\
MATHEMATICAL^
HANDBOOK FOR x
SCIENTISTS AND ENGINEERS Definitions,Theorems, and Formulas for Reference and Review
Granino A. Korn and Theresa M. Korn
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MATHEMATICAL HANDBOOK FOR SCIENTISTS AND ENGINEERS Definitions, Theorems, and Formulas for Reference and Review
Granino A. Korn and
Theresa M. Korn
DOVER PUBLICATIONS, INC. Mineola, New York
Copyright Copyright © 1961,1968 byGranino A.Korn and Theresa M. Korn All rights reserved under Pan American and International Copyright Conventions.
Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario.
Bibliographical Note This Dover edition, first published in 2000, is an unabridged republication of the work originally published in 1968 by McGrawHill, Inc., New York. A number of typographical errors have been corrected.
Library of Congress Cataloging-in-Publication Data Korn, Granino Arthur, 1922-
Mathematical handbook for scientists and engineers : defini tions, theorems, and formulas for reference and review / Granino A. Korn and Theresa M. Korn. p. cm.
Originally published: 2nd, enl.and rev. ed. NewYork: McGrawHill, cl968.
Includes bibliographical references andindex. ISBN 0-486-41147-8 (pbk.) 1. Mathematics—Handbooks, manuals, etc. I. Korn, Theresa M. II. Title.
QA40 .K598 2000 510'.2'1—dc21 00-030318
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
This book is dedicated to the memory of Arthur Korn (1870-1945), who excelled as a mathematician as well as a
theoretical physicist and as a communications engineer.
PREFACE TO THE
SECOND EDITION
This new edition of the Mathematical Handbook has been substantially enlarged, and much of our original material has been carefully, and we hope usefully, revised and expanded. Completelynew sections deal with z transforms, the matrix notation for systems of differential equations (state equations), representation of rotations, mathematical program ming, optimal-control theory, random processes, and decision theory. The chapter on numerical computation was almost entirely rewritten, and the revised appendixes include a discussion of Polya's counting theorem, several new tables of formulas and numerical data, and a new, much larger integral table.
Numerous illustrations have been added
throughout the remaining text.
The handbook is again designed, first, as a comprehensive reference collection of mathematical definitions, theorems, and formulas for scientists, engineers, and students. Subjects of both undergraduate and graduate level are included. The omission of all proofs and the concise tabular presentation of related formulas have made it possible to incorpo rate a relatively large amount of reference material in one volume.
The handbook is, however, not intended for reference purposes alone; it attempts to present a connected survey of mathematical methods useful beyond specialized applications. Each chapter is arranged so as to permit review of an entire mathematical subject. Such a presentation
is made more manageable and readable through the omission of proofs; numerous references provide access to textbook material for more detailed
studies. Special care has been taken to point out, by means of suitable introductions, notes, and cross references, the interrelations of various topics and their importance in scientific and engineering applications. The writers have attempted to meet the individual reader's require ments by arranging the subject matter at three levels: 1. The most important formulas and definitions have been collected
in tables and boxed groups permitting rapid reference and review.
PREFACE TO THE SECOND EDITION
viii
2. The main text presents, in large print, a concise, connected review of each subject.
3. More detailed discussions and advanced topics are presented in
small print. This arrangement makes it possible to include such material without cluttering the exposition of the main review.
We believe that this arrangement has proved useful in the first edition.
The arrangement of the introductory chapters was left unchanged, although additions and changes were made throughout their text. Chap ters 1 to 5 review traditional college material on algebra, analytic geom
etry, elementary and advanced calculus, and vector analysis; Chapter 4 also introducesLebesgue and Stieltjes integrals, and Fourier analysis. Chapters 6, 7, and 8 deal with curvilinear coordinate systems, functions of a complex variable, Laplace transforms, and other functional transforms; new mate rial on finite Fourier transforms and on z transforms was added.
Chapters 9 and 10 deal with ordinary and partial differential equations and include Fourier- and Laplace-transform methods, the method of charac
teristics, and potential theory; eigenvalue problems as such are treated in Chapters 14 and 15. Chapter 11 is essentially new; in addition to ordi nary maxima and minima and the classical calculus of variations, this chapter now contains material on linear and nonlinear programming and on optimal-control theory, outlining both the maximum-principle and dynamic-programming approaches.
Chapter 12, considerably expanded in this edition, introduces the elements of modern abstract language and outlines the construction of mathematical models such as groups, fields, vector spaces, Boolean algebras,
and metric spaces. The treatment of function spaces continues through Chapter 14to permit a modest functional-analysis approach to boundaryvalue problems and eigenvalue problems in Chapter 15, with enough essential definitions to enable the reader to use modern advanced texts and periodical literature.
Chapter 13treats matrices; we have added severalnew sections review ing matrix techniques for systems of ordinary differential equations (state equations of dynamical systems), including an outline of Lyapunov sta bility theory. Chapter 14 deals with the important topics of linear vector spaces, linear transformations (linear operators), introduces eigen value problems, and describes the use of matrices for the representation of mathematical models. The material on representation of rotations was
greatly enlarged because of its importance for space-vehicle design as well as for atomic and molecular physics. Chapter 15 reviews a variety of topics related to boundary-value problems and eigenvalue problems, including Sturm-Liouville problems, boundary-value problems in two and three dimensions, and linear integral equations, considering functions as vectors in a normed vector space.
ix
PREFACE TO THE SECOND EDITION
Chapters 16 and 17 respectively outline tensor analysis and differential geometry, including the description of plane and space curves, surfaces, and curved spaces.
In view of the ever-growing importance of statistical methods, the completely revised Chapter 18 presents a rather detailed treatment of probability theory and includes material on random processes, correlation functions, and spectra. Chapter 19 outlines the principal methods of mathematical statistics and includes extensive tables of formulas involving special sampling distributions. A subchapter on Bayes tests and esti mation was added.
The new Chapter 20 introduces finite-difference methods and difference equations and reviews a number of basic methods of numerical compu tation. Chapter 21 is essentially a collection of formulas outlining the properties of higher transcendental functions; various formulas and many illustrations have been added.
The appendixes present mensuration formulas, plane and spherical trigonometry, combinatorial analysis, Fourier- and Laplace-transform tables, a new, larger integral table, and a new set of tables of sums and series. The treatment of combinatorial analysis was enlarged to outline the use of generating functions and a statement of Polya's counting theorem. Several new tables of formulas and functions were added. As before, there is a glossary of symbols, and a comprehensive and detailed index permits the use of the handbook as a mathematical dictionary. The writers hope and believe that this handbook will give the reader an opportunity to scan the field of mathematical methods, and thus to widen his background or to correlate his specialized knowledge with more general developments. We are very grateful to the many readers who have helped to improve the handbook by suggesting corrections and additions; once again, we earnestly solicit comments and suggestions for improvements, to be addressed to us in care of the publishers. Granino A. Korn
Theresa M. Korn
CONTENTS
Preface
vii
Chapter 1.
Real and Complex Numbers.
Elementary Algebra.
1.1. Introduction. The Real-number System 1.2. Powers, Roots, Logarithms, and Factorials. 1.3. Complex Numbers
Sum and Product Notation
1
2 4 7
1.4. Miscellaneous Formulas
10
1.5. Determinants
12
1.6. Algebraic Equations: General Theorems 1.7. Factoring of Polynomials and Quotients of Polynomials.
15
Partial Frac
tions
19
1.8. Linear, Quadratic, Cubic, and Quartic Equations 1.9. Systems of Simultaneous Equations 1.10. Related Topics, References, and Bibliography
Chapter 2.
2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7.
22 24 27
Plane Analytic Geometry
Introduction and Basic Concepts The Straight Line Relations Involving Points and Straight Lines Second-order Curves (Conic Sections) Properties of Circles, Ellipses, Hyperbolas, and Parabolas Higher Plane Curves Related Topics, References, and Bibliography.
29
...
Chapter 3. Solid Analytic Geometry 3.1. Introduction and Basic Concepts
30 37 39 41 48 53 56
57 58
3.2. The Plane
67
3.3. 3.4. 3.5. 3.6.
69 70 ?4 82
The Straight Line Relations Involving Points, Planes, and Straight Lines Quadric Surfaces Related Topics, References, and Bibliography
Chapter 4. Functions and Limits. Differential and Integral Calculus
83
4.1. Introduction 4.2. Functions
85 85
4.3. Point Sets, Intervals, and Regions 4.4. Limits, Continuous Functions, and Related Topics
87 90
4.5. Differential Calculus
95 si
CONTENTS
xii
4.6. Integrals and Integration 4.7. Mean-value Theorems.
102
Values of Indeterminate Forms.
Weierstrass's
Approximation Theorems 4.8. Infinite Series, Infinite Products, and Continued Fractions. . . . 4.9. Tests for the Convergence and Uniform Convergence of Infinite Series and Improper Integrals
4.10. Representation of Functions by Infinite Series and Integrals. Series and Taylor's Expansion
118 121 127
Power 131 134 144
4.11. Fourier Series and Fourier Integrals
4.12. Related Topics, References, and Bibliography
Chapter 5. Vector Analysis
145
5.1. Introduction
146
5.2. Vector Algebra
147
5.3. Vector Calculus: Functions of a Scalar Parameter 5.4. Scalar and Vector Fields
151 153
5.5. 5.6. 5.7. 5.8.
157 162 164 166
Differential Operators Integral Theorems Specification of a Vector Field in Terms of Its Curl and Divergence. Related Topics, References, and Bibliography
.
Chapter 6. Curvilinear Coordinate Systems
168
6.1. Introduction
168
6.2. Curvilinear Coordinate Systems
169
6.3. Representation of Vectors in Terms of Components 6.4. Orthogonal Coordinate Systems. Vector Relations in Terms of Orthog onal Components 6.5. Formulas Relating to Special Orthogonal Coordinate Systems . . . 6.6. Related Topics, References, and Bibliography
171
Chapter 7. Functions of a Complex Variable 7.1. Introduction
7.2. Functions of a Complex Variable.
174 177 177
187 188
Regions of the Complex-number
Plane
7.3. Analytic (Regular, Holomorphic) Functions 7.4. Treatment of Multiple-valued Functions 7.5. Integral Theorems and Series Expansions 7.6. Zeros and Isolated Singularities 7.7. Residues and Contour Integration 7.8. Analytic Continuation 7.9. Conforms! Mapping 7.10. Functions Mapping Specified Regions onto the Unit Circle . . . . 7.11. Related Topics, References, and Bibliography
188
192 193 196 198 202 205 206 219 219
Chapter 8. The Laplace Transformation and Other Functional Transformations
221
8.1. Introduction
222
8.2. The Laplace Transformation 8.3. Correspondence between Operations on Object and Result Functions . 8.4. Tables of Laplace-transform Pairs and Computation of Inverse Laplace
222 225
Transforms
228
xiii
8.5. 8.6. 8.7. 8.8.
CONTENTS
"Formal" Laplace Transformation of Impulse-function Terms . . . Some Other Integral Transformations Finite Integral Transforms, Generating Functions, and z Transforms . Related Topics, References, and Bibliography
Chapter 9.
Ordinary Differential Equations
233 233 237 240
243
9.1. Introduction
244
9.2. 9.3. 9.4. 9.5. 9.6. 9.7.
247 252 265 277 284 286
First-order Equations Linear Differential Equations Linear Differential Equations with Constant Coefficients Nonlinear Second-order Equations Pfaffian Differential Equations Related Topics, References, and Bibliography
Chapter 10.
Partial Differential Equations
287
10.1. Introduction and Survey 10.2. Partial Differential Equations of the First Order 10.3. Hyperbolic, Parabolic, and Elliptic Partial Differential Equations. Characteristics
10.4. Linear Partial Differential Equations of Physics. 10.5. Integral-transform Methods 10.6. Related Topics, References, and Bibliography
Chapter 11.
288 290 302
Particular Solutions.
Maxima and Minima and Optimization Problems .
330
11.1. Introduction 11.2. Maxima and Minima of Functions of One Real Variable 11.3. Maxima and Minima of Functions of Two or More Real Variables.
321 332 333
11.4. Linear Programming, Games, and Related Topics 11.5. Calculus of Variations. Maxima and Minima of Definite Integrals. 11.6. Extremals as Solutions of Differential Equations: Classical Theory.
311 324 328
335 . .
11.7. Solution of Variation Problems by Direct Methods 11.8. Control Problems and the Maximum Principle 11.9. Stepwise-control Problems and Dynamic Programming
344 346
357 358
11.10. Related Topics, References, and Bibliography
369 371
Chapter 12. Definition of Mathematical Models: Modern (Abstract) Algebra and Abstract Spaces
373
12.1. Introduction
374
12.2. Algebra of Models with a Single Defining Operation: Groups. . . . 12.3. Algebra of Models with Two Defining Operations: Rings, Fields, and Integral Domains 12.4. Models Involving More Than One Class of Mathematical Objects: Linear Vector Spaces and Linear Algebras 12.5. Models Permitting the Definition of Limiting Processes: Topological Spaces
378
382
384 386
12.6. Order 12.7. Combination of Models: Direct Products, Product Spaces, and Direct Sums.
391
12.8. Boolean Algebras 12.9. Related Topics, References, and Bibliography
393 400
392
CONTENTS
Chapter 13.
Matrices.
Quadratic and Hermitian Forms
xiv
....
402
13.1. Introduction
403
13.2. Matrix Algebra and Matrix Calculus 13.3. Matrices with Special Symmetry Properties 13.4. Equivalent Matrices. Eigenvalues, Diagonalization, Topics
403 410
and Related
13.5. Quadratic and Hermitian Forms 13.6. Matrix Notation for Systems of Differential Equations (State Equa tions). Perturbations and Lyapunov Stability Theory 13.7. Related Topics, References, and Bibliography
412 416 420 430
Chapter 14. Linear Vector Spaces and Linear Transformations (Linear Operators). Representation of Mathematical Models in Terms of Matrices
14.1. 14.2. 14.3. 14.4.
14.5. 14.6. 14.7. 14.8. 14.9. 14.10. 14.11.
Introduction. Reference Systems and Coordinate Transformations . Linear Vector Spaces Linear Transformations (Linear Operators) Linear Transformations of a Normed or Unitary Vector Space into Itself. Hermitian and Unitary Transformations (Operators) . . . Matrix Representation of Vectors and Linear Transformations (Opera tors) Change of Reference System Representation of Inner Products. Orthonormal Bases Eigenvectors and Eigenvalues of Linear Operators Group Representations and Related Topics Mathematical Description of Rotations Related Topics, References, and Bibliography
Chapter 15. Linear Integral Equations, Boundary-value Problems, and Eigenvalue Problems Introduction. Functional Analysis Functions as Vectors. Expansions in Terms of Orthogonal Functions. Linear Integral Transformations and Linear Integral Equations . Linear Boundary-value Problems and Eigenvalue Problems Involving Differential Equations 15.5. Green's Functions. Relation of Boundary-value Problems and Eigen value Problems to Integral Equations 15.6. Potential Theory 15.7. Related Topics, References, and Bibliography
15.1. 15.2. 15.3. 15.4.
Chapter 16. Representation of Mathematical Models: Tensor Algebra and Analysis
431
433 435 439
441 447 449 452 457 467 471 482
484 486 487 492 502 515 520 531
533
16.1. Introduction
534
16.2. Absolute and Relative Tensors
537
16.3. 16.4. 16.5. 16.6. 16.7. 16.8.
540 543 543 545 546 549
Tensor Algebra: Definition of Basic Operations Tensor Algebra: Invariance of Tensor Equations Symmetric and Skew-symmetric Tensors Local Systems of Base Vectors Tensors Defined on Riemann Spaces. Associated Tensors . . . . Scalar Products and Related Topics
xv
CONTENTS
16.9. Tensors of Rank Two (Dyadics) Defined on Riemann Spaces.... 16.10. The Absolute Differential Calculus.
Covariant Differentiation.
.
.
16.11. Related Topics, References, and Bibliography Chapter 17.
Differential Geometry
551 552
560 561
17.1. Curves in the Euclidean Plane
562
17.2. 17.3. 17.4. 17.5.
565 569 579 584
Curves in Three-dimensional Euclidean Space Surfaces in Three-dimensional Euclidean Space Curved Spaces Related Topics, References, and Bibliography
Chapter 18.
Probability Theory and Random Processes
18.1. Introduction
18.2. 18.3. 18.4. 18.5. 18.6. 18.7. 18.8. 18.9. 18.10.
Definition and Representation of Probability Models One-dimensional Probability Distributions Multidimensional Probability Distributions Functions of Random Variables. Change of Variables Convergence in Probability and Limit Theorems Special Techniques for Solving Probability Problems Special Probability Distributions Mathematical Description of Random Processes Stationary Random Processes. Correlation Functions and Spectral Densities
585 587
588 593 602 614 620 623 625 637 641
18.11. Special Classes of Random Processes. Examples 18.12. Operations on Random Processes 18.13. Related Topics, References, and Bibliography
650 659 662
Chapter 19.
664
Mathematical Statistics
19.1. Introduction to Statistical Methods
665
19.2. Statistical Description. Definition and Computation of Randomsample Statistics 19.3. General-purpose Probability Distributions
668 674
19.4. Classical Parameter Estimation
676
19.5. Sampling Distributions
680
19.6. Classical Statistical Tests
686
19.7. Some Statistics, Sampling Distributions, and Tests for Multivariate Distributions
19.8. Random-process Statistics and Measurements 19.9. Testing and Estimation with Random Parameters 19.10. Related Topics, References, and Bibliography
Chapter 20. Numerical Calculations and Finite Differences . . . .
697
703 708 713
715
20.1. Introduction
718
20.2. Numerical Solution of Equations 20.3. Linear Simultaneous Equations, Matrix Inversion, and Matrix Eigen
719
value Problems
729
20.4. Finite Differences and Difference Equations
737
20.5. Approximation of Functions by Interpolation 20.6. Approximation by Orthogonal Polynomials, Truncated Fourier Series,
746
and Other Methods
20.7. Numerical Differentiation and Integration
755
770
CONTENTS
xvi
20.8. Numerical Solution of Ordinary Differential Equations 20.9. Numerical Solution of Boundary-value Problems, Partial Differential Equations, and Integral Equations 20.10. Monte-Carlo Techniques 20.11. Related Topics, References, and Bibliography
785 797 800
Chapter 21.
804
Special Functions
777
21.1. Introduction
806
21.2. The Elementary Transcendental Functions 21.3. Some Functions Defined by Transcendental Integrals
806 818
21.4. The Gamma Function and Related Functions
21.5. Binomial Coefficients and Factorial Polynomials.
822
Bernoulli Poly
nomials and Bernoulli Numbers
21.6. Elliptic Functions, Elliptic Integrals, and Related Functions. . 21.7. Orthogonal Polynomials 21.8. Cylinder Functions, Associated Legendre Functions, and Spherical Harmonics
824
827 848 857
21.9. Step Functions and Symbolic Impulse Functions 21.10. References and Bibliography
874 880
Appendix A.
Formulas Describing Plane Figures and S o l i d s . . . .
881
Appendix B.
Plane and Spherical Trigonometry
885
Appendix C.
Permutations, Combinations, and Related Topics.
894
Appendix D.
Tables of Fourier Expansions and Laplace-transform
Pairs
Appendix E.
900
Integrals, Sums, Infinite Series and Products, and
Continued Fractions
Appendix F.
Numerical Tables
925
989
Squares Logarithms Trigonometric Functions Exponential and Hyperbolic Functions Natural Logarithms Sine Integral Cosine Integral
990 993 1010 1018 1025 1027 1028
Exponential and Related Integrals Complete Elliptic Integrals Factorials and Their Reciprocals
1029 1033 1034
Binomial Coefficients Gamma and Factorial Functions Bessel Functions
1034 1035 1037
Legendre Polynomials
1060
Error Function Normal-distribution Areas Normal-curve Ordinates Distribution of t
1061 1062 1063 1064
Distribution of x2
1065
xvii
CONTENTS
Distribution of F
1066
Random Numbers . . . Normal Random Numbers
1070 1075
sin x/x Chebyshev Polynomials
1080 1089
Glossary of Symbols and Notations
1090
Index
1097
MATHEMATICAL HANDBOOK FOR SCIENTISTS AND ENGINEERS
CHAPT
ER 1
REAL AND COMPLEX NUMBERS ELEMENTARY ALGEBRA
1.1. Introduction.
The Real-
1.4. Miscellaneous Formulas
number System 1.4-1. The Binomial Theorem and 1.1-1. Introduction 1.1-2. Real Numbers
1.1-3. Equality Relation 1.1-4. Identity Relation 1.1-5. Inequalities 1.1-6. Absolute Values
Related Formulas
1.4-2. Proportions 1.4-3. Polynomials.
Symmetric
Functions 1.5. Determinants
1.2. Powers, Roots, Logarithms, and Factorials.
Sum and Product
Notation
1.5-1. Definition
1.5-2. Minors and Cofactors.
Expan
sion in Terms of Cofactors 1.2-1. Powers and Roots
1.5-3. Examples: Second- and Third-
1.2-2. Formulas for Rationalizing the Denominators of Fractions
order Determinants
1.2-3. Logarithms
1.5-4. Complementary Minors. Laplace Development
1.2-4. Factorials 1.2-5. Sum and Product Notation
1.5-5. Miscellaneous Theorems
1.2-6. Arithmetic Progression 1.2-7. Geometric Progression
1.5-6. Multiplication of Determinants 1.5-7. Changing the Order of Determi nants
1.3. Complex Numbers 1.6.
1.3-1. Introduction
1.3-2. Representation of Complex
Algebraic Equations: General Theorems
Numbers as Points or Position
1.6-1. Introduction
Vectors.
1.6-2. Solution of an Equation. 1.6-3. Algebraic Equations
Polar Decomposition
1.3-3. Representation of Addition, Multiplication, and Division, Powers and Roots
Roots
1.6-4. Relations between Roots and Coefficients
ELEMENTARY ALGEBRA
1.1-1
1.6-5. Discriminant of an Algebraic Equation 1.6-6. Real Algebraic Equations and Their Roots
(a) (b) (c) (d) (e) (/) (g)
Complex Roots Routh-Hurwitz Criterion Descartes's Rule An Upper Bound Rolle's Theorem Budan's Theorem Sturm's Method
1.8-2. Solution of Quadratic Equations 1.8-3. Cubic Equations: Cardan's Solu tion
1.8-4. Cubic Equations: Trigonometric Solution
1.8-5. Quartic Equations: DescartesEuler Solution
1.8-6. Quartic Equations: Ferrari's Solution
1.9. Systems of Simultaneous Equa tions
1.7. Factoring of Polynomials and Quotients of Polynomials. Par
1.9-1. Simultaneous Equations 1.9-2. Simultaneous Linear Equations: Cramer's Rule
tial Fractions
1.7-1. Factoring of a Polynomial 1.7-2. Quotients of Polynomials. Remainder. Long Division 1.7-3. Common Divisors and Common
Roots of Two Polynomials 1.7-4. Expansion in Partial Fractions 1.8. Linear, Quadratic, Cubic, and Quartic Equations
1.8-1. Solution of Linear Equations
1.9-3. Linear Independence 1.9-4. Simultaneous Linear Equations: General Theory 1.9-5. Simultaneous Linear Equations: n Homogeneous Equations in n Unknowns
1.10. Related Topics, References, and Bibliography
1.10-1. Related Topics 1.10-2. References and Bibliography
1.1. INTRODUCTION THE REAL-NUMBER SYSTEM
1.1-1. This chapter deals with the algebra* of real and complex numbers, i.e., with the study of those relations between real and complex numbers which involve a finite number of additions and multiplications. This is
considered to include the solution of equations based on such relations, even though actual exact numerical solutions may require infinite num bers of additions and/or multiplications.
The definitions and relations
presented in this chapter serve as basic tools in many more general mathematical models (see also Sec. 12.1-1).
1.1-2. Real Numbers. The axiomatic foundations ensuring the selfconsistency of the real-number system are treated in Refs. 1.1 and 1.5, and lead to the acceptance of the following rules governing the addition and multiplication of real numbers. * See also footnote to Sec. 12.1-2.
REAL NUMBERS
1.1-2
a + b and ab are real numbers (algebraic numbers, rational numbers, integers, positive integers) if this is true for both a and b (closure) a+ b= b+ a ab = ba (commutative laws) a+(b + c) = (a + b)+c = a + b + c
a(bc) = (ab)c = abc (associative laws) 0 •1 = a (multiplicative identity) a(b + c) = ab + ac (distributive law)
(1.1-1)
a + c = b + c implies a = b
ca = cb, c^O implies a = b (cancellation laws)
The real number 0 (zero, additive identity) has the properties a + 0 = a
a
0 = 0
(1.1-2)
for every real a.
The (unique) additive inverse —a and the (unique) multiplicative inverse (reciprocal) a-1 = 1/a of a real number a are respectively defined by a + (—a) = a — a = 0
aa~
= 1
(o^O)
(1.1-3)
Division by 0 is not admissible.
In addition to the "algebraic" properties (1), the class of the positive integers 1,2, ... hasthe properties of being simply ordered (Sec. 12.6-2; n is"greater than" m or n > mii and only if n = m + x wherea; is a positive integer) and well-ordered
(every nonempty set of positive integers has a smallest element). A set of positive integers containing (1) 1 and the "successor" n + 1 ofeach ofits elements n, or (2) all integers less than nforany n,contains all positive integers (Principle ofFinite Induction). The properties of positive integers may be alternatively denned by Peano's Five Axioms, viz., (1) 1 is a positive integer, (2) each positive integer n has a unique suc cessor S(n)t (3) S(n) * 1, (4) S(n) = S(m) implies n = m, (5) the principle of finite induction holds. Addition and multiplication satisfying the rules (1) are denned by the "recursive" definitions n + 1 = S(n)} n + S(m) = S(n + m); n •1 = n, n • S(m) — n - m -\- n.
Operations onthe elements m - n of the class of all integers (positive, negative, orzero) are interpreted asoperations oncorresponding pairs (m, n)of positive integers m, n such that (m - n) + n = m, where 0, defined by n + 0 = n, corresponds to (n, n), for all n. An integer is negative if and only if it is neither positive nor zero. The study of the properties of integers is called arithmetic.
Operations on rational numbers m/n (n j* 0) are interpreted as operations on corresponding pairs (m, n) of integers m, n such that (m/n)n = m. m/n is positive if and only if mn is positive.
Real algebraic (including rational and irrational) numbers, corresponding to (real) roots of algebraic equations with integral coefficients (Sec. 1.6-3) and real
1.1-3
ELEMENTARY ALGEBRA
4
transcendental numbers, for which no such correspondence exists, may be intro duced in terms of limiting processes involving rational numbers (Dedekind cuts, Ref. 1.5).
The class of all rational numbers comprises the roots of all linear equations (Sec.
1.8-1) with rational coefficients, and includes the integers. The class of all real algebraic numbers comprises the real roots of all algebraic equations (Sec. 1.6-3) with algebraic coefficients, including the rational numbers. The class of all real numbers contains the real roots of all equations involving a finite or infinite number of additions and multiplications of real numbers and includes real algebraic and tran scendental numbers (see also Sec. 4.3-1).
A real number a is greater than the real number b (a > 6, b < a) if and only if a = b 4- x, where x is a positive real number (see also Sees. 1.1-5 and 12.6-2).
1.1-3. Equality Relation (see also Sec. 12.1-3). An equation a = b implies b = a (symmetry of the equality relation), and a + c = b + c, ac = be [in general, f(a) = f(b) if f(a) stands for an operation having a unique result], a = b and b = c together imply a = c (transitivity of the equality relation), ab j* 0 implies a j& 0, b ^A 0. 1.1-4. Identity Relation.
In general, an equation involving operations on a
quantity x or on several quantities X\, xz, . . . will hold only for special values of x or special sets of values xi, x2, . . • (see also Sec. 1.6-2). // it is desired to stress the fact that an equation holds for all values of x or of xi, xi, . . . within certain ranges of interest, the identity symbol = may be used instead of the equality symbol = [EXAMPLE: (x —l)(x -f 1) = x2 —1], and/or the ranges of the variables in ques tion may be indicated on the right of the equation, a = b (better a — b) is also used with the meaning "a is defined as equal to 6."
1.1-5. Inequalities (see also Sees. 12.6-2 and 12.6-3). a > b implies b < a, a + c> b + cy ac > be (c > 0), -a < -b, 1/a < \/b (a > 0, b > 0). A real number a is positive (a > 0), negative (a < 0), or zero (a = 0). Sums and products of positive numbers are positive, a < A, b < B implies a + b < A + B.
a >b,b > c implies a > c. The absolute
1.1-6. Absolute Values (see also Sees. 1.3-2 and 14.2-5).
value \a\ of a real number a is defined as equal to a if a > 0 and equal to -a if a < 0.
\a\ > 0
Note
\a\ = 0 implies a = 0
I
||a| - |b|| < \a + b\ < \a\ + \b\
\\a\ - |ft|| < \a - b\ < \a\ + \b\ } (1.1-4)
\ab\ = \a\ \b\
|a| < A,
a
b
=g
\b\ '"n"1) (n vafoes) n
In particular,
VI = ±1
(1.3-9)
V11! = ±*
VI =I cos 120° +*sin 120° =M(-l +*V5)
\ cos 120° - t sin 120° =3^(~1 - *V*)
(1.3-10)
{cos 60° + t sin 60° = \i(\ + %V3) -1
cos 60° - i sin 60° = >£(1 - * v3)
1.4. MISCELLANEOUS FORMULAS
1.4-1. The Binomial Theorem and Related Formulas. If a, 6, c are real or complex numbers,
(a ± 6)2 = a2 ± 2ab + b2
la ± by = a3 ± Sa2b + 3ab2 ± 63 (a ± by = a4 ± 4a36 + 6a2&2 ± 4a&3 + 64 (a + b)n with
i
2 (")•"-
(1.4-1)
(n = 1, 2, . . .)
n\
j/ ~j\(n-j)\
(j = 0, 1, 2, . . . < n = 0, 1, 2, . . .)
(Binomial Theorem for integral exponents n; see also Sec. 21.2-12). The
binomial coefficients (n Jare discussed in detail in Sec. 21.5-1. (a + b+ c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc (a2 - b2) = (a + b)(a - b) (a2 + b2) = (a + ib)(a - ib) an _ 5» = (a - &) (a*-1 + an~2b +
(1.4-2) (1.4-3)
+ abn~2 + fc""1)
(1.4-4a)
+ abn~2 - bn~l)
(1.4-46)
If n is an even positive integer,
an _ hn = (a + b)(a«-1 - an"2& +
11
POLYNOMIALS.
SYMMETRIC FUNCTIONS
1.4-3
If n is an odd positive integer,
an + bn = (a + 6) (a*-1 - an~2b d- • • • - abn~2 + b*-1) (1.4-5) Note also
a4 + a2b2 + 64 = (a2 + ab + b2)(a2 - afc + 62) 1.4-2. Proportions.
(1.4-6)
a:b = c:d or a/6 = c/d implies
ma + nb _ mc + nd
pa + qb
(corresponding addition
pc + qd
and subtraction)
(1.4-7)
In particular,
a± b
c± d
a —b __ c — d
b
d
a + b ~~ c + d
(1.4-8)
1.4-3. Polynomials. Symmetric Functions, (a) A polynomial in (integral rational function of) the quantities xlf x2, . . . , xn is a
sum involving a finite number of terms of the form axiklx2k2 • • • xnkn, where each kj is a nonnegative integer. The largest value of ki + k2 + • ' ' + kn occurring in any term is the degree of the polynomial. A polynomial is homogeneous if and only if all its terms are of the same degree (see also Sec. 4.5-5).
(b) A polynomial in xlt x2, . . . , xn (and more generally, any function of xu X2, . . . , xn) is (completely) symmetric if and only if its value is unchanged by permutations of the x1} x2, . . . , xn for any set of values xlf x2, . . . , xn. The elementary symmetric functions Slt S2, . . . , Sn of xu x2, . . . , xn are the polynomials
& = Xi -j- X2 + • • • + Xn
S2 ss XXX2 + XxXz + • • •
SS ss X1X2X3 + 2:1X2X4 +
Sn S3 XlX2 • • • Xn (1.4-9)
n\
where 5* is the sum of all (n _ k^k^ products combining kfactors Xj without repeti tion ofsubscripts (see also Table C-2). Every polynomial symmetric inxlt x2} . . . , xn can be rewritten as a unique polynomial in Slf S2, . . . , Sn; the coefficients in the new polynomial are algebraic sums of integral multiples of the given coefficients.
Every polynomial symmetric in xlf x2, . . . , xn can also be expressed asa polynomial
in a finite numberof the symmetricfunctions n
^ Xi ^l
n
82 s= 2J Xi*
n
' ' ' S* =Y Xik
*^i
(1.4-10)
ffi
The symmetric functions (9) and (10) are related by Newton's formulas
(-l)"kSk + (-1)*-**-*! + (-l)*-%_2s2 + ...» 0
(jfc = 1, 2, • • •) (1.4-11)
where one defines Sk = 0for k > nand k < 0, and So =1(see also Ref. 1.2 for explicit tabulations of Newton's formulas; and see also Sec. 1.6-4). Note that the relations (11) do not involve n explicitly.
12
ELEMENTARY ALGEBRA
1.5-1
1.5. DETERMINANTS
1.5-1. Definition.
The determinant
D — det [aik] =
an
#12
din
a2\
a22
a2n
an\
an2
(1.5-1)
•
of the square array (matrix, Sec. 13.2-1) of n2 (real or complex) numbers (elements) aik is the sum of the n\ terms (—l)raiJfcla2fc8 • • • ankn each corresponding to one of the n\ different ordered sets fci, k2, . . . , kn obtained by r interchanges of elements from the set 1, 2, . . . , n. The number n is the order of the determinant (1). The actual computation of a determinant in terms of its elements is simplified by the use of Sees. 1.5-2 and 1.5-5a.
Note that n
n
|£|2 < 11 / I0**'2 (Hadamard's INEQUALITY) (1.5-•2) = 1 *=1
1.5-2. Minors and Cofactors.
Expansion in Terms of Cofactors.
The (complementary) minor Dik of the element aik in the nth-order determinant (1) is the (n — l)8t-order determinant obtained from (1) on erasing the ith row and the kth column. The cofactor A** of the element ane is the coefficient of aik in the expansion of Z), or
A,, = (-l)-A* = ^
(1.5-3)
A determinant D may be represented in terms of the elements and cofac tors of any one row or column as follows: n
D = det[o,*] == y
n
aijkij = Y ajkAjk
i-1
0" = 1, 2, . . . , n)
(1-5-4)
A=i
(simple Laplace development)
Note also that n
n
y atjAih = T OjtAu = 0 i^\
i = l
k- 1
(i * h)
(1.5-5)
13
MISCELLANEOUS THEOREMS
1.5-5
1.5-3. Examples: Second- and Third-order Determinants. On
I O21 On
012
0i«
(1.5-6)
flllfl&22 — 0>2lQ>l2
0221
Ois
O21
O22
O28
O31
CI32
O33
— a\\a22azz — 011*123032 + 012023031 — #13022031 H~ 018021032 — 012021033
= Oll(022033 —O32O28) —02i(Oi2088 —0820i3) + 03l(Oi2028 ~ O22O13) = Oii(0220»8 —08202s) —Oi2(02l088 —08i028) + Oi8(02l082 —081022) etc.
1.5-4. Complementary Minors.
Laplace Development.
(1.5-7)
The mth-order deter
minant M obtained from the nth-order determinant D by deleting all rows except for the m rows labeled iu i2) . . . , im (m < ri), and all columns except for the m columns labeled klt k2f . . . , km is an ra-rowed minor of D.
The m-rowed minor M
and the (n - w)-rowed minor M' of D obtained by deleting the rows and columns conserved in M are complementary minors; in the special case m = n, M' = 1. The algebraic complement M" of M is defined as ( —l)»'i+-*> &2S eQ.ual to the sum r=l
r=l
r=l
m
y Dr of mnth-order determinants Dr. The elements of each DT are identi cal with those of D, except for the elements of the jth row (or column, respec tively), which are cri, cf2, . . . , crn. EXAMPLE:
On + &11
O12 + &12
Osi
O22
Onl
a„2
• • • Oi» + bm 02n
On
Oi2
flln
6ll
612
6m
021
O22
02n
O21
O22
02n
0„i
0„2
Onl
On2
Onn
(1.6-9)
(e) A determinant is equal to zero if 1. All elements of any row or column are zero.
2. Corresponding elements of any two rows or columns are equal, or proportional with the same proportionality factor.
1.5-6. Multiplication of Determinants (see also Sec. 13.2-2). The product of two nth-order determinants det [aik] and det [bik] is 71
«
det [aik] det [bik] =det [ Ya^J =det [ 2, ^'J y-i
y-i
n
n
- det [^ «fiu] =det [£ a*6,*] (1-5-10) 'y-i
y=i
1.5-7. Changing the Order of Determinants.
A given determinant may be
expressed in terms of a determinant of higher order as follows: On
O12
O21
O22
Onl
On2
*
•
dm
On
O12
*
'
a2n
021
022
'
•
o„„
Onl
On2
0
0
*
OmCKi
•
'
02n 1; see also Sec. 7.6-1) if and only if, for x = xh f(x)/(x - xi)m~l = 0 and f(x)/(x - xt)m ?* 0. A complete solution of Eq. (1) specifies all roots together with their orders. Solutions may be verified by substitution.
1.6-3. Algebraic Equations. An equation (1) of the form
f(x) = aox» + aix*-1 + • • • + an-ix + an = 0 (a„ ^ 0) (1.6-2) where the coefficients a» are real or complex numbers, is called an algebraic equation of degree n in the unknown x. f(x) is a poly nomial of degree n in x (rational integral function; see also Sees.
4.2-2d and 7.6-5). an is the absolute term of the polynomial (2). An algebraic equation ofdegree n has exactly n roots if a root oforder m is counted as m roots (Fundamental Theorem of Algebra). Numbers expressible as roots of algebraic equations with real integral coefficients are algebraic numbers (in general complex, with rational and/or irrational real and
imaginary parts); if the coefficients are algebraic, the roots are still algebraic (see also Sec. 1.1-2). General formulas for the roots of algebraic equations in terms of the coefficients and involving only a finite number of additions, subtractions, multi-
ELEMENTARY ALGEBRA
1.6-4
16
plications, divisions, and root extractions exist only for equations of degree one (linear equations, Sec. 1.8-1), two (quadratic equations, Sec. 1.8-2), three (cubic equations, Sees. 1.8-3 and 1.8-4), and four (quartic equations, Sees. 1.8-5 and 1.8-6).
1.6-4. Relations between Roots and Coefficients. The symmetric functions Sk
and sk (Sec. 1.4-3) of the roots xh x2) . . . , xn of an algebraicequation (2), are related to the coefficients a0, ait . . . , a„ as follows: (k = 0, 1, 2, • • • , n)
(1.6-3)
Go
kak + ajb_isi + ak-2s2 + • • • + Oo«* =0
(k = 1, 2, • • •)
(1.6-4)
where one defines ak = 0 for k > n and k < 0. The equations (1.6-4) are another version of Newton'sformulas (1.4-11). Note also
Ok
(-D*
a0
A;!
o
s,
1
S2
Si
2
0 • • •
Sz
82
Sl
3
8k
8k-i
a\ 2o2
ao ffli
0 Q>o
0 • • •
3fi&3
a2
Q>\
flo 0
kak
ak-i
«-K.)'
0 • Si
(k = 1, 2, • • • , n)
(1.6-5)
ay
1.6-5. Discriminant of an Algebraic Equation.
The discriminant A of an
algebraic equation (2) is the product of a02n~2 and the squares of all differences (xi —Xk)(i > k) between the roots Xi of the equation (a multiple root of order m is considered as m equal roots with different subscripts),
A = a02n~2 PI (Xi - x*)2 == a°2'
= a02n"2
1 1
Xi x2
Xi2 x2*
X\n'
1
Xn
Xn
Xnn
50
Si
s2
Sn_l
51
S2
S3
Sn
Sn-1
Sn
Sn+1
' ' *
xf
n(n-l)
= (-D
i2(/,f)
2
(1.6-6)
ao
S2n-2
where R(f, /') is the resultant (Sec. 1.7-3) offix) andits derivative (Sec. 4.5-1)/'(x). Ais a symmetric function of the roots xh x2, . . . , xn and vanishes if and only if f{x) has at least one multiple root [which is necessarily a common root off(x) and f'(x); see also Sec. 1.6-69]. The second determinant in Eq. (6) is called Vandermondef8 determinant.
1.6-6. Real Algebraic Equations and Their Roots. An algebraic equation (2) is called real if and only if all coefficients a* are real; the corresponding real polynomial f(x) is real for all real values of x. The following theorems are useful for determining the general location of roots (e.g., prior to numerical solution, Sec. 20.2-1; see also Sees. 9.4-4
17
REAL ALGEBRAIC EQUATIONS AND THEIR ROOTS
and 14.8-5).
1.6-6
In theorems (b) through (/), a root of order m is counted
as m roots.
(a) Complex Roots. Complex roots of real algebraic equations occur in pairs"of complex conjugates (Sec. 1.3-1). A real algebraic equation ofodd degree must have at least one real root.
(b) Routh-Hurwitz Criterion. The number of roots with positive real parts of a real algebraic equation (2) is equal to the number of sign changes (disregard vanishing terms) in either one of the sequences m rp
T2 T%
1 0, i l, TjT> 7jT> * ' *
(1.6-7)
To, Th TiT2, T2TZ,
or
where
T0 = a0 > 0
Tx = ax
T2 = a3
«1
T» =
a3 ah
ao
0
a2
ai
a4
a3
Tt =
a2
ai
ao
0
0
a3
a2
ai
ao
a5
at
a3
a2
a7
a6
a5
a\
(1.6-8)
Given ao > 0, all roots have negative real parts if and only if T0, Th T2, . . . , Tn are all positive. This is true if and only if all a{ and either
all even-numbered Tk or all odd-numbered Tk are positive (Lienard-Chipart Test).
alternative formulation.
All the roots of a real nth-degree equation
(2) have negative real parts if and only if this is true for the (n —l)Bt-degree equation
afcn-1 + a[xn~2 + a'2xn-* + a^n~4 + + a3xn~z + aan~A +
'
s aixn~l + a2xn~2 ao
azxn
-2
_
ai
ao
abxn ai
= 0
This theorem may be applied repeatedly and yields a simple recursion
scheme useful, for example, for stability investigations. The number of roots with negative real parts is precisely equal to the number of negative multipliers -atP/ai™ (j = 0,1,2, ... ,n - 1;a0(0) = a0 > 0, ai = ai) encountered in successive applicationsof the theorem.
The method becomes
more complicated if one of the a^ vanishes (see Ref. 1.6, which also indicates an extension to complex equations).
(c) Location of Real Roots: Descartes's Rule of Signs. The number of positive real roots of a real algebraic equation (2) either is equal to the number Na ofsign changes in the sequence a0, ah . . . , an ofcoeffi cients, where vanishing terms are disregarded, or it is less than Na by a positive even integer. Application of this theorem to f(-x) yields a similar theorem for negative real roots.
1.6-6
ELEMENTARY ALGEBRA
18
(d) Location of Real Roots: An Upper Bound (Sec. 4.3-3a) for the Real Roots. // the first k coefficients a0, ah . . . , ak-i in a real
algebraic equation (2) are nonnegative (ak is the first negative coefficient) then all real roots of Eq. (2) are smaller than 1 + y/q/a0, where q is the absolute value of the negative coefficient greatest in absolute value. Applica tion of this theorem to f(—x) may similarly yield a lower bound of the real roots.
(e) Location of Real Roots: Rolle's Theorem (see also Sec. 4.7-la). The derivative (Sec. 4.5-1) f(x) of a real polynomial f(x) has an odd number of real zeros between two consecutive real zeros off(x). f(x) = o hasno real root or one real rootbetween two consecutive real roots o, b of f{x) = 0 if f(a) t* 0 and f(b) j± 0 have equal or opposite signs, respectively. At most, one real root of f(x) = 0 is greater than the greatest root or smaller than the smallest root off'(x) = 0.
(f) Location of Real Roots: Sudan's Theorem. For any real algebraic equation (2), let N(x) be the number of signchanges in the sequence of derivatives (Sec. 4.5-1) f(x), f'(x), f"(x), . . . , fn)(x), if vanishing terms are disregarded. Then the number of real roots of Eq. (2) located between two real numbers a and b > a not themselves roots of Eq. (2) is
either N(a) - N(b), or it is less than N(a) - N(b) by a positive even integer. The number of real roots of Eq. (2) locatedbetween a and b is odd or even if f(a) and f(b) have opposite or equal signs, respectively.
(g) Location of Real Roots: Sturm's Method. Given a real algebraic equation (2) without multiple roots (Sec. 1.6-2), let N(x) be the number of sign changes (disregard vanishing terms) in the sequence of functions
/o = f(x) = go(x)fi(x) - f2(x)
fi = f'(x) = gi(x)f2(x) - U(x) h(x) = g*(x)U(x) - U(x)
• • • (1.6-9)
where for i > 1 eachfi(x) is (-1) times the remainder (Sec. 1.7-2) obtained on dividing fi-2(x) by fi-i(x); fn(x) j* 0 is a constant. Then the number of real roots of Eq. (2) located between two real numbers a and b > a not them selves roots of Eq. (2) is equal to N(a) —N(b). Sturm's method applies even if, for convenience in computation, a function /
|
(2x3 - 4x2 - 2x + 3) + (x - 2) = 2x2 - 2
(1)
w IT w
EXAMPLES X — >5
At *"-» + •••) * (aox« +ai*--» +ajx-"»+•••) =j**»- +(|j - Ao |±)•»—» + •••+nk of order mk are usually combined into x + dki c*i [(» T7Z
,
x + di
„ \2 +I co,2] .. 21 "T" c*2 [(* - a,)2 + co,2]2 - a,)2 ^ "*2
+ ' • • +ckmkI,„_\\2T' 2w *m* [(* - a,)2 + co,2]-*
d.7-5)
The coefficients ckj and dkj may be determined directly by method 2 above. If g(x) and/(x) are real polynomials (Sec. 1.6-6), all coefficients bkj, Ckj, dkj in the resulting partial-fraction expansion are real. Every rational function of x (Sec. 4.2-2c) can be represented as a sum of a
polynomial and a finite set of partial fractions (see also Sec. 7.6-8). Par tial-fraction expansions are important in connection with integration (Sec. 4.6-6c) and integral transforms (Sec. 8.4-5). 1.8. LINEAR, QUADRATIC, CUBIC, AND QUARTIC EQUATIONS
1.8-1. Solution of Linear Equations. The solution of the general equation of the first degree (linear equation) ax = b
or
is
ax - b = 0
(c^O)
x= h
(1.8-1)
(1.8-2)
a
1.8-2. Solution of Quadratic Equations. ax2 + bx + c = 0
The quadratic equation (a ?*Q)
(1.8-3)
has the roots
Xl'2 =
-b ± y/b2 - 4ac Ta
(1.8-4)
The roots Xi and x2 are real and different, real and equal, or complex
conjugates if the discriminant (Sec. 1.6-5) D = b2 - 4ac is, respectively, positive, zero, or negative. Note xi + x2 = —b/a, x\X2 = c/a.
23
CUBIC EQUATIONS: TRIGONOMETRIC SOLUTION
1.8-4
1.8-3. Cubic Equations: Cardan's Solution.
The cubic equation
x3 + ax2 + bx + c = 0
(1.8-5)
is transformed to the "reduced" form
y* + py + q= 0
p= ~j + b a
through the substitution x = y —a/3.
q= 2(%) - ^ + c (1.8-6) "(J)'-* The roots yh y2, y* of the
"reduced" cubic equation (6) are
a ^ t> 2/i = A+ B
2/2,3 =
A + B t .A - B
,g— ± *—g— ^3
with A=^j- | +Ve ^=^"| " VQ -G)'+(f)"
> (1-8-7)
Q
where the real values of the cube roots are used. The cubic equation has one real root and two conjugate complex roots, three real roots of which at least two are equal, or three different real roots, if Q is positive, zero, or negative, respectively. In the latter case ("irreducible" case), the method of Sec. 1.8-4a may be used. Note that the discriminants (Sec. 1.6-5) of Eq. (5) and Eq. (6) are both equal to - 108Q. 1.8-4. Cubic Equations: Trigonometric Solution,
(a) H Q < 0
("irreducible" case)
2/i = 2 V-p/3 cos (a/3)
with
2/2,3 = -2 V-p/3 cos (a/3 ± 60°) , + ) cos a = g 2V-(p/3)3
(1.8-8)
(b) If Q > 0, p > 0
2/i = -2 Vp/3 cot 2a
2/2,3 = \/p73 (cot 2a ± i \/3 cosec 2a)
with
\
1 (1.8-9a)
tan a =v^tan (0/2) (|a| , #2mn
K
y
dm\
am2
.
• •
Omn^
.
. •
(1.9-9)
(system matrix and augmented matrix) are of equalrank (Sec. 13.2-7). Otherwise the equations are inconsistent.
The unique solution of Sec. 1.9-2 applies if r = m = n. If both matrices (9) are of rank r < m, the equations (8) are linearly dependent (Sec. 1.9-3a); m — r equations can be expressed as linear combinations of the remaining r equations and are satisfied by their solution. The r independent equations determine r unknowns as linear functions of the remaining n — r unknowns, which are left arbitrary.
27
REFERENCES AND BIBLIOGRAPHY
1,10-2
1.9-5. Simultaneous Linear Equations: n Homogeneous Equa tions in n Unknowns. In particular, a system of n homogeneous linear equations in n unknowns, n
y OikXk = 0
(i = 1, 2, . . . , n)
(1.9-10)
A= l
has a solution different from the trivial solution X\ = x2 — • • • = xn = 0 if and only if D = det [a.-*] = 0 (see also Sec. 1.9-3a). In this case, there exist exactly n — r linearly independent solutions
*i(1), x2™, . . . , xn™; *i(2), z2(2), . . . , zn; . . . ; *!, . . . , xn{n~r), where r is the rank of the system matrix (Sec. 1.9-4). The most general solution is, then, n
— r
Xi = £ eta®
(i = 1, 2, . . . ,n)
(1.9-11)
j = l
where the Cj are arbitrary constants (see also Sec. 14.8-10). In the important special case where r = n — 1,
xi = cAki
x2 = cAk2
....
xn = cAfcn
(1.9-12)
is a solution for any arbitrary constant c, so that all ratios Xi/xk are uniquely determined; the solutions (12) obtained for different values of k are identi cal (see also Sec. 14.8-6). 1.10. RELATED TOPICS, REFERENCES, AND BIBLIOGRAPHY
1.10-1. Related Topics. The following topics related to the study of elementary algebra are treated in other chapters of this handbook: Quadratic and bilinear forms Abstract algebra Matrix algebra Functions of a complex variable Numerical solution of equations, numerical approximations
Chap. Chap. Chap. Chap. Chap.
13 12 13 7 20
1.10-2. References and Bibliography. 1.1. Aitken, A. C: Determinants and Matrices,8th ed., Interscience, New York, 1956. 1.2. Birkhoff, G., and S. MacLane: A Survey of Modern Algebra, 3d ed., Macmillan, New York, 1965.
1.3. Dickson, L. E.: New First Course in the Theory of Equations, Wiley, New York, 1939.
1.4. Kemeny, J. G., et al.: Introduction to Finite Mathematics, Prentice-Hall, Englewood Cliffs, N.J., 1957.
1.5. Landau, E.: The Foundations of Analysis, Chelsea, New York, 1948. 1.6. Middlemiss, R. R.: College Algebra, McGraw-Hill, New York, 1952. 1.7. Uspensky, J. V.: Theory of Equations, McGraw-Hill, New York, 1948.
1.10-2
ELEMENTARY ALGEBRA
28
Additional Background Material
1.8. Cohen, L. W., et al.: The Structure of the Real Number System, Van Nostrand, Princeton, N.J., 1963. 1.9. Feferman, S.: The Number Systems: Foundations of Algebra and Analysis, Addison-Wesley, Reading, Mass., 1964. 1.10. Landin, J., and N. T. Hamilton: Set Theory: The Structure of Arithmetic, Allyn and Bacon, Boston, 1961. 1.11. Struik, D. J.: A Concise History of Mathematics, 2d ed., Dover, New York, 1948. (See also Sees. 12.9-2 and 13.7-2.)
CHAPTE
r2
PLANE ANALYTIC GEOMETRY
2.1. Introduction and Basic
Concepts 2.1-1. Introduction
2.1-2. Cartesian Coordinate Systems 2.1-3. Right-handed Rectangular Car tesian Coordinate Systems 2.1-4. Basic Relations in Terms of Rec
tangular Cartesian Coordinates 2.1-5. Translation of the Coordinate Axes
2.1-6. Rotation
of
the
Coordinate
Axes
2.1-7. Simultaneous Translation and Rotation of Coordinate Axes 2.1-8. Polar Coordinates
2.1-9. Representation of Curves
2.4. Second-order Curves (Conic Sections)
2.4-1. General Second-degree Equation 2.4-2. Invariants
2.4-3. Classification of Conies
2.4-4. Similarity of Proper Conies 2.4-5. Characteristic Quadratic Form and Characteristic Equation 2.4-6. Diameters and Conic Sections
Centers
of
2.4-7. Principal Axes 2.4-8. Transformation of the Equa tion of a Conic to Standard or
Type Form 2.4-9. Definitions of Proper Conies in Terms of Loci
2.2. The Straight Line
2.2-1. The Equation of the Straight Line
2.2-2. Other Representations of Straight Lines 2.3. Relations Involving Points and Straight Lines
2.3-1. Points and Straight Lines 2.3-2. Two or More Straight Lines 2.3-3. Line Coordinates
2.4-10. Tangents Conic Poles
and
Sections.
Normals Polars
of and
2.4-11. Other Representations of Conies
2.5. Properties of Circles, Ellipses, Hyperbolas, and Parabolas 2.5-1. Special Formulas and Theorems Relating to Circles 2.5-2. Special Formulas and Theorems Relating to Ellipses and Hyperbolas 29
2.1-1
PLANE ANALYTIC GEOMETRY
2.5-3. Construction of Ellipses and Hyperbolas and Their Tangents
30
2.6-2. Examples of Transcendental Curves
and Normals
2.5-4. Construction of Parabolas and
Their Tangents and Normals 2.6. Higher Plane Curves
2.6-1. Examples of Algebraic Curves
2.7. Related Topics, References, and Bibliography
2.7-1. Related Topics 2.7-2. References and Bibliography
2.1. INTRODUCTION AND BASIC CONCEPTS
2.1-1. Introduction (see also Sec. 12.1-1). A geometry is a mathemati cal model involving relations between objects referred to as points. Each geometry is defined by a self-consistent set of defining postulates; the latter may or may not be chosen so as to make the properties of the model correspond to physical space relationships. The study of such models is also called geometry. Analytic geometry represents each point by an ordered set of numbers (coordinates), so that relations between points are represented by relations between coordinates.
Chapters 2 (Plane Analytic Geometry) and 3 (Solid Analytic Geometry) introduce their subject matter in the manner of most elementary courses: the concepts of Euclidean geometry are assumed to be known and are simply translated into analytical language. A more flexible approach, involving actual construction of various geometries from postulates, is
briefly discussed in Chap. 17. The differential geometry of plane curves, including the definition of tangents, normals, and curvature, is outlined in Sees. 17.1-1 to 17.1-6.
2.1-2. Cartesian Coordinate Sys tems.
A
cartesian coordinate
system (cartesian reference system, see also Sec. 17.4-66) associates a unique ordered pair of real numbers (cartesian coordinates), the ab scissa x and the ordinate y, with
every point P = (x, y) in the finite portion of the Euclidean plane by reference to a pair of directed straight
HI
Fig. 2.1-1. Right-handed oblique car tesian coordinate system. The points marked "1" scales used.
define
the
coordinate
lines (coordinate axes) OX, OY intersecting at the origin 0 (Fig. 2.1-1). The parallel to OY through P intersects the x axis OX at the point P'. Similarly, the parallel to OX through P intersects the y axis OY at P".
RECTANGULAR CARTESIAN COORDINATES
31
2.1-4
The directed distances OP' = x (positive in the positive x axis direction) and OP" = y (positive in the positive y axis direction) are the cartesian coordinates of the point P = (x, y). x and y may or may not be measured with equal scales. In a general (oblique) cartesian coordinate system, the angle XOY = a> between the coordinate axes may be between 0 and 180 deg (right-handed car tesian coordinate systems) or between 0 and —180 deg (left-handed cartesian coordinate systems). A system of cartesian reference axes divides the plane into four quadrants (Fig. 2.1-1). The abscissa x is positive for points (x, y) in quadrants I and IV, negative for points in quadrants II and III, and zero for points on the y axis. The ordinate y is positive in quadrants I and II, negative in quadrants III and IV, and zero on the x axis. The origin is the point (0, 0). Note: Euclidean analytic geometry postulates a reciprocal one-to-one correspond ence between the points of a straight line and the real numbers (coordinate axiom, axiom of continuity, see also Sec. 4.3-1).
2.1-3. Right-handed Rectangular Cartesian Coordinate Systems. In a right-handed rectangular cartesian coordinate system, the directions of the coordinate axes are chosen so that a rotation of 90
deg in the positive (counterclock wise) sense would make the posi tive x axis OX coincide with the
positive y axis 0 Y (Fig. 2.1-2). The coordinates x and y are thus equal to the respective directed distances Fig. 2.1-2. Right-handed rectangular cartesian coordinate system and polarcoordinate system.
between the y axis and the point P, and between the x axis and the point P.
Throughout the remainder of this chapter, all cartesian coordi nates x, y refer to right-handed rectangular cartesian coordinate systems, and equal scale units of unit length are used to measure x and y, unless the contrary is specifically stated. 2.1-4. Basic Relations in Terms of Rectangular Cartesian Coordi
nates.
In terms of rectangular cartesian coordinates (x, y), the following
relations hold:
1. The distance d between the points Pi s= (xi, yx) andP2 = (x2, y2) is d -+
-*l)2 + (2/2 -2/i)2
(2.1-1)
2.1-5
PLANE ANALYTIC GEOMETRY
32
2. The oblique angle y between two directed straight-line segments P1P2 and P3P4 is given by COS 7
=
(x2 ~ Xi)(xA - x3) + (t/2 - 7/Q(;/4 - 7/3)
(2.1-2)
V(x2 - xtf + (ij2 - 7/1)2 V(*i - *,)' + (2/4 - y*)2
where the coordinates of the points Pi, P2, P3, P4 are denoted by the respective corresponding subscripts. The direction cosines cos ax and cos ay of a directed line segment F1P2 are the cosines of the angles ax and ay = 90 deg —ax, respectively. Xi — Xi
COS OLx =
V(x2 - si)2 + (2/2 - 2/O2 +
cos oty =
2/2 ~
sin a*
(2.1-3)
2/i
V(*2 - zi)2 + (2/2 - 2/i)2
3. The coordinates a;, 2/ of the point P dividing the directed line segment between the points Pi = (xh y{) and P2 = (x2, y2) in the ratioP\P\PP2 = m:n = ii'.l are x
=
mx2 + nzi
xi + \xx2
m + n
l+n
= m?/2 + nyx = 2/1 + ny2 m + n
1 + m
(— 00 < ix < 00)
(2.1-4)
Specifically, the coordinates of the mid-point of P1P2 are X\ + x2 x
=
V =
2/i + 2/2
(2.1-5)
4. The area S of the triangle with the vertices Pi = (xlf 2/1), P2 S3 (£2, 2/2), P3 S5 (X3, 2/3) iS #1
S = M #2 xz
2/1 2/2 2/3
1 1 1
= }4[x\(y2 - 2/3) + #2(2/3 - 2/1) + #3(2/1 - 2/2)] (2.1-6)
This expression is positive if the circumference P1P2P3 runs around the inside of the triangle in a positive (counterclockwise) direction. Specifically, if #3 = 2/3 = 0, s = y2
X\
x2
2/i
2/2
= M(#i2/2 - #22/1)
2.1-5. Translation of the Coordinate Axes.
(2.1-7)
Let x, y be the coordi
nates of any point P with respect to a right-handed rectangular cartesian
ROTATION OF COORDINATE AXES
33
2.1-7
reference system. Let x, y be the coordinates of the same point P with respect to a second right-handed rectangular cartesian reference system whose axes have the same directions as those of the x, y system, and whose origin has the coordinates x = Xo and y = 2/0 in the x, y system. If equal scales are used to measure the coordinates in both systems, the coordi nates x, y are related to the coordinates x3 y by the transformation equa tions (Fig. 2.1-3a; see also Chap. 14) x = x -
xQ
x = x + xo
or
y =
y -
yQ
y = 2/ + 2/o
(2.1-8)
The equations (8) permit a second interpretation. If x, y are considered as coor dinates referred to the x, y system of axes, then the point defined by x, y is translated by a directed amount —x0 in the x axis direction and by a directed amount —yo in the y axis direction with respect to the point (x, y). Transformations of this type applied to each point x, y of a plane curve may be used to indicate the translation of the entire curve.
2.1-6. Rotation of the Coordinate Axes. Let x, y be the coordinates
of any point P with respect to a right-handed rectangular cartesian reference system. Let x, y be the coordinates of the same point P with respect to a second right-handed rectangular cartesian reference system having the same origin 0 and rotated with respect to the x, y system so that the angle XOX between the x axis OX and x axis OX is equal to #
measured in radians in the positive (counterclockwise) sense (Fig. 2.1-36). If equal scales are used to measure all four coordinates x, y, x, y, the coordinates x, y are related to the coordinates x, y by the trans formation equations x = x cos & + y sin # or x = x cos & — y sin #
y = —x sin & + y cos # 2/ = x sin # + 2/ cos &
(2.1-9)
A second interpretation of the transformation (9) is the definition of a point (x, y) rotated about the origin by an angle —t? with respect to the point (x, y). 2.1-7. Simultaneous Translation and Rotation of Coordinate Axes. If the origin of the x, y system in Sec. 2.1-6 is not the same as
the origin of the x, y system but has the coordinates x = x0 and y = 2/0 in the x, y system, the transformation equations become x = (x — xo) cos & + (2/ — 2/0) sin # y = - (x — xo) sin # + (y - y0) cos & or
x = xo + x cos # — 2/ sin # V = 2/o + x sin # + y cos #
(2.1-10)
2.1-7
PLANE ANALYTIC GEOMETRY
4y
34
Ay
hJP
-*>*
11
-•*
Fig. 2.1-3a. Translation of coordinate axes.
x sin #
Fig. 2.1-36. Rotation of coordinate axes.
*>x
35
POLAR COORDINATES
2.1-8
The relations (10) permit one to relate the coordinates of a point in any two right-handed rectangular cartesian reference systems if the same scales are used for all coordinate measurements.
The transformation (10) may also be considered as the definition of a point (£, y) translated and rotated with respect to the point (x, y). Note: The transformations (8), (9), and (10) do not affect the value of the distance (1) between two points or the value of the angle given by Eq. (2). The relations constituting Euclidean geometry are unaffected by (invariant with respect to) translations and rotations of the coordinate system (see also Sees. 12.1-5, 14.1-4, and 14.4-5).
2.1-8. Polar Coordinates.
A plane polar-coordinate system associ
ates ordered pairs of numbers r,
(b) Transformations a sinh t re =
—r
«
—z
0
=
cosh t — cos o-
a sinh r
cosh t
cosh',
(c) g
=> «>
Pole (to = 1) at z = 0 + t"6
Pole (to => 2) at z = 0
cos x sinh y
(branch
Pole (to = 2) at z = » Pole (to = 1) at z «= 0
sin x cosh 2/
Zero of order 1 at z = 0 point)
TO =
TO =• 2
1
sin z
/
00
00
TO =
to = 2
Isolated singularities Pole (to = 1) at z = «
Essential singularity at z =» »
2
2 a
2 =
00
to=1
ex sin y
1 \
/-*+v*»+»»y
(1/-6) a)2 + (1/ -
-2xy
2 n
2 = 0
2=0
Zeros (order to)
arg/(z) = arctan :
e* cos J/
yf*
/x + v*2 + A^ 1 \ 2 /
(x -a) (X - 0)2 + (1/ -
1
z - (a + ib) (a, 6 real)
(X* + 1/2)2
z*
1/2
x' + ya
z
x* -
X* + l/2
X
1
V
x» -
1
2xy
y
w(x, y)
z»
!/»
m(x, y)
Function /(z)
Note \f(z)\ = Vu2 + v2
}{z) = u(x, y) 4- iv(xt y) of a Complex Variable z = x + iy (see also Sees. 1.3-2 and 21.2-9 to 21.2-11)
Table 7.2-1. Real and Imaginary Parts, Zeros, and Singularities for a Number of Frequently Used Functions
•
r0 (Sec. 4.10-2a). 7.2-2. z Plane and w Plane. Neighborhoods. The Point at Infinity (see also Sec. 4.3-5). Values of the independent variable
z = x + iy are associated with unique points (x, y) of an Argand plane (Sec. 1.3-2), the z plane. Values of w = u + iv are similarly associated with points (u, v) of a w plane.
An (open) 5-neighborhood of the point z = a in the finite portion of the plane is defined as the set of points z such that \z —a\ < 8 for some 5 > 0.
The point at infinity (z = oo) is defined as the point z transformed into the origin (z = 0) by the transformation z = 1/z. A region contain ing the exterior of any circle is a neighborhood of the point z = oo. 7.2-3. Curves and Contours (see also Sees. 2.1-9 and 3.1-13). A con tinuous (arc or segment of a) curve in the z plane is a set of points z = x + iy such that
z = z(t)
or
x = x(t)
y = y(t)
(— oo < ti < t < t2 <
bk(z —a)~k for sufficiently large values of \z\ (see also Sec. 7.5-3).
7.3-4. Properties of Analytic Functions. Letf(z) be analytic through out an open region D. Then, throughout D, * The terms differentiable, analytic, regular, and holomorphic are used interchange ably by some authors.
193
BRANCH POINTS AND BRANCH CUTS
7.4-2
1. The Cauchy-Riemann equations (2) are satisfied (the converse is true). 2. u(x, y) and v(x, y) are conjugate harmonic functions (Sec. 15.6-8). 3. All derivatives of f(z) with respect to z exist and are analytic (see also Sec. 7.5-1).
If the open region D is simply connected (this applies, in particular, to the exterior of a bounded simply connected region),
4. The integral / /(f) df is independent of the path of integration, ca provided that C is a contour of finite length situated entirely in D; the integral is a single-valued analytic function of z, and its derivative is f(z) (see also Sec. 7.5-1).
5. The values of f(z) on a contour arc or a subregion in D define f(z) uniquely throughout D.
All ordinary differentiation and integration rules (Sees. 4.5-4 and 4.6-1) apply to analytic functions of a complex variable. If f(z) is analytic at
z —a andf(a) ?± 0, thenf(z) hasan analytic inverse function (Sec. 4.2-2a) at z = a. If W = F(w) and w = f(z) are analytic, then W is an analytic function of z. If a sequence (or an infinite series, Sec. 4.8-1) of functions fi(z) analytic throughout an open region D converges uniformly to the limit f(z) throughout D, then f(z) is analytic, and the sequence (or series) of the
derivatives f'{(z) converges uniformly to f(z) throughout D. The sequence
(or series) of contour integrals Jcfi(z) dz over any contour Coffinite length in D converges uniformly to l f(z) dz. 7.3-5. The Maximum-modulus Theorem. The absolute value \f(z)\ of a function f(z) analytic throughout a simply connected closed bounded
region D cannot have a maximum in the interior of D.
If \f(z)\ < M on
the boundary of D, then \f(z)\ < M throughout the interior of D unless f(z) is a constant (see also Sec. 15.6-4). 7.4. TREATMENT OF MULTIPLE-VALUED FUNCTIONS
7.4-1. Branches.
One extends the theory of analytic functions to
suitable multiple-valued functions by considering branches fi(z), f2(z), ... off(z) each defined as a single-valued continuous function through out its region of definition. Each branch assumes one set of the function values of f(z) (see also Sec. 7.8-1).
7.4-2. Branch Points and Branch Cuts,
(a) Given a number of
branches off(z) analytic throughout a neighborhood D of z = a, except possibly at z = a, the point z = a is a branch point involving the given
7.4-3
COMPLEX VARIABLES
194
branches* of f(z) if and only if f(z) passes from one of these branches to another as a variable point z in D describes a closed circuit about z = a. The order of the branch point is the number m of branches reached by f(z) before returning to the original or m + lBt branch as z describes successive closed circuits about z = a. If f(z) is defined at a branch point z = a the function value/(o) is common to all branches "joining"
at z = a (EXAMPLE: -\/z has a branch point of order 2 at z = 0). The point z = oo is a branch point of f(z) if and only if the origin is a branch point oif(l/z) (see also Sec. 7.6-3). Given a function w = f(z) whose inverse function $(w) exists and is single-valued throughout a neighborhood of w —f(a), the point z = a j* » is a branch point of order m of f(z) whenever &(w) has a zero of order m (Sec. 7.6-1) or a pole of order m + 2
(Sec. 7.6-2) at w = f(a).
(EXAMPLES: w = y/z and w = 1/y/z, a = 0).
Similarly, if $(w) exists and is single-valued throughout a neighborhood of w = /(©o),
the point z — oo is a branch point of order m of f(z) if &(w) has a zero of order m + 2
or a pole of order matw = /(-x
>
X
«;=v/l-z2
w-Jz~
w=\ogez=J -p-
w = arc sin z = /
y
.
y
z plane
z plane of
-e
*~x
->-*
-l o-*
=>/S^T
u;=\/z2+l w
=arctan*=o/ ^
Fig. 7.4-1. Branch points and branch cuts for some elementary functions.
COMPLEX VARIABLES
7.5-1
196
restriction to a single branch. The construction of Riemann surfaces for arbitrary functions may require considerable ingenuity ,(Refs. 7.8 and 7.15).
Throughout this handbook, statements about analytic func tions refer to single-valued analytic functions or to single-valued branches of monogenic analytic functions unless specific refer ence to multiple-valued functions is made. 7.5. INTEGRAL THEOREMS AND SERIES EXPANSIONS
7.5-1. Integral Theorems. Let zbe a point inside the boundary contour C of a region D throughout which f(z) is analytic, and let f(z) be analytic on C.
Then
9 /(f) d$ = 0 (Cauchy-Goursat integral theorem)
(7.5-1)
f(z) = cT~-
\r - a\> r2) with
1 Qk =
2-m
i
(7.5-7)
/(f) dt
Id,*-- ay-m di The first term ofEq. (7) is analytic and converges uniformly for \z —a\ < ri; the second term [principal part off(z)\ is analytic and converges uniformly for \z —a\> r2. Note: The case a — « is treated by using the transformation 2 = 1/z, which transforms z = oo into the origin.
(b) If the first term in Eq. (7) is terminated with a„_i(z —a)n_1, the remainder Rn(z) is given by
ff M - (* - «)w ^ with
/(f) #
1l&wi' , there exists an integral function f(z) whose only zeros are zeros of given orders mk at the points z = Zk. Let z0 = 0, zk 9* 0 (k > 0); if there is no zero for z = 0, then ra0 = 0. Then f(z) can be represented in the form
m- ~n {(. - a -pii+\ a)'+• ••+san r -«™ k = i
where g(z) is an arbitrary integral function, and the r* are (finite) integers chosen so as to make the product converge uniformly throughout every bounded region (Theorem of Weierslrass; see Sec. 21.2-13 for examples of product expansions).
201
ZERO AND POLES OF MEROMORPHIC FUNCTIONS
7.6-9
7.6-7. Meromorphic Functions. f(z) is meromorphic throughout a region D if and only if its only singularities throughout D are poles. The number of such poles in any finite region D is necessarily finite. Many authors alternatively define a function as meromorphic if and only if its only singularities throughout the finite portion of the plane are poles. Every function meromorphic throughout the finite portion of the plane can be expressed as the quotient of two integral functions without common zeros, and thus as the quotient of two products of the type discussed in Sec. 7.6-6. A function meromorphic throughout the entire plane is a rational algebraic function expressible as the quotient of two polynomials (see also Sec. 4.2-2c).
7.6-8. Partial-fraction Expansion of Meromorphic Functions (see also Sec.
1.7-4). Let f(z) be any function meromorphic in the finite portion of the plane and mk
having poles with given principal parts (Sec. 7.5-3) N bkj(z —zk)~J' at the points y=i
z = Zkof a given finite or infinite set without limit points in the finite portion of the plane. Then it is possible to find polynomials pi(z), p2(z), . . . and an integral function g(z) such that mk
f(z) =£ [£ hki(>z ~Zk)~j +P*W] +9(e) k
(7.6-3)
j= 1
and the series converges uniformly in every bounded region where f(z) is analytic (MittagLeffler's Theorem).
7.6-9. Zeros and Poles of Meromorphic Functions. Let f(z) be meromorphic throughout the bounded region inside and continuous on a
closed contour C on which f(z) ?* 0. Let N be the number of zeros and P
the number of poles off(z) inside C, respectively, where a zero or pole of order m is counted m times.
Then
(7.6-4)
For P = 0, Eq. (4) reduces to the principle of the argument
N=g
(7.6-5)
where Ac# is the variation of the argument # off(z) aroundthe contourC. Equation (5) means that w = f(z) maps a moving point z describing the contour C once into a moving point wwhich encircles the wplane origin N = 0, 1, 2, . . . times
iff(z) has, respectively, 0, 1, 2, . . . zeros inside the contour Cinthe zplane. Equa tions (4) and (5) yield important criteria for locating zeros and poles off(z), such as the famous Nyquist criterion (Ref. 7.6).
7.7-1
COMPLEX VARIABLES
202
7.7. RESIDUES AND CONTOUR INTEGRATION
7.7-1. Residues.
Given a point z = a where f(z) is either analytic or
has an isolated singularity, the residue Res/ (a) of f(z) at z = a is the coefficient of (z —a)-1 in the Laurent expansion (7.5-7), or
Res/ (a) =A, ^/({•)#
(7.7-la)
where C is any contour enclosing z = a but no singularities of /(z) other than z — a.
The residue Res/ (oo) of f(z) at z = «> is defined as
Res/(oo)=^^/(f)#
(7.7-16)
where the integral is taken in the negative sense around any contour
enclosing all singularities off(z) in the finite portion of the plane. Note that
Res, (oo) = lim [-*/(*)]
(7.7-2)
if the limit exists.
If f(z) is either analytic or has a removable singularity at z = a ^ oo,
then Res/ (a) = 0 [see also Eq. (7.5-1)]. If z = a ?* «> is a pole of order m, then
K-'W-^hn^c—w)].-.
(7-7_3)
In particular, let z = a ^ oo be a simple pole off(z) = p(z)/q(z), where p(z) and q(z) are analytic at z = a, and p(a) ^ 0. Then
Res/ (a) =|M>
(7.7-4)
7.7-2. The Residue Theorem (see also Sec. 7.5-1). For every simple closed contour C enclosing at most a finite number of (necessarily isolated) singularities zh z2, . . . , zn of a single-valued function f(z) continuous on C, n
—. $ /(f) # = / Res/ (2*)
(residue theorem)
(7.7-5)
&=i
One of the zk may be the point at infinity. Note carefully that the contour
Cmust not pass through any branch cut (see also Sec. 7.4-2).
203
EVALUATION OF DEFINITE INTEGRALS
7.7-3. Evaluation of Definite Integrals,
7.7-3
(a) One can often evaluate
a real definite integral J^ f(x) dx as a portion of a complex contour
integral oo :
LR^l Is fW # =° whenever the integral exists for all finite values of R, and zf(z) tends uniformly to zero as \z\ —> oo with y > 0
2. Jordan's Lemma: if F(z) is analytic in the upper half plane, except possibly for a finite number of poles, and tends uniformly to zero as \z\ -> oo with y > 0, then for every real number m
lim jfs F(£)e*»t d{ = 0
#_♦„
The contour-integration method may yield the Cauchy principal value (Sec. 4.6-26)
of /_„/(*) dxeven #the integral itselfdoes not exist. Jordan's lemma is particularly useful for the computation of improper integrals of the form f °° F(x)eim* dx and,
because of Eq. (21.2-28), integrals of the form f^ F(x) cos (mx) dx and /_°° F(x) sin (mx) dx (inverse Laplace and Fourier transforms, Sees. 4.11-3 and 82-6) ~" to Iff *« any semicircular arc of the circle \z - a\ =eabout a simple pole z =a
°ff(z), then
r
lim L/(f)^f =^Res/ (a) This fact is used (1) to evaluate integrals over contours "indented" around simple polesand (2) for computing the Cauchy principal values of certain improper integrals, (d) One may apply the residue theorem (5) to integrals of the type
JQ $(cos 0 conformally onto the interior of a polygon in the w plane; the polygon corresponds to the x axis, the vertices Wi, w2, . . . , wn correspond to different points xh x2, . . . , xn on the x axis, and the exterior angle of the polygon at the vertex w3- equals aj (j = 1, 2, . . . , n). For any given polygon in the w plane, three of the x/s can be chosen arbitrarily; the other x/s and the parameters A and B are then uniquely determined.
If xn is chosen to be infinitely large Eq. (9) reduces to
(z - x[) "(z - x'2) - • • • (z - x'n_x)
with
V ay = 2w
- dz + Br (7.9-10)
y=i
where A' and B' are constant parameters, and x[, x2, . . . , x'n_! are new points on the x axis.
7.9-5
COMPLEX VARIABLES
210
Table 7.9-1. Properties of the Transformation 1
(see also Tables 7.9-2 and 7.10-1)
z = x + iy = |z|e»> (|z| > 0);
-*H)
w = w -f iv = re** (|w| > 0)
Point, curve(s), or region in Point, curve(s), or region in z plane w plane
Ellipses, foci at ± 1
Circles about 0
\z\ = e^ = constant ^ 1
u2
v2
cosh2 \p
+ -
sinh2 \p
=
1
Remarks
If
1 and
#
decreases
for \z\ < 1 Straight-line rays through 0
Hyperbolas, foci at ± 1
ip = constant j* 0
v2
cos2 ip
Unit semicircle \z\ = 1, y >0
Unit semicircle \z\ = 1,
sin2 tp
=
1
Straight-line segment (+1, -1)
If ip increases, u de
Straight-line segment
If ip increases, u in
2/
Fig. 10. w — sin z.
V
E F\
A' B'
Fig. 11. w - sin z; BCDiy = k, B'C'D' is \cosh f—^-rY + (-r-^-rY = 1k/ \smh A;/
Fig. 12. w = -—-. z Y 1
Fig. 13. t« =
*+*
7.9-5
7.9-5
COMPLEX VARIABLES
214
Table 7.9-2. Table of Transformations of Regions (Continued) v\
y\ \B
z - a
; a —
1 + xix2 + V(l - xi2)(l - x22)
az — 1
Rq ss
Xi -f- X2
1 -a?isi +V(l -Si2)U -x7) (a > 1 and #0 > 1 when —1 < x2 < Xi < 1). Xi — X2
•>*-*
„
1K
z-a
1 + xix2: + V(si2 - 1)W - 1)., ;
rIG. 15. 10 = • -J ffl = « az — 1
xi + X2
:g'g' - X- V(j'a ~ 1)(aa* ~ 1} fe o
Right half plane x > 0
Transformation
x z — a
w = e%\
z
.v z
w = e6A
Remarks
(x real)
— a*
—
z =s a is trans
a
z + a*
(X real)
formed into the
origin w = 0
Unit circle
I*I 0 where f(t) is continuous (8.2-46) The path of integration in Eq. (4) lies to the right of all singularities of F(s).
inversion integral//(t) reduces to 77—. /
4TTl Jcri —ioo
The
F(s)eatds if the integral exists: other-
wise, fi(t) is a Cauchy principal value (Sec. 4.6-2&).
8.2-7. Existence of the Inverse Laplace Transform. Note carefully that the existence of the limit (4) for a given function F(s) does not in itself imply that F(s) has an inverse Laplace transform [EXAMPLE: F(s) — e*2]. The existence of JB'^M] should be checked [e.g., by Eq. (1)] for every application of the inversion theorem. The following theorems state sufficient (not necessary) conditions for the existenceof£-l[F(s)).
1. If F(s) is analytic for a > 0)
(8.3-1)
Note: The integral on the right is an integral function (Sec. 7.6-5), so that £[f(t)] has no singularities for finite 5 except possibly for simple poles on the imaginary axis. /T/2
|/(«)| dt exists, then
/ (< +tJ) s -/» and £[/(*)] =x+\.Tal2 f0T/2f(t)e-« dt
(«• >0) (8.3-2)
(c) // 0 except for t = t\, h, . . . where f(t) has unilateral limits
(b> 0)
(a > 0)
-/**/i
/i*/^/0B/i(»-r)dr
f(t - b)
f(at)
f^f(r)dr+C
f'(t) /
-tf(t) (-WW
/ /«, a) da Jai
fat
da
lim /(*, a)
Object function
/
F'(s) FM(s)
F(8, a) da
F(s - a)
f"F(s)ds
J ax
Cat
da
ct-*a
lim F(s, a)
Result function
of £[fi * /«].
See also Sec. 8.3-3.
* The abscissa of absolute convergence for £[/(r)(*)] is 0 or
W
»
I
ft
El
8.3-3
THE LAPLACE TRANSFORMATION
8.3-3. Transform of a Product (Convolution Theorem).
Fi(s) m £[/i(0] (a > ax)
228
Given
F2(s) = £[/,(*)] ( (s - s2)mj • • • (s - sn)m% n
(8.4-2)
ink
£->[F(s)] - jB-i [§^] =V VH^** (t >0) with th
H = 1 dh* Us-s^^s)] *' (J - 1) Km - j)!d*-* L W) _L* (see also Table D-6).
(b) It is sometimes convenient to obtain an inverse Laplace transform involving a multipleroot of D(s) = 0 directly by a limiting process leading to the coincidence of distinct simple roots (see also Table 8.3-1,7). EXAMPLE:
and, more generally,
(c) Complex Roots. In Eq. (2), pairs of terms corresponding to complex-con jugate roots 8 = a ± iui maybe combined as follows (see also Sec. 8.4-5):
8.4-5
THE LAPLACE TRANSFORMATION
230
(A + iB)*'e« + (A - iB)t'e <xa; the series } fk(t) converges absolutely to a function f(t) for almost all t (Sec. 4.6-146), and 00
00
00
w)]=£ [ y /*»] = y £[/*(oi = y nw=*•« A= 0
Aj= 0
^ >*.>
* = 0
(8.4-4)
8.4-7. Expansions in Terms of Powers of t. Series expansions of F(s) in descending powers of s,
*•(»> = 7 + if + *• '
(W > r > 0)
(8.4-5)
may frequently be obtained as Laurent expansions (Sec. 7.5-3) about s = 0 or, in the case of rational algebraic functions of the type specified in Sec. 8.4-4, simply by long division (Sec. 1.7-2). If the conditions of Sec. 8.4-6 are satisfied, then, for almost all t in (0, h),
f(t) =eC-TC>] =6! +b2t +^t* +•••+^A_.f**-i+ . . . (h> t> 0)
(8.4-6)
which may furnish (at least) useful approximations to JC_1[y(«)] for t < ti. If 00
F(s) - £ J % [Re («) >0]
(8.4-7)
Aj= 0
converges, then 00
£-'[F(8)) - °
(t >0)
(8.4-8)
A=»0
In particular, if the series on the left converges, 00
£_1 [Aklo°i\ -da[c° +r(2° +(iM(20° +•••] ( 8.4-8. Expansions in Terms of Laguerre Polynomials of t. Every function F(s) = £[/(*)]
( ca)
analytic at s = oo may be expanded into an absolutelyconvergentseriesin powersof
8.4-9
THE LAPLACE TRANSFORMATION
for a > aa, corresponding to a Taylor series convergent for \z\ < 1. for (T0 = 0, F(s) may be expressed in the form
232
In particular,
fW =a-.) I c*>=^ l »(i^y („ >..) ft = 0
fc=>0
(8.4-11)
k
with
ct =2(5)j,' a > 0, and
form, with
Hankel trans
Finite annular
(Hankel series)
transform
Finite Hankel
(Fourier sine series, Sec. 4.11-26)
transform
Finite sine
(Fourier cosine series, Sec. 4.11-26)
transform
Finite cosine
1:
$ix)xBm(\kx) dx
/: $ix)xJmi\kX) dx
/0°*(x:) sin \kX dx
/;
&ix) cos \kX dx
Bmi\nx)
X^JVta'Xt)
Jm2ia\k) -Jm2ia':X*)
2A
fk cos \kX
I_2 V,
00
Jmj\kX) [Jm'i*ka)]'
2Y
/* sin \kx
a2 Li
jfc=i
00
k=i
a Lj
a Jo
ra)
The
(8.7-6)
where z is a complex variable, and the series converges absolutely out side of a circle of absolute convergence of radius ra depending on the given sequence; analytic continuation in the manner of Sec. 8-2.3 can extend the definition. The corresponding inversion integral
fik =gU j)c Fz(z)z^ dz
(k =0, 1, 2, . . .)
(8.7-7)
where C is a closed contour enclosing all singularities of Fz(z), gives the inverse transform Z~1[Fz(z)] = fk for suitable fk (Ref. 8.11). Inversion can then utilize the residue calculus in the manner of Sec. 8.4-3, espe cially if Fz(z) is a rational function expandable in partial fractions. Inversion is even simpler if Fz(z) can be expanded directly in terms of
powers of 1/z. Note that the inverse transform must be unique wherever the series (6) converges absolutely (Sec. 4.10-2c). Table 8.7-2 summarizes the most important properties of z transforms.
Their application to the solution of difference equations and sampleddata systems is treated in Sec. 20.4-6, where the relation of z trans
forms to jump-function Laplace tranforms is also discussed. Table 20.4-1 lists a number of z-transform pairs. The z transform is related to the Mellin transform of Table 8.6-1.
Note the
analogy between power series and Mellin transforms and between Dirichlet series 00
/ 2/Afi"** and Laplace transforms. fc=o
8.8. RELATED TOPICS, REFERENCES, AND BIBLIOGRAPHY
8.8-1. Related Topics. The following topics related to the study of the Laplace transformation and other functional transformations are
241
RELATED TOPICS
8.8-1
Table 8.7-2. Corresponding Operations for z Transforms
The following theorems are valid whenever the ^-transform series in question converge absolutely; limits are assumed to exist (see also Table 8.3-1), and /_i = /_2 = • • • = g-i = 0_2 = • • • = 0 Theo
Operation
rem
1
Object sequence
Result function
otfk + Pgt
aFziz) + pGziz)
Linearity (a, 0 con stant)
fk+i = E/* 2
sequence
3
zFziz) - zfo r-l
Advanced object
(r = 1, 2, . . .) zrFziz) -
/*+r ^ E'/*
Delayed object
fk-i m E~%
sequence
) ftf-i
z-Wziz)
Finite differences 4a
forward difference
fk+i — fk = A/*
46
backward difference
/* - /*-i s Vfk
iz -
l)Fziz) - zf0
*—Fziz) z
k
5
Finite summation of
I"
object sequence
-^—Fziz)
2 — 1
00
6
7
Convolution of
/ /iff*-*
object sequences Corresponding limits (continuity theo rem; a is independ ent of k and z)
lim fkia) ot—*a
Fziz)Gziz)
lim Fziz,a) a—>a
Differentiation and
pendent of A' and z Initial and final*
values of object sequence
10
Differentiation of result function
^Fziz,a)
/ fkia) da J ax
faiFziz,a)da fax
da fa*
respect to a pa rameter a inde
9
-/*(«)
d
integration with 8
/o
lim Fziz)
lim fk
lim" {z- \)Fziz)
k—*»
kfk
k%
da
(r - 1, 2, . . .)
1iz — l)Fziz) is assumed to be analytic for \z\ > 1.
Z->1
d
-z-Fziz) dz
8.8-2
THE LAPLACE TRANSFORMATION
242
treated in other chapters of this handbook: Expansion in partial fractions Limits, integration, and improper integrals Convergent and asymptotic series Functions of a complex variable, contour integration
Chap. Chap. Chap. Chap.
1 4 4 7
Transformations Fourier transforms
Chaps. 12, 14, 15 Chap. 4
Applications of generating functions
Chap. 18, Appendix C
z transforms Tables of Fourier, Hankel, and Laplace-transform pairs
Chap. 20 Appendix D
Applications of the Laplace transformation are discussed in other chap ters of this handbook as follows: Ordinary differential equations Partial differential equations Integral equations
Chaps. 9, 13 Chap. 10 Chap. 15
8.8-2. References and Bibliography. 8.1. Campbell, G. A., and R. M. Foster: Fourier Integrals for Practical Applications, Van Nostrand, Princeton, N.J., 1958. 8.2. Churchill, R. V.: Operational Mathematics, 2d ed., McGraw-Hill, New York, 1958.
8.3. Ditkin, V. A., and A. P. Prudnikov: Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965. 8.4. Doetsch, G.: Anleitung zum Praktischen Gebranch der Laplace-Transformation, Oldenburg, Munich, 1956. 8.5. : Handbuch der Laplace-Transformation, 3 vols., Birkhauser, Basel, 1950. 8.6. and D. Voelker: Die Zweidimensionale Laplace-Transformation, Birk hauser, Basel, 1950.
8.7.
, H. Ejieiss, and D. Voelker: Tabellen zur Laplace-Transformation und Anleitung zum Gebrauch, Springer, Berlin, 1947. 8.8. Erdelyi, A., et al.: Tables of Integral Transforms (Bateman Project), 2 vols., McGraw-Hill, New York, 1954.
8.9. Jury, E. I.: Theory and Application of the z-transform Method, Wiley, New York, 1964.
8.10. McLachlan, N. W.: Modern Operational Calculus, Macmillan, New York, 1948. 8.11. Miles, J. W.: Integral Transforms, in E. F. Beckenbach, Modern Mathematics for the Engineer, 2d series, McGraw-Hill, New York, 1961. 8.12. Nixon, F. E.: Handbook of Laplace Transformation, 2d ed., Prentice-Hall, Englewood Cliffs, N.J., 1965.
8.13. Papoulis, A.: The Fourier Integral and Its Applications, McGraw-Hill, New York, 1962.
8.14. Scott, E. J.: Transform Calculus, Harper, New York, 1955. 8.15. Sneddon, I. N.: Fourier Transforms, McGraw-Hill, New York, 1951. 8.16. Tranter, C. J.: Integral Transforms in Mathematical Physics, 2d ed., Wiley, New York, 1956.
8.17. Van Der Pol, B., and H. Bremmer: Operational Calculus Based on the Two-sided Laplace Integral, Cambridge University Press, London, 1950. 8.18. Widder, D. V.: The Laplace Transform, Princeton University Press, Princeton, N.J., 1941.
CHAPTER 9
ORDINARY
DIFFERENTIAL EQUATIONS
9.1. Introduction
9.1-1. Survey 9.1-2. Ordinary Differential Equations 9.1-3. Systems of Differential Equa tions
9.1-4. Existence and Desirable Prop erties of Solutions 9.1-5. General Hints
9.2. First-order Equations 9.2-1. Existence and Uniqueness of Solutions
9.2-2. Geometrical Interpretation. Singular Integrals 9.2-3. Transformation of Variables
9.2-4. Solution of Special Types of First-order Equations 9.2-5. General Methods of Solution
(a) Picard's Method of Successive Approximations (6) Taylor-series Expansion 9.3. Linear Differential Equations
9.3-1. Linear Differential Equations. Superposition Theorems 9.3-2. Linear Independence and Fun damental Systems of Solutions 9.3-3. Solution by Variation of Con stants.
Green's Functions
9.3-4. Reduction of Two-point Boundary-value Problems to Initial-value Problems
9.3-5. Complex-variable Theory of Linear Differential Equations. Taylor-series Solution and Ef fects of Singularities 9.3-6. Solution of Homogeneous Equations by Series Expansion about a Regular Singular Point 9.3-7. Integral-transform Methods 9.3-8. Linear Second-order Equations 9.3-9. Gauss's Hypergeometric Dif ferential Equation and Riemann's Differential Equation 9.3-10. Confluent Hypergeometric Functions 9.3-11. Pochhammer's Notation
9.4. Linear
Differential
Equations
with Constant Coefficients
9.4-1. Homogeneous
Linear
Equa
tions with Constant Coefficients
9.4-2. Nonhomogeneous Equations. Normal Response, Steady-state Solution, and Transients 9.4-3. Superposition Integrals and Weighting Functions 9.4-4. Stability 243
ORDINARY DIFFERENTIAL EQUATIONS
9.1-1
9.4-5. The Laplace-transform Method of Solution
9.4-6. Periodic Forcing Functions and Solutions.
The Phasor
Method
(a) Sinusoidal
Forcing
and Solutions.
Functions
Sinusoidal
Steady-state Solutions (b) The Phasor Method (c) Rotating Phasors
and
Fre
quency-response Functions (a) Transfer Functions (6) Frequency-response Functions (c) Relations between Transfer Functions or Frequencyresponse Functions and Weight ing Functions 9.4-8. Normal Coordinates and Nor mal-mode Oscillations
(a) Free Oscillations (6) Forced Oscillations
9.5. Nonlinear Second-order Equa tions
9.5-1. Introduction
Solution
9.5-3. Critical Points and Limit Cycles (a) Ordinary and Critical Phaseplane Points (6) Periodic Solutions and Limit Cycles son's Theorems
Functions
Functions
9.5-2. The Phase-plane Representation. Graphical Method of
(c) Poincare^s Index and Bendix-
(d) More General Periodic Forcing 9.4-7. Transfer
244
9.5-4. Poincare-Lyapounov Theory of Stability 9.5-5. The Approximation Method of Krylov and Bogoliubov (a) The First Approximation (6) The Improved First Approxima tion
9.5-6. Energy-integral Solution
9.6. Pfaffian Differential Equations 9.6-1. Pfaffian Differential Equations 9.6-2. The Integrable Case 9.7. Related Topics, References, and Bibliography
9.7-1. Related Topics 9.7-2. References and Bibliography
9.1. INTRODUCTION
9.1-1. Survey. Differential equations are used to express relations between changes in physical quantities and are thus of great importance in many applications. Sections 9.1-2 to 9.3-10 present a straight forward classical introduction to ordinary differential equations, including some complex-variable theory.
Sections 9.4-1 to 9.4-8 introduce the linear
differential equations with constant coefficients used in the analysis of vibra tions, electric circuits, and control systems, with emphasis on solutions by Laplace-transform methods. Sections 9.5-1 to 9.5-6 deal with non linear second-order equations. Sections 9.6-1 and 9.6-2 introduce Pfaff ian differential equations, although these are not ordinary differential equations. Some naturally related material is treated in other chapters of this handbook, particularly in Chap. 8 and Sees. 13.6-1 to 13.6-7. Boundary-
245
ORDINARY DIFFERENTIAL EQUATIONS
9.1-2
value problems, eigenvalue problems, and orthogonal-function expansions of solutions are discussed in Chap. 15, and a number of differential equa tions defining special functions are treated in Chap. 21. The notation used in the various subdivisions of this chapter has been chosen so as to
simplify reference to standard textbooks in different special fields.
Thus the usually
real variables in Sees. 9.2-1 to 9.2-5 are denoted by x, y = yix); the frequently com
plex variables encountered in the general theory of linear ordinary differential equa tions (Sees. 9.3-1 to 9.3-10) are denoted by z, w = wiz). The variables in Sees. 9.4-1 to 9.5-6 usually represent physical time and various mechanical or electrical variables and are thus introduced as t, yk = ykit).
9.1-2. Ordinary Differential Equations. equation of order r is an equation
An ordinary differential
F[x, y(x), y'(x), y"(x), . . . , y^(x)] = 0
(9.1-1)
to be satisfied by the function y = y(x) together with its derivatives
y'(x), y"(x), . . . , y(r)(x) with respect to a single independent variable x. To solve (integrate) a given differential equation (1) means to find functions (solutions, integrals) y(x) which satisfy Eq. (1) for all values of x in a specified bounded or unbounded interval (a, b). Note that solutions can be checked by resubstitution. The complete primitive (complete integral, general solution) of an ordinary differential equation of order r has the form y = y(x, ft, ft, . . . , ft)
(9.1-2)
where ft, ft, . . . , ft are r arbitrary constants (constants of integra tion, see also Sec. 4.6-4). Each particular choice of these r constants yields a particular integral (2) of the given differential equation. Typical problems require one to find the particular integral (2) subject to r initial conditions
2/(zo) = 2/0
y'(xo) = y'0
y"(x0) = y'i
• • • y(r-»(X9) = ^(-d
(9.1-3)
which determine the r constants ft, ft, . . . , ft. Alternatively, one may be given r boundary conditions on y(x) and its derivatives for x = a and x = b (see also Sec. 9.3-4).* Many ordinary differential equations admit additional solutions known
as singular integrals which are not included in the complete primitive (2) (see also Sec. 9.2-26). A differential equation is homogeneous if and only if ay(x) is a solu tion for all a whenever y(x) is a solution (see also Sees. 9.1-5 and 9.3-4). * Strictly speaking, initial and boundary conditions refer to unilateral derivatives (Sec. 4.5-1).
9.1-3
ORDINARY DIFFERENTIAL EQUATIONS
246
Given an r-parameter family of suitably differentiate functions (2), one can elimi nate Ci, C2, . . . , Cr from the r -f 1 equations y™ = y^ix, Ch C2, . . . , Cr) (j = 0, 1, 2, . . . , r) to obtain an rth-order differential equation describing the family. Note: An ordinary differential equation is a special instance of a functional equation imposing conditions on the functional dependence y = yix) for a set of values of x. OTHER EXAMPLES OF FUNCTIONAL EQUATIONS: yix&2) = y(xx) + 2/(^2) [logarithmic property, satisfied by yix) = A logfl re], partial differential equations (Sec. 10.1-1), integral equations (Sec. 15.3-2), and difference equations (Sec. 20.4-3).
9.1-3. Systems of Differential Equations (see also Sees. 13.6-1 to 13.6-7). A system of ordinary differential equations
F
might be easy to solve for y' » y'{x) or y' = y'fo), respectively; substitution of this result intothe given Eq. (16) yields the desired relation of x and y. If the solution of
Eq. (17) takes the form uix, y') = 0 or uiy, y') * 0, the desired relation of x and y
is given in terms of a parameter p = y'.
EXAMPLES: Clairaut's differential equation y = y'x + fiy') yields the complete primitive y 0,
6=zr^o
* - - ^44
(9-2-226)
The procedureis repeated until (after A; steps) the right side of the differential equation is constant.
Similarly, for k < 0, the transformation x = £-i/(«+D, y =
• m 1 n yields
a differential equation of the form (22a) with ra + 1
5= - m—5_ + 1
9.2-5. General Methods of Solution,
ra = - ?^-±i ra + 1
(9.2-22c)
(a) Picard's Method of
Suecessive Approximations. To solve the differential equation y' = f(Xy y) for a given initial value y(x0) = y0, start with a trial solution y[0] (x) and compute successive approximations
yu+u(x) = 2/0 + fXf[x, yM(x)] dx J xo
(j = 0, 1, 2, . . .) (9.2-23)
to the desired solution y (x). The process converges subject to the conditions of Sec. 9.2-1. Picard's method is useful mainly if the integrals in Eq.
(23) can be evaluated in closed form, although numerical integration can, in principle, be used. A completely analogous procedure applies to systems (3) of first-order differential equations.
9.3-1
ORDINARY DIFFERENTIAL EQUATIONS
252
(b) Taylor-series Expansion (see also Sec. 4.10-4). If the given function f(x,y) is suitably differentiable, obtainthe coefficients y(m)(xo)/m\ of the Taylor series
y(x) =y(x0) +y'(x0)(x - x0) +^ y"(x*)(x - z0)2 + ••• (9.2-24) by successive differentiations of the given differential equation: y'(x) = f(x, y)
with x = xo, y = y(x0) = y0.
An analogous procedure applies to systems of first-order equations. 9.3. LINEAR DIFFERENTIAL EQUATIONS
9.3-1. Linear Differential Equations. Superposition Theorems (see also Sees. 10.4-2, 13.6-2, 13.6-3, 14.3-1, and 15.4-2). A linear ordi
nary differential equation of orderr relating the real or complex variables z and w = w(z) has the form
Lw = a0(z) -^r + at(z) -^ + • • • + ar(z)w =f(z)
(9.3-1)
where the ak(z) and f(z) are real or complex functions of z. The general solution (Sec. 9.1-2) of a linear differential equation (1) can be expressed as the sum of any particular integral and the general solution of the homo geneous linear differential equation (Sec. 9.1-2)
lw ma0(z) ^ +ai(«) ~ +' *•+ar(z)w =0 (9.3-2) For any given nonhomogeneous or "complete" linear differential equa tion (1), the homogeneous equation (2) is known as the complementary equation or reduced equation, and its general solution as the com plementary function.
Let Wi(z) and W2(z) be particular integrals of the linear differential equa tion (1) for the respective "forcing functions" f(z) = fi(z) and f(z) s f2(z). Then awi(z) + Pw2(z) is a particular integral for the forcing function f(z) = afi(z) + 0/2(2) (Superposition Principle). In particular, every linear combination of solutions of a homogeneous linear differential equation (2) is also a solution.
253
SOLUTION BY VARIATION OF CONSTANTS
9.3-3
The superposition theorems often represent some physical superposition principle. Mathematically, they permit one to construct solutions of Eq. (1) or (2) subject to given initial or boundary conditions by linear superposition.
Analogous theorems apply to systems of linear differential equations (see also Sec. 9.4-2).
9.3-2. Linear Independence and Fundamental Systems of Solu tions (see also Sees. 1.9-3, 14.2-3, and 15.2-la). (a) Let wx(z), w2(z), . . . , wr(z) be r — 1 times continuously differentiable solutions of a homogeneous linear differential equation (2) with continuous coefficients in a domain D of values of z.
The r solutions wk(z) are linearly inde-
r
pendent in D if and only if Y \kwk(z) = 0 inD implies Xi = X2 = * • • Jk = l
= \r = 0 (Sec. 1.9-3). This is true if and only if the Wronskian deter minant (Wronskian)
W[wh w2, . . . , wr] s
Wi(z)
W2(Z)
' ' •
WT(Z)
w[(z)
w2(z)
•••
wr(z)
w^-»(z)
w2^l\z)
differs from zero throughout D.
(9.3-3)
• • • w*-»(z)
W = 0 for any z in D implies W == 0 for
all z in D*
(b) A homogeneous linear differential equation (2) of order r has at most r linearly independent solutions, r linearly independent solutions Wi(z), w2(z), . . . , wk(z) constitute a fundamental system of solutions r
whose linear combinations ) akwk(z) include all particular integrals of Eq. (2). (c) Use of Known Solutions to Reduce the Order. If ra < r linearly inde pendent solutions wiiz), 102(3), . . . , wmiz) of the homogeneous equation (2) are known, then the transformation w = W[wi, wt, . . . , wm, w]
C'2(X) =HZL
Wl{x)
,_
a0(x) wiix)w2ix) —w2ix)w[ix)
\
aQix) wiix)w'2ix) —Wiix)w[ix)
G( .,
' }"
L_U>1JX)W2JZ) - W2ix)Wii^)
acta «i(e»;(© - «,(©»;«) ^
__
?;
(9.3.9)
255
THEORY OF LINEAR DIFFERENTIAL EQUATIONS
9.3-5
(c) While the general solution (7) obtained with the aid of the par ticular Green's function (8) is only another way of writing Eq. (4),
it is often possible to construct a Greenes function G(x, £) such that the par ticular integral (6) satisfies the specific initial or boundary conditions of a given problem. Assuming boundary conditions linear and homogeneous in w(x) and its derivatives, the required Green's function G(x, £) must satisfy the given boundary conditions and
LG(x, 0 =0 {x*&
f* LG(x, &d£ =1 (9.3-10)
for x in (a, b), with dr~2G/dxr-2 continuous in (a, b), and
^1
_£W]
1
(a
