0.3992 0.4026 0.4018 ;0.3971 0.3887 0.3766 0.3610 0.3423 ;0.3205 0.2961 0.2693 0.2404 0.2097
0.0499 0.0995 0.1483 0.1960 +0.2423 0.2867 0.3290 0.3688 0.4059 +0.4401 0.4709 0.4983 0.5220 0.5419 +0.5579 0.5699 0.5778 0.5815 0.5812 +0.5767 0.5683 0.5560 0.5399 0.5202 +0.4971 0.4708 0.4416 0.4097 0.3754 +0.3391 0.3009 0.2613 0.2207 0.1792 +0.1374 0.0955 0.0538 +0.0128 ;0.0272 ;0.0660 0.1033 0.1386 0.1719 0.2028 ;0.2311 0.2566 0.2791 0.2985 0.3147
0.0 +1.0000 +0.0000 0.5 0.6 0.7 0.8 0.9
1.0 1.1 1.2 1.3 1.4
1.5 1.6 1.7 1.8 1.9
2.0 2.1 2.2 2.3 2.4
2.5 2.6 2.7 2.8 2.9
3.0 3.1 3.2 3.3 3.4
3.5 3.6 3.7 3.8 3.9
4.0 4.1 4.2 4.3 4.4
4.5 4.6 4.7 4.8 4.9
Y0(x)
Y1(x) I0(x)
I1(x)
K0(x)
K1(x)
0.1005 0.1517 0.2040 0.2579 0.3137 0.3719 0.4329 0.4971 0.5652 0.6375 0.7147 0.7973 0.8861 0.9817 1.085 1.196 1.317 1.448 1.591 1.745 1.914 2.098 2.298 2.517 2.755 3.016 3.301 3.613 3.953 4.326 4.734 5.181 5.670 6.206 6.793 7.436 8.140 8.913 9.759 10.69 11.71 12.82 14.05 15.39 16.86 18.48 20.25 22.20
1.7527 1.3725 1.1145 0.9244 0.7775 0.6605 0.5653 0.4867 0.4210 0.3656 0.3185 0.2782 0.2437 0.2138 0.1880 0.1655 0.1459 0.1288 0.1139 0.1008 0.08927 0.07914 0.07022 0.06235 0.05540 0.04926 0.04382 0.03901 0.03474 0.03095 0.02759 0.02461 0.02196 0.01960 0.01750 0.01563 0.01397 0.01248 0.01116 0.009980 0.008927 0.007988 0.007149 0.006400 0.005730 0.005132 0.004597 0.004119
9.8538 4.7760 3.0560 2.1844 1.6564 1.3028 1.0503 0.8618 0.7165 0.6019 0.5098 0.4346 0.3725 0.3208 0.2774 0.2406 0.2094 0.1826 0.1597 0.1399 0.1227 0.1079 0.09498 0.08372 0.07389 0.06528 0.05774 0.05111 0.04529 0.04016 0.03563 0.03164 0.02812 0.02500 0.02224 0.01979 0.01763 0.01571 0.01400 0.01248 0.01114 0.009938 0.008872 0.007923 0.007078 0.006325 0.005654 0.005055 0.004521
;1 ;1 +1.000 0.0000 1 ;1.5342 ;6.4590 1.003 +0.0501 2.4271 1.0181 0.8073 0.6060 ;0.4445 0.3085 0.1907 ;0.0868 +0.0056 +0.0883 0.1622 0.2281 0.2865 0.3379 +0.3824 0.4204 0.4520 0.4774 0.4968 +0.5104 0.5183 0.5208 0.5181 0.5104 +0.4981 0.4813 0.2605 0.4359 0.4079 +0.3769 0.3431 0.3070 0.2691 0.2296 +0.1890 0.1477 0.1061 0.0645 +0.0234 ;0.0169 0.0561 0.0938 0.1296 0.1633 ;0.1947 0.2235 0.2494 0.2723 0.2921
3.3238 2.2931 1.7809 ;1.4715 1.2604 1.1032 0.9781 0.8731 ;0.7812 0.6981 0.6211 0.5485 0.4791 ;0.4123 0.3476 0.2847 0.2237 0.1644 ;0.1070 ;0.0517 +0.0015 0.0523 0.1005 +0.1459 0.1884 0.2276 0.2635 0.2959 +0.3247 0.3496 0.3707 0.3879 0.4010 +0.4102 0.4154 0.4167 0.4141 0.4078 +0.3979 0.3846 0.3680 0.3484 0.3260 +0.3010 0.2737 0.2445 0.2136 0.1812
1.010 1.023 1.040 1.063 1.092 1.126 1.167 1.213 1.266 1.326 1.394 1.469 1.553 1.647 1.750 1.864 1.990 2.128 2.280 2.446 2.629 2.830 3.049 3.290 3.553 3.842 4.157 4.503 4.881 5.294 5.747 6.243 6.785 7.378 8.028 8.739 9.517 10.37 11.30 12.32 13.44 14.67 16.01 17.48 19.09 20.86 22.79 24.91
1
21.11 Bessel Functions (Cylindrical Functions) 1065
x
J0(x)
J1(x)
Y0(x)
Y1(x) I0(x) I1(x) K0(x) K1(x)
5.0 ;0.1776 ;0.3276 ;0.3085 +0.1479 5.1 5.2 5.3 5.4
5.5 5.6 5.7 5.8 5.9
6.0 6.1 6.2 6.3 6.4
6.5 6.6 6.7 6.8 6.9
7.0 7.1 7.2 7.3 7.4
7.5 7.6 7.7 7.8 7.9
8.0 8.1 8.2 8.3 8.4
8.5 8.6 8.7 8.8 8.9
9.0 9.1 9.2 9.3 9.4
9.5 9.6 9.7 9.8 9.9
10.0
0.1443 0.1103 0.0758 0.0412 ;0.0068 +0.0270 0.0599 0.0917 0.1220 +0.1506 0.1773 0.2017 0.2238 0.2433 +0.2601 0.2740 0.2851 0.2931 0.2981 +0.3001 0.2991 0.2951 0.2882 0.2786 +0.2663 0.2516 0.2346 0.2154 0.1944 +0.1717 0.1475 0.1222 0.0960 0.0692 +0.0419 +0.0146 ;0.0125 0.0392 0.0653 ;0.0903 0.1142 0.1367 0.1577 0.1768 ;0.1939 0.2090 0.2218 0.2323 0.2403 ;0.2459
0.3371 0.3432 0.3460 0.3453 ;0.3414 0.3343 0.3241 0.3110 0.2951 ;0.2767 0.2559 0.2329 0.2081 0.1816 ;0.1538 0.1250 0.0953 0.0652 0.0349 ;0.0047 +0.0252 0.0543 0.0826 0.1096 +0.1352 0.1592 0.1813 0.2014 0.2192 +0.2346 0.2476 0.2580 0.2657 0.2708 +0.2731 0.2728 0.2697 0.2641 0.2559 +0.2453 0.2324 0.2174 0.2004 0.1816 +0.1613 0.1395 0.1166 0.0928 0.0684 +0.0435
0.3216 0.3313 0.3374 0.3402 ;0.3395 0.3354 0.3282 0.3177 0.3044 ;0.2882 0.2694 0.2483 0.2251 0.1999 ;0.1732 0.1452 0.1162 0.0864 0.0563 ;0.0259 +0.0042 0.0339 0.0628 0.0907 +0.1173 0.1424 0.1658 0.1872 0.2065 +0.2235 0.2381 0.2501 0.2595 0.2662 +0.2702 0.2715 0.2700 0.2659 0.2592 +0.2499 0.2383 0.2245 0.2086 0.1907 +0.1712 0.1502 0.1279 0.1045 0.0804 +0.0557
0.1137 0.0792 0.0445 +0.0101 ;0.0238 0.0568 0.0887 0.1192 0.1481 ;0.1750 0.1998 0.2223 0.2422 0.2596 ;0.2741 0.2857 0.2945 0.3002 0.3029 ;0.3027 0.2995 0.2934 0.2846 0.2731 ;0.2591 0.2428 0.2243 0.2039 0.1817 ;0.1581 0.1331 0.1072 0.0806 0.0535 ;0.0262 +0.0011 0.0280 0.0544 0.0799 +0.1043 0.1275 0.1491 0.1691 0.1871 +0.2032 0.2171 0.2287 0.2379 0.2447 +0.2490
27.24 29.79 32.58 35.65 39.01 42.69 46.74 51.17 56.04 61.38 67.23 73.66 80.72 88.46 96.96 106.3 116.5 127.8 140.1 153.7 168.6 185.0 202.9 222.7 244.3 268.2 294.3 323.1 354.7 389.4 427.6 469.5 515.6 566.3 621.9 683.2 750.5 824.4 905.8 995.2 1094 1202 1321 1451 1595 1753 1927 2119 2329 2561 2816
24.34 26.68 29.25 32.08 35.18 38.59 42.33 46.44 50.95 55.90 61.34 67.32 73.89 81.10 89.03 97.74 107.3 117.8 129.4 142.1 156.0 171.4 188.3 206.8 227.2 249.6 274.2 301.3 331.1 363.9 399.9 439.5 483.0 531.0 583.7 641.6 705.4 775.5 852.7 937.5 1031 1134 1247 1371 1508 1658 1824 2006 2207 2428 2671
0.00 3691 3308 2966 2659 2385 2139 1918 1721 1544 1386 1244 1117 1003 09001 08083 07259 06520 05857 05262 04728 04248 03817 03431 03084 02772 02492 02240 02014 01811 01629 01465 01317 01185 01066 009588 008626 007761 006983 006283 005654 005088 004579 004121 003710 003339 003036 002706 002436 002193 001975 001778
0.00 4045 3619 3239 2900 2597 2326 2083 1866 1673 1499 1344 1205 1081 09691 08693 07799 06998 06280 05636 05059 04542 04078 03662 03288 02953 02653 02383 02141 01924 01729 01554 01396 01255 01128 01014 009120 008200 007374 006631 005964 005364 004825 004340 003904 003512 003160 002843 002559 002302 002072 001865
1066 21. Tables
21.12 Legendre Polynomials of the First Kind
For the Di erential equation and the formulas to the Legendre Polynomials see 9.1.2.6,3., p. 511.
P0(x) = 1 P2(x) = 12 (3x2 ; 1) P4(x) = 81 (35x4 ; 30x2 + 3) 1 (231x6 ; 315x4 + 105x2 ; 5) P6(x) = 16
x = P1 ( x)
P2(x)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
;0:5000 ;0:4962 ;0:4850 ;0:4662 ;0:4400 ;0:4062 ;0:3650 ;0:3162 ;0:2600 ;0:1962 ;0:1250 ;0:0462
+0:0400 0.1338 0.2350 0.3438 0.4600 0.5838 0.7150 0.8538 1.0000
P3(x) 0.0000
;0:0747 ;0:1475 ;0:2166 ;0:2800 ;0:3359 ;0:3825 ;0:4178 ;0:4400 ;0:4472 ;0:4375 ;0:4091 ;0:3600 ;0:2884 ;0:1925 ;0:0703 +0:0800 0.2603 0.4725 0.7184 1.0000
P1(x) = x P3(x) = 12 (5x3 ; 3x) P5(x) = 18 (63x5 ; 70x3 + 15x) 1 (429x7 ; 693x5 + 315x3 ; 35x): P7(x) = 16
P4(x)
P5(x)
P6(x)
0.3750 0.3657 0.3379 0.2928 0.2320 0.1577 +0:0729 ;0:0187 ;0:1130 ;0:2050 ;0:2891 ;0:3590 ;0:4080 ;0:4284 ;0:4121 ;0:3501 ;0:2330 ;0:0506 +0:2079 0.5541 1.0000
0.0000 0.0927 0.1788 0.2523 0.3075 0.3397 0.3454 0.3225 0.2706 0.1917 +0:0898 ;0:0282 ;0:1526 ;0:2705 ;0:3652 ;0:4164 ;0:3995 ;0:2857 ;0:0411 +0:3727 1.0000
;0:3125 ;0:2962 ;0:2488 ;0:1746 ;0:0806
+0:0243 0.1292 0.2225 0.2926 0.3290 0.3232 0.2708 0.1721 +0:0347 ;0:1253 ;0:2808 ;0:3918 ;0:4030 ;0:2412 +0:1875 1.0000
P7(x) 0.0000
;0:1069 ;0:1995 ;0:2649 ;0:2935 ;0:2799 ;0:2241 ;0:1318 ;0:0146
+0:1106 0.2231 0.3007 0.3226 0.2737 +0:1502 ;0:0342 ;0:2397 ;0:3913 ;0:3678 +0:0112 1.0000
21.13 Laplace Transformation 1067
21.13 Laplace Transformation (see 15.2.1.1, p. 710)
Z1 F (p) = e;ptf (t) dt f (t) = 0 for t < 0: 0
C is the Euler constant: C = 0:577216 (see 8.2.5, 2., p. 460).
F (p)
No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 1 p 1 pn 1 (p ; ) n 1 (p ; )(p ; ) p (p ; )(p ; ) 1 p2 + 2p + 2 p2 + 2 cos + p sin p2 + 2 p p2 + 2p + 2 p p2 + 2
p cos ; sin p2 + 2 p2 ; 2 p p2 ; 2 1 (p ; )(p ; )(p ; )
f (t)
0 1
tn;1 (n ; 1)! tn;1 e t (n ; 1)! e t ; e t ; e t ; e t ; q ; t pe2 ; 2 sin 2 ; 2 t sin t sin(t + )
!
q p cos 2 ; 2 t ; p 2 2 sin 2 ; 2 t e; ; cos t cos(t + ) sinh t cosh t t t t ; ( ; )e( ;+()( ;;)e )(+;(); )e
t
1068 21. Tables No.
F (p)
16 (p ; )(1p ; )2 17 (p ; )(pp ; )2 2 18 (p ; )(p p ; )2 1 19 (p2 + 2)(p2 + 2) p 20 (p2 + 2)(p2 + 2) p2 + 22 21 p(p2 + 42) 2 2 22 2 p (p + 4 2 ) p2 ; 22 23 p(p2 ; 42) 2 24 p(p22; 42) 2 25 p42+ 4p4
(p2 + 22) p4 + 44 2 2 27 (pp4 +;424 ) 3 28 p4 +p 44 29 (p2 +p2)2 p 30 (p2 ; 2)2 26
31
(p2 ; 2)(p2 ; 2)
f (t) e t ; 1 + ( ; )t] e t ( ; )2 e t ; + ( ; )t]e t ( ; )2 2e t ; 2 ; + ( ; )t] e t ( ; )2 sin t ; sin t (2 ; 2) cos t ; cos t ( 2 ; 2 ) cos2 t sin2 t cosh2 t sinh2 t sin t sinh t sin t cosh t cos t sinh t cos t cosh t
t 2 sin t t sinh t 2 sinh t ; sinh t 2 ; 2
21.13 Laplace Transformation 1069
F (p)
No. 32 33 34 35 36
p (p2 ; 2)(p2 ; 2) p1p 1 ppp 1 pnpp pp1+
37
pp + ; pp +
38
qp
39 40 41 42
p2 + 2 ; p
sp
p2 + 2 ; p p2 + 2 sp 2 2 p + +p p2 + 2 sp 2 2 p ; ;p p2 ; 2 sp 2 2 p ; +p p2 ; 2
43
1 ppp +
44
1p (p + ) p +
45 46 47
pp +
p pp2 1+ 2 pp2 1; 2
f (t) cosh t ; cosh t 2 ; 2 p1 t s 2 t n! p4n tn; 12 (n > 0 integer) (2n)! p1 e; t t p1 e; t ; e; t 2t t sin p t t 2t
s
2 t sin t s 2 t cos t s 2 t sinh t s 2 t cosh t p Zt 2 p e; 2 d 0
p( ; )t
Z ; t q 2e e; 2 d ( ; ) p0 e; t + 2r Z te; 2 d p 0 t J0 (t) (Bessel function of order 0, p. 509) I0(t) (modi ed Bessel function of order 0, p. 509)
1070 21. Tables
F (p)
No. 48 49 50 51 52 53 54 55 56 57 58 59 60 61
f (t) !
1 (p + )(p + ) pp2 + 21p + 2 e1=p ppp arctan p arctan p2 ;2p 2 + 2
e; +2 t I0 ;2 t
2 arctan p ; p + ln p p ln p pn+1 (ln p)2 p ln pp ; ; + = 2artanh ln pp ; p 2 2 ln p2 + 2 p + 2 2 p ln 2 ; 2 p ;
e t ; 1 sin t t
q
e;
p
Re > 0
p
e; t J0
p
p
2 ; 2 t
sinhp2 t sin t t 2 sin t cos t t
;C ; ln t tn (n) ; ln t] (n) = 1 + 1 + + 1 ; C n! 2 n 2 (ln t + C )2 ; 6 1 e t ; e t t 2 sinh t t 2 cos t ; cos t t cosh t ; cosh t 2 t ; 2 =4t p e p 2 t t e;p 2=4t t
p1p e; p Re 0 p 2 2 p + ;p pp2 + 2 63 Re > ;1 J (t) (see Bessel function, p. 509) 62
p
21.13 Laplace Transformation 1071
No.
F (p)
p 2 2 p; p ; pp2 ; 2
f (t) Re > ;1
65
1 e; p ( > 0 reell) p
I (t) (see Bessel function, p. 509) 0 for t < 1 for t >
66
ep; p2+ 2 p2 + 2
(0 p for t < J0 t2 ; 2 for t >
67
ep; p2; 2 p2 ; 2
68
qe
69
e; p2+ 2 + p 1 p2 + 2 p2 + 2
70
e; p2; 2 + p 1 p2 ; 2 p2 ; 2
71
e; p ; e;
pp2+ 2
72
e;
pp2; 2
; e; p
73
1 ; e; p
74
e; p ; e; p p
64
p p
;
p(p+ )(p+ )
(p + )(p + )
p
!
p
!
p
(0 p for t < I0 t2 ; 2 for t > 80 > < ! for t < ; qt2 ; 2 for t > ;( + ) 2t > e I 0 : 2 8 > for t < < 0p 2 2 q t ; J t2 ; 2 for t > > : 1 8 > for t < < 0p 2 2 q t ; I t2 ; 2 for t > > : 1 8 > for t < p > : t2 ; 2 1 8 >0 for t < < I qt2 ; 2 for t > p > : t2 ; 2 1 0 for t > 1 for 0 < t < 8 0 for 0 < t < < 0 (Gamma function, see 8.2.5, 6., p. 461), 0 1 1 +2 n n X (;1) ( 2 z) J (z) = (Bessel functions, see 9.1.2.6, 2., p. 509), n=0 n ! ; ( + n + 1) K (z) = 21 (sin( ));1 I; (z) ; I (z)] with I (z) = e; 12 i J (z e 12 i ) (modi ed Bessel functions, see 9.1.2.6, 3., p. 509), Z x cos t 9 1 > p dt > C (x) = p 2 Z0x t = (Fresnel integrals, see 14.4.3.2, 5., p. 697), sin p t dt > S (x) = p1 2 0 t " 9 Z x sin t > = Si(x) = dt 0 Z 1t (Integral sine, see 14.4.3.2, 2., p. 696), sin t > " si(x) = ; dt = Si( x ) ; t 2 x Z 1 cos t Ci(x) = ; (Integral cosine, see 14.4.3.2, 2., p. 696). t dt x The abbreviations for functions occurring in the table correspond to those introduced in the corresponding chapters.
21.14.1 Fourier Cosine Transformation 1
No. f (t) 1. 2. 3. 4. 5.
1, 0,
0 2a 1 n(! + 3a) ln (! + 3a) 8 +(! ; 3a) ln j! ; 3oaj ; (! + a) ln (! + a) ;(! ; a) ln j! ; aj
(3a2 ; !2) , 0 3a 2 1 ln 1 ; a 2 !2 (a ; ! ) !a ; ab e cosh (b!) !a 2 b "
1 1 b + 2 b2 + (a ; !)2 b2 + (a + !)2 a2 + !2 1 r e; 4b cosh a! 2 b 2b
54. b2 +t t2 tan (at)
cosh (b!) (1 + e2 ab); 1
t b2 + t2 cot (at)
cosh (b!) (e2 ab ; 1); 1
55.
21.14 Fourier Transformation 1077
1
Fc(!) = 0R f (t) cos(t !) dt
No. f (t)
!
56. sin (at2 )
1 r cos !2 ; sin !2 2 2a 4a 4a
57. sin a(1 ; t2 )]
r 2! ; 12 a cos a + 4 + !4a
2 58. sin (2at ) t 2 59. sin (tat )
60. e
; at2
sin (bt2 )
61. cos (at2 ) 62. cos a(1 ; t2 )]
!!
" ! ! ! ! S !2 ; C !2 + p2a sin + !2 2 4a 4a 4 4a 8 " 9 ! " ! < 1 ; C !2 2 ; S !2 2= 2 :2 4a 4a "
p
2 2 ; 4 ; 4 a! (a + b ) 2 (a + b ) e" !
2 sin 12 arctan ab ; 4 (ab! 2 + b2 ) " ! ! 1 r cos !2 + sin !2 2 2a 4a 4a ! r 2 1 sin a + + ! 2 a 4 4a 1
1
2
2
2 ;1
p
63. e; at2 cos (bt2 )
64. 1t sin at 65. p1 sin at t !3 66. p1 sin at t 67. p1 cos at t !3 68. p1 cos at t
2 2 ; 14 ; 14 a! 2 (a2 + b2 );1 2 (a + b ) e" ! 2 1 arctan b cos 4 (ab! ; 2 + b2 ) 2 a
J (2 pa!) 2 0 1 r hsin (2 pa!) + cos (2 pa!) ; e; 2pa! i 2 2! 1 r hsin (2 pa!) + cos (2 pa!) + e; 2pa! i 2 2a 1 r hcos (2 pa!) ; sin (2 pa!) + e; 2pa! i 2 2! 1 r hcos (2 pa!) ; sin (2 pa!) + e; 2pa! i 2 2a
1078 21. Tables
1
Fc(!) = 0R f (t) cos(t !) dt
No. f (t)
"
70. e; bt sin (a t)
p
71. sin (a t) t
p (a2 + !2); 14 e; 14 ab2 (a2 + b2 );1
; at p 73. ep cos (b t) t
;a 75. e p
p
t
"
p
t
p
p
cos (a t) ; sin (a t)]
r a2 ; 2! 2! e
1
No. f (t)
2. 3. 4.
2
p a (2!); 32 e; 2a!2
21.14.2 Fourier Sine Transformation 1.
cos 4 (ab2 +! !2) ; 21 arctan !a
cos (a t)
t
!
r 2! sin + a ! 4 4!
p 72. p1 cos (a t) t
p
!
a p (a2 + b2 ) 34 e; 14 a2 b (b2 + !2);1 2 " 2 x cos 4 (b2a +! !2) ; 32 arctan !b " 2 ! 2! S 4a! + C 4a!
p
74. e; a
!
r 2 2 2 2 2 2! C 4a! sin 4a! ; S 4a! cos 4a!
p 69. p1 sin a t t
1, 0,
0
1b 1 1 ; 2 b2 + (a ; !)2 b2 + (a + !)2 ! 1 ln b2 + (! + a)2 2 2 4 b + ( ! ; a) 1 r e; 14 a2 +b !2 sinh a! 2 b 2b , 0 < ! < 2a 4 , ! = 2a 8 0,
! > 2a
0
53. sin (at)t sin (bt)
0, , 4 0,
2 54. sin t2(at)
1 (! + 2a) ln (! + 2a) 4 +(! ; 2a) ln j! ; 2aj ; 12 ! ln !
! >a+b
21.14 Fourier Transformation 1083
1
Fs(!) = 0R f (t) sin(t !) dt
No. f (t)
! 2a ; ! , 4 2 a2 , 2
2 55. sin t3(at)
0
e;b! cosh(ab) , !>a 2 r " 2! 2! 2! 2 ! cos ! C ! + sin ! S ! 2a 4a 4a 4a 4a " ! ! C !2 ; S ( !2 2 4a 4a " ! ! ! ! r !2 C !2 ; cos !2 S !2 sin 2a 4a 4a 4a 4a " ! ! 2 2 C ! +S ( ! 2 4a 4a
58. sin (at2 ) 2 59. sin (tat )
60. cos (at2 ) 2 61. cos (at ) t p
!>a
; 2 e;ab sinh (b!) ,
(at) 57. t cos b2 + t2
! > 2a
0
0, , 4 , 2
56. cos t(at)
0 < ! < 2a
p
62. e;a t sin (a t)
r
ap ; 2a!2 2 2! ! e
21.14.3 Fourier Transformation
Although F (!) can be represented by the Fourier cosine transformation Fc and the Fourier sine transformation Fs according to (15.75a), here we give some direct transforms F (!).
No. f (t)
1
R e;i!tf (t) dt F (!) = ;1
1.
(t) (Dirac function)
2.
(n) (t)
(i !)n
3.
(n) (t ; a)
(i !)ne;ia! (n = 0 1 2 : : :)
1
1084 21. Tables No. f (t) 4.
1
5. 6.
tn H (t) = 1 for t > 0 H (t) = 0 for t < 0 (Heaviside unit step function)
7.
tn H (t)
8. 9. 10. 11. 12. 13.
e;at H (t) = e;at for t > 0 e;at H (t) = 0 for t < 0 p 1 e;t2 =(4a) 4a 1 ;ajtj 2a e 1 t2 + a2 t t2 + a2 H (t + a) ; H (t ; a) = 1 for jtj < a H (t + a) ; H (t ; a) = 0 for jtj > a
1
R e;i!t f (t) dt F (!) = ;1 2 (!)
2in (n)(!) (n = 1 2 : : :) 1 i ! + (!)
n! n (n) (i !)n+1 + i (!) (n = 1 2 : : :) 1 a + i ! (a > 0) e;a!2 (a > 0) 1 !2 + a2 (a > 0) e;aj!j a
;ie;aj!jsign ! 2 sin a ! !
14. ei at
2 (! ; a)
15. cos at
(! + a) + (! ; a)]
16. sin at 1 17. cosh t 1 18. sinh t
i (! + a) ; (! ; a)] cosh ! 2 ;i tanh ! 2 r !2 ! a 4a + 4 (a > 0) r !2 ! a 4a ; 4 (a > 0)
19. sin at2 20. cos at2
21.14 Fourier Transformation 1085
21.14.4 Exponential Fourier Transformation
Although the exponential Fourier transformation Fe(!) can be represented by the Fourier transformation F (!) according to (15.77), i.e., Fe(!) = 1 F (;!), here we give some direct transforms. 2
1
No. f (t) 1.
f (t) = A f (t) = 0
for a t b otherwise
2.
f (t) = tn f (t) = 0
for 0 t b otherwise (n = 1 2 : : :)
3.
1 (a + i t)
Re > 0
4.
1 (a ; i t)
Re > 0
R f (t) eit! dt Fe(!) = 12 ;1 i A (ei a! ; ei b! ) 2!
"
n n! 1 n ! (;i !);(n+1) ; ei b! X (;i !)m;n;1 bm 2 m=0 m !
;1 ;a! ; ( ) ! e 0
for ! > 0 for ! < 0
0
for ! > 0 for ! < 0
;1 a! ; ( ) (;!) e
1086 21. Tables
21.15 Z Transformation
For de nition see 15.4.1.2, p. 734, for rules of calculations see 15.4.1.3, p. 735, for inverses see p. 737
No. Original Sequence fn
Transform F (z) = Z (fn)
Convergence Region
1
1
z z;1
jzj > 1
2
(;1)n
z z+1
jzj > 1
3
n
z (z ; 1)2
jzj > 1
4
n2
z (z + 1) (z ; 1)3
jzj > 1
5
n3
z (z + 4 z + 1) (z ; 1)4
jzj > 1
6
ean
z z ; ea
jzj > jea j
7
an
z z;a
jzj > jaj
8
an n!
e az
jzj > 0
9
n an
za (z ; a)2
jzj > jaj
az(z + a) (z ; a)3
jzj > jaj
z (z ; 1)k+1
jzj > 1
10 n2 an 11
n k
12
k n
! !
k 1 + 1z
jzj > 0
13 sin bn
z sin b z2 ; 2z cos b + 1
jzj > 1
14 cos bn
z(z ; cos b) z2 ; 2z cos b + 1
jzj > 1
21.15 Z Transformation 1087
Transform F (z) = Z (fn)
Convergence Region
15 ean sin bn
zea sin b z2 ; 2zea cos b + e2a
jzj > jea j
16 ean cos bn
z(z ; ea cos b) z2 ; 2zea cos b + e2a
jzj > jea j
17 sinh bn
z sinh b z2 ; 2z cosh b + 1
jzj > max(jebj je;bj)
18 cosh bn
z(z ; cosh b) z2 ; 2z cosh b + 1
jzj > max(jebj je;bj)
19 an sinh bn
za sinh b z2 ; 2za cosh b + a2
jzj > max(jaebj jae;bj)
20 an cosh bn
z(z ; a cosh b) z2 ; 2za cosh b + a2
jzj > max(jaebj jae;bj)
1 zk
jzj > 0
2z z2 ; 1
jzj > 1
No. Original Sequence fn
21
fn = 0 f "ur n 6= k fk = 1
22 f2n = 0
f2n+1 = 2
23
f2n = 0 f2n+1 = 2(2n + 1)
2z(z2 + 1) (z2 ; 1)2
jzj > 1
24
f2n = 0 f2n+1 = 2n 2+ 1
1 ln zz ; +1
jzj > 1
25 cos n 2
z2 z2 + 1
jzj > 1
26 (n + 1) ean
z2 (z ; ea )2
jzj > jea j
b(n+1) ; ea(n+1) 27 e eb ; ea
z2 a (z ; e ) (z ; eb)
jzj > max(jeaj jebj) a 6= b
28 1 (n ; 1) n(n + 1) 6
z2
(z ; 1)4
jzj > 1
1088 21. Tables No. Original Sequence fn 29 f0 = 0 30
(;1)n (2n + 1) !
n 31 ((2;n1)) !
fn = n1
n 1
Transform F (z) = Z (fn)
Convergence Region
ln z ;z 1
jzj > 1
pz sin p1
jzj > 0
cos p1z
jzj > 0
z
21.16 Poisson Distribution 1089
21.16 Poisson Distribution
For the formula of the Poisson distribution see 16.2.3.3, p. 757.
k 0 1 2 3 4 5 6 7
k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.1 0.904837 0.090484 0.004524 0.000151 0.000004
0.7 0.496585 0.347610 0.121663 0.028388 0.004968 0.000696 0.000081 0.000008 0.000001
0.2 0.818731 0.163746 0.016375 0.001091 0.000055 0.000002
0.3 0.740818 0.222245 0.033337 0.003334 0.000250 0.000015 0.000001
0.4 0.670320 0.268128 0.053626 0.007150 0.000715 0.000057 0.000004
0.5 0.606531 0.303265 0.075816 0.012636 0.001580 0.000158 0.000013 0.000001
0.6 0.548812 0.329287 0.098786 0.019757 0.002964 0.000356 0.000035 0.000003
0.8 0.449329 0.359463 0.143785 0.038343 0.007669 0.001227 0.000164 0.000019 0.000002
0.9 0.406570 0.365913 0.164661 0.049398 0.011115 0.002001 0.000300 0.000039 0.000004
1.0 0.367879 0.367879 0.183940 0.061313 0.015328 0.003066 0.000511 0.000073 0.000009 0.000001
2.0 0.135335 0.270671 0.270671 0.180447 0.090224 0.036089 0.012030 0.003437 0.000859 0.000191 0.000038 0.000007 0.000001
3.0 0.049787 0.149361 0.224042 0.224042 0.168031 0.100819 0.050409 0.021604 0.008102 0.002701 0.000810 0.000221 0.000055 0.000013 0.000003 0.000001
1090 21. Tables (continuation)
k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
4.0 0.018316 0.073263 0.146525 0.195367 0.195367 0.156293 0.104194 0.059540 0.029770 0.013231 0.005292 0.001925 0.000642 0.000197 0.000056 0.000015 0.000004 0.000001
5.0 0.006738 0.033690 0.084224 0.140374 0.175467 0.175467 0.146223 0.104445 0.065278 0.036266 0.018133 0.008242 0.003434 0.001321 0.000472 0.000157 0.000049 0.000014 0.000004 0.000001
6.0 0.002479 0.014873 0.044618 0.089235 0.133853 0.160623 0.160623 0.137677 0.103258 0.068838 0.041303 0.022529 0.011264 0.005199 0.002228 0.000891 0.000334 0.000118 0.000039 0.000012 0.000004 0.000001
7.0 0.000912 0.006383 0.022341 0.052129 0.091126 0.127717 0.149003 0.149003 0.130377 0.101405 0.070983 0.045171 0.026350 0.014188 0.007094 0.003311 0.001448 0.000596 0.000232 0.000085 0.000030 0.000010 0.000003 0.000001
8.0 0.000335 0.002684 0.010735 0.028626 0.057252 0.091604 0.122138 0.139587 0.139587 0.124077 0.099262 0.072190 0.048127 0.029616 0.016924 0.009026 0.004513 0.002124 0.000944 0.000397 0.000159 0.000061 0.000022 0.000008 0.000003 0.000001
9.0 0.000123 0.001111 0.004998 0.014994 0.033737 0.060727 0.091090 0.117116 0.131756 0.131756 0.118580 0.097020 0.072765 0.050376 0.032384 0.019431 0.010930 0.005786 0.002893 0.001370 0.000617 0.000264 0.000108 0.000042 0.000016 0.000006 0.000002 0.000001
21.17 Standard Normal Distribution 1091
21.17 Standard Normal Distribution
For the formula of the standard normal distribution see 16.2.4.2, p. 759.
21.17.1 Standard Normal Distribution for 0.00 x
(x)
x
(x)
x
(x)
x
(x)
0:01 0:02 0:03 0:04 0:05 0:06 0:07 0:08 0:09 0:10 0:11 0:12 0:13 0:14 0:15 0:16 0:17 0:18 0:19
0:5040 0:5080 0:5120 0:5160 0:5199 0:5239 0:5279 0:5319 0:5359 0:5398 0:5438 0:5478 0:5517 0:5557 0:5596 0:5636 0:5675 0:5714 0:5753
0:21 0:22 0:23 0:24 0:25 0:26 0:27 0:28 0:29 0:30 0:31 0:32 0:33 0:34 0:35 0:36 0:37 0:38 0:39
0:5832 0:5871 0:5910 0:5948 0:5987 0:6026 0:6064 0:6103 0:6141 0:6179 0:6217 0:6255 0:6293 0:6331 0:6368 0:6406 0:6443 0:6480 0:6517
0:41 0:42 0:43 0:44 0:45 0:46 0:47 0:48 0:49 0:50 0:51 0:52 0:53 0:54 0:55 0:56 0:57 0:58 0:59
0:6591 0:6628 0:6664 0:6700 0:6736 0:6772 0:6808 0:6844 0:6879 0:6915 0:6950 0:6985 0:7019 0:7054 0:7088 0:7123 0:7157 0:7190 0:7224
0:61 0:62 0:63 0:64 0:65 0:66 0:67 0:68 0:69 0:70 0:71 0:72 0:73 0:74 0:75 0:76 0:77 0:78 0:79
0:7291 0:7324 0:7357 0:7389 0:7422 0:7454 0:7486 0:7517 0:7549 0:7580 0:7611 0:7642 0:7673 0:7704 0:7734 0:7764 0:7794 0:7823 0:7852
x
1.99
x (x) 0:00 0:5000 0:20 0:5793 0:40 0:6554 0:60 0:7257 0:80 0:7881 0:81 0:82 0:83 0:84 0:85 0:86 0:87 0:88 0:89 0:90 0:91 0:92 0:93 0:94 0:95 0:96 0:97 0:98 0:99
0:7910 0:7939 0:7967 0:7995 0:8023 0:8051 0:8079 0:8106 0:8133 0:8159 0:8186 0:8212 0:8238 0:8264 0:8289 0:8315 0:8340 0:8365 0:8389
x (x) x (x) x (x) x (x) x (x) 1:00 0:8413 1:20 0:8849 1:40 0:9192 1:60 0:9452 1:80 0:9641 1:01 1:02 1:03 1:04 1:05 1:06 1:07 1:08 1:09 1:10 1:11 1:12 1:13 1:14 1:15 1:16 1:17 1:18 1:19
0:8438 0:8461 0:8485 0:8508 0:8531 0:8554 0:8577 0:8599 0:8621 0:8643 0:8665 0:8686 0:8708 0:8729 0:8749 0:8770 0:8790 0:8810 0:8830
1:21 1:22 1:23 1:24 1:25 1:26 1:27 1:28 1:29 1:30 1:31 1:32 1:33 1:34 1:35 1:36 1:37 1:38 1:39
0:8869 0:8888 0:8907 0:8925 0:8944 0:8962 0:8980 0:8997 0:9015 0:9032 0:9049 0:9066 0:9082 0:9099 0:9115 0:9131 0:9147 0:9162 0:9177
1:41 1:42 1:43 1:44 1:45 1:46 1:47 1:48 1:49 1:50 1:51 1:52 1:53 1:54 1:55 1:56 1:57 1:58 1:59
0:9207 0:9222 0:9236 0:9251 0:9265 0:9279 0:9292 0:9306 0:9319 0:9332 0:9345 0:9357 0:9370 0:9382 0:9394 0:9406 0:9418 0:9429 0:9441
1:61 1:62 1:63 1:64 1:65 1:66 1:67 1:68 1:69 1:70 1:71 1:72 1:73 1:74 1:75 1:76 1:77 1:78 1:79
0:9463 0:9474 0:9484 0:9495 0:9505 0:9515 0:9525 0:9535 0:9545 0:9554 0:9564 0:9573 0:9582 0:9591 0:9599 0:9608 0:9616 0:9625 0:9633
1:81 1:82 1:83 1:84 1:85 1:86 1:87 1:88 1:89 1:90 1:91 1:92 1:93 1:94 1:95 1:96 1:97 1:98 1:99
0:9649 0:9656 0:9664 0:9671 0:9678 0:9686 0:9693 0:9699 0:9706 0:9713 0:9719 0:9726 0:9732 0:9738 0:9744 0:9750 0:9756 0:9761 0:9767
1092 21. Tables
21.17.2 Standard Normal Distribution for 2.00 x
(x)
x
(x)
x
(x)
x
(x)
2:01 2:02 2:03 2:04 2:05 2:06 2:07 2:08 2:09 2:10 2:11 2:12 2:13 2:14 2:15 2:16 2:17 2:18 2:19
0:9778 0:9783 0:9788 0:9793 0:9798 0:9803 0:9808 0:9812 0:9817 0:9821 0:9826 0:9830 0:9834 0:9838 0:9842 0:9846 0:9850 0:9854 0:9857
2:21 2:22 2:23 2:24 2:25 2:26 2:27 2:28 2:29 2:30 2:31 2:32 2:33 2:34 2:35 2:36 2:37 2:38 2:39
0:9864 0:9868 0:9871 0:9875 0:9878 0:9881 0:9884 0:9887 0:9890 0:9893 0:9896 0:9894 0:9901 0:9904 0:9906 0:9909 0:9911 0:9913 0:9916
2:41 2:42 2:43 2:44 2:45 2:46 2:47 2:48 2:49 2:50 2:51 2:52 2:53 2:54 2:55 2:56 2:57 2:58 2:59
0:9920 0:9922 0:9925 0:9927 0:9929 0:9931 0:9932 0:9934 0:9936 0:9938 0:9940 0:9941 0:9943 0:9945 0:9946 0:9948 0:9949 0:9951 0:9952
2:61 2:62 2:63 2:64 2:65 2:66 2:67 2:68 2:69 2:70 2:71 2:72 2:73 2:74 2:75 2:76 2:77 2:78 2:79
0:9955 0:9956 0:9957 0:9959 0:9960 0:9961 0:9962 0:9963 0:9964 0:9965 0:9966 0:9967 0:9968 0:9969 0:9970 0:9971 0:9972 0:9973 0:9974
x
3.90
x
(x) 2:00 0:9773 2:20 0:9861 2:40 0:9918 2:60 0:9953 2:80 0:9974 2:81 2:82 2:83 2:84 2:85 2:86 2:87 2:88 2:89 2:90 2:91 2:92 2:93 2:94 2:95 2:96 2:97 2:98 2:99
0:9975 0:9976 0:9977 0:9977 0:9978 0:9979 0:9979 0:9980 0:9981 0:9981 0:9982 0:9983 0:9983 0:9984 0:9984 0:9985 0:9985 0:9986 0:9986
x (x) x (x) x (x) x (x) x (x) 3:00 0:9987 3:20 0:9993 3:40 0:9997 3:60 0:9998 3:80 0:9999 3:10 0:9990
3:30 0:9995
3:50 0:9998
3:70 0:9999
3:90 0:9999
21.18 2 Distribution 1093
21.18
2 Distribution
For the formula of the 2 distribution see 16.2.4.6, p. 762.
2 Distribution: Quantile 2 m
Degree of Freedom
m
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 70 80 90 100
Probability 0:99 0:00016 0:020 0:115 0:297 0:554 0:872 1:24 1:65 2:09 2:56 3:05 3:57 4:11 4:66 5:23 5:81 6:41 7:01 7:63 8:26 8:90 9:54 10:2 10:9 11:5 12:2 12:9 13:6 14:3 15:0 22:2 29:7 37:5 45:4 53:5 61:8 70:1
0:975 0:00098 0:051 0:216 0:484 0:831 1:24 1:69 2:18 2:70 3:25 3:82 4:40 5:01 5:63 6:26 6:91 7:56 8:23 8:91 9:59 10:3 11:0 11:7 12:4 13:1 13:8 14:6 15:3 16:0 16:8 24:4 32:4 40:5 48:8 57:2 65:6 74:2
0:95 0:05 0:0039 3:8 0:103 6:0 0:352 7:8 0:711 9:5 1:15 11:1 1:64 12:6 2:17 14:1 2:73 15:5 3:33 16:9 3:94 18:3 4:57 19:7 5:23 21:0 5:89 22:4 6:57 23:7 7:26 25:0 7:96 26:3 8:67 27:6 9:39 28:9 10:1 30:1 10:9 31:4 11:6 32:7 12:3 33:9 13:1 35:2 13:8 36:4 14:6 37:7 15:4 38:9 16:2 40:1 16:9 41:3 17:7 42:6 18:5 43:8 26:5 55:8 34:8 67:5 43:2 79:1 51:7 90:5 60:4 101:9 69:1 113:1 77:9 124:3
0:025 5:0 7:4 9:4 11:1 12:8 14:4 16:0 17:5 19:0 20:5 21:9 23:3 24:7 26:1 27:5 28:8 30:2 31:5 32:9 34:2 35:5 36:8 38:1 39:4 40:6 41:9 43:2 44:5 45:7 47:0 59:3 71:4 83:3 95:0 106:6 118:1 129:6
0:01 6:6 9:2 11:3 13:3 15:1 16:8 18:5 20:1 21:7 23:2 24:7 26:2 27:7 29:1 30:6 32:0 33:4 34:8 36:2 37:6 38:9 40:3 41:6 43:0 44:3 45:6 47:0 48:3 49:6 50:9 63:7 76:2 88:4 100:4 112:3 124:1 135:8
1094 21. Tables
21.19 Fisher F Distribution
For the formula of the Fisher F distribution see 16.2.4.7, p. 763. Fisher F Distribution: Quantile f
m2
1 1 161:4 2 18:51 3 10:13 4 7:71 5 6:61 6 5:99 7 5:59 8 5:32 9 5:12 10 4:96 11 4:84 12 4:75 13 4:67 14 4:60 15 4:54 16 4:49 17 4:45 18 4:41 19 4:38 20 4:35 21 4:32 22 4:30 23 4:28 24 4:26 25 4:24 26 4:23 27 4:21 28 4:20 29 4:18 30 4:17 40 4:08 60 4:00 125 3:92 1 3:84
2 199:5 19:00 9:55 6:94 5:79 5:14 4:74 4:46 4:26 4:10 3:98 3:89 3:81 3:74 3:68 3:63 3:59 3:55 3:52 3:49 3:47 3:44 3:42 3:40 3:39 3:37 3:35 3:34 3:33 3:32 3:23 3:15 3:07 3:00
3 4 215:7 224:6 19:16 19:25 9:28 9:12 6:59 6:39 5:41 5:19 4:76 4:53 4:35 4:12 4:07 3:84 3:86 3:63 3:71 3:48 3:59 3:36 3:49 3:26 3:41 3:18 3:34 3:11 3:29 3:06 3:24 3:01 3:20 2:96 3:16 2:93 3:13 2:90 3:10 2:87 3:07 2:84 3:05 2:82 3:03 2:80 3:01 2:78 2:99 2:76 2:98 2:74 2:96 2:73 2:95 2:71 2:93 2:70 2:92 2:69 2:84 2:61 2:76 2:53 2:68 2:44 2:60 2:37
5 230:2 19:30 9:01 6:26 5:05 4:39 3:97 3:69 3:48 3:33 3:20 3:11 3:03 2:96 2:90 2:85 2:81 2:77 2:74 2:71 2:68 2:66 2:64 2:62 2:60 2:59 2:57 2:56 2:55 2:53 2:45 2:37 2:29 2:21
m1
6 234:0 19:33 8:94 6:16 4:95 4:28 3:87 3:58 3:37 3:22 3:09 3:00 2:92 2:85 2:79 2:74 2:70 2:66 2:63 2:60 2:57 2:55 2:53 2:51 2:49 2:47 2:46 2:45 2:43 2:42 2:34 2:25 2:17 2:10
m1 m2
8 238:9 19:37 8:85 6:04 4:82 4:15 3:73 3:44 3:23 3:07 2:95 2:85 2:77 2:70 2:64 2:59 2:55 2:51 2:48 2:45 2:42 2:40 2:37 2:36 2:34 2:32 2:31 2:29 2:28 2:27 2:18 2:10 2:01 1:94
for = 0:05
12 243:9 19:41 8:74 5:91 4:68 4:00 3:57 3:28 3:07 2:91 2:79 2:69 2:60 2:53 2:48 2:42 2:38 2:34 2:31 2:28 2:25 2:23 2:20 2:18 2:16 2:15 2:13 2:12 2:10 2:09 2:00 1:92 1:83 1:75
24 249:0 19:45 8:64 5:77 4:53 3:84 3:41 3:12 2:90 2:74 2:61 2:51 2:42 2:35 2:29 2:24 2:19 2:15 2:11 2:08 2:05 2:03 2:00 1:98 1:96 1:95 1:93 1:91 1:90 1:89 1:79 1:70 1:60 1:52
30 250:0 19:46 8:62 5:75 4:50 3:81 3:38 3:08 2:86 2:70 2:57 2:47 2:38 2:31 2:25 2:19 2:15 2:11 2:07 2:04 2:01 1:98 1:96 1:94 1:92 1:90 1:88 1:87 1:85 1:84 1:74 1:65 1:55 1:46
40 251:0 19:47 8:59 5:72 4:46 3:77 3:34 3:05 2:83 2:66 2:53 2:43 2:34 2:27 2:20 2:15 2:10 2:06 2:03 1:99 1:96 1:94 1:91 1:89 1:87 1:85 1:84 1:82 1:80 1:79 1:69 1:59 1:49 1:39
1 254:3 19:50 8:53 5:63 4:36 3:67 3:23 2:93 2:71 2:54 2:40 2:30 2:21 2:13 2:07 2:01 1:96 1:92 1:88 1:84 1:81 1:78 1:76 1:73 1:71 1:69 1:67 1:65 1:64 1:62 1:51 1:39 1:25 1:00
21.19 Fisher F Distribution 1095
Fisher F Distribution: Quantile f
m2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 125
1
1 4052 98:50 34:12 21:20 16:26 13:74 12:25 11:26 10:56 10:04 9:65 9:33 9:07 8:86 8:68 8:53 8:40 8:29 8:18 8:10 8:02 7:95 7:88 7:82 7:77 7:72 7:68 7:64 7:60 7:56 7:31 7:08 6:84 6:63
2 4999 99:00 30:82 18:00 13:27 10:92 9:55 8:65 8:02 7:56 7:21 6:93 6:70 6:51 6:36 6:23 6:11 6:01 5:93 5:85 5:78 5:72 5:66 5:61 5:57 5:53 5:49 5:45 5:42 5:39 5:18 4:98 4:78 4:60
3 5403 99:17 29:46 16:69 12:06 9:78 8:45 7:59 6:99 6:55 6:22 5:95 5:74 5:56 5:42 5:29 5:18 5:09 5:01 4:94 4:87 4:82 4:76 4:72 4:68 4:64 4:60 4:57 4:54 4:51 4:31 4:13 3:94 3:78
4 5625 99:25 28:71 15:98 11:39 9:15 7:85 7:01 6:42 5:99 5:67 5:41 5:21 5:04 4:89 4:77 4:67 4:58 4:50 4:43 4:37 4:31 4:26 4:22 4:18 4:14 4:11 4:07 4:04 4:02 3:83 3:65 3:48 3:32
5 5764 99:30 28:24 15:52 10:97 8:75 7:46 6:63 6:06 5:64 5:32 5:06 4:86 4:70 4:56 4:44 4:34 4:25 4:17 4:10 4:04 3:99 3:94 3:90 3:86 3:82 3:78 3:76 3:73 3:70 3:51 3:34 3:17 3:02
m1
6 5859 99:33 27:91 15:21 10:67 8:47 7:19 6:37 5:80 5:39 5:07 4:82 4:62 4:46 4:32 4:20 4:10 4:01 3:94 3:87 3:81 3:76 3:71 3:67 3:63 3:59 3:56 3:53 3:50 3:47 3:29 3:12 2:95 2:80
m1 m2
8 5981 99:37 27:49 14:80 10:29 8:10 6:84 6:03 5:47 5:06 4:74 4:50 4:30 4:14 4:00 3:89 3:79 3:71 3:63 3:56 3:51 3:45 3:41 3:36 3:32 3:29 3:26 3:23 3:20 3:17 2:99 2:82 2:66 2:51
for = 0 01
12 6106 99:42 27:05 14:37 9:89 7:72 6:47 5:67 5:11 4:71 4:40 4:16 3:96 3:80 3:67 3:55 3:46 3:37 3:30 3:23 3:17 3:12 3:07 3:03 2:99 2:96 2:93 2:90 2:87 2:84 2:66 2:50 2:33 2:18
24 6235 99:46 26:60 13:93 9:47 7:31 6:07 5:28 4:73 4:33 4:02 3:78 3:59 3:43 3:29 3:18 3:08 3:00 2:92 2:86 2:80 2:75 2:70 2:66 2:62 2:58 2:55 2:52 2:49 2:47 2:29 2:12 1:94 1:79
30 6261 99:47 26:50 13:84 9:38 7:23 5:99 5:20 4:65 4:25 3:94 3:70 3:51 3:35 3:21 3:10 3:00 2:92 2:84 2:78 2:72 2:67 2:62 2:58 2:54 2:50 2:47 2:44 2:41 2:38 2:20 2:03 1:85 1:70
40 6287 99:47 26:41 13:74 9:29 7:14 5:91 5:12 4:57 4:17 3:86 3:62 3:43 3:27 3:13 3:02 2:92 2:84 2:76 2:69 2:64 2:58 2:54 2:49 2:45 2:42 2:38 2:35 2:33 2:30 2:11 1:94 1:75 1:59
1 6366 99:50 26:12 13:46 9:02 6:88 5:65 4:86 4:31 3:91 3:60 3:36 3:16 3:00 2:87 2:75 2:65 2:57 2:49 2:42 2:36 2:31 2:26 2:21 2:17 2:13 2:10 2:06 2:03 2:01 1:80 1:60 1:37 1:00
1096 21. Tables
21.20 Student t Distribution
For the formula of the Student t distribution see 16.2.4.8, p. 763. Student t Distribution: Quantile t
Degree of Freedom
m
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120
1
m
or t
=2m
Probability for Two{Sided Problem 0:10 0:05 0:02 0:01 0:002 0:001 6:31 12:7 31:82 63:7 318:3 637:0 2:92 4:30 6:97 9:92 22:33 31:6 2:35 3:18 4:54 5:84 10:22 12:9 2:13 2:78 3:75 4:60 7:17 8:61 2:01 2:57 3:37 4:03 5:89 6:86 1:94 2:45 3:14 3:71 5:21 5:96 1:89 2:36 3:00 3:50 4:79 5:40 1:86 2:31 2:90 3:36 4:50 5:04 1:83 2:26 2:82 3:25 4:30 4:78 1:81 2:23 2:76 3:17 4:14 4:59 1:80 2:20 2:72 3:11 4:03 4:44 1:78 2:18 2:68 3:05 3:93 4:32 1:77 2:16 2:65 3:01 3:85 4:22 1:76 2:14 2:62 2:98 3:79 4:14 1:75 2:13 2:60 2:95 3:73 4:07 1:75 2:12 2:58 2:92 3:69 4:01 1:74 2:11 2:57 2:90 3:65 3:96 1:73 2:10 2:55 2:88 3:61 3:92 1:73 2:09 2:54 2:86 3:58 3:88 1:73 2:09 2:53 2:85 3:55 3:85 1:72 2:08 2:52 2:83 3:53 3:82 1:72 2:07 2:51 2:82 3:51 3:79 1:71 2:07 2:50 2:81 3:49 3:77 1:71 2:06 2:49 2:80 3:47 3:74 1:71 2:06 2:49 2:79 3:45 3:72 1:71 2:06 2:48 2:78 3:44 3:71 1:71 2:05 2:47 2:77 3:42 3:69 1:70 2:05 2:46 2:76 3:40 3:66 1:70 2:05 2:46 2:76 3:40 3:66 1:70 2:04 2:46 2:75 3:39 3:65 1:68 2:02 2:42 2:70 3:31 3:55 1:67 2:00 2:39 2:66 3:23 3:46 1:66 1:98 2:36 2:62 3:17 3:37 1:64 1:96 2:33 2:58 3:09 3:29 0:05 0:025 0:01 0:005 0:001 0:0005
Probability for One{Sided Problem
21.21 Random Numbers 1097
21.21 Random Numbers
For the meaning of random numbers see 16.3.5.2, p. 783. 4730 0612 0285 7768 4450 7332 4044 0067 5358 0038 8344 7164 7454 3454 0401 6202 8284 9056 9747 2992 6170 3265 0179 1839 2276 4146 3526 3390 4806 7959 8245 7551 5903 9001 0265
1530 2278 1888 9078 8085 6563 1643 7697 5256 4772 2271 7492 7616 6292 7414 0195 0338 0151 3840 8836 4595 8619 3949 6042 8078 9952 3809 7825 9286 5983 9611 4915 2744 4521 3305
8004 8634 9284 3428 8931 4013 9005 9278 7574 0449 4689 5157 8021 0067 3186 1077 4286 7260 7921 3342 2539 0814 6995 9650 9973 7945 5523 7012 5051 0204 0641 2913 7318 5070 3814
7993 2549 3672 2217 3162 7406 5969 4765 3219 6906 3835 8731 2995 5579 3081 7406 5969 4765 3219 6906 7592 5133 3170 3024 4398 5207 0648 9934 4651 4325 7024 9031 7614 4150 0973
3141 3737 7033 0293 9968 4439 9442 9647 2532 8859 2938 4980 7868 9028 5876 4439 9442 9647 2532 8859 1339 7995 9915 0680 3121 1967 3326 7022 1580 5039 3899 9735 5999 5059 4958
0103 7686 4844 3978 6369 5683 7696 4364 7577 5044 2671 8674 0683 5660 8150 5683 7696 4364 7577 5044 4802 8030 6960 1127 7749 7325 1933 2260 5004 7342 8981 7820 1246 5178 4830
4528 0723 0149 5933 1256 6877 7510 1037 2815 8826 4691 4506 3768 5006 1360 6877 7510 1037 2815 8826 5751 7408 2621 8088 8191 7584 6265 0190 8981 7252 1280 2478 9759 7130 6297
7988 4505 7412 1032 0416 2920 1620 4975 8696 6218 0559 7262 0625 8325 1868 2920 1620 4975 8696 6218 3785 2186 6718 0200 2087 3485 0649 1816 1950 2800 5678 9200 6565 2641 0575
4635 6841 6370 5192 4326 9588 4973 1998 9248 3206 8382 8127 9887 9677 9265 9588 6973 1998 9248 3206 7125 0725 4059 5868 8270 5832 6177 7933 2201 4706 8096 7269 1012 7812 4843
8478 1379 1884 1732 7840 3002 1911 1359 9410 9034 2825 2022 7060 2169 3277 3002 1911 1359 9410 9034 4922 5554 9919 0084 5233 8118 2139 2906 3852 6881 7010 6284 0059 1381 3437
9094 6460 0717 2137 6525 2869 1288 1346 9282 0843 4928 2178 0514 3196 8465 2869 1288 1346 9282 0843 8877 5664 1007 6362 3980 8433 7236 3030 6855 8828 1435 9861 2419 6158 5629
9077 1869 5740 9357 2608 3746 6160 6125 6572 9832 5379 7463 0034 0357 7502 3746 6160 6125 6572 9832 9530 6791 6469 6808 6774 0606 0441 6032 5489 2785 7631 2849 0036 9539 3496
5306 5700 8477 5941 5255 3690 9797 5078 3940 2703 8635 4842 8600 7811 6458 3690 9797 5078 3940 2703 6499 9677 5410 3727 8522 2719 1352 1685 6386 8375 7361 2208 2027 3356 5406
4357 5339 6583 6564 4811 6931 8755 6742 6655 8514 8135 4414 3727 5434 7195 2705 1547 3424 8969 5225 6432 3085 0246 8710 5736 2889 1499 3100 3736 7232 8903 8616 5467 5861 4790
8353 6862 0717 2171 3763 1230 6120 3443 9014 4124 7299 0127 5056 0314 9869 6251 4972 1354 3659 8898 1516 8319 3687 6065 3132 2765 3068 1929 0498 2483 8684 5865 5577 9371 9734
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Mathematical Sciences. Springer-Verlag 1991. Frank, Ph. Mises, R. v.: Die Di erential- und Integralgleichungen der Mechanik und Physik, Bd. 1, 2. | Verlag Vieweg 1961. Greiner, W.: Quanten Mechanics. An Introduction. | Springer-Verlag 1994. Kamke, E.: Di erentialgleichungen, Losungsmethoden und Losungen, Teil 1, 2. | BSB B. G. Teubner 1977. Landau, L.D. Lifschitz, E.M.: Quantenmechanik. | Verlag H. Deutsch 1992. Poljanin, A.D. Saizew, V.F.: Sammlung gewohnlicher Di erentialgleichungen. | Verlag H. Deutsch 1996. Reissig, R. Sansone, G. Conti, R.: Nichtlineare Di erentialgleichungen hoherer Ordnung. | Edizioni Cremonese 1969. Smirnow, W.I.: Lehrbuch der hoheren Mathematik, Teil 2. | Verlag H. Deutsch 1994. Sommerfeld, A.: Partielle Di erentialgleichungen der Physik. | Verlag H. Deutsch 1992.
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9.16] Dodd, R.K. Eilbeck, J.C. Gibbon, J.D. Morris, H.C.: Solitons and Non-Linear Wave Equations. | Academic Press 1982. 9.17] Drazin, P.G., Johnson, R.: Solitons. An Introduction. | Cambridge University Press 1989. 9.18] Gu Chaohao (Ed.): Soliton Theory and its Applications. | Springer-Verlag 1995. 9.19] Lamb, G.L.: Elements of Soliton Theory. | John Wiley 1980. 9.20] Makhankov, V.G.: Soliton Phenomenology. | Verlag Kluwer 1991. 9.21] Remoissenet, S.: Waves Called Solitons. Concepts and Experiments. | Springer-Verlag 1994. 9.22] Toda, M.: Nonlinear Waves and Solitons. | Verlag Kluwer 1989. 9.23] Vvedensky, D.: Partical Di erential Equations with Mathematica. | Addison Wesley 1993.
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10.1] Blanchard, P. Bruning, E.: Variational Methods in Mathematical Physics. | SpringerVerlag 1992. 10.2] Giaquinta, M. Hildebrandt, S.: Calculus of Variations. | Springer-Verlag 1995. 10.3] Klingbeil, E.: Variationsrechnung. | BI-Verlag 1988. 10.4] Klotzler, R.: Mehrdimensionale Variationsrechnung. | Birkhauser 1970. 10.5] Kosmol, P.: Optimierung und Approximation. | Verlag W. de Gruyter 1991. 10.6] Michlin, S.G.: Numerische Realisierung von Variationsmethoden. | Akademie-Verlag 1969. 10.7] Rothe, R.: Hohere Mathematik fur Mathematiker, Physiker, Ingenieure, Teil VII. | B. G. Teubner 1960.
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11.1] Corduneanu, I.C.: Integral Equations and Applications. | Cambridge University Press 1991. 11.2] Estrada, R. Kanwal, R.P.: Singular Integral Equations. | John Wiley 1999. 11.3] Hackbusch, W.: Integral Equations: Theory and Numerical Treatment. | Springer-Verlag 1995. 11.4] Kanwal, R.P.: Linear Integral Equations. | Springer-Verlag 1996. 11.5] Kress, R.: Linear Integral Equations. | Springer-Verlag 1999. 11.6] Michlin, S.G. Prossdorf, S.: Singular Integral Operators. | Springer-Verlag 1986. 11.7] Michlin, S.G.: Integral Equations and their Applications to Certain Problems in Mechanics. | MacMillan 1964. 11.8] Muskelishvili, N.I.: Singular Integral Equations: Boundary Problems of Functions Theory and their Applications to Mathematical Physics. | Dover 1992. 11.9] Pipkin, A.C.: A Course on Integral Equations. | Springer-Verlag 1991. 11.10] Polyanin, A.D. Manzhirov, A.V.: Handbook of Integral Equations. | CRC Press 1998.
12. Functional Analysis
12.1] Achieser, N.I. Glasmann, I.M.: Theory of Linear Operators in Hilbert Space. | M. Nestell. Ungar. 1961. 12.2] Aliprantis, C.D. Burkinshaw, O.: Positive Operators. | Academic Press 1985. 12.3] Aliprantis, C.D. Border, K.C. Luxemburg, W.A.J.: Positive Operators, Riesz Spaces and Economics. | Springer-Verlag 1991. 12.4] Alt, H.W.: Lineare Funktionalanalysis. { Eine anwendungsorientierte Einfuhrung. | Springer-Verlag 1976.
1103 12.5] Balakrishnan, A.V.: Applied Functional Analysis. | Springer-Verlag 1976. 12.6] Bauer, H.: Ma
