The Mathematica Guidebook: Programming

Contents PREFACE to The Mathematica GuideBooks CHAPTER 0 Introduction and Orientation 0.1 Overview 0.1.1 0.1.2 0.1.3 ...

Author: Michael Trott

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Contents PREFACE

to The Mathematica GuideBooks CHAPTER 0

Introduction and Orientation 0.1

Overview 0.1.1 0.1.2 0.1.3 0.1.4

0.2

Requirements 0.2.1 0.2.2

0.3

Exercises Solutions

The Books Versus the Electronic Components 0.5.1 0.5.2 0.5.3

0.6

Doing Computer Mathematics Programming Paradigms

Exercises and Solutions 0.4.1 0.4.2

0.5

Hardware and Software Reader Prerequisites

What the GuideBooks Are and What They Are Not 0.3.1 0.3.2

0.4

Content Summaries Relation of the Four Volumes Chapter Structure Code Presentation Style

Working with the Notebook Reproducibility of the Results Earlier Versions of the Notebooks

Style and Design Elements 0.6.1 0.6.2 0.6.3 0.6.4 0.6.5 0.6.6 0.6.7

Text and Code Formatting References Variable Scoping, Input Numbering, and Warning Messages Graphics Notations and Symbols Units Cover Graphics

0.7

Production History

0.8

Four General Suggestions

© 2004, 2005 Springer Science+Business Media, Inc.

Printed from THE MATHEMATICA GUIDEBOOKS

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CHAPTER 1

Introduction to Mathematica 1.0

Remarks

1.1

Basics of Mathematica as a Programming Language 1.1.1

General Background In and Out Numbering † General Naming, Spelling, and Capitalization Conventions for Symbols † Options and Option Settings † Messages † Add-On Packages

1.1.2

Elementary Syntax Common Shortcuts † Parentheses, Braces, and Brackets † Comments Inside Code † Font Usage † Referring to Outputs † Functional Programming Style † “Ideal” Formatting

1.2

Introductory Examples 1.2.0 1.2.1

Remarks Numerical Computations Periodic Continued Fractions † Pisot Numbers † Fast Integer Arithmetic † Digit Sums † Numerical Integration † Numerical ODE Solving † Burridge–Knopoff Earthquake Model † Trajectories in a Random Two-Dimensional Potential † Numerical PDE Solving † Benney PDE † Sierpinski Triangle-Generating PDE † Monitoring Numerical Algorithms † Hilbert Matrices † Distances between Matrix Eigenvalues † Special Functions of Mathematical Physics † Sums and Products † Computing a High-Precision Value for Euler’s Constant g † Numerical Root-Finding † Roots of Polynomials † Jensen Disks † De Rham’s Function † Logistic Map † Built-in PseudoCompiler † Forest Fire Model † Iterated Digit Sums † Modeling a Sinai Billiard

1.2.2

Graphics Gibbs Phenomena † Fourier Series of Products of Discontinuous Functions † Dirichlet Function † Counting Digits † Apollonius Circles † Generalized Weierstrass Function † 3D Plots † Plotting Parametrized Surfaces † Plotting Implicitly Defined Surfaces † Graphics-Objects as Mathematica Expressions † Kepler Tiling † Fractal Post Sign † Polyhedral Flowers † Gauss Map Animation † Random Polyehdra


THE MATHEMATICA GUIDEBOOKS to PROGRAMMING—GRAPHICS—NUMERICS—SYMBOLICS

1.2.3

Symbolic Calculations Differentiation † Integration † Symbolic Solutions of ODEs † Vandermonde Matrix † LU Decomposition of a Vandermonde Matrix † Redheffer Matrix † Symbolic Representations of Polynomial Roots † Solving Systems of Polynomials † Eliminating Variables from Polynomial Systems † Series Expansions † L’Hôspital’s Rule † Radical Expressions of Trigonometric Function Values † Prime Factorizations † Symbolic Summation † Proving Legendre’s Elliptic Integral Identity † Geometric Theorem Proofs Using Gröbner Bases † Medial Parallelograms † Inequality Solving † Symbolic Description of a Thickened Lissajous Curve † Simplifications under Assumptions † Numbers with Identical Digits in the Decimal and Continued Fraction Expansions † Conformal Map of a Square to the Unit Disk † Vortex Motion in a Rectangle † Magnetic Field of a Magnet with Air Gap † Localized Propagating Solution of the Maxwell Equation † Customized Notations † Schmidt Decomposition of a Two-Particle State

1.2.4

Programming Large Calculations † Partitioning Integers † Binary Splitting-Based Fast Factorial † Bolyai Expansion in Nested Radicals † Defining Pfaffians † Bead Sort Algorithm † Structure of Larger Programs † Making Platonic Solids from Tori † Equipotential Surfaces of a Charged Icosahedral Wireframe † Tube along a 3D Hilbert Curve

1.3

What Computer Algebra and Mathematica 5.1 Can and Cannot Do What Mathematica Does Well † What Mathematica Does Reasonably Well † What Mathematica Cannot Do † Package Proposals † What Mathematica Is and What Mathematica Not Is † Impacts of Computer Algebra † Relevant Quotes † Computer Algebra and Human Creativity † New Opportunities Opened by Computer Algebra † Computer Mathematics—The Joy Now and the Joy to Come

@ @ Exercises @ @ Solutions

Computing Wishes and Proposals † Computer Algebra Systems 100 Proposals for Problems to Tackle † Sources of Interesting and Challenging Problems † ISSAC Challenge Problems † 100$–100Digit Challenge


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CHAPTER 2

Structure of Mathematica Expressions 2.0

Remarks

2.1

Expressions Everything Is an Expression † Hierarchical Structure of Symbolic Expressions † Formatting Possibilities † Traditional Mathematics Notation versus Computer Mathematics Notation † Typeset Forms † Heads and Arguments † Symbols † Nested Heads † Input Form and the Formatting of Programs

2.2

Simple Expressions 2.2.1

Numbers and Strings Formatting Fractions † Integers † Autosimplifications † Rational Numbers † Approximate Numbers † Real Numbers † Complex Numbers † Autonumericalization of Expressions † Strings † HighPrecision Numbers † Inputting Approximate Numbers † Inputting High-Precision Numbers † Approximate Zeros

2.2.2

Simplest Arithmetic Expressions and Functions Basic Arithmetic Operations † Reordering Summands and Factors † Precedences of Simple Operators † Algebraic Numbers † Domains of Numeric Functions † Autoevaluations of Sums, Differences, Products, Quotients, and Powers

2.2.3

Elementary Transcendental Functions Exponential and Logarithmic Functions † Trigonometric and Hyperbolic Functions † Exponential Singularities † Picard’s Theorem † Secants Iterations † Exact and Approximate Arguments † Postfix Notation † Infix Notation

2.2.4

Mathematical Constants Imaginary Unit † p † Autoevaluations of Trigonometric Functions † Base of the Natural Logarithm † Golden Ratio † Euler’s Constant g † Directed and Undirected Infinities † Indeterminate Expressions

2.2.5

Inverse Trigonometric and Hyperbolic Functions Multivalued Functions † Inverse Trigonometric Functions † Inverse Hyperbolic Functions † Complex Number Characteristics † Real and Imaginary Parts of Symbolic Expressions † Branch Points and Branch Cuts † Branch Cuts Not Found in Textbooks

2.2.6

Do Not Be Disappointed Real versus Complex Arguments † Seemingly Missing Simplifications † Principal Sheets of Multivalued Functions

2.2.7

Exact and Approximate Numbers Symbols and Constants † Numericalization to Any Number of Digits † Precision of Real Numbers † Precision of Complex Numbers

2.3

Nested Expressions 2.3.1

An Example Constructing Nested Expressions † Canonical Order † Displaying Outlines of Expressions † Displaying Nested Expressions



2.3.2

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Analysis of a Nested Expression A Large Expression † Parts of Expressions † Recursive Part Extraction † Depths of Expressions † Extracting Multiple Parts † Extracting Parts Hierarchically † Locating Subexpressions in Expressions † Level Specifications † Length of Expressions † Leaves of Expressions

2.4

Manipulating Numbers 2.4.1

Parts of Fractions and Complex Numbers Rational Numbers as Raw Objects † Numerators and Denominators † Complex Numbers as Raw Objects † Real and Imaginary Parts

2.4.2

Digits of Numbers Digits of Integers † Digits of Real Numbers † Writing Numbers in Any Base † Counting Digits of Numbers † Fibonacci Chain Map Animation

@ @ Overview @ @ Exercises

Analyzing the Levels of an Expression † Branch Cuts of Nested Algebraic Functions † Analyzing the Branch Cut Structure of Inverse Hyperbolic Functions † “Strange” Analytic Functions

@ @ Solutions

Principal Roots † Analyzing a Large Expression † Levels Counted from Top and Bottom † Branch Cuts of Iz4 M

Branch Cuts of z+ 1êz z- 1êz † Riemann Surface of arctanHtanHz ê 2L ê 2L Singularities

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1ê4

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Repeated Mappings of

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CHAPTER 3

Definitions and Properties of Functions 3.0

Remarks

3.1

Defining and Clearing Simple Functions 3.1.1

Defining Functions Immediate and Delayed Function Definitions † Expansion and Factorization of Polynomials † Expansion and Factorization of Trigonometric Expressions † Patterns † Nested Patterns † Patterns in Function Definitions † Recursive Definitions † Indefinite Integration † Matching Patterns † Definitions for Special Values † Functions with Several Arguments † Ordering of Definitions

3.1.2

Clearing Functions and Values Clearing Symbol Values † Clearing Function Definitions † Clearing Specific Definitions † Removing Symbols † Matching Names by Name Fragments † Metacharacters in Strings



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3.1.3

Applying Functions Univariate and Multivariate Functions † Prefix Notation † Postfix Notation † Infix Notation

3.2

Options and Defaults Meaning and Usage of Options † Lists as Universal Containers † Options of Functions † Plotting Simple Functions † Extracting Option Values † Setting Option Values

3.3

Attributes of Functions Meaning and Usage of Attributes † Assigning Attributes to Functions † Commutative Functions † Associative Functions † Functions Operating Naturally on Lists † Numerical Functions † Differentiation of Functions † Protected Functions † Preventing the Evaluation of Expressions † Forcing the Evaluation of Expressions

3.4

Downvalues and Upvalues Function Definitions Associated with Heads † Function Definitions Associated with Specific Arguments † Downvalues and Upvalues † Timing for Adding and Removing Definitions † Caching † Values of Symbols † Numerical Values of Symbols

3.5

Functions that Remember Their Values Caching Function Values † Multiple Assignments † Simplification of Expressions † Timings of Computations † Takeuchi Function

3.6

Functions in the l-Calculus l-Calculus † Functions as Mappings † Functions without Named Arguments † Self-Reproducing Functions † Splicing of Arguments † Sequences of Arguments † Pure Functions with Attributes † Nested Pure Functions

3.7

Repeated Application of Functions Applying Functions Repeatedly † Iterative Maps † Solving an ODE by Iterated Integration † Iterated Logarithm in the Complex Plane † Fixed Points of Maps † Fixed Point Iterations † Newton’s Method for Square Root Extraction † Basins of Attractions † Cantor Series

3.8

Functions of Functions Compositions of Functions † Applying Lists of Heads † Inverse Functions † Differentiation of Inverse Functions


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Predicting Results of Inputs † Nice Polynomial Expansions † Laguerre Polynomials † Puzzles † Unexpected Outputs † Power Tower † Cayley Multiplication Matching Unevaluated Arguments † Equality of Pure Functions † Invalid Patterns † Counting Function Applications



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CHAPTER 4

Meta-Mathematica 4.0

Remarks

4.1

Information on Commands 4.1.1

Information on a Single Command Built-in Function Definitions as Outputs † Information about Functions † Listing of All Built-in Commands † Messages † Printing Text and Cells † Warnings and Error Messages † Wrong and “Unexpected” Inputs † Suppressing Messages † Carrying out Multiple Calculations in One Input

4.1.2

A Program that Reports on Functions Converting Strings to Expressions † Converting Expressions to Strings † String Form of Typeset Expressions

4.2

Control over Running Calculations and Resources 4.2.1

Intermezzo on Iterators Do Loops † Multiple Iterators † Possible Iterator Constructions † Iterator Step Sizes

4.2.2

Control over Running Calculations and Resources Aborting Calculations † Protecting Calculations from Aborts † Interrupting and Continuing Calculations † Collecting Data on the Fly † Time-Constrained Calculations † Memory-Constrained Calculations † Time and Memory Usage in a Session † Expressions Sharing Memory † Memory Usage of Expressions

4.3

The $-Commands 4.3.1

System-Related Commands Mathematica Versions † The Date Function † Smallest and Largest Machine Real Numbers

4.3.2

Session-Related Commands In and Out Numbering † Input History † Collecting Messages † Display of Graphics † Controlling Recursions and Iterations † Deep Recursions † Ackermann Function

4.4

Communication and Interaction with the Outside 4.4.1

Writing to Files Extracting Function Definitions † Writing Data and Definitions to Files † Reading Data and Definitions from Files † File Manipulations

4.4.2

Simple String Manipulations Concatenating Strings † Replacing Substrings † General String Manipulations † Case Sensitivity and Metacharacters † A Program that Prints Itself

4.4.3

Importing and Exporting Data and Graphics Importing and Exporting Files † Importing Web Pages † Importing From and To Strings † Making Low-Resolution JPEGs


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4.5

Debugging Displaying Steps of Calculations † Evaluation Histories as Expressions † Recursion versus Iteration † Interactive Inputs

4.6

Localization of Variable Names 4.6.1

Localization of Variables in Iterator Constructions Sums and Products † Scoping of Iterator Variables

4.6.2

Localization of Variables in Subprograms Scoping Constructs † Lexical Scoping † Dynamic Scoping † Local Constants † Temporary Variables † Variable Scoping in Pure Functions † Creating Unique Variables † Nonlocal Program Flow

4.6.3

Comparison of Scoping Constructs Delayed Assignments in Scoping Constructs † Temporarily Changing Built-in Functions † Variable Localization in Iterators † Scoping in Nested Pure Functions † Nesting Various Scoping Constructs † Timing Comparisons of Scoping Constructs

4.6.4

Localization of Variables in Contexts Contexts † Variables in Contexts † Searching through Contexts † Manipulating Contexts † Beginning and Ending Contexts

4.6.5

Contexts and Packages Loading Packages † General Structure of Packages † Private Contexts † Analyzing Context Changes

4.6.6

Special Contexts and Packages Developer Functions † Special Simplifiers † Bit Operations † Experimental Functions † Standard Packages

4.7

The Process of Evaluation Details of Evaluating an Expression † Analyzing Evaluation Examples † Standard Evaluation Order † Nonstandard Evaluations † Held Arguments


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Frequently Seen Messages † Unevaluated Arguments † Predicting Results of Inputs † Analyzing Context Changes † Evaluated versus Unevaluated Expressions Shortcuts for Functions † Functions with Zero Arguments † Small Expressions that Are Large † Localization of Iterator Variables † Dynamical Context Changes † Local Values



@ @ References

CHAPTER 5

Restricted Patterns and Replacement Rules 5.0 5.1

Remarks Boolean and Related Functions 5.1.1

Boolean Functions for Numbers Truth Values † Predicates † Functions Ending with Q † Numbers and Numeric Quantities † Integer and Real Numbers † Compound Numeric Quantities † Exact and Inexact Numbers † Primality † Gaussian Primes † Stating Symbolic and Verifying Numeric Inequalities † Comparisons of Numbers † Ordering Relations † Positivity

5.1.2

Boolean Functions for General Expressions Testing Expressions for Being a Polynomial † Vectors and Matrices † Mathematical Equality † Equality and Equations † Structural Equality † Identity of Expressions † Equality versus Identity † Canonical Order † Membership Tests

5.1.3

Logical Operations Boolean Operations † And, Or, Not, and Xor † Rewriting Logical Expressions † Precedences of Logical Operators

5.1.4

Control Structures Branching Constructs † The If Statement † Undecidable Conditions † While and For Loops † Prime Numbers in Arithmetic Progression

5.1.5

Piecewise Functions Piecewise Defined Functions † Canonicalization of Piecewise Functions † Composition of Piecewise Functions † Interpreting Functions as Piecewise Functions † Specifying Geometric Regions † Endpoint Distance Distribution of Random Flights

5.2

Patterns 5.2.1

Patterns for Arbitrary Variable Sequences Simple Patterns † Patterns for Multiple Arguments † Testing Patterns † Named Patterns † Trace of Products of Gamma Matrices † Shortcuts for Patterns † Avoiding Evaluation in Patterns † Literal Patterns

5.2.2

Patterns with Special Properties Optional Arguments † Default Values for Optional Arguments † Repeated Arguments † Excluding Certain Patterns † Alternative Arguments † Restricted Patterns † Pattern Tests † Conditional Patterns † Recursive Definitions † Pattern-Based Evaluation of Elliptic Integrals † Generating Tables † Selecting Elements from Lists † All Syntactically Correct Shortcuts


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5.2.3

Attributes of Functions and Pattern Matching Pattern Matching in Commutative and Associative Functions † Arguments in Any Order † Nested Functions † Automatic Use of Defaults † Analyzing Matchings and Recursions in Pattern and Attribute Combinations

5.3

Replacement Rules 5.3.1

Replacement Rules for Patterns Immediate and Delayed Rules † One-Time and Repeated Replacements † Unevaluated Replacements † Common Pattern Matching Pitfalls † Finding All Possible Replacements † Scoping in Rules † Replacements and Attributes † Modeling Function Definitions † Options and Rules † Replacing Position-Specified Parts of Expressions

5.3.2

Large Numbers of Replacement Rules Optimized Rule Application † Complexity of Optimized Rule Application

5.3.3

Programming with Rules Examples of Rule-Based Programs † Splitting Lists † Cycles of Permutations † Sorting of Complex Numbers † Cumulative Maxima † Dividing Lists † House of the Nikolaus † Polypaths † Rule-Based versus Other Programming Styles

5.4

String Patterns Strings with Pattern Elements † Patterns for Character Sequences † String-Membership Tests † Shortest and Longest Possible Matches † Overlapping Matches † Counting Characters † Replacing Characters † All Possible Replacements † Analyzing the Online Documentation † Cumulative Letter Frequencies


Rule-Based Expansion of Polynomials † All Possible Patterns from a Given Set of Shortcuts † Extending Built-in Functions † General Finite Difference Weights † Zeta Function Derivatives † Operator Products † q-Binomial Theorem † q-Derivative † Ordered Derivatives † Differentiating Parametrized Matrices † Ferrer Conjugates † Hermite Polynomial Recursions † Peakons † Puzzles † Catching Arguments and Their Head in Calculations † Nested Scoping

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Modeling Noncommutative Operations † Campbell–Baker–Hausdorff Formula † Counting Function Calls Using Side Effects † q-Deformed Pascal Triangle † Ordered Derivative † Avoiding Infinite Recursions in Pattern Matchings † Dynamically Generated Definitions



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CHAPTER 6

Operations on Lists, and Linear Algebra 6.0

Remarks Prevalence of List Manipulations † Building Polyhedra by Reflecting Polygons Iteratively † Animating the Folding Process Based on Iterated Reflections

6.1

Creating Lists 6.1.1

Creating General Lists Lists and Nested Lists as Arrays, Tables, Vectors, and Matrices † Timings of Creating Nested Lists † Changing Heads of Expressions † Summing Elements of Lists

6.1.2

Creating Special Lists Kronecker Symbol and Identity Matrix † Levi-Civita Symbol and Antisymmetric Tensors † Creating Multiple Iterators † Stirling Numbers † Subsets and Tuples

6.2

Representation of Lists 2D Formatting of Tables and Matrices † Aligning Rows and Columns † Formatting Higher-Dimensional Tensors † Tensors and Arrays

6.3

Manipulations on Single Lists 6.3.1

Shortening Lists Extracting Elements from Lists † Deleting Elements by Specifying Position, Pattern, or Property † Prime Sieving

6.3.2

Extending Lists Prepending, Appending, and Inserting List Elements † Working with Named Lists

6.3.3

Sorting and Manipulating Elements Rotating Lists Cyclically † Sorting Lists † Sorting Criteria † Analyzing the Built-in Sorting Algorithm † Splitting Lists † Mapping Functions over Lists † Listable Functions † Mapping Functions to Expressions and Parts of Expressions † Extracting Common Subexpressions † Optimized Expressions

6.3.4

Arithmetical Properties of Lists Average Value of a List † Sum of a List † Variance of a List † Quantiles of a List

6.4

Operations with Several Lists or with Nested Lists 6.4.1

Simple Operations Hadamard Arithmetic on Lists † Transposing Tensors † Permutations † Using Side Effects for Monitoring List Algorithms † Joining Lists † Intersections and Complements of Lists † Finding Approximately Identical Elements


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6.4.2

List of All System Commands Working with Unevaluated Expressions † Options and Attributes of All Built-in Functions † Analyzing All Built-in Function Names † Dependencies of Definitions

6.4.3

More General Operations Contractions and Kronecker Products—Inner and Outer Products † Rotations in 3D † Cross Products † Threading Functions over Lists

6.4.4

Constructing a Crossword Puzzle A Large, List-Based Calculation † Example Construction † Manipulating Function Definitions through Downvalues † Crossword Array of All Built-in Functions † Crossword Array of All Package Functions † Crossword Array of All Named Characters

6.5

Mathematical Operations with Matrices 6.5.1

Linear Algebra Inverse Matrices † Determinants † Timing Comparisons for Various Element Types † Traces of Matrices † Modeling Trace Calculations † Eigenvalues and Eigenvectors † Pauli Matrices † Properties of Eigenprojectors † Power Method for Finding the Largest Eigenvalue † Generalized Eigenvalue Problem † Solving Systems of Linear Equations † Siamese Sisters † Lorentz Transformations in Matrix Form † Moore–Penrose Inverse † Best Solutions to Overdetermined Linear Systems † Algorithms of Linear Algebra † Quantum Cellular Automata † Extending Linear Algebra Functions

6.5.2

Constructing and Solving Magic Squares Underdetermined Linear Systems † Integer Solutions of Linear Systems † Decoding and Encoding Magic Squares † Finding All Solutions of a Magic Square

6.5.3

Powers and Exponents of Matrices Integer and Fractional Powers of Matrices † Exponential Function of a Matrix † Trigonometric Functions of Matrices † Fractional Powers and Matrix Spectral Decompositions † Matrix Evolution Equations † Time-Development of a Linear Chain † Cayley–Hamilton Theorem † Characteristic Polynomials

6.6

The Top Ten Built–in Commands Finding Filenames † Working with Unevaluated Expressions † Counting Function Uses † Reading Packages † Zipf’s Law † Analyzing Notebooks, Cell Types, References, Typeset Structures, and Text


Benford’s Rule † Timing Comparisons for List Operations † SumFree Sets † Generating an Index for This Book † Consistency of References † Line Length Distribution † Spacing Check † Moessner’s Process † Ducci’s Iterations † Stieltjes Iterations † Pseudorandom trees † Levi–Civita Tensor Contractions † Dirac Matrices Products † Determinants of Multidimensional Arrays † Mediants † d’Hondt Voting † Identifying Approximate Vectors Efficiently † Unsorted Complements † All Arithmetic Expressions † Ideal Formatting † Functions with Method Options † Functions with Level Specifications † Changing Formatting by Programs † Pattern Instances † Matrix Identities † Amitsur–Levitzky Identity † Frobenius Formula for Block Matrices † Iterative Matrix Square Root † Differential Matrix Identities † Matrix Derivatives † Autoloaded Functions † Precedences of All Operators † One-Liners † Changing $1 † Meissel Formula † Binary Bracketing † Kolakoski Sequence † Puzzles † Cloning Functions † Hash Values † Permutation Digit Sets



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Chemical Element Data † Population Data of US Cities and Villages † Caching versus List-Lookup † Electronic Publication Growth † Statistics of Author Initials † Analyzing Bracket Frequencies † Word Neighbor Statistics † Weakly Decreasing Sequences † Finding All Built-in Symbols with Values † Automated Custom Code Formatting † Making Dynamically Formatted Inputs † Working with Symbolic Matrices † Downvalues and Autoloading † Determining Precedence Automatically † Permutation Polynomials † Working with Virtual Matrices

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CHAPTER 1

Two–Dimensional Graphics 1.0

Remarks Role of Visualization in and of Mathematics

1.1

Fundamentals 1.1.1

Graphics Primitives Points, Lines, and Polygons † Text in Graphics † Creating and Displaying Graphics † Complex Cantor Sets † Dimension Transitions Animation † Tree of Pythagoras † Generalized Pythagoras Theorem † 2D Graphics Sampler with 100 Examples † Constructing a Caustic † Pedal Curve † Projection into 2D † Pentagon Tree † Meyer Quasicrystal † Poincaré Model of the Hyperbolic Plane † Böttcher Function of the Quadratic Map † Complex Continued Fractions † From Graphics to Animations † Phyllotaxis Spiral † Julia Sets † Farey Tree † Deposition Modeling † Rauzy Tessellations † Islamic Wicker

1.1.2

Directives for Graphics Primitives Absolute and Relative Sizes of Points and Lines † Color Schemes and Color Values † Circles Rolling on Circles † An Optical Illusion: The Bezold Effect

1.1.3

Options for 2D Graphics Max Bill’s Picture of Nested n-gons

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Aspect Ratios † Adding Axes to Graphics † Labeling Axes † Fonts and Typeset Expressions in Graphics † Framing Graphics † Adding Labels to Graphics † Overlaying Graphics † Specifying Tick Marks † Repeatedly Displaying Graphics

1.1.4

A First Graphics Application: Voderberg Nonagon Polygons that Enclose Each Other † Reinhardt’s Conjecture † Finding Matching Polygons

1.2

Plots of Functions 1.2.1

Plots of Functions Given Analytically The Process of Making a Plot † Controlling Smoothness and Resolutions of Plots † Iterated Trigonometric Functions † Plotting Multiple Functions † Absolute Value Approximation † Distribution of Bend Angles † Fooling the Plotting Function † Visualizing High-Order Taylor Series † Plotting Parametrized Curves † Lissajous Figures † Hedgehogs of Curve Families † Astroid

1.2.2

Plots of Functions Defined Only at Discrete Points Digit Distributions in Various Bases † Nowhere Differentiable Continuous Functions † Riemann’s Continuous Nondifferentiable Function † Minkowski’s Function † Periodic Continued Fractions Made Continuous



1.3

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Combining Several Images 1.3.1

Arrays of Graphics Spirals † Arrays of Graphics † Inverting Graphics † Polyspirals † Inscribing Graphics into Rectangles † Graphing a Mouse † Manipulating Given Graphics † Puzzles Made from Subdivided Graphics † Clipping Polygons † Absolute Size of Text

1.3.2

Animations Vibrating Linear Chain † Perron Tree Construction † Circles on Circles † Microscopic Moiré Pattern † Tangential Circles in Regular Polygons † Julia Set Evolution from Pullbacks of the Quadratic Map † Polygonal Radix Representation † Lattice Interpolations † Pólya’s Orchard Problem † Dragon Generation Animation

1.4

Packages Graphics Packages † Visualizing Graphs † Hypercube Wireframe † Graphing Implicit Curves † Graphing Vector Fields

1.5

Graphics of Iterative Mappings 1.5.0 1.5.1

Remarks Sierpinski Triangle Iteratively Subdividing Triangles † Overlaying Graphics † Inverted Sierpinski Triangle † Applying Nonlinear Transformations

1.5.2

Peano Curves Space-Filling Curves † Filling a Triangle with a Curve † Connecting Subdivided Triangles

1.5.3

Lebesgue’s Mapping of the Cantor Set Curves Based on Digit Expansions † Filling Fractal Curves † General Digit Expansions

1.5.4

Subdivision of an L–Shaped Domain Aperiodic Tilings † Applying Transformations to Graphics † Triangle Subdivisions

1.5.5

Penrose and Substitution Tilings Tilings Using Rhombii † Coloring and Painting Tilings † Tilings Based on Kites and Darts † Manipulating Existing Graphics † Fractal Tilings † Cut-and-Project Method

1.5.6

Barnsley’s Fern, Mazes, and Other Random Images Random Numbers † Random Number Generators † Generating Random Expressions † Law of the Iterated Logarithm † Random Sums † Random Replacements † Bak–Sneppen Model † Samples of 2D Graphics that Contain Randomness † Eigenvalues of Random Matrices † Randomly Nested Radicals † Making Concave Polygons Convex † Strange Nonchaotic Attractors † Random Circle Segment Patterns † Kaleidoscopes † Mazes † Square and Hexagonal Truchet Images † Randomly Bent Ropes † Iterated Function Systems † Barnsley’s Fern † Searching for Iterated Function Systems † Bahar Systems

1.5.7

Koch Curves Koch Curve Generator † Random and Deterministic Koch Curves † Filling Koch Curves † Manipulating Koch Curves

1.5.8

Honeycombs and Escher Drawings Constructing and Coloring Hexagon Lattices † Interlocking Lizards † Hyperbolic Triangles and Hyperbolic Tilings † Inversion on a Circle

1.5.9

Lindenmayer Systems, Monster Curves, Grasses, and Herbs L-System Syntax: Axioms and Replacement Rules † Examples of LSystems † Space Filling Curves † Filled Gosper Curve † L-Systems with Branching † L-Systems that Model Plants † Random L-Systems



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1.6

Coloring Closed Curves Coloring Plots † Finding Curve Intersections † Sorting 2D Line Segments † Loop Construction † Constructing the Clusters † Checkerboard Coloring † Some Examples † Checking if Polygons are Disjoint


Game of Life † Langton’s Ant † Brillouin Zones † Maxwell–Helmholtz Color Triangle † Conformal Maps † Cornet Isogons † Jarník Polygons † Light Ray Reflections in a Water Drop † Warped Patterns † Moiré Patterns † Triptych Fractal † Multiple Reflected Pentagons † Random Lissajous Figures † Walsh Function † Sorting Game † Ball Moves † Rectangle Packings † Smoothed L-Systems † Polygonal Billiards † Random Walk on a Sierpinski Fractal † Voronoi Tessellations † Lévy Flights † Random Supersymmetric Potential † Common Plotting Problems † Nomogram for Quadratic Equation † Clusters on Square Grids † Aperiodic Triangle Tilings

@ @ Solutions

Random Cluster Generation † Leath Clusters † Midsector Lines † Analyzing Mathematica Code † Visualizing Piecewise Linear Approximations † Cartesian Ray † Kepler Cubes † Modulated SinCurves † Superimposed Lattices † Triptych Fractals † Two Superimposed Bumps Forming Three Bumps † Repeatedly Mirrored Decagons † Smoothly Connected Curves † Randomly Deformed Graphics † Random Expressions

@ @ References

CHAPTER 2

Three–Dimensional Graphics 2.0

Remarks

2.1

Fundamentals 2.1.1

Graphics Primitives Points, Lines, and Polygons † Cuboids † Projecting a Hypercube into 3D † Nonplanar and Nonconvex Polygons † Translating 3D Shapes † Escher’s Cube World

2.1.2

Directives for Three-Dimensional Graphics Primitives Absolute and Relative Sizes of Points and Lines † Constructing an Icosahedron from Quadrilaterals † Coloring Polygons in the Presence of Light Sources † Diffuse and Specular Reflection † Edges and Faces of Polygons † Rotating 3D Shapes † Random Rotations † Stacked Tubes † Text in 3D Graphics

2.1.3

Options for 3D Graphics The 34 Options of 3D Graphics † Relative and Absolute Coordinate Systems † Space Curves versus Space Tubes

2.1.4

The Structure of Three-Dimensional Graphics Resolving Automatic Option Settings † Nested Primitives and Directives † Converting 3D Graphics to 2D Graphics



2.1.5

17

Discussion of Selected Options Platonic Solids † Choosing the Viewpoint † Simple 3D Shapes † Light Sources and Colored Polygons † Cluster of Dodecahedra † Views on an Octant Filled with Cubes † Restricting the Plot Range † The 3D Graphics Enclosing Box † View Direction † Sizing Identical Graphics Independently of the Viewpoint † Rendering All versus Rendering Only Visible Polygons † Intersecting Polygons † Colliding Platonic Solids † A Scale with Platonic Solids † Diamond Faces † Rolled Checkered Paper † Woven Tubes † Smooth Dodecahedron– Icosahedron Transition † Platonic Solid Metamorphosis † Slicing a Cube

2.2

Display of Functions 2.2.1

Functions Given Analytically Graphing Functions of Two Variables † Special Plotting Options † Wireframes † Showing Multiple Plots † Parametrized Vector Functions † Cubed Torus † Klein Bottle † Parametrized Surfaces Samples † Using Symmetries to Construct Graphics † Constructing a Candelabra † Surfaces of Revolution † Emission of an Accelerated Point Charge † Borromaen Rings † Spiraling Spiral † Constructing a Birthday Bow

2.2.2

Functions Given at Data Points Visualizing 2D Arrays of Data † Visualizing Computation Timings † Time Evolution on a Torus † 3D Bar Charts † Randomized Geode

2.3

Some More Complicated Three-Dimensional Graphics 2.3.0 2.3.1

Remarks 3D Graphics of Iterative Mappings Rauzy Fractal From a 3D Projection † 3D Sierpinski Sponge † Exercising a Sierpinski Sponge † Kepler Tiling † 3D Iterated Function System † Random Clusters of Tetrahedra † Quaquaversal Tiling † 3D Truchet Graphics † 3D Space Fillers

2.3.2

Tubular Curves Frenet Frame † Tangents, Normals, and Binormals of Space Curves † Tubes around Space Curves † Knots † Mapping Textures to Knots † Tubes around Piecewise Straight Curves † Biased Random Walk † Osculating Circles of Curves

2.3.3

Recursively Colored Easter Eggs Recursively Subdividing Surfaces † Deformed Spheres † Mapping Patterns to Spheres † Rough Surfaces

2.3.4

Klein Bottles Making Surfaces by Gluing the Edges of a Square † Spine Curves † Cross Sections of Klein Bottles † Slicing and Coloring Klein Bottles † Deformed Klein Bottles † Cubistic Klein Bottles

2.3.5

A Hypocycloidal Torus Triangulating Quadrilaterals † Rotating Curves to Sweep out Surfaces † Triangulations † Surfaces with Holes

2.3.6

The Penrose Tribar Constructing a Tribar † Coordinate System Transformations † Choosing the Right View Point † Calculating the Optimal Viewpoint † An Impossible Crate

2.3.7

Riemann Surfaces of Simple Functions Plotting Multivalued Functions † Riemann Surfaces of Algebraic Functions † Cutting Surfaces along Branch Cuts † Surfaces Subdivided Using Tilings † A Family of Polynomial Riemann Surfaces † Implicit Parametrizations † Riemann Surfaces of Nested Logarithms † Riemann Surfaces over the Riemann Sphere



18

2.3.8

Interwoven Polygonal Frames Planes Intersecting Convex Bodies † Calculating All Intersections † Creating Frames † Interweaving Frames † Examples of Interwoven Frames

2.3.9

Selfintersecting Origami and 4D Hilbert Curves Paper Folding Models † Goffinet Kite † Folding Animation † Hilbert Curves in Higher Dimensions

2.3.10

The Cover Image: Hyperbolic Platonic Bodies Triangulating Platonic Solids † Symmetry Considerations † Compact Code † Evolution of the Cover Graphics from Version 2 to Version 5 † Nonplanar Contraction and Expansion of Polyhedra

2.4

Brillouin Zones of Cubic Lattices Higher Degree Voronoi Regions † Simple Cubic Lattice † Bisector Planes † Intersection of Planes † Symmetry of a Cube † Forming Brillouin Zones from Polygons † Gluing Polygons Together † BodyCentered Lattice † Face-Centered Lattice


3D Surface Sampler † Warped, Twisted, and Interlocked Tori † Dodecahedra Iteratively Reflected on its Faces † Snail † Trinomial Theorem Visualization † Ball Blending Method † Loop Subdivision † 3 -Subdivision Algorithm † Averaging Closed Curves † Projective Plane Model † Counting Surfaces for a Given Genus † Lattice Pyramids † Fractal Mountains † Random Walk on a Sphere † Projecting onto Polyhedra † Alexander’s Horned Sphere † Polyhedral Caustic † Sliced Möbius Strip † Perspective Modeling † Displaying Hidden Edges † Generating Platonic Solid Clusters † A 4D Platonic Solid—The 120-Cell † Folding a Dodecahedron † Continuously Changing Polyhedra † Inscribing Five Cubes in a Dodecahedron † Interwoven Bands around a Dodecahedron † Knot Made from Knots † Knot with Escher Tiling † Gear Chain Animation † 3D Peano Polygon † Tetraview Riemann Surface Animation † Riemann Surface of Kepler Equation † Sierpinski Plant

@ @ Solutions

Cayley Cusp † Boy Surface † Möbius Strip † Steiner’s Cross Cap † Henneberg Surface † Flying Saucer Construction † Random Parametrized Surfaces † Dodecahedral Flowers † Extruded Platonic Solids † Smoothing through Graph Plotting † Staggered Trefoil Knots † Field Lines of Two Charged Spheres † Random Symmetric Polyhedra † Graphics of a Screw † Arranging Worn Stones Tightly † Random Cones † Broken Tube † Weaving a Torus † Constructing Double and Triple Tori from Torus Pieces † Massive Wireframes of Platonic Solids † Smoothing a Cube Wireframe † Smoothing a Stellated Icosahedron † Pyramids on Lattices † Closed Random Walks † Slicing and Coloring a Möbius Strip † Coordinate System Transformations † Kochen–Specker Theorem † Smooth Random Functions † Subdividing Concave Polygons



@ @ References

CHAPTER 3

Contour and Density Plots 3.0

Remarks

3.1

Contour Plots Contour Graphics † Converting Contour Graphics † Options of Contour Graphics † Cassini Curve † Various Sample Contour Plots † Functions Varying Strongly † Homogeneous Contour Line Density † Coloring Contour Plots † Contour Graphics in Nonrectangular Domains † Speckles and Scarlets from Superimposing 2D Waves † Smoothing Contour Lines † Superimposed 2D Waves in Symmetric Directions † Comparing Options and Option Settings of Plotting Functions † Algebraic Description of Polygons † Blaschke Products † Charged Goffinet Dragon † Square Well-Scattering Amplitude

3.2

Density Plots Density Graphics † Converting Density Graphics † Arrays of Gray or Color Values † Lifting Color Value Arrays to 3D † Earth Graphics † Array Plots † Gauss Sums † Visualizing Difference Equation Solutions † Visualizing Matrices † Saunders Pictures † Making Photomosaics from Density Plots

3.3

Plots of Equipotential Surfaces Visualizing Scalar Functions of Three Variables † Marching Cubes † Plots of Implicitly Defined Algebraic Surfaces † Implicit Descriptions of Riemann Surfaces † Gluing Implicitly Defined Surfaces Smoothly Together † Using Reflection and Rotation Symmetries to Visualize Algebraic Surfaces † Examples of Surfaces from Spheres, Tubes, and Tori Glued Together † An Algebraic Candelabra † Joining Three Cylinders Smoothly † Zero-Velocity Surfaces † Implicit Form of an Oloid † Isosurfaces of Data


Clusters of Irreducible Fractions † Chladny Tone Figures in Rectangles and Triangles † Helmholtz Operator Eigenfunctions of a Tetrahedron † Liénard–Wiechert Potential of a Rotating Point Charge † Shallit–Stolfi–Barbé Plots † Random Fractals † Functions with the Symmetry of Cubes and Icosahedra † Icosahedron Equation † Belye Functions † Branch Cuts of Hyperelliptic Curves † Equipotential Plots of Charged Letters † Charged Random Polygon † Gauss–Bonnet Theorem † Interlocked Double and Triple Tori † Inverse Elliptic Nome † Contour Plots of Functions with Boundaries of Analyticity † Isophotes on a Supersphere † Structured Knots † Textures on a Double Torus


19


20

@ @ Solutions

Visualizing Saddle Points † Outer Products † Repeatedly Mirrored Matrix † Halley Map † Generating Random Functions † Weierstrass ƒ Function Based Fractal † Contour Plots in Non-Cartesian Coordinate Systems † Spheres with Handles † Cmutov Surfaces † Random Surfaces with Dodecahedral Symmetry † Polynomials over the Riemann Sphere † Random Radial-Azimuthal Transition † Contour Lines in 3D Plots † Lines on Polygons † Slicing Surfaces † Euler–Poincaré Formula † Mapping Disks to Polygons † Statistics of n-gons in 3D Contour Plots

@ @ References



N

U

M

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S

CHAPTER 1

Numerical Computations 1.0

Remarks Summing Machine Numbers † Klein’s Modular Function and Chazy Equation † Discretizing the Rössler System † Modeling the Ludwig– Soret Effect

1.1

Approximate Numbers 1.1.0 1.1.1

Remarks Numbers with an Arbitrary Number of Digits Machine Arithmetic versus High-Precision Arithmetic † Modified Logistic Map † Numerical Calculation of Weierstrass Functions † High-Precision Arithmetic System Parameters † Fixed-Precision Arithmetic † Random Fibonacci Recursion † Smart Numericalization † Precision and Accuracy of Real Numbers † Precision and Accuracy of Complex Numbers † Precision Loss and Gain in Calculations † Error Propagation in Numerical Calculations † Principles of Significance Arithmetic † Error Propagation for Multivariate Functions † Collapsing Numeric Expressions † Setting Precision and Accuracy of Numbers † Guard Digits in High-Precision Numbers † The Bits of a Number † Sum-Based Methods of Calculating p † Comparing High-Precision Numbers † Automatic Switching to High-Precision Arithmetic

1.1.2

Interval Arithmetic Rigorous Arithmetic † Notion of an Interval † Joining and Intersecting Intervals † Modeling Error Propagation † Global Relative Attractor of Rationals Maps

1.1.3

Converting Approximate Numbers to Exact Numbers Rational Numbers from Approximate Numbers † Continued Fractions † Liouville Constant † Periodic Continued Fractions † Numbers with Interesting Continued Fraction Expansions † Continued Fraction Convergents † Pseudoconvergents † Gauss– Kusmin Distribution † Khinchin Constant † Khinchin–Lévy Theorem † Lochs’ Theorem † Canonical Continued Fractions † Minkowski Function † Generalized Expansions † Rounding Numbers † Frisch Function † Egyptian Fractions

1.1.4

When N Does Not Succeed Using Extra Precision † Undecidable Numerical Comparisons † Caching High-Precision Results † Recursive Prime Number Definition † Sylvester Expansion


21


22

1.1.5

Packed Arrays Machine Numbers, Tensors, and Packed Arrays † Developer Functions for Packed Arrays † Invisibility of Packed Arrays † Controlling Automatic Packed Array Generation † Counting Sums and Products of Sets of Integers † Long-Range Correlations in Natural Texts † Analyzing Shakespeare’s “Hamlet” † Zipf’s Law † Mean Square Fluctuation of a Random Walk † Analyzing a Chapter of This Book † Analyzing a PostScript Graphic

1.2

Fitting and Interpolating Functions Fitting Data † Least Squares and Pseudoinverses † Approximate Solution of the Helmholtz Equation by Plane Wave Expansion † Nonlinear Fits † File Size Distribution † Polynomial Interpolation of Data † Neville Algorithm † Convergence and Divergence of Polynomial Interpolations † Runge Phenomena † Newton–Cotes Weights † Interpolating Functions † Smoothness of Interpolating Functions † Curvature Driven Evolution † Dissecting an Interpolating Function † Splines

1.3

Compiled Programs Compiling a Calculation † Compiled Functions † Julia Set of the Quadratic Map † Timing Comparisons for Compiled Procedural and Functional Programs † Randomized Fibonacci Iterations † Products of partial Sums of Random Variables † Hansen–Patrick Root-Finding Method † Distances in Truchet Images † Cycles in Iterated Exponentiation † Ikeda Map † 3D Period-Doubling Animation † Sandpiles † Identity Sandpile † Nonlocal Cellular Automata † Caustics from Refraction

1.4

Linear Algebra Finite Resistor Network † Exact versus Approximate Solutions † Avoiding Numericalization of Indicies † Calculating Resistances Through Eigenvalues † Tagaki Function † Numerical Solution of a Functional Equation † Fixed-Precision Arithmetic in Linear Algebra † Modular Equation for Klein’s Modular Function † Null Spaces of Linear Systems † Bound State in a Waveguide Crossing † Sparse Matrices † Square Network with Random Resistance Values † Anderson Model

1.5

Fourier Transforms Discretized Periodic Functions † Fourier Transform † Amplitude and Frequency Modulation † Approximating a Function † Uncertainty Relations † Strang’s Strange Figures † Timing Comparisons of Numerical Fourier Transforms † Inverse Fourier Transforms † Fourier Transforms of Arrays † Approximating the Gosper Curve † Fourier Transforms of Aperiodic Tilings † Fractional Fourier Transform † High-Precision Frequency Approximation of Data † Approximating the Continuous Fourier transform † List Convolutions and Correlations † Manipulating Bitmap Graphics † Visualizing Trigonometric Identities

1.6

Numerical Functions and Their Options Common Options of Numerical Functions † Precision To Be Used in Calculations † Machine Precision versus High-Precision † Precision Goal for a Numerical Calculation † Accuracy Goal for a Numerical Calculation † Accuracy Goals for Independent and Dependent Variables † Monitoring Numerical Calculations † Evaluation Order in Numerical Function † Avoiding the Evaluation of the First Argument † Using Vector-Valued Variables † Dummy Variable-Free Function Calls



1.7

23

Sums and Products Numerical Products † Options of Numerical Product Calculations † Compensated Summation † Order Sensitivity in Floating Point Summations † Numerical Sums † Options of Numerical Summation † Verifying Convergence † Borel–Tanner Distribution † Sequence Transformations † Numerically Summing Divergent Series † Continuous Integer Spiral

1.8

Integration Numerically Integrating a Function † Introductory Examples † Integrable Singularities † Dealing with Singularities along the Integration Path † Contour Integration † Constructing Integration Path Iterators † Monitoring Numerical Integration † Matrix Functions Defined through Integrals † Options of Numerical Integration † Accuracy and Precision of Results † Termination Conditions † Methods of Numerical Integration † Integrating Discontinuous Functions † Comparison of Basic Integration Methods † Visualization of the Sample Points † Gauss Linking Number † Area of a Supersphere † Comparing Multidimensional Integration Methods † Double Exponential Method † Monte-Carlo and Quasi Monte-Carlo Integration † Distribution of Monte Carlo Sample Points † van Der Corput Sequences † Integration of Piecewise Continuous Functions † Using Symmetries of the Integrands † Picard–Lindelöf Iteration

1.9

Solution of Equations Numerical Solution of Polynomials, Polynomial Systems, and Arbitrary Functions † Sensitivity of Polynomial Roots to Changes in a Coefficient † Iterated Roots † Distances between Polynomial Roots † Hofstadter’s Butterfly † Schrödinger Equation for Periodic Potential and Applied Magnetic Field † Farey Sequences † Hofstadter Butterfly on a Finite Lattice † Kohmoto Model † Bézout and Bernstein Bounds for the Number of Roots of Polynomial Equations † Quadrature Weights † Root Finding of General Functions † Monitoring the Search Path † Adapative Precision Raising † Termination Conditions † Root-Finding Methods † Methods of Numerical Equation Solving † Calculating Jacobians † Multiple Roots and Roots of Noninteger Order † Variable-Free Minimization † Voderberg Spiral † Nested Touching Circles

1.10

Minimization Finding the Minimum † Methods of Numerical Minimization † Visualizing Search Paths † Method Option Choices for Numerical Optimization † Minimizing Sums of Squares † Sliding Down a Spiral Slide † Finding Global Minima † Minimum Energy Configuration of n Electrons in a Disk † Iterative Minimizations



24

1.11

Solution of Differential Equations 1.11.1

Ordinary Differential Equations Boundary and Initial Value Problems † Interpolating Functions as Solutions † Differential-Algebraic Equations † Pendulum ODE † Anharmonic Oscillator with Random Forcing † Squatting on a Swing † Newton Vector Field † Spiral Waves † 4D Chaotic Attractor † Energy Bands in a Random Complex Potential † Stiff and Nonstiff Systems † Precision Control † Nonlinear Differential Equation with Isochronous Solutions † Buchstab Function † Higher Order ODEs † Ablowitz–Ladik Chain † Particle Motion in a Wave Field † Chazy Equation † Boundaries of Analyticity † Generalized Airy Functions † Monitoring Numerical Differential Equation Solving † Stepsize Control † Coupled Pendulums † Restricting the Solutions † Stopping the Solution Process † Calculating and Visualizing Pursuits † Finding the Initial Slope for the Thomas–Fermi Equation † Forced Coupled Oscillators † Chaotic Scattering on a Four-Hill Potential † Events in Differential Equation Solving † Vector and Matrix Differential Equations † Method Option Choices † Integrated Brownian Motion † Modified Lorenz System † Calculating Contour Curves Through Differential Equations † Geodesics on a Triple-Periodic Surface † Using Homotopies to Solve Polynomial Systems † Modeling Newton’s Cradle † Trajectories in Central Force Fields † Three-Body Scattering † Interacting Vortices † Periodic Orbits of the Restricted Three-Body Problem † Combining Numerical Functions † Periodic Orbits of the Lorenz System † Bohm’s Quantum Trajectories † Continuous Time Random Walks on Graphs † Sparse Arrays in Differential Equations

1.11.2

Partial Differential Equations Parabolic and Hyperbolic PDEs † 1D Schrödinger Equation with Dirichlet Boundary Conditions † Scattering on a Potential Wall † 1D Wave Equation † PDE-Specific Options † Singular Initial Conditions † Wave Function Shredding in an Infinite Well of Time-Dependent Width † Fokker–Planck Equation for a Damped Anharmonic Oscillator † Liouville Equation for an Anharmonic Oscillator † Klein– Gordon Equation † Differential Equations with Mixed Derivatives † Nonlinear Schrödinger Equation † Complex Ginzburg–Landau Equation † Zakharov Equations † Prague Reaction-Diffusion Model

1.12

Two Applications 1.12.0 1.12.1

Remarks Visualizing Electric and Magnetic Field Lines Differential Equations for Field Lines † Field Lines of 2D Charge Configurations † Reusing Programs † Stopping Criteria for Field Lines † Field Lines for 3D Charge Configurations † Field Lines as Tubes † Field Lines of Magnetic Fields † Biot–Savart Rule † Magnetic Field Lines of a Peano Curve-Shaped Wire † Nonclosed Magnetic Field Lines † Field Lines of a Ring Coil

1.12.2

Riemann Surfaces of Algebraic Functions Algebraic Functions as Bivariate Polynomials † Faithful Riemann Surfaces † Implicit Parametrizations † Branch Cuts and Branch Points † Discriminant † First Order ODEs for Algebraic Functions † Sheets of Riemann Surfaces † Samples of Riemann Surfaces




25

Logistic Map † Randomly Perturbed Iterative Maps † Functions with Boundaries of Analyticity † q-Trigonometric Functions † Franel Identity † Bloch Oscillations † Courtright Trick † Hannay Angle † Harmonic Nonlinear Oscillators † Orbits Interpolating Between Harmonic Oscillator and Kepler Potential † Shooting Method for Quartic Oscillator † Eigenvalues of Symmetric Tridiagonal Matrices † Optimized Harmonic Oscillator Expansion † Diagonalization in the Schwinger Representation † Möbius Potential † Bound States in the Continuum † Wynn’s Epsilon Algorithm † Aitken Transformation † Numerical Regularization † Scherk’s Fifth Surface † Clebsch Surface † Smoothed Dodecahedron Wireframe † Standard Map † Stochastic Webs † Forced Logistic Map † Web Map † Strange Attractors † Hénon Map † Triangle Map Basins † Trajectories in 2D Periodic Potentials † Egg Crate Potential † Pearcey Integral † Charged Square and Hexagonal Grids † Ruler on Two Fingers † Branched Flows in Random Potentials † Maxwell Line † Iterated Secant Method Steps † Unit Sphere Inside a Unit Cube † IsingModel Integral † Random Binary Trees † Random Matrices † Iterated Polynomial Roots † Weierstrass Root Finding Method † Animation of Newton Basins † Lagrange Remainder of Taylor Series † Nodal Lines † Bloch Equations † Branch Cuts of Hyperelliptic Curves † Strange 4D Attractors † Billiard with Gravity † Schwarz–Riemann Minimal Surface † Jorge–Meeks Trinoid † Random Minimal Surfaces † Precision Modeling † Infinite Resistor Networks † AutoCompiling Functions † Card Game Modeling † Charges With Cubical Symmetry on a Sphere † Tricky Questions † Very High-Precision Quartic Oscillator Ground State † 1D Ideal Gas † Odlyzko-Stanley Sequences † Tangent Products † Thompson’s Lamp † Parking Cars † Seceder Model † Avoided Patterns in Permutations † Cut Sequences † Exchange Shuffles † Frog Model † Second Arcsine Law † Average Brownian Excursion Shape † ABC-System † Vortices on a Sphere † Oscillations of a Triangular Spring Network † Lorenz System † Fourier Differentiation † Fourier Coefficients of Klein’s Function † Singular Moduli † Curve Thickening † Random Textures † Random Cluster Growth † First Digit Frequencies in Mandelbrot Set Calculation † Interesting Jerk Functions † Initial Value Problems for the Schrödinger Equation † Initial Value Problems for 1D, 2D, and 3D Wave Equation † Continued Inverse Square Root Expansion † Lüroth Expansion † Lehner Expansion † Brjuno Function † Sum of Continued Fraction Convergents Errors † Average Scaled Continued Fraction Errors † Bolyai Expansion † Symmetric Continued Fraction Expansion

@ @ Solutions

Solving Polynomials Using Differential Equations † Stabilizing Chaotic Sequences † Oscillator Clustering † Transfer Matrices † Avoided Eigenvalue Crossings † Hellmann–Feynman Theorem † Scherk Surface Along a Knot † Time-Evolution of a Localized Density Under a Discrete Map † Automatic Selection of “Interesting” Graphic † Gradient Fields † Static and Kinematic Friction † Smoothing Functions † Eigenvalues of Random Binary Trees † Basins of Attraction Fractal Iterations † Calculating Contour Lines Through Differential Equations † Manipulating Downvalues at Runtime † Path of Steepest Descent † Fourier Series Arc Length † Poincaré Sections † Random Stirring † Heegner Numbers † Quantum Random Walk † Quantum Carpet † Coherent State in a Quantum Well



26

@ @ References

CHAPTER 2

Computations with Exact Numbers 2.0

Remarks Using Approximate Numerics in Exact Calculations † Integer Part Map † Misleading Patterns † Primes in Quadratic Polynomials

2.1

Divisors and Multiples Factoring Integers † Number of Prime Factors † Divisors † Sum of Squares † Derivative of an Integer † mod Function † Rotate and Mod † nth Digit of a Proper Fraction † Schönberg’s Peano Curve † Greatest Common Divisors and Least Common Multiples † Euclidean Algorithm † Classical and Generalized Maurer Roses † de Bruijn Medallions and Friezes

2.2

Number Theory Functions Prime Numbers † Prime Number Spiral † Prime Counting Function † Euler’s Totient Function † Absolutely Abnormal Number † Möbius Function † Redheffer Matrix † Möbius Inversion † Calculating Fourier Transforms through Möbius Inversion † Jacobi Symbol † Reciprocity Law

2.3

Combinatorial Functions Factorials † Digits of Factorials † Stirling’s Formula † Binomials and Multinomials † Nested Triangle Patterns † Stirling Numbers † Counting Partitions † Generating Partitions † Partition Identities

2.4

Euler, Bernoulli, and Fibonacci Numbers Akiyama–Tanigawa Algorithm † Euler–Maclaurin Formula † Lidstone Approximations † Boole Summation Formula † Divide-and-Conquer Algorithm for Calculating Large Fibonacci Numbers † FibonacciBinomial Theorem † Discretized Cat Map


Sum of Divisor Powers † Recurrence Relation for Primes † Arcsin Law for Divisors † Average Length of Continued Fractions of Rationals † Isenkrahe Algorithm † Prime Divisors † Kimberling Sequence † Cantor Function Integral † Cattle Problem of Archimedes † Mirror Charges in a Wedge † Periodic Decimal Numbers † Digit Sequences in Numbers † Numbered Permutations † Binomial Coefficient Values † Smith’s Sturmian Word Theorem † Modeling a Galton Board † Ehrenfest Urn Model † Ring Shift Modeling † Sandpile Model † Longest Common Subsequence † Riffle Shuffles † Weekday from Date † Easter Dates † Lattice Points in Disks † Binomial Digits † Average of Partitions † Partition Moments † 15 and 6174 † Selberg Identity † Kluyver Identities † Ford Circles † Farey–Brocot Interval Coverings † Sum of Primes † Visualizing Eisenstein Series † Magnus Expansion † Rademacher Identity † Goldbach Conjecture † Zeckendorf Representation † Sylvester– Fibonacci Expansion † Ramanujan t Function † Cross-Number Puzzle † Cyclotomic Polynomials † Generalized Bell Polynomials † Online Bin Packings † Composition Multiplicities † Subset Sums



@ @ Solutions

Nested Iterators † Being Prime Expressed Analytically † Legendre Symbol † Pell Equation † Nested Radicals Identity † Recognizing Algebraic Numbers † Iterated Digit Sum of Divisors † Guiasu Prime Counting Formula † Divisor Sum Identities † Choquet Approximation † Optical Factoring † Generalized Multinomial Theorem † Sums with Constraints † Faà di Bruno Formula † Symbolic Tables

@ @ References


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28

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CHAPTER 1

Symbolic Computations 1.0

Remarks

1.1

Introduction General Assumptions about Variables † Simplifying Expressions † Type Declarations for Simplifications † Evaluating Expressions Under Assumptions

1.2

Operations on Polynomials 1.2.0 1.2.1

Remarks Structural Manipulations on Polynomials Expanding and Factoring Polynomials † Factors of Random Polynomials † Irreducible Polynomials † Constructing Irreducible Polynomials from Primes † Factorization over Extension Fields † Reordering Multivariate Polynomials † Indeterminates of Polynomials † Extracting Coefficients from Polynomials † Decomposing Polynomials

1.2.2

Polynomials in Equations Polynomial Division † Resultants † Sylvester Matrix † Differential Equation for the Elliptic Nome † Gröbner Bases † Applications of Gröbner Bases † Equation Solving Using Gröbner Basis † Approximative Gröbner Bases † Monomial Orders † Showing Inconsistency of Equations Using Gröbner Bases † FiniteDimensional Representation of the Canonical Commutation Relations † Eliminating Variables Using Gröbner Bases † Geometric Theorem Proving † All Square Roots of Square Matrices † Bound States in Spherical Symmetric Potentials † Gröbner Walks † Reducing Polynomials

1.2.3

Polynomials in Inequalities Cylindrical Algebraic Decompositions † Solving Inequalities † Locally Parametrizing a Squeezed Torus † Arnold Cat Map † Generic Cylindrical Algebraic Decomposition † Quantifier Elimination † Generally Proving Inequalities † Proving Triangle Inequalities † Deriving New Geometry Theorems † Restricting Polynomial Roots † Proving the Sendov–Iliev Conjecture for Quadratic Polynomials † Deriving Clauser–Horn Inequalities † Algebraic Blending † Minkowski Sums

1.3

Operations on Rational Functions Numerators and Denominators † Expanding Parts of Nested Fractions † Partial Fraction Decomposition † Writing Rational Functions over Common Denominators † Gale–Robinson Sequence † The Power of “Togethering” † Mapping of the Fundamental Domain



1.4

Operations on Trigonometric Expressions Expansion and Factorization of Trigonometric Expressions † Addition Theorems for Trigonometric Functions † Converting Trigonometric Functions to Exponential Form † Real and Imaginary Parts of Symbolic Expressions

1.5

Solution of Equations The Notion of Generic Solutions † Solving Univariate Polynomials in Radicals † Cubic Polynomials with Three Real Roots † Symbolic Roots as Solutions of Univariate Polynomials of Any Degree † Exact Operations on Polynomial Roots † Matrix Eigenvalues † Canonicalization of Symbol-Free Algebraic Expression † Hölder’s Theorem about Real Roots of Cubics † Solving Systems of Polynomials † Vieta Relations † Solving Systems of Algebraic Equations † Solving Nonpolynomial Equations † Using Inverse Functions † Solving Trigonometric Equations † Solving Transcendental Equations † Verifying Parametric Solutions † Superposition of Damped Oscillations † Finding Degenerate Solutions † Elimination of Variables † Universal Differential Equation † Guidelines for Solving Equations and Systems of Equations

1.6

Classical Analysis 1.6.1

Differentiation Multivariate Differentiation † Numericalization of Unevaluated Derivatives † Numerical Differentiation † Differentiating in the Complex Plane † Schwarz Theorem † Differential Algebraic Constants † High-Order Derivatives † Derivatives of Inverse Functions † Differentiation With Respect to Vectors † Derivatives of Pure Functions † Adding New Differentiation Rules † Differential Equations for n -Nomials † Generalized Taylor Expansion † Differentials † Metric Tensors, Christoffel Symbols, and Geodesics † Iterated Evolutes † Phase Integral Approximation

1.6.2

Integration Algorithms for Symbolic Integration † Assumptions on Variables Having Generic Values † Integrating Abstract Functions † Korteweg– deVries Equation Hierarchy † Indefinite Integration Samples † Integrals and Special Functions † Integrating Rational Functions † Integrating Algebraic Functions † Assumptions of Parameter Variables † Assumptions in Indefinite Integrals † Generating Conditions for Convergence † Divergent and Hadamard-Regularized Integrals † Cauchy Principal Value Integrals † Multidimensional Integrals † Robbin’s Integral Identity † Definite Integrals from Indefinite Integrals † Piecewise Continuous Antiderivatives † Continuity of Indefinite Integrals † Weierstrass Parametrization of Minimal Surfaces † Infinite Resistor Network † Timings of Indefinite versus Definite Integration † d’Alembert Solution of the OneDimensional Wave Equation † Schrödinger Equation with a Timedependent Linear Potential † Definite Integrals and Branch Cuts

1.6.3

Limits Indeterminate Expressions and Limits † Limit Samples † Direction Dependence of Limits † Evaluating Limits Under Assumptions † Limits of Analytic Functions † Schwarz Derivative † Extracting Leading Terms † Limits of Iterative Function Applications † Multiple Limits


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30

1.6.4

Series Expansions Internal Structure of a Series-Object † Taylor Series † Continued Fraction with Three Limit Points † Laurent Series † Puiseux Series † Series Expansions at Branch Points and Branch Cuts † Series of Special Functions † Essential Singularities † Numerov–Mickens Scheme † Multivariate Series † Roots of Truncated Series † q -Taylor Series † Arithmetic of Series † Change for $1 † Iterated Constant Terms † Inverse Series † Higher-Order Newton and Chebyshev Methods † Fractional Iterations † Cumulant Expansions † Laurent Series for Mandelbrot Set † Approximating Linear Functionals

1.6.5

Residues Symbolic Residues at Poles † Generalized Residues † Residues of Special Functions

1.6.6

Sums Sum of Powers † Numericalization of Symbolic Expressions † Procedural versus Symbolic Finite Summations † Riemann Surface of the Square Root Function † Weierstrass’s Method of Analytic Continuation

1.7

Differential Equations 1.7.0 1.7.1

Remarks Ordinary Differential Equations Solutions as Rules † Pure Functions as Solutions † Degenerate Solutions † Differential Equation for Free Fall Including the Coriolis Force † Integration Constants † Linear Inhomogeneous ODE with Constant Coefficients † ODEs with Separated Variables † Homogeneous ODEs † Exact ODEs † Bernoulli ODE † Jacobi ODEs † Special Riccati ODEs † Abel ODEs of the First Kind † Abel ODEs of the Second Kind † Chini ODEs † Lagrange ODEs † Clairaut ODEs † ODEs with Shifted Argument † Cayley ODE † Second Order ODEs † Differential Equations of Special Functions † Schrödinger Equations for Various Smooth Potentials † Schrödinger Equations for Piecewise-Defined Potentials † Higher-Order Differential Equations † Implicit Solutions † Monitoring Differential Equation Solving † d-Expansion

1.7.2

Partial Differential Equations Hamilton–Jacobi Equation † Szebehely’s Equation † Solutions with Arbitrary Functions

1.7.3

Difference Equations Linear Difference Equations † Calculating Casoratians † Linear Difference Equations with Nonconstant Coefficients † Some Nonlinear Difference Equations † Difference Equations Corresponding to Differential Equations † Systems of Difference Equations

1.8

Integral Transforms and Generalized Functions Generalized Functions and Linear Functionals † Heaviside Theta Function and Dirac Delta Function † Integrals Containing Generalized Functions † Multivariate Heaviside Theta and Dirac Delta Function † Time Dilation † Derivatives of the Dirac Delta Function † Simplifying Generalized Functions † Sequence Representations of Generalized Functions † Green’s Function of Linear Differential Operators † Generalized Solutions of Differential Equations † Compactons † Fourier Transforms † Self-Fourier Transform † Principle Value Distribution † Sokhotsky–Plemelj Formula † Poincaré–Bertrand Identity † Laplace Transforms † Borel Summation of Divergent Sums † Adomian Decomposition



1.9

Additional Symbolics Functions Variational Calculus † Symbolic Series Terms † Ramanujan’s Master Theorem

1.10

Three Applications 1.10.0 1.10.1

Remarks Area of a Random Triangle in a Square A Quote from M. W. Crofton † Generalizations † Generic Cylindrical Algebraic Decompositions † Six-Dimensional Definite Integrals from Indefinite Integrals † Monte Carlo Modeling † Calculating the Probability Distribution of the Area

1.10.2

cosI 257 M à la Gauss 2p

The Morning of March 29 in 1796 † Gauss Periods † Primitive Roots † Splitting and Combining Periods † Thousands of Square Roots † cosJ

1.10.3

2p 65 537

N

†

Fermat Primes

Implicitization of a Trefoil Knot Parametric versus Implicit Description of Surfaces † Envelope Surface of a Moving Ball † Polynomialization of Trigonometric Expressions † Calculating a Large Resultant † Smoothing the Trefoil Knot † Inflating a Trefoil Knot † Implicit Klein Bottle


31


32


Heron’s formula † Tetrahedron Volume † Apollonius Circles † Proving Trigonometric Identities † Icosahedron Inequalities † TwoPoint Taylor Expansion † Horner Form † Nested Exponentials and Logarithms † Minimal Distance between Polynomial Roots † Dynamical Determimants † Appell–Nielsen Polynomials † Scoping in Iterated Integrals † Rational Solution of Painlevé II † Differential Equation for Products and Quotients of Linear Second Order ODEs † Singular Points of First-Order ODEs † Fredholm Integral Equation † Inverse Sturm–Liouville Problem † Graeffe Method † Lagrange Interpolation in 2D Triangles † Finite Element Matrices † Hermite Interpolation-Based Finite Element Calculations † Hylleraas– Undheim Helium Ground State Calculation † Variational Calculations † Hyperspherical Coordinates † Constant Negative Curvature Surfaces † Optimal Throw Angle † Jumping from a Swing † Normal Form of Sturm–Liouville Problems † Noncentral Collisions † Envelope of the Bernstein Polynomials † Eigensystem of the Bernstein Operator † A Sensitive Linear System † Bisector Surfaces † Smoothly Connecting Three Half-Infinite Cylinders † Nested Double Tori † Changing Variables in PDEs † Proving Matrix Identities † A Divergent Sum † Casimir Effect Limit † Generating Random Functions † Numerical Techniques Used in Symbolic Calculations † Series Solution of the Thomas–Fermi Equation † Majorana Form of the Thomas–Fermi Equation † Yoccoz Function † Lagrange–Bürmann Formula † Divisor Sum Identities † Eisenstein Series † Product Representation of exp † Multiple Differentiation of Vector Functions † Expressing Trigonometric Values in Radicals † First Order Modular Transformations † Forced Damped Oscillations † Series for Euler’s Constant † q-Logarithm † Symmetrized Determinant † High Order WKB Approximation † Greenberger–Horne–Zeilinger State † Entangled Four Particle State † Integrating Polynomial Roots † Riemann Surface of a Cubic † Series Solution of the Kepler Equation † Short Time-Series Solution of Newton’s Equation † Lagrange Points of the Three-Body Problem † Implicitization of Lissajou Curves † Evolutes † Orthopodic Locus of Lissajous Curves † Cissoid of Lisssajou Curves † Multiple Light Ray Reflections † Hedgehog Envelope † Supercircle Normal Superpositions † Discriminant Surface † Periodic Surface † 27 Lines on the Clebsch Surface † 28 Bitangents of a Plane Quartic † Pentaellipse † Galilean Invariance of Maxwell Equations † Relativistic Field Transformations † X-Waves † Thomas Precession † Liénard–Wiechert Potential Expansion † Spherical Standing Wave † Ramanujan’s Factorial Expansion † q-Series to q-Products † q-Binomial

†

Multiplicative Series † gcd-Free Partitions † Single

Differential Equations for Nonlinear Systems † Lattice Green’s Function Differential Equation † Puzzles † Newton–Leibniz Theorem in 2D † Square Root of Differential Operator † Polynomials with Identical Coefficients and Roots † Amoebas † Cartesian Leaf Area † Average Distance between Random Points † Series Solution for Duffing Equation † Secular Terms † Implicitization of Various Surfaces † Kronig–Penney Model Riemann Surface † Ellipse Secants Envelope † Lines Intersecting Four Lines † Shortest Triangle Billiard Path † Weak Measurement Identity † Logarithmic Residue † Geometry Puzzle † Differential Equations of Bivariate Polynomials † Graph Eigenvalues † Change of Variables in the Dirac Delta Function † Probability Distributions for Sums † Random Determinants † Integral Representation of Divided Differences † Fourier Transform and Fourier Series † Functional Differentiation † Operator Splitting Formula † Tetrahedron of Maximal Volume



@ @ Solutions

33

ODE for Circles † Modular Equations † Converting Trigonometric Expressions into Algebraic Expressions † Matrix Sign Function † Integration with Scoping † Collecting Powers and Logarithms † Bound State in Continuum † Element Vectors, Mass Matrices, and Stiffness Matrices † Multivariate Minimization † Envelopes of Throw Trajectories † Helpful Warning Messages † Using Ansätze † Schanuel’s Conjecture † Matrix Derivatives † Lewis–Carroll Identities † Abel and Hölder Summation † Extended Poisson Summation Formula † Integration Testing † Detecting the Hidden Use of Approximate Numbers † Functions with Nontrivial Derivatives † Expressing ODEs as Integral Equations † Finding Modular Null Spaces † Canonicalizing Tensor Expressions † Nonsorting “Unioning” † Linear Diophantine Equations † Ramanujan Trigonometric Identities † Cot Identities † Solving the Fokker–Planck Equation for the Forced Damped Oscillator † Implementing Specialized Integrations † Bras and Kets † Density Matrices † Recognizing Algebraic Numbers † Differentiation of Symbolic Vectors † Visualizing the Lagrange Points † Gröbner Walk † Piecewise Parametrizations of Implicit Surfaces † Generalized Clebsch Surfaces † Algorithmic Rewriting of Covariant Equations in 3D Vectors † Darboux–Halphen System † Cubed Sphere Equation † Numerically Checking Integrals Containing Derivatives of Dirac Delta Functions † Lagrange Multipliers † Elementary Symmetric Polynomials

@ @ References

CHAPTER 2

Classical Orthogonal Polynomials 2.0

Remarks

2.1

General Properties of Orthogonal Polynomials Orthogonal Polynomials as Solutions of Sturm–Liouville Eigenvalue Problems † General Properties of Orthogonal Polynomials † Expansion of Arbitrary Functions in Orthogonal Polynomials

2.2

Hermite Polynomials Definition † Graphs † ODE † Orthogonality and Normalization † Harmonic Oscillator Eigenfunctions † Density of States † Shifted Harmonic Oscillator

2.3

Jacobi Polynomials Definition † Graphs † ODE † Orthogonality and Normalization † Electrostatic Interpretation of the Zeros † Pöschl–Teller Potential

2.4

Gegenbauer Polynomials Laplace Equation in n D † Definition † Graphs † ODE † Orthogonality and Normalization † Smoothing the Gibbs Phenomenon

2.5

Laguerre Polynomials Definition † Graphs † ODE † Orthogonality and Normalization † Expanding Riemann Spheres † Summed Atomic Orbitals



34

2.6

Legendre Polynomials Definition † Graphs † ODE † Orthogonality and Normalization † Associated Legendre Polynomials † Modified Pöschl–Teller Potential

2.7

Chebyshev Polynomials of the First Kind Definition † Graphs † ODE † Orthogonality and Normalization † Trigonometric Form † Special Properties

2.8

Chebyshev Polynomials of the Second Kind Definition † Graphs † ODE † Orthogonality and Normalization † Trigonometric Form

2.9

Relationships Among the Orthogonal Polynomials Gegenbauer Polynomials as Special Cases of Jacobi Polynomials † Hermite Polynomials as Special Cases of Associated Laguerre Polynomials † Relations between the Chebyshev Polynomials † Calogero–Sutherland Model † Schmeisser Companion Matrix † Iterated Roots of Orthogonal Polynomials

2.10

Ground-State of the Quartic Oscillator Harmonic and Anharmonic Oscillators † Matrix Elements in the Harmonic Oscillator Basis † High-Precision Eigenvalues from Diagonalizing the Hill Matrix † Lagrange Interpolation-Based Diagonalization † Complex Energy Surfaces † Time-Dependent Schrödinger Equation † 7;-Invariant Oscillators


@ @ Solutions

Mehler’s Formula † Addition Theorem for Hermite Polynomials † Sums of Zeros of Hermite Polynomials † Spherical Harmonics † Sums of Zeros † General Orthogonal Polynomials † Gram-Schmidt Orthogonalization † Power Sums † Elementary Symmetric Polynomials † Newton Relations † Waring Formula † Generalized Lissajous Figures † Hyperspherical Harmonics † Hydrogen Orbitals † Zeros of Hermite Functions for Varying Order † Ground State Energy of Relativistic Pseudodifferential Operator † Moments of Hermite Polynomial Zeros † Coherent States † Smoothed Harmonic Oscillator States † Darboux Isospectral Transformation † Forming Wave Packets from Superpositions † Multidimensional Harmonic Oscillator † High-Order Perturbation Theory † Differential Equation System for Eigenvalues † Time-Dependent Sextic Oscillator † Time Dependent Schrödinger Equation with Calogera Potential Bauer–Rayleigh Identity † Parseval Identity † Transmission through Periodic Structures † Freud’s Weight Function † Wronski Polynomials † Root-Finding Using Differential Equations † Finding Ramification Indices Numerically † Classical and Quantum Mechanical Probabilities for the Harmonic Oscillator † Root Approximant † Using Recursion Relations to Calculate Orthogonal Polynomials



35

@ @ References

CHAPTER 3

Classical Special Functions 3.0

Remarks Information Sources about Special Functions † Experimental Mathematics † Generalized Harmonic Numbers † Position and Momentum Eigenfunctions and Wigner Function of the Liouville Potential † Ramanujan Theta Functions † Modular Identities

3.1

Introduction Simplifying Expressions Containing Special Functions † Expressing Special Functions through Simpler Ones † Indefinite Integrals of Compositions of Elementary Functions † Volume of a Supersphere † 7;-Symmetric Oscillator † Monitoring Simplifying Transformations

3.2

Gamma, Beta, and Polygamma Functions Definitions † Exact Values † Graphs † Riemann Surface of the Incomplete Gamma Function † Pochhammer Symbol

3.3

Error Functions and Fresnel Integrals Definitions † Error Function in the Complex Plane † Iterated Integrals of Error Functions † Free Particle Schrödinger Equation with Piecewise Constant Initial Conditions † Moshinsky Function † Harmonic Oscillator Green’s Function † Fresnel Diffraction on a HalfPlane

3.4

Exponential Integral and Related Functions Definitions † Graphs † Logarithmic Integral and Prime Counting Function

3.5

Bessel and Airy Functions Definitions † Random Walk on a 2D Square Lattice † Fractal Based on Bessel Function † Weber–Schafheitlin Integrals † Bessel Zeros as a Function of the Index † Oscillation of a Circular Drum † Oscillation of a Drum of General Shape † 2D Helmholtz Equation † Eigenvalues and Eigenfunctions of the Stadium Billiard † Free Nonspreading Wave “Packet” † Airy Functions in the Uniform Approximation of Linear Turning Point Problem † Harmonic Oscillator Approximations

3.6

Legendre Functions Definitions † Graphs † Electrostatic Potential in a Conducting Cone

3.7

Hypergeometric Functions Gauss Hypergeometric Function and Generalized Hypergeometric Functions † Some Special Cases † Closed Form of Partial Sums of Taylor Series for Trigonometric Functions † Closed Form Padé Approximations of exp and sign † Generalized Fresnel Integrals † Generalized Exponential Functions † Point Charge Outside a Dielectric Sphere † Finding Contiguous Relations † Regularized Hypergeometric Functions † Solutions of the Hypergeometric Differential Equation † Meijer G Function † Eigenfunctions of the Inverse Harmonic Oscillator † Bivariate Hypergeometric Functions



36

3.8

Elliptic Integrals Integrals Containing Square Roots of Cubics and Quartics † Definitions † Complete and Incomplete Elliptic Integrals † Graphs † Deriving Differential Equations for Incomplete Elliptic Integrals † Green’s Function of the Zeilon Operator † Finding Modular Equations for Ratios of Elliptic Integrals

3.9

Elliptic Functions Inverting Elliptic Integrals † Definitions † Jacobi’s Amplitude Function † Minimal Surface in a Cube Wireframe † Applications of Elliptic Functions † Pendulum Oscillations † Current Flow through a Rectangular Conducting Plate † Arithmetic–Geometric Mean

3.10

Product Log Function Definition † Solving Transcendental Equations † Riemann Surface of the Product Log Function

3.11

Mathieu Functions Differential Equation with Periodic Coefficients † Definition † Characteristic Values † Resonance Tongues † Branch Cuts and Branch Points † Oscillation of an Ellipsoidal Drum † Degenerate Eigenfunctions † Wannier Functions

3.12

Additional Special Functions Expressing Other Special Functions through Built-in Special Functions † More Elliptic Functions † Zeta Functions and Lerch Transcendents

3.13

Solution of Quintic Polynomials Solving Polynomials in Radicals † Klein’s Solution of the Quintic † Tschirnhaus Transformation † Principal Quintic † Belyi Function and Stereographic Projection of an Icosahedron Projection † Solving a Polynomial of Degree 60 through Hypergeometric Functions † Numerical Root Calculation Based on Klein’s Formula




@ @ Solutions

37

Asymptotic Expansions of Bessel Functions † Carlitz Expansion † Meissel’s Formula † Rayleigh Sums † Gumbel Distribution † Generalized Bell Numbers † Borel Summation † Bound State in Continuum † ODEs for Incomplete Elliptic Integrals † Addition Formulas for Elliptic Integrals † Magnetic Field of a Helmholtz Coil † Identities, Expansions, ODEs, and Visualizations of the Weierstrass ƒ Function † Sutherland–Calogero Model † Weierstrass Zeta and Sigma Functions † Lamé Equation † Vortex Lattices † ODEs, Addition Formulas, Series Expansions for the Twelve Jacobi Elliptic Functions † Schrödinger Equations with Potentials that are Rational Functions of the Wave Functions † Periodic Solutions of Nonlinear Evolution Equations † Complex Pendulum † Harmonic Oscillator Eigenvalues † Contour Integral Representation of Bessel Functions † Large Order and Argument Expansion for Bessel Functions † Aperture Diffraction † Circular Andreev Billiard † Contour Integral Representation for Beta Functions † Beta Distribution † Euler’s Constant Limit † Time-Evolution in a Triple-Well Oscillator † Eigenvalues of a Singular Potential † Dependencies in the Numerical Calculation of Special Functions † Hidden Derivative Definitions † Perturbation Theory for a Square Well in an Electric Field † Oscillations of a Pendulum with Finite Mass Cord † Approximation and Asymptotics of Fermi–Dirac Integrals † Sum of All 9-Free Reciprocal Numbers † Green’s Function for 1D Heat Equation † Green’s Function for the Laplace Equation in a Rectangle † Addition Theorems for Theta Functions † Series Expansion of Theta Functions † Bose Gas in a 3D Box † Scattering on a Conducting Cylinder † Poincaré Waves † Scattering on a Dielectric Cylinder † Coulomb Scattering † Spiral Waves † Scattering on a Corrugated Wall † Random Helmholtz Equation Solutions † Toroidal Coordinates † Riemann-Siegel Expansion † Zeros of the Hurwitz Zeta Function † Zeta Zeta Function † Harmonic Polylogarithms † Riemann Surface of Gauss Hypergeometric Functions † Riemann Surface of the Ratio of Complete Elliptic Integrals † Riemann Surface of the Inverse Error Function † Kummer’s 24 Solutions of Gauss Hypergeometric Equation † Differential Equation for Appell Function † Gauss–Lucas Theorem † Roots of Differentiated Polynomials † Coinciding Bessel Zeros † Ramanujan p Formulas † Force-Free Magnetic Fields † Bessel Beams † Gauge Transformation for a Square † Riemann Surface of the Bootstrap Equation † Differential Equations for Powers of Airy Functions † Asymptotic Expansions for the Zeros of Airy Functions † Map-Airy Distribution † Dedekind h ODE † Darboux–Halphen System † Ramanujan Identities for j and l Functions † Generating Identities in Gamma Functions † Modular Equations for Dedekind h Function Truncation of Asymptotic Series † Contour Plots of the Gamma Function † Series of a Gamma Function Ratio † Partial Sums of Taylor Series for sin † Area and Volume of a Hypersphere † All Integrals of Three Compositions of Elementary Functions † Binomial at Negative Integers † Contour Lines of zz † Weierstrass ƒ Function over the Riemann Sphere † Using Gröbner Bases to Derive ODEs † Riemann Surface of Inverse Weierstrass ƒ Function † Rocket with Discrete Propulsion † Monitoring All Internal Calculations † Machine versus High-Precision Evaluations of Special Functions † Checking Numerical Function Evaluation † Zeta Regularized Divergent Products † Fractional Derivatives † Identifying Algebraic Numbers



38

@ @ References

A P P E N D I C E S

APPENDIX A

General References to Computer Algebra and to Mathematica A.0 A.1

Remarks References and Other Sources of Information A.1.1

General References on Algorithms for Computer Algebra General Computer Algebra Books, References, and Websites † Sources of Algorithms † Computer Algebra Journals and Conferences

A.1.2

Comparison of Various Systems Benchmarks and Timing Comparisons

A.1.3

References on Mathematica Books † Journals and Websites † Conferences † Package Libraries † Dedicated Newsgroups † Timing Comparisions

A.1.4

Applications of Computer Algebra Systems Article Samples † Further Information Sources



@ @ References

APPENDIX B (from http://www.mathematicaguidebooks.org)

The Front End, the Help Browser, Notebooks, Stylesheets, Cells, Typesetting, Buttons, Boxes, and All That B.0

Remarks

B.1

Notebooks and Cells as Expressions B.1.1 B.1.2 B.1.3 B.1.4

B.2

The Structure of Notebook The Appearance of Cells Stylesheets Selected Cell Options

Front End Functions and Operations B.2.1 B.2.2

B.3

Navigating and Manipulating Notebook Performing Menu and Keyboard Operations Programmatically

Typesetting and Boxes B.3.1 B.3.2 B.3.3

B.4

Two-Dimensional Formatting Tweaking Formula Appearances Creating Typesetting Rules

Buttons, Hyperlinks, and Palettes B.4.1 B.4.2 B.4.3

B.5

General Buttons Hyperlinks Palettes

Dynamic Boxes B.5.1 B.5.2

B.6

Automatic Numbering Displaying Values Automatically

Special Notebooks B.6.1 B.6.2

Help Browser Notebooks The Message Notebook

B.7

MathLink-Related Operations

B.8

Three Applications B.8.0 B.8.1 B.8.2 B.8.3

Remarks Analyzing the Notebook Version of The Mathematica Book Incorporating the GuideBooks into the Help Browser Evaluating a Complete GuideBooks Chapter Programmatically

@ @ References


39


40

A D D I T I O N S

ADDITIONS FROM THE WEBSITE http://www.mathematicaguidebooks.org

Additional Exercises and Solutions W.0

Remarks

W.7

Additions to Chapter 1 of the Graphics Volume Repeated Breaking of a Stick † Animation of Rotating Tiles of an Aperiodic Tiling † Animation of Circles on Lissajou Figures

W.8

Additions to Chapter 2 of the Graphics Volume Animation of Rotating Textured Möbius Bands † Animation Of Rotating Interlocked Tori † Klein Bottle with Hexagonal Massive Wireframe † Many Random Walkers in 3D † Bivariate Minkowski Function † Farey and Bary Addition † Projections from 4D

W.9

Additions to Chapter 3 of the Graphics Volume Animation of Equal-Eigenvalue Chladny Figures † Animation of Moving Charged Regular Polygons † Graphics of Charged Truchet Patterns

W.10

Additions to Chapter 1 of the Numerics Volume Random Walks with Variable Stepsize † Chaotic Scattering on Three Disks † Vibrating 2D Hilbert Curve † Optimal Projections of Hypercubes † Currents Through a Penrose Tiling † Numerical Solutions of Various Partial Differential Equations † Brain Growth Modeling † Step Bunching Modeling † Swift–Hohenberg Equation † Meinhardt Equations † Complex Ginzburg–Landau Equation Hierarchy † Splitting Localized Structures † Wave Equation with Piecewise-Constant Phase Velocities † Local Induction Approximation † Born–Infeld Wave Equation † Peakon Trains † Vibrations of a Square Koch Drum † Weyl–Berry Law † Diverging Gradients at Inner Corners † Classical and Quantum Mechanical Treatment of a Duffing Oscillator † Calculating Wigner Functions Through Fractional Fourier Transforms † Sub-Ñ Structures in the Wigner Function † Circular Aperture Diffraction Integral † Checking the Cauchy–Born Hypothesis † Schwarz–Christoffel Map for Some Symmetric Polygons † Normalized Banzhaf Indices for the European Union Countries † Wave Propagation on a Torus Surface

W.11

Additions to Chapter 2 of the Numerics Volume A Special Infinite Product of Cosines † A Special p-Related Continued Fraction † Plots of the Argument of Cyclotomic Polynomials



W.12

41

Additions to Chapter 1 of the Symbolics Volume Convergence Radius of the Virial Series for the Bose Gas † Midpoint Quadrature Formula † MacMahon Master Theorem † Adler–Moser Polynomials † Differential Equation for Yablonskii–Vorob’ev Polynomials † Implicit Polynomial Description of A Hypocycloidal Torus † Calculating the Second Feigenbaum Constant † Green’s Function for a Sequence of Delta Function Potentials † Implicit Form of Poynting Vector Equisurfaces † Symmetrically Arranged Points on Spheres † The Isospectral Polygons Bilby and Hawk † Probability Distribution of a Quotient † Vibrations of Springs on a Gosper Curve † Probability Distribution for the Distance Between Two Points from the Unit Square † Animation of the Nodal Lines of a Dirichlet– Neumann Boundary Conditions Transition † Checking Higher Order Generalized WKB Approximation for the Harmonic Oscillator † Evaluating an Iterated Integral † The Kobussen–Leubner–Lopez Lagrangian for the Harmonic Oscillator

W.13

Additions to Chapter 2 of the Symbolics Volume Eigenfunctions of the Hénon–Heiles Potential

W.14

Additions to Chapter 3 of the Symbolics Volume Rational Values of the Gauss Hypergeometric Function † Eigenfunctions of the Discrete Harmonic Oscillator † Average Length of Smallest Component of Multidimensional Unit Vectors † Differential Equation of the Jacobi Elliptic Function sn with Respect to the Modul † A Certain Sum of Zeta Functions † High-Order Series Expansion of Harmonic Numbers of a Given Size † Movement of a Sliding Spinning Disk † Identities of Jacobi Theta Function for Special Argument Values

@ @ References


INTRODUCTION AND ORIENTATION

to The Mathematica GuideBooks Language Concepts—Programming Examples— Visualization Demos—Scientific Applications

0.1 Overview à 0.1.1 Content Summaries The Mathematica GuideBooks are published as four independent books: The Mathematica GuideBook to Programming, The Mathematica GuideBook to Graphics, The Mathematica GuideBook to Numerics, and The Mathematica GuideÖ Book to Symbolics. † The Programming volume deals with the structure of Mathematica expressions and with Mathematica as a programming language. This volume includes the discussion of the hierarchical construction of all Mathematica objects out of symbolic expressions (all of the form head[argument]), the ultimate building blocks of expressions (numbers, symbols, and strings), the definition of functions, the application of rules, the recognition of patterns and their efficient application, the order of evaluation, program flows and program structure, the manipulation of lists (the universal container for Mathematica expressions of all kinds), as well as a number of topics specific to the Mathematica programming language. Various programming styles, especially Mathematica’s powerful functional programming constructs, are covered in detail. † The Graphics volume deals with Mathematica’s two-dimensional (2D) and three-dimensional (3D) graphics. The chapters of this volume give a detailed treatment on how to create images from graphics primitives, such as points, lines, and polygons. This volume also covers graphically displaying functions given either analytically or in discrete form. A number of images from the Mathematica Graphics Gallery are also reconstructed. Also discussed is the generation of pleasing scientific visualizations of functions, formulas, and algorithms. A variety of such examples are given. † The Numerics volume deals with Mathematica’s numerical mathematics capabilities—the indispensable sledgehammer tools for dealing with virtually any “real life” problem. The arithmetic types (fast machine, exact integer and rational, verified high-precision, and interval arithmetic) are carefully analyzed. Fundamental numerical operations, such as compilation of programs, numerical Fourier transforms, minimization, numerical solution of equations, and ordinary/partial differential equations are analyzed in detail and are applied to a large number of examples in the main text and in the solutions to the exercises. † The Symbolics volume deals with Mathematica’s symbolic mathematical capabilities—the real heart of Mathematica and the ingredient of the Mathematica software system that makes it so unique and powerful. Structural and mathematical operations on systems of polynomials are fundamental to many symbolic calculations and are covered in detail. The



2

solution of equations and differential equations, as well as the classical calculus operations, are exhaustively treated. In addition, this volume discusses and employs the classical orthogonal polynomials and special functions of mathematical physics. To demonstrate the symbolic mathematics power, a variety of problems from mathematics and physics are discussed. The four GuideBooks contain about 25,000 Mathematica inputs, representing more than 75,000 lines of commented Mathematica code. (For the reader already familiar with Mathematica, here is a more precise measure: The Leaf Count of all inputs would be about 900,000 when collected in a list.) The GuideBooks also have more than 4,000 graphics, 150 animations, 11,000 references, and 1,000 exercises. More than 10,000 hyperlinked index entries and hundreds of hyperlinks from the overview sections connect all parts in a convenient way. The evaluated notebooks of all four volumes have a cumulative file size of about 20 GB. Although these numbers may sound large, the Mathematica GuideBooks actually cover only a portion of Mathematica’s functionality and features and give only a glimpse into the possibilities Mathematica offers to generate graphics, solve problems, model systems, and discover new identities, relations, and algorithms. The Mathematica code is explained in detail throughout all chapters. More than 13,000 comments are scattered throughout all inputs and code fragments.

à 0.1.2 Relation of the Four Volumes The four volumes of the GuideBooks are basically independent, in the sense that readers familiar with Mathematica programming can read any of the other three volumes. But a solid working knowledge of the main topics discussed in The Mathematica GuideBook to Programming—symbolic expressions, pure functions, rules and replacements, and list manipulations—is required for the Graphics, Numerics, and Symbolics volumes. Compared to these three volumes, the Programming volume might appear to be a bit “dry”. But similar to learning a foreign language, before being rewarded with the beauty of novels or a poem, one has to sweat and study. The whole suite of graphical capabilities and all of the mathematical knowledge in Mathematica are accessed and applied through lists, patterns, rules, and pure functions, the material discussed in the Programming volume. Naturally, graphics are the center of attention of the The Mathematica GuideBook to Graphics. While in the Programming volume some plotting and graphics for visualization are used, graphics are not crucial for the Programming volume. The reader can safely skip the corresponding inputs to follow the main programming threads. The Numerics and Symbolics volumes, on the other hand, make heavy use of the graphics knowledge acquired in the Graphics volume. Hence, the prerequisites for the Numerics and Symbolics volumes are a good knowledge of Mathematica’s programming language and of its graphics system. The Programming volume contains only a few percent of all graphics, the Graphics volume contains about two-thirds, and the Numerics and Symbolics volume, about one-third of the overall 4,000+ graphics. The Programming and Graphics volumes use some mathematical commands, but they restrict the use to a relatively small number (especially Expand, Factor, Integrate, Solve). And the use of the function N for numericalization is unavoidable for virtually any “real life” application of Mathematica. The last functions allow us to treat some mathematically not uninteresting examples in the Programming and Graphics volumes. In addition to putting these functions to work for nontrivial problems, a detailed discussion of the mathematics functions of Mathematica takes place exclusively in the Numerics and Symbolics volumes. The Programming and Graphics volumes contain a moderate amount of mathematics in the examples and exercises, and focus on programming and graphics issues. The Numerics and Symbolics volumes contain a substantially larger amount of mathematics. Although printed as four books, the fourteen individual chapters (six in the Programming volume, three in the Graphics volume, two in the Numerics volume, and three in the Symbolics volume) of the Mathematica GuideBooks form one organic whole, and the author recommends a strictly sequential reading, starting from Chapter 1 of the Programming volume and ending with Chapter 3 of the Symbolics volume for gaining the maximum benefit. The electronic compo-



3

nent of each book contains the text and inputs from all the four GuideBooks, together with a comprehensive hyperlinked index. The four volumes refer frequently to one another.

à 0.1.3 Chapter Structure A rough outline of the content of a chapter is the following: † The main body discusses the Mathematica functions belonging to the chapter subject, as well their options and attributes. Generically, the author has attempted to introduce the functions in a “natural order”. But surely, one cannot be axiomatic with respect to the order. (Such an order of the functions is not unique, and the author intentionally has “spread out” the introduction of various Mathematica functions across the four volumes.) With the introduction of a function, some small examples of how to use the functions and comparisons of this function with related ones are given. These examples typically (with the exception of some visualizations in the Programming volume) incorporate functions already discussed. The last section of a chapter often gives a larger example that makes heavy use of the functions discussed in the chapter. † A programmatically constructed overview of each chapter functions follows. The functions listed in this section are hyperlinked to their attributes and options, as well as to the corresponding reference guide entries of The Mathematica Book. † A set of exercises and potential solutions follow. Because learning Mathematica through examples is very efficient, the proposed solutions are quite detailed and form up to 50% of the material of a chapter. † References end the chapter. Note that the first few chapters of the Programming volume deviate slightly from this structure. Chapter 1 of the Programming volume gives a general overview of the kind of problems dealt with in the four GuideBooks. The second, third, and fourth chapters of the Programming volume introduce the basics of programming in Mathematica. Starting with Chapters 5 of the Programming volume and throughout the Graphics, Numerics, and Symbolics volumes, the above-described structure applies. In the 14 chapters of the GuideBooks the author has chosen a “we” style for the discussions of how to proceed in constructing programs and carrying out calculations to include the reader intimately.



4

à 0.1.4 Code Presentation Style The typical style of a unit of the main part of a chapter is: Define a new function, discuss its arguments, options, and attributes, and then give examples of its usage. The examples are virtually always Mathematica inputs and outputs. The majority of inputs is in InputForm are the notebooks. On occasion StandardForm is also used. Although Stan dardForm mimics classical mathematics notation and makes short inputs more readable, for “program-like” inputs, InputForm is typically more readable and easier and more natural to align. For the outputs, StandardForm is used by default and occasionally the author has resorted to InputForm or FullForm to expose digits of numbers and to TraditionalForm for some formulas. Outputs are mostly not programs, but nearly always “results” (often mathematical expressions, formulas, identities, or lists of numbers rather than program constructs). The world of Mathematica users is divided into three groups, and each of them has a nearly religious opinion on how to format Mathematica code [1÷], [2÷]. The author follows the InputForm cult(ure) and hopes that the Mathematica users who do everything in either StandardForm or TraditionalForm will bear with him. If the reader really wants to see all code in either StandardForm or TraditionalForm, this can easily be done with the Convert To item from the Cell menu. (Note that the relation between InputForm and StandardForm is not symmetric. The InputForm cells of this book have been line-broken and aligned by hand. Transforming them into StandardForm or TraditionalForm cells works well because one typically does not line-break manually and align Mathematica code in these cell types. But converting StandardForm or TraditionalForm cells into InputForm cells results in much less pleasing results.) In the inputs, special typeset symbols for Mathematica functions are typically avoided because they are not monospaced. But the author does occasionally compromise and use Greek, script, Gothic, and doublestruck characters. In a book about a programming language, two other issues come always up: indentation and placement of the code. † The code of the GuideBooks is largely consistently formatted and indented. There are no strict guidelines or even rules on how to format and indent Mathematica code. The author hopes the reader will find the book’s formatting style readable. It is a compromise between readability (mental parsabililty) and space conservation, so that the printed version of the Mathematica GuideBook matches closely the electronic version. † Because of the large number of examples, a rather imposing amount of Mathematica code is presented. Should this code be present only on the disk, or also in the printed book? If it is in the printed book, should it be at the position where the code is used or at the end of the book in an appendix? Many authors of Mathematica articles and books have strong opinions on this subject. Because the main emphasis of the Mathematica GuideBooks is on solving problems with Mathematica and not on the actual problems, the GuideBooks give all of the code at the point where it is needed in the printed book, rather than “hiding” it in packages and appendices. In addition to being more straightforward to read and conveniently allowing us to refer to elements of the code pieces, this placement makes the correspondence between the printed book and the notebooks close to 1:1, and so working back and forth between the printed book and the notebooks is as straightforward as possible.



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0.2 Requirements à 0.2.1 Hardware and Software Throughout the GuideBooks, it is assumed that the reader has access to a computer running a current version of Mathematica (version 5.0/5.1 or newer). For readers without access to a licensed copy of Mathematica, it is possible to view all of the material on the disk using a trial version of Mathematica. (A trial version is downloadable from http://www.wolfram.com/products/mathematica/trial.cgi.) The files of the GuideBooks are relatively large, altogether more than 20 GB. This is also the amount of hard disk space needed to store uncompressed versions of the notebooks. To view the notebooks comfortably, the reader’s computer needs 128 MB RAM; to evaluate the evaluation units of the notebooks 1 GB RAM or more is recommended. In the GuideBooks, a large number of animations are generated. Although they need more memory than single pictures, they are easy to create, to animate, and to store on typical year-2005 hardware, and they provide a lot of joy.

à 0.2.2 Reader Prerequisites Although prior Mathematica knowledge is not needed to read The Mathematica GuideBook to Programming, it is assumed that the reader is familiar with basic actions in the Mathematica front end, including entering Greek characters using the keyboard, copying and pasting cells, and so on. Freely available tutorials on these (and other) subjects can be found at http://library.wolfram.com. For a complete understanding of most of the GuideBooks examples, it is desirable to have a background in mathematics, science, or engineering at about the bachelor’s level or above. Familiarity with mechanics and electrodynamics is assumed. Some examples and exercises are more specialized, for instance, from quantum mechanics, finite element analysis, statistical mechanics, solid state physics, number theory, and other areas. But the GuideBooks avoid very advanced (but tempting) topics such as renormalization groups [6÷], parquet approximations [27÷], and modular moonshines [14÷]. (Although Mathematica can deal with such topics, they do not fit the character of the Mathematica GuideBooks but rather the one of a Mathematica Topographical Atlas [a monumental work to be carried out by the Mathematica–Bourbakians of the 21st century]). Each scientific application discussed has a set of references. The references should easily give the reader both an overview of the subject and pointers to further references.

0.3 What the GuideBooks Are and What They Are Not à 0.3.1 Doing Computer Mathematics As discussed in the Preface, the main goal of the GuideBooks is to demonstrate, showcase, teach, and exemplify scientific problem solving with Mathematica. An important step in achieving this goal is the discussion of Mathematica functions that allow readers to become fluent in programming when creating complicated graphics or solving scientific problems. This again means that the reader must become familiar with the most important programming, graphics, numerics, and symbolics functions, their arguments, options, attributes, and a few of their time and space complexities. And the reader must know which functions to use in each situation.



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The GuideBooks treat only aspects of Mathematica that are ultimately related to “doing mathematics”. This means that the GuideBooks focus on the functionalities of the kernel rather than on those of the front end. The knowledge required to use the front end to work with the notebooks can easily be gained by reading the corresponding chapters of the online documentation of Mathematica. Some of the subjects that are treated either lightly or not at all in the GuideBooks include the basic use of Mathematica (starting the program, features, and special properties of the notebook front end [16÷]), typesetting, the preparation of packages, external file operations, the communication of Mathematica with other programs via MathLink, special formatting and string manipulations, computer- and operating system-specific operations, audio generation, and commands available in various packages. “Packages” includes both, those distributed with Mathematica as well as those available from the Mathematica Information Center (http://library.wolfram.com/infocenter) and commercial sources, such as MathTensor for doing general relativity calculations (http://smc.vnet.net/MathTensor.html) or FeynCalc for doing high-energy physics calculations (http://www.feyncalc.org). This means, in particular, that probability and statistical calculations are barely touched on because most of the relevant commands are contained in the packages. The GuideBooks make little or no mention of the machine-dependent possibilities offered by the various Mathematica implementations. For this information, see the Mathematica documentation. Mathematical and physical remarks introduce certain subjects and formulas to make the associated Mathematica implementations easier to understand. These remarks are not meant to provide a deep understanding of the (sometimes complicated) physical model or underlying mathematics; some of these remarks intentionally oversimplify matters. The reader should examine all Mathematica inputs and outputs carefully. Sometimes, the inputs and outputs illustrate little-known or seldom-used aspects of Mathematica commands. Moreover, for the efficient use of Mathematica, it is very important to understand the possibilities and limits of the built-in commands. Many commands in Mathematica allow different numbers of arguments. When a given command is called with fewer than the maximum number of arguments, an internal (or user-defined) default value is used for the missing arguments. For most of the commands, the maximum number of arguments and default values are discussed. When solving problems, the GuideBooks generically use a “straightforward” approach. This means they are not using particularly clever tricks to solve problems, but rather direct, possibly computationally more expensive, approaches. (From time to time, the GuideBooks even make use of a “brute force” approach.) The motivation is that when solving new “real life” problems a reader encounters in daily work, the “right mathematical trick” is seldom at hand. Nevertheless, the reader can more often than not rely on Mathematica being powerful enough to often succeed in using a straightforward approach. But attention is paid to Mathematica-specific issues to find time- and memory-efficient implementations—something that should be taken into account for any larger program. As already mentioned, all larger pieces of code in this book have comments explaining the individual steps carried out in the calculations. Many smaller pieces of code have comments when needed to expedite the understanding of how they work. This enables the reader to easily change and adapt the code pieces. Sometimes, when the translation from traditional mathematics into Mathematica is trivial, or when the author wants to emphasize certain aspects of the code, we let the code “speak for itself”. While paying attention to efficiency, the GuideBooks only occasionally go into the computational complexity ([8÷], [40÷], and [7÷]) of the given implementations. The implementation of very large, complicated suites of algorithms is not the purpose of the GuideBooks. The Mathematica packages included with Mathematica and the ones at MathSource (http://library.wolfram.com/database/MathSource ) offer a rich variety of selfstudy material on building large programs. Most general guidelines for writing code for scientific calculations (like descriptive variable names and modularity of code; see, e.g., [19÷] for a review) apply also to Mathematica programs. The programs given in a chapter typically make use of Mathematica functions discussed in earlier chapters. Using commands from later chapters would sometimes allow for more efficient techniques. Also, these programs emphasize the use of commands from the current chapter. So, for example, instead of list operation, from a complexity point of view, hashing techniques or tailored data structures might be preferable. All subsections and sections are “self-contained” (meaning that no other code than the one presented is needed to evaluate the subsections and sections). The



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price for this “self-containedness” is that from time to time some code has to be repeated (such as manipulating polygons or forming random permutations of lists) instead of delegating such programming constructs to a package. Because this repetition could be construed as boring, the author typically uses a slightly different implementation to achieve the same goal.

à 0.3.2 Programming Paradigms In the GuideBooks, the author wants to show the reader that Mathematica supports various programming paradigms and also show that, depending on the problem under consideration and the goal (e.g., solution of a problem, test of an algorithm, development of a program), each style has its advantages and disadvantages. (For a general discussion concerning programming styles, see [3÷], [41÷], [23÷], [32÷], [15÷], and [9÷].) Mathematica supports a functional programming style. Thus, in addition to classical procedural programs (which are often less efficient and less elegant), programs using the functional style are also presented. In the first volume of the Mathematica GuideBooks, the programming style is usually dictated by the types of commands that have been discussed up to that point. A certain portion of the programs involve recursive, rule-based programming. The choice of programming style is, of course, partially (ultimately) a matter of personal preference. The GuideBooks’ main aim is to explain the operation, limits, and efficient application of the various Mathematica commands. For certain commands, this dictates a certain style of programming. However, the various programming styles, with their advantages and disadvantages, are not the main concern of the GuideBooks. In working with Mathematica, the reader is likely to use different programming styles depending if one wants a quick one-time calculation or a routine that will be used repeatedly. So, for a given implementation, the program structure may not always be the most elegant, fastest, or “prettiest”. The GuideBooks are not a substitute for the study of The Mathematica Book [45÷] http://documents.wolfram.com/mathematica). It is impossible to acquire a deeper (full) understanding of Mathematica without a thorough study of this book (reading it twice from the first to the last page is highly recommended). It defines the language and the spirit of Mathematica. The reader will probably from time to time need to refer to parts of it, because not all commands are discussed in the GuideBooks. However, the story of what can be done with Mathematica does not end with the examples shown in The Mathematica Book. The Mathematica GuideBooks go beyond The Mathematica Book. They present larger programs for solving various problems and creating complicated graphics. In addition, the GuideBooks discuss a number of commands that are not or are only fleetingly mentioned in the manual (e.g., some specialized methods of mathematical functions and functions from the Developer` and Experimen tal` contexts), but which the author deems important. In the notebooks, the author gives special emphasis to discussions, remarks, and applications relating to several commands that are typical for Mathematica but not for most other programming languages, e.g., Map, MapAt, MapIndexed, Distribute, Apply, Replace, ReplaceAll, Inner, Outer, Fold, Nest, NestList, FixedPoint, FixedPointList, and Function. These commands allow to write exceptionally elegant, fast, and powerful programs. All of these commands are discussed in The Mathematica Book and others that deal with programming in Mathematica (e.g., [33÷], [34÷], and [42÷]). However, the author’s experience suggests that a deeper understanding of these commands and their optimal applications comes only after working with Mathematica in the solution of more complicated problems. Both the printed book and the electronic component contain material that is meant to teach in detail how to use Mathematica to solve problems, rather than to present the underlying details of the various scientific examples. It cannot be overemphasized that to master the use of Mathematica, its programming paradigms and individual functions, the reader must experiment; this is especially important, insightful, easily verifiable, and satisfying with graphics, which involve manipulating expressions, making small changes, and finding different approaches. Because the results can easily be visually checked, generating and modifying graphics is an ideal method to learn programming in Mathematica.



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0.4 Exercises and Solutions à 0.4.1 Exercises Each chapter includes a set of exercises and a detailed solution proposal for each exercise. When possible, all of the purely Mathematica-programming related exercises (these are most of the exercises of the Programming volume) should be solved by every reader. The exercises coming from mathematics, physics, and engineering should be solved according to the reader’s interest. The most important Mathematica functions needed to solve a given problem are generally those of the associated chapter. For a rough orientation about the content of an exercise, the subject is included in its title. The relative degree of difficulty is indicated by level superscript of the exercise number ( L1 indicates easy, L2 indicates medium, and L3 indicates difficult). The author’s aim was to present understandable interesting examples that illustrate the Mathematica material discussed in the corresponding chapter. Some exercises were inspired by recent research problems; the references given allow the interested reader to dig deeper into the subject. The exercises are intentionally not hyperlinked to the corresponding solution. The independent solving of the exercises is an important part of learning Mathematica.



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à 0.4.2 Solutions The GuideBooks contain solutions to each of the more than 1,000 exercises. Many of the techniques used in the solutions are not just one-line calls to built-in functions. It might well be that with further enhancements, a future version of Mathematica might be able to solve the problem more directly. (But due to different forms of some results returned by Mathematica, some problems might also become more challenging.) The author encourages the reader to try to find shorter, more clever, faster (in terms of runtime as well complexity), more general, and more elegant solutions. Doing various calculations is the most effective way to learn Mathematica. A proper Mathematica implementation of a function that solves a given problem often contains many different elements. The function(s) should have sensibly named and sensibly behaving options; for various (machine numeric, high-precision numeric, symbolic) inputs different steps might be required; shielding against inappropriate input might be needed; different parameter values might require different solution strategies and algorithms, helpful error and warning messages should be available. The returned data structure should be intuitive and easy to reuse; to achieve a good computational complexity, nontrivial data structures might be needed, etc. Most of the solutions do not deal with all of these issues, but only with selected ones and thereby leave plenty of room for more detailed treatments; as far as limit, boundary, and degenerate cases are concerned, they represent an outline of how to tackle the problem. Although the solutions do their job in general, they often allow considerable refinement and extension by the reader. The reader should consider the given solution to a given exercise as a proposal; quite different approaches are often possible and sometimes even more efficient. The routines presented in the solutions are not the most general possible, because to make them foolproof for every possible input (sensible and nonsensical, evaluated and unevaluated, numerical and symbolical), the books would have had to go considerably beyond the mathematical and physical framework of the GuideBooks. In addition, few warnings are implemented for improper or improperly used arguments. The graphics provided in the solutions are mostly subject to a long list of refinements. Although the solutions do work, they are often sketchy and can be considerably refined and extended by the reader. This also means that the provided solutions to the exercises programs are not always very suitable for solving larger classes of problems. To increase their applicability would require considerably more code. Thus, it is not guaranteed that given routines will work correctly on related problems. To guarantee this generality and scalability, one would have to protect the variables better, implement formulas for more general or specialized cases, write functions to accept different numbers of variables, add typechecking and error-checking functions, and include corresponding error messages and warnings. To simplify working through the solutions, the various steps of the solution are commented and are not always packed in a Module or Block. In general, only functions that are used later are packed. For longer calculations, such as those in some of the exercises, this was not feasible and intended. The arguments of the functions are not always checked for their appropriateness as is desirable for robust code. But, this makes it easier for the user to test and modify the code.



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0.5 The Books Versus the Electronic Components à 0.5.1 Working with the Notebooks Each volume of the GuideBooks comes with a multiplatform DVD, containing fourteen main notebooks tailored for Mathematica 4 and compatible with Mathematica 5. Each notebook corresponds to a chapter from the printed books. (To avoid large file sizes of the notebooks, all animations are located in the Animations directory and not directly in the chapter notebooks.) The chapters (and so the corresponding notebooks) contain a detailed description and explanation of the Mathematica commands needed and used in applications of Mathematica to the sciences. Discussions on Mathematica functions are supplemented by a variety of mathematics, physics, and graphics examples. The notebooks also contain complete solutions to all exercises. Forming an electronic book, the notebooks also contain all text, as well as fully typeset formulas, and reader-editable and reader-changeable input. (Readers can copy, paste, and use the inputs in their notebooks.) In addition to the chapter notebooks, the DVD also includes a navigation palette and fully hyperlinked table of contents and index notebooks. The Mathematica notebooks corresponding to the printed book are fully evaluated. The evaluated chapter notebooks also come with hyperlinked overviews; these overviews are not in the printed book. When reading the printed books, it might seem that some parts are longer than needed. The reader should keep in mind that the primary tool for working with the Mathematica kernel are Mathematica notebooks and that on a computer screen and there “length does not matter much”. The GuideBooks are basically a printout of the notebooks, which makes going back and forth between the printed books and the notebooks very easy. The GuideBooks give large examples to encourage the reader to investigate various Mathematica functions and to become familiar with Mathematica as a system for doing mathematics, as well as a programming language. Investigating Mathematica in the accompanying notebooks is the best way to learn its details. To start viewing the notebooks, open the table of contents notebook TableOfContents.nb. Mathematica notebooks can contain hyperlinks, and all entries of the table of contents are hyperlinked. Navigating through one of the chapters is convenient when done using the navigator palette GuideBooksNavigator.nb. When opening a notebook, the front end minimizes the amount of memory needed to display the notebook by loading it incrementally. Depending on the reader’s hardware, this might result in a slow scrolling speed. Clicking the “Load notebook cache” button of the GuideBooksNavigator palette speeds this up by loading the complete notebook into the front end. For the vast majority of sections, subsections, and solutions of the exercises, the reader can just select such a structural unit and evaluate it (at once) on a year-2005 computer (¥512 MB RAM) typically in a matter of minutes. Some sections and solutions containing many graphics may need hours of computation time. Also, more than 50 pieces of code run hours, even days. The inputs that are very memory intensive or produce large outputs and graphics are in inactive cells which can be activated by clicking the adjacent button. Because of potentially overlapping variable names between various sections and subsections, the author advises the reader not to evaluate an entire chapter at once. Each smallest self-contained structural unit (a subsection, a section without subsections, or an exercise) should be evaluated within one Mathematica session starting with a freshly started kernel. At the end of each unit is an input cell. After evaluating all input cells of a unit in consecutive order, the input of this cell generates a short summary about the entire Mathematica session. It lists the number of evaluated inputs, the kernel CPU time, the wall clock time, and the maximal memory used to evaluate the inputs (excluding the resources needed to evaluate the Program cells). These numbers serve as a guide for the reader about the to-be-expected running times and memory needs. These numbers can deviate from run to run. The wall clock time can be substantially larger than the CPU time due to other processes



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running on the same computer and due to time needed to render graphics. The data shown in the evaluated notebooks came from a 2.5 GHz Linux computer. The CPU times are generically proportional to the computer clock speed, but can deviate within a small factor from operating system to operating system. In rare, randomly occurring cases slower computers can achieve smaller CPU and wall clock times than faster computers, due to internal time-constrained simplification processes in various symbolic mathematics functions (such as Integrate, Sum, DSolve, …). The Overview Section of the chapters is set up for a front end and kernel running on the same computer and having access to the same file system. When using a remote kernel, the directory specification for the package Overview.m must be changed accordingly. References can be conveniently extracted from the main text by selecting the cell(s) that refer to them (or parts of a cell) and then clicking the “Extract References” button. A new notebook with the extracted references will then appear. The notebooks contain color graphics. (To rerender the pictures with a greater color depth or at a larger size, choose Rerender Graphics from the Cell menu.) With some of the colors used, black-and-white printouts occasionally give low-contrast results. For better black-and-white printouts of these graphics, the author recommends setting the Color Output option of the relevant graphics function to GrayLevel. The notebooks with animations (in the printed book, animations are typically printed as an array of about 10 to 20 individual graphics) typically contain between 60 and 120 frames. Rerunning the corresponding code with a large number of frames will allow the reader to generate smoother and longer-running animations. Because many cell styles used in the notebooks are unique to the GuideBooks, when copying expressions and cells from the GuideBooks notebooks to other notebooks, one should first attach the style sheet notebook GuideBooksStylesheet.nb to the destination notebook, or define the needed styles in the style sheet of the destination notebook.

à 0.5.2 Reproducibility of the Results The 14 chapter notebooks contained in the electronic version of the GuideBooks were run mostly with Mathematica 5.1 on a 2 GHz Intel Linux computer with 2 GB RAM. They need more than 100 hours of evaluation time. (This does not include the evaluation of the currently unevaluatable parts of code after the Make Input buttons.) For most subsections and sections, 512 MB RAM are recommended for a fast and smooth evaluation “at once” (meaning the reader can select the section or subsection, and evaluate all inputs without running out of memory or clearing variables) and the rendering of the generated graphic in the front end. Some subsections and sections need more memory when run. To reduce these memory requirements, the author recommends restarting the Mathematica kernel inside these subsections and sections, evaluating the necessary definitions, and then continuing. This will allow the reader to evaluate all inputs. In general, regardless of the computer, with the same version of Mathematica, the reader should get the same results as shown in the notebooks. (The author has tested the code on Sun and Intel-based Linux computers, but this does not mean that some code might not run as displayed (because of different configurations, stack size settings, etc., but the disclaimer from the Preface applies everywhere). If an input does not work on a particular machine, please inform the author. Some deviations from the results given may appear because of the following: † Inputs involving the function Random[…] in some form. (Often SeedRandom to allow for some kind of reproducibility and randomness at the same time is employed.) † Mathematica commands operating on the file system of the computer, or make use of the type of computer (such inputs need to be edited using the appropriate directory specifications). † Calculations showing some of the differences of floating-point numbers and the machine-dependent representation of these on various computers. † Pictures using various fonts and sizes because of their availability (or lack thereof) and shape on different computers. † Calculations involving Timing because of different clock speeds, architectures, operating systems, and libraries. † Formats of results depending on the actual window width and default font size. (Often, the corresponding inputs will contain Short.)



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Using anything other than Mathematica Version 5.1 might also result in different outputs. Examples of results that change form, but are all mathematically correct and equivalent, are the parameter variables used in underdetermined systems of linear equations, the form of the results of an integral, and the internal form of functions like Interpolat ingFunction and CompiledFunction. Some inputs might no longer evaluate the same way because functions from a package were used and these functions are potentially built-in functions in a later Mathematica version. Mathematica is a very large and complicated program that is constantly updated and improved. Some of these changes might be design changes, superseded functionality, or potentially regressions, and as a result, some of the inputs might not work at all or give unexpected results in future versions of Mathematica.

à 0.5.3 Earlier Versions of the Notebooks The first printing of the Programming volume and the Graphics volumes of the Mathematica GuideBooks were published in October 2004. The electronic components of these two books contained the corresponding evaluated chapter notebooks as well as unevaluated versions of preversions of the notebooks belonging to the Numerics and Symbolics volumes. Similarly, the electronic components of the Numerics and Symbolics volume contain the corresponding evaluated chapter notebooks and unevaluated copies of the notebooks of the Programming and Graphics volumes. This allows the reader to follow cross-references and look up relevant concepts discussed in the other volumes. The author has tried to keep the notebooks of the GuideBooks as up-to-date as possible. (Meaning with respect to the efficient and appropriate use of the latest version of Mathematica, with respect to maintaining a list of references that contains new publications, and examples, and with respect to incorporating corrections to known problems, errors, and mistakes). As a result, the notebooks of all four volumes that come with later printings of the Programming and Graphics volumes, as well with the Numerics and Symbolics volumes will be different and supersede the earlier notebooks originally distributed with the Programming and Graphics volumes. The notebooks that come with the Numerics and Symbolics volumes are genuine Mathematica Version 5.1 notebooks. Because most advances in Mathematica Version 5 and 5.1 compared with Mathematica Version 4 occurred in functions carrying out numerical and symbolical calculations, the notebooks associated with Numerics and Symbolics volumes contain a substantial amount of changes and additions compared with their originally distributed version.

0.6 Style and Design Elements à 0.6.1 Text and Code Formatting The GuideBooks are divided into chapters. Each chapter consists of several sections, which frequently are further subdivided into subsections. General remarks about a chapter or a section are presented in the sections and subsections numbered 0. (These remarks usually discuss the structure of the following section and give teasers about the usefulness of the functions to be discussed.) Also, sometimes these sections serve to refresh the discussion of some functions already introduced earlier. Following the style of The Mathematica Book [45÷], the GuideBooks use the following fonts: For the main text, Times; for Mathematica inputs and built-in Mathematica commands, Courier plain (like Plot); and for user-supplied arguments, Times italic (like userArgument1 ). Built-in Mathematica functions are introduced in the following style:



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MathematicaFunctionToBeIntroduced[typeIndicatingUserSuppliedArgument(s)]

is a description of the built-in command MathematicaFunctionToBeIntroduced upon its first appearance. A definition of the command, along with its parameters is given. Here, typeIndicatingUserSuppliedArgument(s) is one (or more) user-supplied expression(s) and may be written in an abbreviated form or in a different way for emphasis. The actual Mathematica inputs and outputs appear in the following manner (as mentioned above, virtually all inputs are given in InputForm). In[5]:=

Out[6]=

(* A comment. It will be/is ignored as Mathematica input: Return only one of the solutions *) Last[Solve[{x^2 - y == 1, x - y^2 == 1}, {x, y}]] :x → −

J9 +

2ê3

1 3

+4

3 J9 +

2

177 N

+

1ê3

y → −2

3 J9 +

2

177 N

+

177 N

2ê3

, 3 22ê3 31ê3

J 12 J9 +

177 NN

1ê3

32ê3

>

When referring in text to variables of Mathematica inputs and outputs, the following convention is used: Fixed, nonpattern variables (including local variables) are printed in Courier plain (the equations solved above contained the variables x and y). User supplied arguments to built-in or defined functions with pattern variables are printed in Times italic. The next input defines a function generating a pair of polynomial equations in x and y. In[7]:=

equationPair[x_, y_] := {x^2 - y == 1, x - y^2 == 1}

x and y are pattern variables (usimng the same letters, but a different font from the actual code fragments x_ and y_)

that can stand for any argument. Here we call the function equationPair with the two arguments u + v and w z. In[8]:= Out[8]=

equationPair[u + v, w - z] 9Hu + vL2 − w + z

1, u + v − Hw − zL2

1=

Occasionally, explanation about a mathematics or physics topic is given before the corresponding Mathematica implementation is discussed. These sections are marked as follows:

Mathematical Remark: Special Topic in Mathematics or Physics A short summary or review of mathematical or physical ideas necessary for the following example(s). 1

From time to time, Mathematica is used to analyze expressions, algorithms, etc. In some cases, results in the form of English sentences are produced programmatically. To differentiate such automatically generated text from the main text, in most instances such text is prefaced by “ë” (structurally the corresponding cells are of type "PrintText" versus "Text" for author-written cells). Code pieces that either run for quite long, or need a lot of memory, or are tangent to the current discussion are displayed in the following manner.



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Make Input

mathematicaCodeWhichEitherRunsVeryLongOrThatIsVeryMemoryIntensive OrThatProducesAVeryLargeGraphicOrThatIsASideTrackToTheSubjectUnder Discussion (* with some comments on how the code works *)

To run a code piece like this, click the Make Input button above it. This will generate the corresponding input cell that can be evaluated if the reader’s computer has the necessary resources. The reader is encouraged to add new inputs and annotations to the electronic notebooks. There are two styles for readeradded material: "ReaderInput" (a Mathematica input style and simultaneously the default style for a new cell) and "ReaderAnnotation" (a text-style cell type). They are primarily intended to be used in the Reading environment. These two styles are indented more than the default input and text cells, have a green left bar and a dingbat. To access the "ReaderInput" and "ReaderAnnotation" styles, press the system-dependent modifier key (such as Control or Command) and 9 and 7, respectively.

à 0.6.2 References Because the GuideBooks are concerned with the solution of mathematical and physical problems using Mathematica and are not mathematics or physics monographs, the author did not attempt to give complete references for each of the applications discussed [38÷], [20÷]. The references cited in the text pertain mainly to the applications under discussion. Most of the citations are from the more recent literature; references to older publications can be found in the cited ones. Frequently URLs for downloading relevant or interesting information are given. (The URL addresses worked at the time of printing and, hopefully, will be still active when the reader tries them.) References for Mathematica, for algorithms used in computer algebra, and for applications of computer algebra are collected in the Appendix A. The references are listed at the end of each chapter in alphabetical order. In the notebooks, the references are hyperlinked to all their occurrences in the main text. Multiple references for a subject are not cited in numerical order, but rather in the order of their importance, relevance, and suggested reading order for the implementation given. In a few cases (e.g., pure functions in Chapter 3, some matrix operations in Chapter 6), references to the mathematical background for some built-in commands are given—mainly for commands in which the mathematics required extends beyond the familiarity commonly exhibited by non-mathematicians. The GuideBooks do not discuss the algorithms underlying such complicated functions, but sometimes use Mathematica to “monitor” the algorithms. References of the form abbreviationOfAScientificField/yearMonthPreprintNumber (such as quant-ph/0012147) refer to the arXiv preprint server [43÷], [22÷], [30÷] at http://arXiv.org. When a paper appeared as a preprint and (later) in a journal, typically only the more accessible preprint reference is given. For the convenience of the reader, at the end of these references, there is a Get Preprint button. Click the button to display a palette notebook with hyperlinks to the corresponding preprint at the main preprint server and its mirror sites. (Some of the older journal articles can be downloaded free of charge from some of the digital mathematics library servers, such as http://gdz.sub.uni-goettingen.de, http://www.emis.de, http://www.numdam.org, and http://dieper.aib.uni-linz.ac.at.) As much as available, recent journal articles are hyperlinked through their digital object identifiers (http://www.doi.org).



15

à 0.6.3 Variable Scoping, Input Numbering, and Warning Messages Some of the Mathematica inputs intentionally cause error messages, infinite loops, and so on, to illustrate the operation of a Mathematica command. These messages also arise in the user’s practical use of Mathematica. So, instead of presenting polished and perfected code, the author prefers to illustrate the potential problems and limitations associated with the use of Mathematica applied to “real life” problems. The one exception are the spelling warning messages General::spell and General::spell1 that would appear relatively frequently because “similar” names are used eventually. For easier and less defocused reading, these messages are turned off in the initialization cells. (When working with the notebooks, this means that the pop-up window asking the user “Do you want to automatically evaluate all the initialization cells in the notebook?” should be evaluated should always be answered with a “yes”.) For the vast majority of graphics presented, the picture is the focus, not the returned Mathematica expression representing the picture. That is why the Graphics and Graphics3D output is suppressed in most situations. To improve the code’s readability, no attempt has been made to protect all variables that are used in the various examples. This protection could be done with Clear, Remove, Block, Module, With, and others. Not protecting the variables allows the reader to modify, in a somewhat easier manner, the values and definitions of variables, and to see the effects of these changes. On the other hand, there may be some interference between variable names and values used in the notebooks and those that might be introduced when experimenting with the code. When readers examine some of the code on a computer, reevaluate sections, and sometimes perform subsidiary calculations, they may introduce variables that might interfere with ones from the GuideBooks. To partially avoid this problem, and for the reader’s convenience, sometimes Clear[sequenceOfVariables]and Remove[sequenceOfVariables] are sprinkled throughout the notebooks. This makes experimenting with these functions easier. The numbering of the Mathematica inputs and outputs typically does not contain all consecutive integers. Some pieces of Mathematica code consist of multiple inputs per cell; so, therefore, the line numbering is incremented by more than just 1. As mentioned, Mathematica should be restarted at every section, or subsection or solution of an exercise, to make sure that no variables with values get reused. The author also explicitly asks the reader to restart Mathematica at some special positions inside sections. This removes previously introduced variables, eliminates all existing contexts, and returns Mathematica to the typical initial configuration to ensure reproduction of the results and to avoid using too much memory inside one session.

à 0.6.4 Graphics In Mathematica 5.1, displayed graphics are side effects, not outputs. The actual output of an input producing a graphic is a single cell with the text Graphics or Graphics3D or GraphicsArray and so on. To save paper, these output cells have been deleted in the printed version of the GuideBooks. Most graphics use an appropriate number of plot points and polygons to show the relevant features and details. Changing the number of plot points and polygons to a higher value to obtain higher resolution graphics can be done by changing the corresponding inputs. The graphics of the printed book and the graphics in the notebooks are largely identical. Some printed book graphics use a different color scheme and different point sizes and line and edge thicknesses to enhance contrast and visibility. In addition, the font size has been reduced for the printed book in tick and axes labels. The graphics shown in the notebooks are PostScript graphics. This means they can be resized and rerendered without loss of quality. To reduce file sizes, the reader can convert them to bitmap graphics using the CellöConvert Toö Bitmap menu. The resulting bitmap graphics can no longer be resized or rerendered in the original resolution. To reduce file sizes of the main content notebooks, the animations of the GuideBooks are not part of the chapter notebooks. They are contained in a separate directory.



16

0.6.5 Notations and Symbols The symbols used in typeset mathematical formulas are not uniform and unique throughout the GuideBooks. Various mathematical and physical quantities (such as normals, rotation matrices, and field strengths) are used repeatedly in this book. Frequently the same notation is used for them, but depending on the context, also different ones are used, e.g. ” sometimes bold is used for a vector (such as r) and sometimes an arrow (such as r). Matrices appear in bold or as doublestruck letters. Depending on the context and emphasis placed, different notations are used in display equations and in the Mathematica input form. For instance, for a time-dependent scalar quantity of one variable yHt; xL, we might use one of many patterns, such as ψ[t][x] (for emphasizing a parametric t-dependence) or ψ[t, x] (to treat t and x on an equal footing) or ψ[t, {x}] (to emphasize the one-dimensionality of the space variable x). Mathematical formulas use standard notation. To avoid confusion with Mathematica notations, the use of square brackets is minimized throughout. Following the conventions of mathematics notation, square brackets are used for three cases: a) Functionals, such as -t @ f HtLD HwL for the Fourier transform of a function f HtL. b) Power series coefficients, Axk E H f HxLL denotes the coefficient of xk of the power series expansion of f HxL around x = 0. c) Closed intervals,

like @a, bD (open intervals are denoted by Ha, bL). Grouping is exclusively done using parentheses. Upper-case double-

struck letters denote domains of numbers,  for integers,  for nonnegative integers,  for rational numbers,  for braces 8c1 , …, cn ! outputFileName & at the prompt to run the kernel in batch mode.) The following example shows the first input and output lines of an initial Mathematica session [611÷]. (In[n]:= and are generated by Mathematica, and not input by the user.) 1 + 1


Out[n]=


2

(-2) * (-2)

Everything done to this point in a given Mathematica session is saved in the values of the variables In and Out. The following list provides the basic rules for the use of Mathematica as a programming language. † Almost all built-in commands (we will use the words “command” and “function” interchangeably in the GuideBooks) begin with a capital letter and are nonabbreviated, standard English words. If a command consists of several words, the first letter of each word comprising the command is also a capital letter. The complete word is written without spaces, (e.g., AxesLabel, ContourSmoothing, and TeXForm). If the name of a person is involved, for example, in the special functions of mathematical physics, the name comes first, followed immediately by the usual symbol for this function, represented by a capital letter (e.g., JacobiP, HermiteH, BesselJ, and RiemannSiegelZeta). Two classes of exceptions exist to this general rule. The first class concerns mathematical notation: Shorter symbols are used—such as E for the number e, I for i = -1 , Det for determinant, Sin for sine, and LCM for the least common multiple. The second class includes the abbreviation N for numerical operations (e.g., N for the computation of numerical values themselves, such as N[Sqrt[2]], which evaluates and prints as 1.41421); and NSolve for the numerical solution of equations); the abbreviation D for operations involving differentiation (e.g., D for differentiation and DSolve for solving differential equations); and the abbreviation Q (question) for functions asking questions (e.g., EvenQ for testing if something is an even number). Mathematica knows about one thousand executable commands. † Symbols defined by the user usually begin with lowercase letters. Variable names can be arbitrarily long and include both uppercase and lowercase letters, $, and numbers (but numbers cannot be used as the first character). Only complete, well-developed routines should be given names starting with capital letters (as mentioned in the preface, we will not strictly follow this convention). Names of the form name1_name2 are not allowed in Mathematica (one can input an expression of the form name1_name2, but Mathematica does not interpret this as one name). Users should never introduce symbols of the form name$ or name$number because Mathematica produces symbols in this form to make names unique (see Chapter 4). † The operation of many Mathematica functions can be influenced by a variety of options of the form optionName -> specialOptionSetting (e.g., PlotPoints -> 25 and Method -> GaussKronrod). The possible settings for the options of a command depend on the command and include numbers, lists, or such things as All, None, Automatic, True, False, Bottom, Top, Left, GaussKronrod, and CofactorExpansion. Around 450 differently named options exist. For simple options, these names are Mathematica expressions, for more specialized options they are typically strings. Options can sometimes contain suboption settings. † About 120 commands work together with Mathematica as a general programming (computer-dependent) system and begin with $ (e.g., $MachineEpsilon and $MachineType). † Mathematical functions rarely used, or used only for special purposes, are not implemented in the kernel, which is written in C. They are often available in external packages, which are written in Mathematica. To use these functions, one must first load the appropriate package. The same naming conventions apply. For operating systems allowing arbitrarily long file names, these packages have names of the form Subject`SpecialTopic` (e.g., Algebra`Quater nions`, DiscreteMath`CombinatoricalFunctions`, and NumericalMath`BesselZeros`) and are loaded using Needs["Subject`SpecialTopic`"] or Get["Subject`SpecialTopic`"]. † Error messages of the form command::nameOfTheError:RoughSpecificationOfTheError result when syntactically incorrect source code is input, the wrong number of arguments is given, the wrong type of argument is given for a particular command, or errors arise in the calculation. For example, the input



3

Plot[Sin[x], {x, 0, soFarThatANicePictureComesOut}] produces the following message: Plot::plln: Limiting value soFarThatANicePictureComesOut in {x, 0, soFarThatANicePictureComesOut} is not a machine-size real number.

Σ (* session summary *) TMGBs`PrintSessionSummary[]

à 1.1.2 Elementary Syntax The algebraic operations addition, subtraction, multiplication, and division are denoted as usual by +, -, *, and /. The * for multiplication can be omitted by using a blank space instead. Parentheses () are used exclusively for grouping, and brackets [] are used for enclosing arguments in functions. Braces {} are used to enclose components of vectors and elements of sets (here, any number of elements of arbitrary type are allowed, which can be nested to any level). Mathematical Expression Addition c + b Subtraction d - e Multiplication 3 x Division 4 ê r

ö ö ö ö

Mathematica Form c + b d − e 3 x or 3∗x 4êy H4êx y is H4êxL∗y L

Exponentiation hl Grouping H2 + 3L 4 Function with an argument f HxL Discrete iterator i = 1, 2, 3, …, 9, 10 Continuous range x = 0 … 1

ö ö ö ö ö

h^l H2 + 3L 4 f@xD 8i, 1, 10, 1< or 8i, 10< 8x, 0, 1
[Rule] for replacement, != [Unequal] for inequality). No blank spaces are allowed between the symbols in these short forms. Relatively short Mathematica inputs representing mathematical expressions often look better in StandardForm notation (in StandardForm no additional spaces should be added). Because this book contains a lot of code and to maintain uniformity, we will use InputForm throughout this book. In some rare cases, we will use StandardForm, mainly for demonstration purposes. In procedural programs, we will typically use one line per procedural statement. If possible and appropriate, we will carry out multiple assignments at once (for instance {one, two} = {1, 2} instead of one = 1; two = 2). Below is an example of the general rules for Mathematica source code. In addition to the formatting, note that named temporary auxiliary variables can be largely dispensed with using Mathematica’s functional programming capabilities. In the following code only, the variables armed, numberOfPoints, and rotation in the function definition appear; no further user-defined variables exist. Starting from now, we will display user-changeable arguments in italic. For the function RotatedBlackWhiteStrips below the three arguments armed, numberOfPoints, and rotation are user-changeable arguments. The frequent appearance of # and & are parts of so-called pure functions; we discuss them in detail in Chapter 3. It is a common convention in Mathematica that, whenever possible, a “typical” mathematical symbol (character sequence) for a quantity should be used. If not, a notation should be chosen to reflect the effect of the corresponding command or the contents of the corresponding list. Readers will probably not understand the following code initially. However, after reading this book and looking at this code again, they will have no problem understanding how it works.



5

RotatedBlackWhiteStrips[ armed_Integer?((# >= 4 && EvenQ[#])&), numberOfPoints_Integer?(# > 3&), rotation_?(Im[#] == 0&)] := Graphics[ (* black or white? *) MapIndexed[{If[(-1)^Total[#2] == 1, GrayLevel[0], GrayLevel[0.8]], (* make polygons *) Polygon[Join[#1[[1]], Reverse[#1[[2]]]]]}&, Partition[ Partition[(* calculate vertices *) Distribute[{N[{{+Cos[#], Sin[#]}, {-Sin[#], Cos[#]}}]& /@ Range[0, 2Pi, 2Pi/armed], N[( 1 - (#/(2Pi)))* {Cos[rotation #], Sin[rotation #]} ]& /@ Range[0, 2Pi, 2Pi/numberOfPoints] }, List, List, List, Dot], numberOfPoints + 1], {2, 2}, 1], {2}], (* options for a nice-looking graphic *) AspectRatio -> Automatic, PlotRange -> All]

We now look at three short examples of RotatedBlackWhiteStrips [762÷], [489÷]. Show[GraphicsArray[{RotatedBlackWhiteStrips[ 4, 24, 1/4], RotatedBlackWhiteStrips[12, 36, -1/8], RotatedBlackWhiteStrips[72, 36, 1/4]}]]

In the programming code, we will try adhere to the aforementioned formatting conventions. But because of both horizontal and vertical space limitations on the pages of the book, it will not always be possible to follow the conventions exactly in every piece of code. Closing parentheses, brackets, and braces will not often be aligned vertically with the corresponding opening ones. Successive arguments of functions will either be written in one line or sometimes aligned vertically. This is in particular the case when a program uses many nested (pure) functions such as following. Here we partition a regular n-gon (n even) into rhombuses (once again, we make no use of temporary auxiliary variables). GrayRhombusPartition[n_?(EvenQ[#] && # > 4&), opts___] := Graphics[ (* make gray colors *) {MapIndexed[{GrayLevel[(#2[[1]] - 1)/(n/2 - 2)], #1}&, MapThread[Polygon[ (* make polygons *) Join[#1, Reverse[#2]]]&, #]& /@ ((Partition[#, 3, 2]& /@ #&) /@ ({Drop[Drop[#[[1]], 1], -1], #[[2]]}& /@ Partition[#, 2, 1]))], (* make lines *) {Thickness[0.15/n], MapIndexed[{GrayLevel[1 - (#2[[1]] - 1)/(n/2 - 1)], #1}&, Line /@ #]}}&[ (* the points calculated by iteration *) Drop[Flatten[Transpose[{#1, Join[#2, {{}}]}], 1], -1]& @@ #& /@ NestList[{Last[#], 2(Total[#]/2& /@ Partition[Last[#], 2, 1]) Drop[Drop[First[#], 1], -1]}&, N[{Array[{0, 0}&, {n/2 + 1}], Array[{Cos[Pi/n(1 + 2#)], Sin[Pi/n(1 + 2#)]}&, n/2, 0]}], n/2 - 1]], AspectRatio -> Automatic, opts]

Here are two examples using GrayRhombusPartition.



6

Show[GraphicsArray[ {GrayRhombusPartition[ 8, Background -> Hue[0.12]], GrayRhombusPartition[28, Background -> Hue[0.12]]}]]

Obeying strictly the above-formulated guidelines, this routine is quite big and nearly “ununderstandable” if formatted “properly” on paper. GrayRhombusPartition[n_?(EvenQ[#] && # > 4&), opts___] := Graphics[ Function[ (* º 100 lines deleted for brevity *) ][ (* º another 120 lines deleted for brevity *) ], Rule[AspectRatio, Automatic] ]


1.2 Introductory Examples à 1.2.0 Remarks In this section, we will give a short overview of the mathematical, graphical, and numerical possibilities built into Mathematica. The examples are largely unrelated to each other. We discuss all graphics-related commands in the Graphics volume of the GuideBooks [1283÷] and mathematics-related Mathematica commands in detail in the Numerics [1284÷] and Symbolics [1285÷] volumes. Mathematica also contains a fully developed programming language. We will discuss programming-related features in detail in the next five chapters. The meaning of some of the inputs will be clear to readers without prior Mathematica experience. Some of the inputs will use commands that are not immediately recognizable; others will use “cryptic” shortcuts. In the following chapters, we will discuss the meaning of all the commands, as well as their aliases, in detail. The division into programming, graphics, numerics, and symbolics does not reflect the structure of Mathematica. Just the opposite: The harmonic and fluent connection between all functions makes Mathematica an integrated environment where all parts can be used together in a smooth way. Also, the division into numerics and symbolics is not a strict one: To derive efficient numerical methods, one needs symbolic techniques, and for carrying out complicated symbolic calculations, one frequently needs validated numeric decision procedures. The examples of this chapter form a “random” collection. By no means are they intended to give up a complete, coherent, and logically built overview of Mathematica. Its capabilities are much too many and too diverse to even try to give such an overview inside one chapter.

à 1.2.1 Numerical Computations sinHp ê 3L gives an “exact number”. Sin[Pi/3]

We can compute this number to machine accuracy. Six digits are usually displayed. N[Sin[Pi/3]]

Here are 18 digits in the result. N[Sin[Pi/3], 18]

We can also compute and display a result with 180 digits.



7

N[Sin[Pi/3], 180]

This input calculates the first 1000 terms of the simple continued fraction expansion of semicolon at the end of an input avoids that the result is printed.)

3

5 . (As mentioned above, the

cf = ContinuedFraction[5^(1/3), 1000];

This result shows the number of times various integers appear in the continued fraction expansion. Map[(* count occurrences *) Function[digit, {digit, Count[cf, digit]}], (* all occurring integers *) Union[cf]]

The next input counts the number of occurrences of the number 1 in the first million continued fraction digits of This can be done in a few seconds.

3

5.

Count[ContinuedFraction[5^(1/3), 1000000], 1] // Timing

Continued fractions of square roots are ultimately periodic. ContinuedFraction[66^(1/2), 20]

Is

3

expIp

163 M - 744 the integer 640320? The answer is no, but it “almost” is [1249÷]. Element[(E^(Sqrt[163] Pi) - 744)^(1/3), Integers] N[(E^(Sqrt[163] Pi) - 744)^(1/3) - 640320, 60] 3

To find out that

expIp

163 M - 744 is less than 640320, one does not have to use explicitly a numerical approxima3

tion. Just evaluating the comparison

expIp

163 M - 744 < 640 320 causes Mathematica to carry out all necessary

calculations to answer this question. (E^(Sqrt[163] Pi) - 744)^(1/3) < 640320

For an explanation of why this number is almost an integer, see [294÷], [1338÷], and [1141÷]; for similar identities, see [922÷]. Much more extreme cases exist of numbers that are almost integers. They are called Pisot numbers (see [138÷], [139÷], [711÷], [408÷], [868÷], [185÷], and [887÷]). Consider 3

3

3

2

27 + 3

+ 69

27 369

27 + 3 3

3

69

.

2

The result is not an integer, but it nearly is.



8

N[(2^(1/3)/(27 + 3 Sqrt[69])^(1/3) + (27 + 3 Sqrt[69])^(1/3)/(3 2^(1/3)))^27369, (* numericalize to 5030 digits *) 5030 (* why 5030? $MaxExtraPrecision = Infinity; (* left of decimal point 3342 *) Floor[Log[10, Round[#]] (* right of decimal point 1671 *) Floor[Log[10, # - Round[#]]] + (* some more non-9-digits 0017 *)]&[theRadical] *)] 248872083860566242801488633985778816168566582615463984666186327177996889794 302876969944745816129045615885143011927101923791713997993058914014883941331 9658866585963617988675636547948407631504856110204145022057101449742807283745 349044713489229346181918805096874878013575556923353742673696224778320245988 540213301883484666470466149889402655143734621040204402439497074243583844435 8085722840358097062929679889933382659868624398785471672437476033581010058232 770325288671140498237982079089990431287680958041449065611648473793797460006 542685289106532890742345783983687027507936729079442473934078360160815378816 494153662235479538964578833871970301073249242325586046493271959208073441641 9408849950012979654395273385341095562256314722477722302818244400186545582913 013684116069229948450508385560502376379491505913877574694543067098950233734 875259586944931660657861461142958051706161345801562687419677892445722586732 5513485511448982113074128616447024942770432196754923847050903086833932583983 456210775092840495926289398412204946622896060874294857076651762085967637510 0775376705660134601877102706808623385083704763163416133841647181234902568523 014554906330744898465446950034570811433400237285702426141033340407021679373 889901563587912181986503488932240588333472792264516219643268144193209629883 6704587273618997970936633010894468362292302548038860927089257990505837606565 4372722673382421099596652032752423655970286505879088423573116299843248723992 370681856106228825253081951335763606805097314767756009989894248180226689216 887125546603079786776420339174335241777036123462355674280571688628682637154 744918786523022395903717847865060788592985252403020060537542636129564913749 579902728693786036767203892699418847034973900792486513050707875184722293046 6835523411784976227884753642738424032537593171006892800308328208350825894165 757110641854633899165463352000712509400393706057751324434941912458367864031 438044741715469307650984762987113625655095113341410659514797573216487308588 207929723616047980118369534484150697741703276041764283828990373663679698758 3830362244613565592323446457417387836546707590791148857442335097804365308142 758237796222541372347526347511124157083242577253654864546653468558226069365 215604513857702802435076942062477624009724087750511435288253440943800323682 814500906873898893269944000616164741243202139992999892419706344951703777826 055705878691043258271291941546764790768702904202815388755953467402952252786 242105372182173621873752243352251007748639891006060850310559871809504335746 4009505526256479756716140052888061921437953507269705531834507752244853777872 8480751496694305142481208434058663054256649588333816952893118732756129038115 625316839963397212327107969696245976920848255222591348999445674453161441801 1492624723899611977533345482296723851296876182987982763612903081830406428255 761789360866674785134042824865250319983289744838888137526494195021927158720 980424579870985098762439838255243931303193820158912431012986549938720840348 6505853704619531981994143584471102830065857739428507878016585984828808526342 8870383309534828233466065660553398382006320312599424684146205166069028788982 959050373271686613923208614965923844927939159262755102043035136468782747102 192779859301117801065439219569499299420368424993003990461640112615325982631 808971152916585811064172283699654029309129460623214205826005262694547534088

The last output has 1670 consecutive 9s. StringLength[First[StringCases[ToString[%], "9" ..]]]

Infinitely many such numbers exists, whose high powers are almost integers.



9

Arithmetic operations with integers always lead to exact results. 111^111

Even 11111111 can be computed in a (nearly) vanishing amount of time. (We use Short to suppress printing the entire number and give only some of its first and last digits.) Short[Timing[1111^1111] // OutputForm]

Mathematica can deal quickly with large integers. Here are two integers, both having more than one million digits. int1 = 111111^222222; int2 = 222222^333333; N[{int1, int2}]

Multiplying these two integers can be done in a few seconds on a modern computer. (The actual calculation was carried out on a 2 GHz computer.) Timing[int3 = int1 int2;]

The resulting number has more than 2.9 million digits. N[int3]

Here is the total number of digits in base 2—nearly ten million digits. Length[IntegerDigits[int3, 2]]

But the reader should keep in mind that Mathematica is an interpreted language. It does not carry out any meaningand/or result-changing optimization automatically. So the following simple loop takes a few seconds. Do[1, {10^8}] // Timing

The following picture shows the distribution of the digitsums of 1000 random integers between 1 and 1010 . Each color represents the digitsum in base b, where 2 § b § 50. With[{(* 1000 random numbers *) randomNumberList = Table[Random[Integer, {1, 10^10}], {1000}]}, Show[Graphics[{PointSize[0.005], (* different color for each base *) MapIndexed[{Hue[#2[[1]]/60], #}&, MapIndexed[Point[{#2[[2]], #1}]&, (* digitsums of the random numbers *) Table[Sort[Total[IntegerDigits[#, b]]& /@ randomNumberList], {b, 2, 50}], {2}]]}], PlotRange -> All, Frame -> True]]

Here is a simple numerical integration: Ÿ0 x3 dx. 1

NIntegrate[x^3, {x, 0, 1}]

In the following numerical integration Ÿ0 1 ë 1

x dx, the function is integrable, but it has a singularity at x = 0.

NIntegrate[1/Sqrt[x], {x, 0, 1}]

Here is a contour integral in the complex z-plane (by the Residue theorem, its value is 2 p i). ‡

1

i1

z

dz + ‡

i

-1 1

z

dz + ‡

-i 1

-1

z

dz + ‡

11

-i

z

dz



10

NIntegrate[1/z, {z, 1, I, -1, -I, 1}]

The small, real part comes from the use of an approximating method and approximate numbers. Using Mathematica’s high-precision arithmetic, we can get more correct digits. NIntegrate[1/z, {z, 1, I, -1, -I, 1}, WorkingPrecision -> 50]

Of course, Mathematica can carry out this integral also exactly. Integrate[1/z, {z, 1, I, -1, -I, 1}]

Next, we numerically solve the differential equation: x≥ HtL + x£ HtL3 ë 20 + xHtL ê 5 = cosHe tL ê 3, with initial conditions xH0L = 1 and x£ H0L = 0 (a forced nonlinear oscillator with damping [703÷], [587÷]).

sol = NDSolve[{(* differential equation *) x''[t] + 1/20 x'[t]^3 + 1/5 x[t] == 1/3 Cos[E t], (* initial conditions *) x[0] == 1, x'[0] == 0}, x[t], {t, 0, 360}]

The result is an approximate solution represented in Mathematica as an InterpolatingFunction-object that is embedded in a replacement rule {x -> solution}. We can now plot it. Plot[Evaluate[x[t] /. sol], {t, 0, 100}]

The next picture shows a phase-portrait of the oscillations. ParametricPlot[Evaluate[{x[t] /. sol[[1]], D[x[t] /. sol[[1]], t]}], {t, 0, 360}, Frame -> True, Axes -> False, PlotPoints -> 3600]

Here is a more complicated system of differential equations—the so-called Burridge–Knopoff model for earthquakes [217÷], [460÷], [976÷], [1080÷], [372÷], [1352÷], [371÷], [598÷], [1397÷]. n points xi HtL on a straight line, each of mass m interact with each other via springs of stiffness kc , all masses are subject to a force that is proportional to the

distance of the masses from their equilibrium position and to a friction force - HvL. (The equilibrium position of mass i

is i a.)

m x££i HtL = kc Hxi-1 HtL - 2 xi HtL + xi+1 HtLL - k p Hxi HtL - i a - t vL - f - IM x£i HtLM odeSystem[n_, {m_, kc_, kp_, v_, a_, f_, M_}] := Table[m x[i]''[t] == kc (x[i + 1][t] - 2 x[i][t] + x[i - 1][t]) -kp (x[i][t] - i a - v t) - f -[M x[i]'[t]], {i, n}] /. (* remove first and last masses *) {x[0][t] :> x[1][t] - 1, x[n + 1][t] :> x[n][t] + 1}

We choose sgnHvL †v§1ê2 e-†v§ for - HvL.

-[v_?NumberQ] := Sign[v] Sqrt[Abs[v]] Exp[-Abs[v]];

The function solveODEsAndShowSolutions solves the system of equations for given values of the parameters under certain initial conditions.



11

solveODEsAndShowSolutions[{n_, T_}, {m_, kc_, kp_, v_, a_, f_, M_}, opts___] := Module[{nsol}, (* solve differential equations *) nsol = NDSolve[Flatten[{odeSystem[n, {m, kc, kp, v, a, f, M}], (* initial conditions *) Flatten[Table[{x[i][0] == i a + a Cos[i]/3, x[i]'[0] == 0}, {i, n}]]}], Table[x[i], {i, n}], {t, 0, T}, opts, MaxSteps -> 10^5, PrecisionGoal -> 5, AccuracyGoal -> 5]; (* display solutions *) Plot[Evaluate[Table[x[i][t] - v t, {i, n}] /. nsol], {t, 0, T}, PlotRange -> All, PlotStyle -> {Thickness[0.002]}, Frame -> True, Axes -> False, PlotPoints -> 500]]

Here is the solution for a numerical set of parameters shown. One clearly sees collective motions of the particles caused by their nonlinear coupling. solveODEsAndShowSolutions[{50, 50}, (* parameter values *) {-0.826801, -8.710866, -0.195864, -0.709007, -9.852322, 1.596424, -3.359798}]

Next, we consider a particle in a two-dimensional potential that has confining quadratic part and a random, smoothly oscillating part: V Hx, yL = x2 + y2 + ‚ ri, j cosIi x + ji,HxLj M cosI j y + ji, j M. o

HyL

i, j=0

Here the ri, j are random variables from the interval @-1, 1D and the ji,HxLj , ji, j random phases from the interval @0, 2 pD. HyL

We assume frictionless motion and solve the equations of motions; four coupled nonlinear ordinary differential equations of first order, for a time T . Instead of explicitly specifying the 3 Ho + 1L2 random parameters, we seed the random number generator using a 20-digit seed seed. The code for random2DPotentialParticlePath is longer than the above inputs because, in addition to solving the equations of motions and plotting the particle path, we color the path according to the particle’s velocity (red being slow and blue being fast), show the zero-velocity contour as a guide for the eye of the reachable configuration space, and show the potential itself as a contour plot underneath.



12

random2DPotentialParticlePath[o_, seed_, T_, pp_, opts___] := Module[{V, x, y, vx0, vy0, nsol, pathData, path, xMin, xMax, yMin, yMax, zeroVelocityContour, potentialLandscape}, (* seed random number generator *) SeedRandom[seed]; (* generate random potential *) V[x_, y_] = x^2 + y^2 + (* random part of the potential *) Sum[Random[Real, {-1, 1}]* Cos[i x + 2Pi Random[]] Cos[j y + 2Pi Random[]], {i, 0, o - 1}, {j, 0, o - 1}]; (* random initial velocity components *) {vx0, vy0} = Table[Random[Real, {-2, 2}], {2}]; (* solve Newton's equations *) nsol = NDSolve[{x'[t] == vx[t], y'[t] == vy[t], vx'[t] == -D[V[x[t], y[t]], x[t]], vy'[t] == -D[V[x[t], y[t]], y[t]], (* initial conditions *) x[0] == 0, y[0] == 0, vx[0] == vx0, vy[0] == vy0}, {x, y, vx, vy}, {t, 0, T}, MaxSteps -> 10^5]; (* position and velocity data *) pathData = Table[Evaluate[{{x[t], y[t]}, {vx[t], vy[t]}} /. nsol[[1]]], {t, 0, T, T/pp}]; (* particle path; colored according to velocity *) path = {Hue[0.5 ArcTan[Sqrt[#.#]&[(#1[[2]] + #2[[2]])/2]]], Line[{#1[[1]], #2[[1]]}]}& @@@ Partition[pathData, 2, 1]; (* maximal x,y-extensions *) {{xMin, xMax}, {yMin, yMax}} = {#1 - #3/12, #2 + #3/12}&[ Min[#], Max[#], Max[#] - Min[#]]& /@ Transpose[First /@ pathData]; (* zero-velocity contour and contour plot of the potential *) {zeroVelocityContour, potentialLandscape} = ContourPlot[Evaluate[V[x, y]], {x, xMin, xMax}, {y, yMin, yMax}, DisplayFunction -> Identity, PlotPoints -> 240, ##]& @@@ (* set options for contour plot *) {{Contours -> {(vx0^2 + vy0^2)/2 + V[0, 0]}, ContourShading -> False, ContourStyle -> {{GrayLevel[0], Thickness[0.002]}}}, {Contours -> 100, ColorFunction -> (GrayLevel[1 - #]&), PlotRange -> All, ContourLines -> False}}; (* show potential, zero-velocity contour, and particle path *) Show[{potentialLandscape, zeroVelocityContour, Graphics[{Thickness[0.002], path}]}, opts, AspectRatio -> 1, PlotRange -> All, Frame -> False, DisplayFunction -> $DisplayFunction]]

Here are three example potentials and particle paths for o = 3, o = 12, and o = 10. The first two motions are pseudoperiodic. The third motion is chaotic and the particle samples the accessible configuration space in a complicated manner [704÷]. The potential used in the last graphic has 3 μ 112 = 363 random parameters. Show[GraphicsArray[ Block[{$DisplayFunction = Identity}, random2DPotentialParticlePath[##, 10 #3]& @@@ (* pseudoperiodic motion *) {{ 3, 21598974805925082378, 200}, {12, 60923097090049506424, 50}, (* chaotic motion *) {10, 58211857412104937056, 200}}]]]

Mathematica can also solve partial differential equations. Here is the so-called Benney equation in 1 + 1 dimensions [1148÷], [1059÷], a nonlinear partial differential equation.



∑ yHx, tL ∑t

+ yHx, tL

∑ yHx, tL ∑x

+

∑2 yHx, tL ∑ x2

+¶

∑3 yHx, tL ∑ x3

+

∑4 yHx, tL ∑ x4

13

=0

We will solve the Benney equation for ¶ = 0.001167, periodic boundary conditions, and the following initial condition (a “random”, oscillating function of magnitude > 100 ): 1 px 38 px 11 px 21 px 79 3px cos cos cos cos cos yHx, 0L = + + 25 191 116 23 140 20 25 40 50 50 7 px 4 3px 95 px 115 3px cos cos cos cos 55 100 131 100 101 200 166 200 3 7px 9 9px 100 11 p x 12 13 p x cos cos cos cos + . 5 200 16 200 123 200 19 200 Solving a partial differential equation is more time-consuming than solving an ordinary differential equation; the following inputs need longer than the above one to complete. (* the differential equation *) pde = D[ψ[x, t], t] + ψ[x, t] D[ψ[x, t], x] + D[ψ[x, t], {x, 2}] + ∂ D[ψ[x, t], {x, 3}] + D[ψ[x, t], {x, 4}]; (* the initial condition *) ψ0[x_] = 1/25 Cos[Pi x/20] - 38/191 Cos[Pi x/25] 11/116 Cos[Pi x/40] - 21/23 Cos[Pi x/50] + 79/140 Cos[3 Pi x/50] + 7/55 Cos[Pi x/100] 4/131 Cos[3 Pi x/100] - 95/101 Cos[Pi x/200] 115/166 Cos[3 Pi x/200] - 3/5 Cos[7 Pi x/200] 9/16 Cos[9 Pi x/200] - 100/123 Cos[11 Pi x/200] + 12/19 Cos[13 Pi x/200]; (* system parameters *) xM = 100; T = 80; ∂ = 0.001167; (* solve the differential equation *) nsol = NDSolve[{pde == 0, ψ[x, 0] == ψ0[x], ψ[xM, t] == ψ[-xM, t]}, ψ[x, t], {x, -xM, xM}, {t, 0, T}, (* set options for a solution appropriate for visualization *) AccuracyGoal -> 2, PrecisionGoal -> 2, Method -> {"MethodOfLines", "SpatialDiscretization" -> {"TensorProductGrid", "DifferenceOrder" -> 10, "MaxPoints" -> {1200}, "MinPoints" -> {1200}}}];

We visualize the solution as a density plot as well as a 3D plot. We see the “birth and death” processes for soliton-like structures [781÷] typical for this equation. Show[GraphicsArray[{ (* density plot *) DensityPlot[Evaluate[ψ[x, t] /. nsol[[1]]], {x, -xM, xM}, {t, 0, T}, Mesh -> False, PlotRange -> All, ColorFunction -> (Hue[0.78 #]& DisplayFunction -> Identity, PlotPoints -> 400], (* 3D plot *) Plot3D[Evaluate[ψ[x, t] /. nsol[[1]]], {x, -xM, xM}, {t, 0, T}, Mesh -> False, PlotRange -> All, PlotPoints -> 400, DisplayFunction -> Identity]}]]

The solution of the last partial differential equation was quite complicated. In general, solutions of nonlinear partial differential equations can have “any possible” shape (see [713÷] for some examples). One solution of the following coupled system of two partial differential equations (of reaction-diffusion type) has a conjectured solution exhibiting the symmetry of a Sierpinski triangle [612÷], [613÷], [614÷], [615÷], [709÷].



14

t

∑ uHx, tL ∑t ∑ vHx, tL ∑t

= Du = Dv

∑2 uHx, tL ∑ x2 ∑2 vHx, tL ∑ x2

+ f HuHx, tLL - vHx, tL + uHx, tL.

Here f HuL is the following nonlinear function: f HuL = 1 ê 2 HtanhHHu - aL ê dL + tanhHa ê dLL - u.

The initial condition is uHx, 0L = expI-x2 M, vHx, 0L = 0, periodic spatial boundary conditions are imposed, and the parameter values are a = 0.1, t = 0.34, d = 0.05, Du = 1, and Dv = 10 [612÷]. We use the function NDSolve to numerically solve the system and show a density plot of vHx, tL. (A magnified view of the solution would show a complicated fine structure [775÷], [1025÷].) Module[{a = 0.1, τ = 0.34, δ = 0.05, Dv = 10, Du = 1, xM = 240, T = 130, pp = 700, nsol, pdeU, pdeV, u, v, x, t}, (* avoid use of high-precision arithmetic *) Developer`SetSystemOptions["CatchMachineUnderflow" -> False]; (* nonlinear term *) f[u_, a_, δ_] := 1/2(Tanh[(u - a)/δ] + Tanh[a/δ]) - u; (* differential equations *) pdeU = τ D[u[x, t], t] == Du D[u[x, t], {x, 2}] + f[u[x, t], a, δ] - v[x, t]; pdeV = 1 D[v[x, t], t] == Dv D[v[x, t], {x, 2}] + u[x, t]; (* initial conditions *) u0[x_] := Exp[-x^2]; v0[x_] := 0; (* solve differential equations numerically *) nsol = NDSolve[{pdeU, pdeV, u[x, 0] == u0[x], v[x, 0] == v0[x], u[+xM, t] == u[-xM, t], v[+xM, t] == v[-xM, t]}, {u, v}, {x, -xM, xM}, {t, 0, T}, (* set options appropriate for the specific problem and the visualization purpose *) MaxSteps -> 10^5, PrecisionGoal -> 2.8, AccuracyGoal -> 2.8, Method -> "BDF", Method -> {"MethodOfLines", "SpatialDiscretization" -> {"TensorProductGrid", "DifferenceOrder" -> 5, "MaxPoints" -> {2xM/pp}, "MinPoints" -> {2xM/pp}}}]; (* display density plot of v[x, t] *) DensityPlot[Evaluate[v[x, t] /. nsol[[1]]], {x, -xM, xM}, {t, 0, T}, Mesh -> False, PlotPoints -> 200, PlotRange -> All, ColorFunction -> (Hue[0.78 #]&)]]

Because of the unified underlying language of Mathematica, it is not only possible to perform calculations, but also to monitor the methods and algorithms used to perform the calculations. We solve the following differential equation numerically. (This differential equation is related to the s-wave phase shift of a quantum mechanical scattering problem [249÷], [258÷], [1196÷].) 1 d£ HrL = HsinHk rL cosHdHrLL + cosHk rL sinHdHrLLL2 kr For small k (we use k = 10-4 ) the solution dHrL will have wide flat plateaus and short steep walls. We display the solution dHrL as a solid line (with the corresponding ticks to the left) and the number of cumulative steps taken in the numerical solution process as a dashed line (with the corresponding ticks to the right). The correlation between the steep increases of dHrL with the number of steps taken is obvious. (In this case, the r-values used in the solution process are easily extractable from the solution itself; in more complicated situations one can use side effects to monitor details of the algorithm and method; see Chapter 1 of the Numerics volume of the GuideBooks [1284÷] for more examples.)



15

(* the differential equation *) ode[k_] = δ'[r] == (Sin[k r] Cos[δ[r]] + Cos[k r] Sin[δ[r]])^2/(k r); (* numerical solution of the differential equation *) nsol = NDSolve[{ode[10^-4], δ[60] == 15.70789703}, δ, {r, 1, 60}, MaxSteps -> 10^4, Method -> "BDF"]; Show[Graphics[{ {GrayLevel[0], Dashing[{0.01, 0.01}], (* extract and number r-points *) Line[MapIndexed[{#1, #2[[1]]/50}&, nsol[[1, 1, 2, 3, 1]]]]}, {GrayLevel[0], Line[Table[{r, δ[r] /. nsol[[1]]}, {r, 1, 60, 1/10}]]}}], (* make left and right ticks *) Frame -> True, Frame -> True, FrameTicks -> {Automatic, Automatic, False, Table[{2 k, 2 k 50}, {k, 0, 8}]}]

The command Table can be used to generate a matrix. Here is a Hilbert matrix aij = 1 ë Hi + j + 1L [280÷], [1139÷]. hilbert = Table[1/(i + j + 1), {i, 4}, {j, 4}]

Here, it is in the usual form. hilbert // MatrixForm

Next, we find its eigenvalues exactly. The use of Short prevents a large amount of output from being printed. The structure shows the number of terms left out. Eigenvalues[hilbert] // Short[#, 12]&

If we numerically evaluate the eigenvalues, they are, of course, much more compact. Eigenvalues[N[hilbert]]

We get the same result if we numerically evaluate the above exact formulas for the eigenvalues. N[%%]

Here is a slightly larger example from linear algebra. We take two random symmetric 12 μ 12 matrices /1 and /2 ,

form /a = H1 - aL /0 + a /1 , and calculate the minimal distance between the eigenvalues lk HaL of /a as a function of the complex variable a. The peaks in the graphics are the branch points of the multivalued function lHaL. (* two symmetric matrices *) {/0, /1} = With[{n = 12}, (* generate random symmetric matrix *) Table[Developer`ToPackedArray[(# + Transpose[#])&[ Table[If[i > j, 0., 2 Random[] - 1], {i, n}, {j, n}]]], {2}]]; (* minimal distance between eigenvalues *) minEigenvalueDistance = Compile[{{/, _Complex, 2}}, Module[{evs = Eigenvalues[/], n = Length[/]}, (* distance between all pairs *) Min[Table[Min[Table[Abs[evs[[i]] - evs[[j]]], {j, i + 1, n}]], {i, 1, n - 1}]]], {{Eigenvalues[_], _Complex, 1}}];



16

(* display eigenvalue distances over complex a-plane *) ParametricPlot3D[{αr Cos[αϕ], αr Sin[αϕ], (* use logarithm *) -Log[minEigenvalueDistance[(* the matrix *) N[(1 - αr Exp[I αϕ]) /0 + αr Exp[I αϕ] /1]]], {EdgeForm[]}}, {αr, 0, 2.5}, {αϕ, 0, 2Pi}, PlotPoints -> 6{30, 60}, Compiled -> False, BoxRatios -> {1, 1, 1/2}, PlotRange -> {-1, 5}]

Here is the numerical value of the Gamma function at 1/2. N[Gamma[1/2]]

Here is the numerical value of the Bessel function J3.3 H6.7L. N[BesselJ[3.3, 6.7]]

We can also evaluate the Bessel function for a complex argument and a complex index, for instance, I3.3+0.6 i H6.7 - 9.5 iL.

N[BesselI[3.3 + 0.6 I, 6.7 - 9.5 I]]

Here is a 100-digit value of I3.3+0.6 i H6.7 - 9.5 iL (to get 100 digits, the input must have enough digits and one cannot use machine numbers as input). N[BesselI[33/10 + 6/10 I, 67/10 - 95/10 I], 100]

The next input has 100-digit arguments. The result has fewer digits now. All digits displayed are guaranteed to be correct. BesselI[N[33/10 + 6/10 I, 100], N[67/10 - 95/10 I, 100]]

Special functions can be evaluated for all complex arguments. This makes it possible to numerically solve the differential equation z≥ HtL = J2 HzHtL, qL where J2 Hz, qL is an elliptic theta function. The left picture shows a solution curve of this differential equation in the complex plane and the right picture shows the corresponding recurrence plot [427÷], [242÷], [516÷], [908÷].



17

Module[{q = -0.483069 - 0.482155 I, ζ0 = 0.514593 - 0.909303 I, ζp0 = -0.784268 - 0.773652 I, T = 300, pp, line, λ}, (* solve differential equation numerically *) nsol = NDSolve[{z''[τ] == EllipticTheta[2, z[τ], q], z[0] == ζ0, z'[0] == ζp0}, {z}, {τ, 0, T}, MaxSteps -> 10^5]; (* maximal solution time *) T = nsol[[1, 1, 2, 1, 1, 2]]; pp = ParametricPlot[Evaluate[{Re @ z[τ], Im @ z[τ]} /. nsol[[1]]], {τ, 0, T}, PlotPoints -> 9000, DisplayFunction -> Identity]; (* extract curve *) line = pp[[1, 1, 1, 1]]; λ = Length[line]; Show[GraphicsArray[ (* solution curve in the complex plane *) {Graphics[MapIndexed[(* color curve *) {Hue[0.8 #2[[1]]/λ], Line[#1]}&, Partition[line, 2, 1]], Frame -> True], (* recurrence plot for t § 100 *) ContourPlot[Evaluate[Abs[z[t] - z[τ]] /. nsol[[1]]], {t, 0, 100}, {τ, 0, 100}, PlotPoints -> 300, PlotRange -> All, ContourLines -> False, (* use red-blue coloring scheme *) ColorFunction -> (RGBColor[#, 1 - #, 0]&), DisplayFunction -> Identity]}]]]

Next, we interpolate the data 881, 2 Infinity, Method -> Fit, NSumExtraTerms -> 50, WorkingPrecision -> 200]

Now n = 60 yields already more than 100 correct digits for Euler’s constant. γ[n_] := nSum[(n^k/k!)^2 HarmonicNumber[k], {k, 0, Infinity}]/ nSum[(n^k/k!)^2, {k, 0, Infinity}] - Log[n] γ[60] % - EulerGamma

-2 The infinite product ¤¶ s=2 I1 - s M has an exact value of 1 ê 2. Here, we compute it numerically.

NProduct[1 - 1/s^2, {s, 2, Infinity}]

Mathematica can also solve the following simple minimization problem. min IHx - 2.56L2 + Hy - 3.78L4 + 3.1M x,y

FindMinimum[(x - 2.56)^2 + (y - 3.78)^4 + 3.1, {x, 1}, {y, 1}]

Here is the computation using more digits (the detailed meaning of the options PrecisionGoal and Accuracy Goal will be discussed in Chapter 1 of the Numerics volume of the GuideBooks [1284÷]). This time we use an objective function with exact numbers instead of machine numbers. FindMinimum[(x - 256/100)^2 + (y - 378/100)^4 + 31/10, {x, 1}, {y, 1}, PrecisionGoal -> 30, AccuracyGoal -> 30, WorkingPrecision -> 100] // (* shorten output *) N[#, 30]&

FindRoot solves implicit equations. Here, we solve the simple equation cosHxL = sinHxL. FindRoot[Sin[x] == Cos[x], {x, 1}]

The result compares well with the exact root. N[Pi/4]

Next, we look at a higher degree polynomial: x + 2 x2 + 3 x3 + 4 x4 + ∫ + 66 x66 = 0. poly = Sum[i xî, {i, 66}]

It has 66 zeros. (By the fundamental theorem of algebra, every polynomial of nth degree has exactly n possibly complex zeros.) NSolve[poly == 0] // Short[#, 10]& Length[%]

Here are the Jensen disks (disks whose diameter is the segment joining complex conjugate roots) for this polynomial and all of its derivatives. (Jensen’s theorem asserts that all nonreal roots of the derivative of a polynomial with real coefficients lie inside the Jensen disks of the polynomial itself [645÷], [1341÷], [1108÷], [1090÷].) JensenDisks[poly_, x_] := Disk[{Re[#[[1]]], 0}, Abs[Im[#[[1]]]]]& /@ (* pairs of complex conjugate roots *) Partition[Cases[x /. NSolve[poly == 0, x], _Complex], 2]



19

Show[Graphics[MapIndexed[{GrayLevel[Mod[#2[[1]], 2]], #}&, (* form disks from roots *) JensenDisks[#, x]& /@ NestList[D[#, x]&, poly, 66]]], Frame -> True, AspectRatio -> Automatic]

Generally, a large amount of data, as in the last example, is better expressed graphically. Here, we compute all of the zeros of all polynomials of degree less than or equal to maxDegree with nonzero integer coefficients between -maxIntÖ Coeff and maxIntCoeff. (For the zeros of related polynomials, see [142÷], [1017÷], [179÷], and [1016÷].) allRoots[maxDegree_, maxIntCoeff_] := Module[{x, allMonomials, allIntegers, allCoefficientLists}, (* the monomials *) allMonomials = Table[xî, {i, 0, maxDegree}]; (* the coefficients *) allIntegers = Range[-maxIntCoeff, maxIntCoeff]; (* all possible lists of coefficients *) allCoefficientLists = Flatten[Outer[List, Sequence @@ Table[allIntegers, {maxDegree + 1}]], maxDegree]; (* showing all roots in the complex plane *) Graphics[{PointSize[0.003], Point[{Re[#], Im[#]}]& /@ (* solving all polynomials, taking roots *) Flatten[(Cases[NRoots[#, x], _Real | _Complex, {-1}])& /@ DeleteCases[allMonomials.# == 0& /@ allCoefficientLists, False]]}, PlotRange -> {{-3, 3}, {-2, 2}}, Frame -> True, AspectRatio -> Automatic]]

For allRoots[2, 14], we have 24361 different polynomials and for allRoots[5, 2], we have 15621 different polynomials with the following roots in the complex plane. Show[GraphicsArray[{allRoots[2, 14], allRoots[5, 2]}]]

Now, we solve a large system of linear equations. The de Rham’s function ja HxL fulfills the following functional equations [126÷], [691÷], [127÷], [367÷]: x ja K O = a ja HxL 2 x+1 = a + H1 - aL ja HxL. ja 2 1

2

Discretizing the functional equations at x = 0, n , n , …, ja H0L, ja I 2 n M, …, ja I 1

2 n-1 M, 2n

n-1 , n

1 yields 2 n + 2 linear equations for 2 n + 1 unknowns

ja H1L. The function deRhamϕPoints solves the linear equations for a given a.

deRhamϕPoints[α_, n_] := Module[{ϕ, ϕs, eqs}, (* avoid numericalization of arguments of j *) SetAttributes[ϕ, NHoldAll]; (* the unknowns *) ϕs = Table[ϕ[x], {x, 0, 1 - 1/(2n), 1/(2n)}]; (* the linear equations *) eqs = N[Flatten[Table[{ϕ[x/2] - α ϕ[x] == 0, ϕ[(x + 1)/2] - α - (1 - α) ϕ[x] == 0}, {x, 0, 1 - 1/n, 1/n}]]]; (* 2n + 1 points of the de Rahm's function j *) Apply[{First[#1], #2}&, First[Solve[eqs, ϕs]], {1}]]

The next graphic shows de Rahm’s functions for various values of a. Each curve has 401 points.



20

Show[Graphics[Table[ {Hue[0.8 α], Line[deRhamϕPoints[α, 200]]}, {α, 1/20, 19/20, 1/20}]], Frame -> True]

The ability to calculate with numbers of arbitrary precision allows for straightforward investigations that otherwise would be very difficult. The following graphic shows how two orbits of the logistic map xn+1 = 1 - a x2n move apart

with increasing iteration number. We choose x0 = 3 ê 7 and x£0 = 3 ê 7 + 10-300 , follow 500 iterations for 200 values of a,

and use 500 digits in all calculations. The resulting contour plot shows that the distance between xn and x£n is a sensitive function of a. (For more about the Liapunov exponent [1111÷], [123÷] of the logistic map, see [100÷], [312÷], [1286÷], [435÷], and [820÷].) Module[{∂ = 10^-6}, (* distance between two orbits *) δList[a_, x0_, δx0_, n_, prec_] := NestList[(1 - a #^2)&, N[x0 + δx0, prec], n] NestList[(1 - a #^2)&, N[x0, prec], n]; (* data for different a *) data = Table[Log[10, Abs[δList[a, 3/7, 10^-300, 500, 500]]], {a, ∂, 2, (2 - ∂)/200}]; (* visualize distance *) ListContourPlot[data, MeshRange -> {{1, 500}, {0, 2}}, Contours -> 120, ContourLines -> False, PlotRange -> All, ColorFunction -> (Hue[Random[]]&), FrameLabel -> {None, "a"}]]

Next, we carry out a fast Fourier transform. How does one find a built-in function that does this? The question mark ? stands for a request for information, whereas * after the letters replaces any sequence of lowercase or capital letters or other characters. For example, we can find out what Fourier is (of course, another possibility is to look under Fourier in on-line version of The Mathematica Book, in the Help Browser). ?Four*

The following creates the values of two superimposed sine waves with different frequencies and different amplitudes. The semicolon prevents the printing of the 1000 values generated. sinTable = Table[N[Sin[10 n 2Pi/1024] + 2 Sin[5 n 2Pi/1024]], {n, 1, 1024}]

Here is the Fourier transform (we visualize the result in the next subsection). fourierTable = Fourier[sinTable];

We display the first 20 elements of fourierTable. Take[fourierTable, 20] // Chop

Here, the two-dimensional (2D) Fourier transforms of the reciprocals of the greatest common divisor of two integers gcdHi, jL and the least common multiple lcmHi, jL for 1 § i, j § 256 is shown. Show[GraphicsArray[ Block[{$DisplayFunction = Identity}, (* display absolute value of 2D Fourier transform *) ListDensityPlot[Abs[Fourier[Table[1/#[i, j], {i, 256}, {j, 256}]]], Mesh -> False, ColorFunction -> (Hue[0.8 #]&)]& /@ {GCD, LCM}]]]

Starting with Mathematica Version 4, in addition to the ease of programming, the fact that computations are done immediately, and the ability to plot numerical functions, Mathematica provides the possibility to carry out larger numerical calculations very efficiently. Larger, compiled (using the function Compile) numerical calculations in Mathematica can within a small factor achieve the speed of corresponding Fortran programs, such as those in NAG



21

(http://www.nag.com), IMSL (http://www.imsl.com), and netlib (http://www.netlib.org) ([1093÷], [1240÷], [821÷], and [400÷]) or can even be faster. (For a rough survey of these kinds of programs, see [1137÷] and [776÷]). Mathematica has a built-in (pseudo-)compiler. It generates machine-independent pseudo-code. For many numerical calculations, the use of the compiler will speed up calculations by a factor 2 to 20. Here is an example: the calculation of the Fourier spectrum of the quantum-mechanical energy spectrum of a 2D square well [1142÷], [166÷], [990÷]. According to the Gutzwiller–Maslov theory, the Fourier spectrum contains information about the length of the classical periodic orbits [593÷], [190÷], [398÷], [340÷], [1378÷], [1144÷], [1157÷], [1355÷], [884÷], [150÷], [151÷], and [1143÷]). This is the list of eigenvalues taken into account. evList = Select[Sort[Flatten[Table[Sqrt[n^2 + m^2], {n, 60}, {m, 60}]]], # {0, 30000}, PlotPoints -> 80, Frame -> True, FrameTicks -> {Automatic, None}, Compiled -> False]

By using compiled functions, many numerical calculations can be carried out quite fast. Here is a modeling problem: the forest fire (model) [68÷], [975÷], [1177÷], [889÷], [1096÷]. We consider a 1D array. Each element represents either a burning tree, a nonburning tree or an empty site. At each time step, a burning tree burns down creating an empty site and ignites trees that are direct neighbors and which have trees. All empty sites grow a tree with probability p. The implementation of a compiled version of a time step of the forest fire model is straightforward. In the array with

the fires, trees, and empty sites, 1 stands for fire, -1 for a tree and 0 for an empty site (and ashes). forestFireStepC = Compile[{{l, _Integer, 1}, p}, Module[(* use periodic boundary conditions *) {l1 = Append[Prepend[l, Last[l]], First[l]], l2}, l2 = l1; Do[(* burn trees and ignite neighbors *) l1[[k]] = Which[l2[[k]] == 1, 0, l2[[k]] == -1 && (l2[[k - 1]] == 1 || l2[[k + 1]] == 1), 1, True, l2[[k]]], {k, 2, Length[l2] - 1}]; (* grow new trees *) If[# == 0, If[Random[] < p, -1, 0], #]& /@ Take[l1, {2, -2}]]];

For visualizing the forest fire, we implement a function forestFirePlot. Fires are shown in red, trees in green, and empty sites in white.



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forestFirePlot[data_] := ListDensityPlot[data, Mesh -> False, FrameTicks -> None, ColorFunctionScaling -> False, (* red for fire; green for trees; white for empty *) ColorFunction -> (Which[# == 1, RGBColor[1, 0, 0], # == -1, RGBColor[0, 1, 0], # == 0, RGBColor[1, 1, 1]]&)]

For p = 0.22, p = 0.32, and p = 0.42 we show the resulting fires and trees over 500 time steps for an initial array length

of L = 500. (In average, we need p t 1 ê lnHLL to keep the fire burning.) On a year-2005 computer, the calculation takes

a fraction of a second for each p. With[{L = 500, T = 500}, Show[GraphicsArray[ Block[{$DisplayFunction = Identity}, (* start with same initial fires and trees *) Function[p, SeedRandom[111]; forestFirePlot[NestList[forestFireStepC[#, p]&, Table[Random[Integer, {-1, 1}], {L}], T]]] /@ {0.22, 0.32, 0.42}]]]]

Here is a more-complicated calculation within integer arithmetic. The sum sb HnL of the digits of an integer n in base b can be calculated in Mathematica in the following way. digitSum[n_Integer?Positive, base_Integer /; base > 2] := Total[IntegerDigits[n, base]]

If one iterates sb HnL until a fixed point is reached, one gets a new function yHnL. We call it IteratedDigitSum[n]. IteratedDigitSum[n_Integer?Positive, base_Integer /; base > 2] := FixedPoint[digitSum[#, base]&, n]

y is an arithmetic function [56÷], which means yHn + mL = yHyHnL + yHmLL and yHn mL = yHyHnL yHmLL. Here, this property for two large integers is tested. x = 9218359834598298562984567230456723624068502495865409134; y = 3109579823049090378621220813796509245672098567203496722; b = 13; {{IteratedDigitSum[x + y, b], IteratedDigitSum[IteratedDigitSum[x, b] + IteratedDigitSum[y, b], b]}, {IteratedDigitSum[x * y, b], IteratedDigitSum[IteratedDigitSum[x, b] * IteratedDigitSum[y, b], b]}}

The following pictures visualize the values of the function yHn mL in the n, m-plane for base 100 and base 26. Show[GraphicsArray[ ListDensityPlot[Table[IteratedDigitSum[x y, #], {x, 100}, {y, 100}], Mesh -> False, ColorFunction -> Hue, DisplayFunction -> Identity]& /@ (* two integer bases *) {26, 100}]]

As mentioned, high-precision arithmetic is a very useful tool for scientific computations. We will end this subsection with a slightly larger example. Let us deal with a simple mechanical system: a billiard ball bouncing between two types of regularly arranged circular scatterers (or a light ray reflected by perfectly mirroring circles, also called a Sinai billiard with finite horizon or a Lorentz gas [1259÷], [164÷], [1164÷]). Here are some of the scatterers shown.



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With[{o = 3}, Show[gr = Graphics[{Thickness[0.002], (* array of large and small circles *) Table[If[(-1)^(i + j) == 1, Circle[{i, j}, 5/8], Circle[{i, j}, 1/4]], {i, -o, o}, {j, -o, o}]}, AspectRatio -> Automatic]]]

The following functions implement the elastic scattering process of a point-shaped billiard ball between the scatterers. (* nearest intersection (if any) of a ray with a circle *) nearestIntersection[Ray[p_, d_], Circle[q_, r_]] := Module[{eqs = (p - q + t d).(p - q + t d) - r^2, sol}, sol = Select[t /. Solve[eqs == 0, t], (Im[#] == 0 && # > 0)&]; If[sol === {}, {}, p + t d /. t -> Min[sol]]] (* reflection of a ray at the point s at a circle *) reflect[Ray[_, d_], s_, Circle[q_, _]] := Module[{n = #/Sqrt[#.#]&[s - q]}, Ray[s, d - 2d.n n]] (* the circle of next reflection for a ray *) nextCircle[ray_, lastCircle_] := Module[{is}, circles = If[(-1)^Total[lastCircle[[1]]] === 1, (* big circles *) Join[Circle[lastCircle[[1]] + #, 5/8]& /@ {{2, 0}, {0, 2}, {-2, 0}, {0, -2}, {1, 1}, {-1, 1}, {1, -1}, {-1, -1}, {3, 1}, {1, 3}, {-3, 1}, {-1, 3}, {3, -1}, {1, -3}, {-3, -1}, {-1, -3}}, Circle[lastCircle[[1]] + #, 1/4]& /@ {{1, 0}, {0, 1}, {-1, 0}, {0, -1}, {2, 1}, {1, 2}, {-2, 1}, {-1, 2}, {2, -1}, {1, -2}, {-2, -1}, {-1, -2}}], (* small circles *) Join[Circle[lastCircle[[1]] + #, 5/8]& /@ {{1, 0}, {0, 1}, {-1, 0}, {0, -1}, {2, 1}, {1, 2}, {-2, 1}, {-1, 2}, {2, -1}, {1, -2}, {-2, -1}, {-1, -2}}, Circle[lastCircle[[1]] + #, 1/4]& /@ {{1, 1}, {-1, 1}, {1, -1}, {-1, -1}}]]; is = DeleteCases[{nearestIntersection[ray, #], #}& /@ circles, {{}, _}]; is[[Position[#, Min[#]]&[ #.#& /@ ((First[#] - First[ray])& /@ is)][[1, 1]], 2]]]

The next function calculates o reflections of a billiard ball that starts at the angle f0 of the central scatterer in direction 8cosHj0L, sinHj0L 3.7 {{-1, 1}, {-1, 1}}, AspectRatio -> Automatic]] Show[{(* high-precision path *) rayPathGraphic[{rayPath[127/426 Pi, 121/291 Pi, 25, 100]}], (* machine-precision path *) rayPathGraphic[{rayPath[127/426 Pi, 121/291 Pi, 25]}] /. (* make blue path *) Hue[x_] :> Hue[x + 0.75]}]

The following animation shows the extreme sensitivity of the billiard path as a function of its starting direction. The trajectory starts at the rightmost point of the central circle. We color the pieces of the trajectory from red to blue. Show[GraphicsArray[ Function[ϕ0, rayPathGraphic[{rayPath[0, ϕ0, 30, 120]}] /. Line[l_] :> (* color line segments *) MapIndexed[{Hue[0.78(#2[[1]] - 1)/30], Line[#]}&, Partition[l, 2, 1]]] /@ {Pi/40, Pi/4, 3Pi/8}]] Make Input

Show Animation

Do[Show[rayPathGraphic[{rayPath[0, ϕ0, 30, 120]}] /. Line[l_] :> (* color line segments *) MapIndexed[{Hue[0.78(#2[[1]] - 1)/30], Line[#]}&, Partition[l, 2, 1]]], {ϕ0, 0, Pi/2, Pi/2/90}];

By slightly changing the function rayPath, we can implement a function fixedFinalTimeRayPath that does not carry out a fixed number of reflections, but calculates each path for a fixed time. (* carry out reflections of a ray starting at {Cos[f0], Sin[f0]} with direction {Cos[j0], Sin[j0]} until time T *) fixedFinalTimeRayPath[φ0_, ϕ0_, T_, prec___] := Module[{ray = Ray[5/8 {Cos[φ0], Sin[φ0]}, N[{Cos[ϕ0], Sin[ϕ0]}, prec]], λ, Λ = 0, δL, nC = Circle[{0, 0}, 5/8], nI, nIO, rayBag, finalPoint}, rayBag = {{ray, Λ}}; nIO = ray[[1]]; (* reflect until time T has gone by *) While[Λ < T, nC = nextCircle[ray, nC]; nI = nearestIntersection[ray, nC]; ray = reflect[ray, nI, nC]; λ = Sqrt[#.#]&[nI - nIO]; Λ = Λ + λ; nIO = nI; rayBag = {rayBag, {ray, Λ}}]; (* the flight segments *) rays = Partition[Flatten[rayBag], 2]; (* cut last flight segment so that it end at time T *) δL = T - rays[[-2, 2]]; finalPoint = rays[[-2, 1, 1]] + δL rays[[-2, 1, 2]]; (* return list of flight segments *) Append[#[[1, 1]]& /@ Most[rays], finalPoint]]

We now follow 192 paths for the time 12 (we assume unit speed). The machine number calculation is about three times faster than the high-precision calculation with 120 digits.



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Module[{ppφ = 48, ppϕ = 3, T = 12, prec = 120, K}, (* calculate paths for machine precision and high precision *) K = Timing[Table[fixedFinalTimeRayPath[φ0, ϕ0, T, #], {φ0, 0, 2Pi (1 - 1/ppφ), 2Pi/ppφ}, {ϕ0, φ0 - Pi/2, φ0 + Pi/2, Pi/ppϕ}]]& /@ {MachinePrecision, prec}; (* make path segment assignments and return timings *) {pathData, pathDataHP} = Last /@ K; First /@ K]

We color the paths and display them. The left graphic shows the machine arithmetic results and the right graphic shows high–precision results. The two graphic are qualitatively similar, but have different detailed paths. Show[GraphicsArray[ Graphics[{Thickness[0.002], MapIndexed[{Hue[#2[[1]]/8], Line[#1]}&, #, {2}]}, PlotRange -> All, AspectRatio -> Automatic, Frame -> True, FrameTicks -> None]& /@ (* machine number and high-precision data *) {pathData, pathDataHP}]]


à 1.2.2 Graphics We have already used various graphics types in the last subsection for visualizing some of the numerical results. In this subsection, we will concentrate on the graphics. We begin with a simple plot in the plane. Plot[Sin[x], {x, 0, 10}]

The Gibbs phenomenon (see [768÷], [681÷], [1166÷], [1205÷], [850÷], [480÷], [1272÷], [554÷], [555÷], [632÷], [556÷], [548÷], [557÷], [300÷], and [1075÷]) involves the “overshoots” that occur in replacing a given function by the partial sums of its Fourier series. It is L2 -convergent, which means that the integral of the squared difference of the approximation to the given function goes to zero, but in general no pointwise convergence to the original function is achieved. If the original function f HxL is of bounded variation, the series will converge pointwise to f HxL at every point of continuity of f HxL. Here, we examine this phenomenon for the expansion of the function qIHp ê 2L2 - x2 M (qHzL is the

Heaviside function) in terms of 9H2 ê pL1ê2 sinHi xL=i=1,2,… . The series converge at x = 0 to 0 and at x = p ê 2 to 1 ê 2. We see

“overshoots” near x = 0 and x = p ê 2. The left graphic shows the original step function and the first 18 partial sums over

the interval @0, pD. The right graphic shows the first 120 partial sums, colored from black to white in the interval @0, p ê 2D near f HxL º 1.

partialSum[n_, x_] := Sum[Sqrt[2/Pi] (1 - Cos[i Pi/2])/i Sqrt[2/Pi] Sin[i x], {i, n}]



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Show[GraphicsArray[ Block[{opts = Sequence[DisplayFunction -> Identity, Frame -> True, Axes -> False]}, {(* the left plot *) Plot[Evaluate[Table[partialSum[j, x], {j, 18}]], {x, 0, Pi}, Evaluate[opts], PlotRange -> All, PlotStyle -> {{Thickness[0.002], GrayLevel[0]}}, Prolog -> {Thickness[0.02], GrayLevel[1/2], Line[{{0, 0}, {0, 1}, {Pi/2, 1}, {Pi/2, 0}, {Pi, 0}}]}], (* the right plot *) Plot[Evaluate[Table[partialSum[j, x], {j, 10, 120}]], {x, 0, Pi/2}, Evaluate[opts], PlotPoints -> 1000, PlotRange -> {0.88, 1.2}, PlotStyle -> Table[{Thickness[0.002], GrayLevel[k/120]}, {k, 120}]]}]]]

A related, not less interesting, but much less known phenomenon happens for the Fourier series description of the product of two discontinuous functions, whose product is a continuous function [844÷], [1348÷], [252÷], [1349÷]. Here are two such functions f1 and f2. The left graphic shows the function f1 in red, the function f2 in blue, and the right product shows the product of f1 and f2. (* two functions with concurrent jumps *) f1[x_] := (1 + x) UnitStep[x] f2[x_] := (2 + x) UnitStep[x]

+ (2 - x) UnitStep[-x] + (1 - x) UnitStep[-x]

Show[GraphicsArray[ Block[{$DisplayFunction = Identity}, {(* the two functions in red and blue *) Plot[{f1[x], f2[x]}, {x, -Pi, Pi}, AxesOrigin -> {0, 0}, PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 0, 1]}], (* the product of the two functions in black *) Plot[f1[x] f2[x], {x, -Pi, Pi}, AxesOrigin -> {0, 0}]}]]]

The partial sums of the two functions give Gibbs oscillations near the origin. The Fourier series coefficients of the product c12 HkL is the convolution ⁄oj=-o c1 Hk - jL c2 H jL of the Fourier series coefficients c1 HkL and c2 HkL of the two factors (we truncate the series at order o). But although the product is a continuous function, the truncated Fourier series è exhibits strong oscillations (next left graphic). The right graphic shows Li’s corrected version c12 HkL in which the

Fourier series coefficient c1 Hk - jL is replaced by the k , j element of the inverse of the Toeplitz matrix of the Fourier series coefficients of the inverse of f1.

(* Fourier coefficients of f1 and f2 *) c1[k_] = If[k == 0, (3 + Pi)/2, ((2 - I k)(-1 + Cos[k Pi]) + k (3 + 2 Pi) Sin[k Pi])/(2 k^2 Pi) c2[k_] = If[k == 0, (3 + Pi)/2, (Sin[(k Pi)/2] * (k (3 + 2 Pi) Cos[(k Pi)/2] + (-2 - I k) Sin[(k Pi)/2]))/(k^2 P



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(* classical convolution for product coefficient: Table[Sum[If[Abs[k - j] > o, 0, c1[k - j] c2[j]], {j, -o, o}], {k, -o, o}] *) (* calculated all coefficients at once *) c12List[o_] := ListConvolve[Table[c1[j], {j, -o, o}], Table[c2[j], {j, -o, o}], o + 1, 0] (* Fourier series coefficient for 1/f1 *) c1Inv[k_] := c1Inv[k] = N @ If[k == 0, Log[(1 + Pi) (2 + Pi)/2]/(2 Pi), (Gamma[0, -2 I k] + E^(3 I k)(Gamma[0, I k] - Gamma[0, I k (1 + Pi)]) Gamma[0, (-I) k (2 + Pi)])/(E^(2 I k)(2 Pi))]; (* concurrent jump-corrected convolution for product coefficient *) c12SmoothedList[o_] := Module[{c1InvToeplitz, invMat}, (* form Toeplitz matrix *) c1InvToeplitz = Table[c1Inv[n - m], {n, -o, o}, {m, -o, o}]; invMat = Inverse[c1InvToeplitz]; (* Fourier series coefficients *) invMat.Table[c2[k], {k, -o, o}]] Show[GraphicsArray[Function[fourierSeriesList, Plot[Evaluate[Flatten[{f1[x] f2[x], (Re @ fourierSeriesList)& /@ {16, 32, 64, 128, 256}}]], {x, -Pi/16, Pi/16}, PlotRange -> All, DisplayFunction -> Identity, PlotStyle -> (* product in black, approximations of degree 16, 32, 64, 128, 256 from red to blue *) {{GrayLevel[0.8], Thickness[0.01]}, Sequence @@ ({#, Thickness[0.004]} {Hue[0], Hue[0.22], Hue[0.3], Hue[0.55], Hue[0.7]})}], {HoldAll}] @@ (* classical Laurent convolution and Li's convolution *) {Hold[c12List[#].Table[Exp[I k x], {k, -#, #}]], Hold[c12SmoothedList[#].Table[Exp[I k x], {k, -#, #}]]}]]

The following two pictures are a visualization of the interesting limit [636÷], [485÷], [654÷]. 0 if x is irrational 1 ¶ p f HxL = lim ‚ HcosHk p xLL2 k = 1 if x = q is rational, gcdH p, qL = 1 nØ¶ n q k=0

The left picture represents the right-hand side (with points at all rational x with denominator less than or equal 200), and the right picture shows the convergence of the first 40 partial sums of the left-hand side. With[{maxDenominator = 200, maxSeriesTerms = 40}, Show[GraphicsArray[ (* left graphics *) {Graphics[{PointSize[0.002], Union[Flatten[Table[Point[{i/j, 1/j}], {j, maxDenominator}, {i, 0, j}]]]}, PlotRange -> All, Frame -> True], (* generate the right plot *) Plot[Evaluate[Table[1/n Sum[Cos[k Pi x]^(2k), {k, 1, n}], {n, 1, maxSeriesTerms}]], {x, 0, 1}, PlotPoints -> 200, PlotRange -> All, DisplayFunction -> Identity, (* different colors for n terms *) PlotStyle -> Table[{Thickness[0.002], Hue[0.8 i/maxSeriesTerms]}, {i, maxSeriesTerms}]]}]]]

We consider the sum ⁄nk=0 H-1Ls2 Hl kL , where s2 HnL counts the 1’s in the binary representation of the integer n. This function is called DigitCount in Mathematica and exhibits fractal properties for many integers l [541÷]. The follow-



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ing picture shows the behavior of such sums. Show[Graphics3D[{Thickness[0.002], Table[{Hue[l/120], Line[ MapIndexed[{#2[[1]] - 1, l, #1}&, (* the data *) FoldList[Plus, 0, Table[(-1)^DigitCount[l k, 2][[1]], {k, 0, 250}]]]]}, {l, 0, 100}]}], BoxRatios -> {2, 1, 1}, ViewPoint -> {-0.6, -3.0, 0.8}, Axes -> True]

The following plot shows a collection of half circles overlapping in a hierarchical structure. Here, we make direct use of the graphics primitive Circle. (* upper circles *) twoNewCircles[Circle[{x0_, y0_}, r_, {0, Pi}]] := {Circle[{x0 - r, y0 - r/2}, r/2, {0, Pi}], Circle[{x0 + r, y0 - r/2}, r/2, {0, Pi}]}; (* lower circles *) twoNewCircles[Circle[{x0_, y0_}, r_, {Pi, 2Pi}]] := {Circle[{x0 - r, y0 + r/2}, r/2, {Pi, 2Pi}], Circle[{x0 + r, y0 + r/2}, r/2, {Pi, 2Pi}]}; Show[Graphics[{Thickness[0.002], (* iterate generation *) NestList[Flatten[twoNewCircles /@ #]&, {#}, 7]& /@ {Circle[{0, +1}, 1, {0, Pi}], Circle[{0, -1}, 1, {Pi, 2Pi}]}}], AspectRatio -> 1, PlotRange -> All]

Here is a slightly more complicated example: the recursive filling of the area between three touching circles with circles (per Apollonius). We use an iterative rather than a direct method to calculate the new circle data via the Soddy formula [1223÷], [319÷], [128÷], [1224÷], [1255÷], [467÷], [907÷], [565÷], [566÷], [567÷], [568÷], [519÷] (for touching spheres, see [174÷], [78÷], [79÷], [1421÷]).



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makeTouchingCircles[{p1_, p2_, p3_}, iter_, minRadius_:10^-3] := Module[{newton, newCircleData, rMax}, (* the derivative for the Newton method *) newton[{{{x1_, y1_}, r1_}, {{x2_, y2_}, r2_}, {{x3_, y3_}, r3_}}, {{xn_, yn_}, rn_}] := {{#[[1]], #[[2]]}, #[[3]]}&[{xn, yn, rn} - 1/2* Inverse[{{(xn - x1), (yn - y1), -(rn + r1)}, {(xn - x2), (yn - y2), -(rn + r2)}, {(xn - x3), (yn - y3), -(rn + r3)}}]. {(xn - x1)^2 + (yn - y1)^2 - (r1 + rn)^2, (xn - x2)^2 + (yn - y2)^2 - (r2 + rn)^2, (xn - x3)^2 + (yn - y3)^2 - (r3 + rn)^2}]; (* the next smaller circle *) newCircleData[{pp1_, pp2_, pp3_}] := {Circle @@ #, {{pp1, pp2, #[[1]]}, {pp1, pp3, #[[1]]}, {pp2, pp3, #[[1]]}}}&[ Module[{r12, r23, r13, r1, r2, r3, startx, starty, startr, ∂ = 10^-10}, (* radii *) {r12, r23, r13} = Sqrt[#.#]& /@ N[{pp1 - pp2, pp2 - pp3, pp1 - pp3}]; r1 = ( r12 + r13 - r23)/2; r2 = ( r12 - r13 + r23)/2; r3 = (-r12 + r13 + r23)/2; startr = Sqrt[#.#]&[N[pp1 - ({startx, starty} = N[(pp1 + pp2 + pp3)/3])]] - r1; (* iterating the Newton method *) FixedPoint[newton[{{pp1, r1}, {pp2, r2}, {pp3, r3}}, #]&, {{startx, starty}, startr}, SameTest -> (#.#&[Flatten[#1 - #2]] < ∂&)]]]; Join[Module[{r1, r2, r3}, (* the start circles *) {r12, r23, r13} = Sqrt[#.#]& /@ N[{p1 - p2, p2 - p3, p1 - p3}]; r1 = ( r12 + r13 - r23)/2; r2 = (r12 - r13 + r23)/2; r3 = (-r12 + r13 + r23)/2; rMax = Max[r1, r2, r3]; {Circle[p1, r1], Circle[p2, r2], Circle[p3, r3]}], (* iterating the calculation of new circles *) Map[First, NestList[newCircleData /@ Flatten[Map[Last, Select[#, (#[[1, 2]]/rMax > minRadius)&], {1}], 1]&, {newCircleData[{p1, p2, p3}]}, iter], {2}]]] (* display calculated circles *) Show[GraphicsArray[ {#, (* color circles according to radius *) # /. Circle[mp_, r_?(# < 0.2&)] :> {Hue[Log[10, r]], Disk[mp, r]}}&[ Graphics[{Thickness[0.001], makeTouchingCircles[{{-1, 0}, {1, 0}, {0, -1}}, 9, 0.0005]}, PlotRange -> {(Sqrt[2] - 1) {-1, 1}, {-Sqrt[2]/2, 0}}, AspectRatio -> Automatic]]]]

Here is the distribution of the logarithms of the radii of the circles [184÷], [393÷] from the last picture shown. ListPlot[Reverse[Log[10, Sort[#1[[2]]& /@ (* extract all circles from the last picture *) Cases[%[[1]], _Circle, Infinity]]]], PlotRange -> All]

Discrete data can also be displayed. For example, here is again a plot of a Fourier transform of a superposition of sin functions. We clearly see the frequency and amplitude ratios according to the signal. fourierTable = Fourier[Table[(* the signal *) Sum[Sin[16. k n 2Pi/1024]/k, {k, 20}], {n, 1, 1024}]];



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ListPlot[(* add x-values *) Flatten[MapIndexed[{{#2[[1]], 0}, {#2[[1]], #1}, {#2[[1]], 0}}&, Abs[Take[fourierTable, 512]]], 1], PlotRange -> All, PlotJoined -> True]

The next picture shows the first 30 partial sums of the generalized Weierstrass function ⁄nk=1 k -2 expIi k 3 zM in the complex plane. In the limit n Ø ¶, the resulting curve is nowhere differentiable [261÷]. Module[{l = 10000, cf, lines}, (* fast calculation of the cumulative sums *) cf = Compile[{{z, _Complex}}, Rest[FoldList[Plus, 0, Table[Exp[I k^3 z]/k^2, {k, 30}]]]]; (* the lines *) lines = Line /@ Transpose[Table[{Re[#], Im[#]}& /@ cf[ϑ], {ϑ, 0., 2.Pi, 2.Pi/l}]]; (* the graphics *) Show[Graphics[{Reverse[ MapIndexed[{Hue[3 #2[[1]]/40], (* smoother lines are thicker *) Thickness[0.002 #2[[1]]], #1}&, Reverse[lines]]]}], PlotRange -> All, Frame -> True, FrameTicks -> None, AspectRatio -> Automatic, Background -> GrayLevel[0.8]]]

Here is a typical three-dimensional (3D) plot. By default, the surface is illuminated with three colored light sources. Plot3D[Sin[x^2 + y^2]/(x^2 + y^2), {x, -4, 4}, {y, -4, 4}, PlotPoints -> 35, PlotRange -> All]

The coloring and many other details can be varied as desired. Plot3D[{Sin[x^2 + y^2]/(x^2 + y^2), Hue[Sqrt[x^2 + y^2]/Sqrt[32]]}, {x, -4, 4}, {y, -4, 4}, PlotPoints -> 40, PlotRange -> All, Axes -> None, Boxed -> False, Mesh -> False]

We now plot a sphere given in a parametric form analogous to the one for the circle above. ParametricPlot3D[{Cos[ϕ] Sin[ϑ], Sin[ϕ] Sin[ϑ], Cos[ϑ]}, {ϕ, 0, 2 Pi}, {ϑ, 0, Pi}]

Here is a more complicated surface. It is based on the parametrization of a torus. This time we do not show the edges of the polygons. torus[ϕ_, ϑ_, R_, r_, color_] := {R Cos[ϕ] + r Cos[ϕ] Cos[ϑ], R Sin[ϕ] + r Sin[ϕ] Cos[ϑ], r Sin[ϑ], color} ParametricPlot3D[Evaluate[(* modify torus parametrization *) torus[ϕ + Sin[7 ϕ]/3, 4 ϕ + ϑ, 3 + Sin[ϕ]/3 + Sin[ϑ]/5, 1 + Sin[11 ϕ]/3, (* surface coloring *) {EdgeForm[], SurfaceColor[RGBColor[0.9, 0, 0.4], RGBColor[0.3, 0.4, 0], 2.3]}]], {ϕ, 0, 2 Pi}, {ϑ, 0, Pi}, (* set options *) PlotPoints -> {300, 40}, Boxed -> False, Axes -> False, ViewPoint -> {0, 0, 0.51}]

Next, we visualize a complicated closed surface with infinitely many holes. It is implicitly defined by cos

x+ y x2 + y2 + z2 sin

+ cos

x- y x2 + y2 + z 2

x+z x2 + y2 + z 2

- sin

+ cos

x-z x2 + y2 + z 2

y+z x2 + y2 + z 2

+ sin

+

y-z x2 + y2 + z2

= 0.

Because the denominators of the arguments of the trigonometric functions vanish faster than the numerators when



31

approaching the origin, the surface becomes quite complicated near the origin. The following code generates an approximation of this surface. We use the function ContourPlot3D from the package Graphics`ContourPlot3D`. Needs["Graphics`ContourPlot3D`"] Module[{n = 1, pp0 = 32, ppR = 22, cp, polys}, (* define 3D contour plot of function with {x, y, z} --> {x, y, z}/(x^2 + y^2 + z^2) *) cp[pp_] := cp[pp] = Cases[ ContourPlot3D[Cos[x + y] + Cos[x + z] + Cos[y + z] + Sin[x - y] - Sin[x - z] + Sin[y - z], {x, -Pi, Pi}, {y, -Pi, Pi}, {z, -Pi, Pi}, MaxRecursion -> 0, PlotPoints -> pp, Contours -> {0}, DisplayFunction -> Identity], _Polygon, Infinity]; (* the polygons *) polys = Table[Map[# + 2.Pi{i, j, k}&, If[i == j == k == 0, cp[pp0], cp[ppR]], {-2}], {i, -n, n}, {j, -n, n}, {k, -n, n}] // Flatten; (* display polygons *) Show[Graphics3D[{EdgeForm[], SurfaceColor[Hue[0.22], Hue[0.02], 2.6], polys} /. (* invert *) Polygon[l_] :> Polygon[#/#.#& /@ l]], PlotRange -> All, Boxed -> False]]

Cutting the surface along the x, y-plane and removing the upper part shows its complicated structure near the origin. Show[%, PlotRange -> {All, All, {-3/4, 0}}, ViewPoint -> {0, 0, 3}]

The results of such functions as Plot, Plot3D, and ParametricPlot3D are composed of graphics primitives that can be further manipulated by Mathematica. In the following plot, we use ParametricPlot3D to subdivide the sides of a cube (leftmost image). These side faces are then pulled toward the center of the cube by an amount corresponding to their distance to the center. The upper right image shows the resulting surface reflected in a sphere. To see inside these surfaces, we have made holes in the polygons.



32

Show[GraphicsArray[Map[ Function[α, Graphics3D[{EdgeForm[Thickness[0.001]], SurfaceColor[Hue[0.12], Hue[0], 2.2], (* fit cube in unit cube *) Function[polys, Module[{rMax = Max[ Sqrt[#.#]& /@ Level[Cases[polys, _Polygon, Infinity], {-2}]]}, Map[#/rMax&, polys, {-1}]]] @ (* making holed polygons on deformed surfaces *) (Function[x, Module[{mp = Mean[x[[1]]]}, {Polygon[(mp + 0.2 (# - mp))& /@ x[[1]]], MapThread[Polygon[Join[#1, Reverse[#2]]]&, {Partition[Append[#, First[#]]&[ (mp + 0.8 (# - mp))& /@ x[[1]]], 2, 1], Partition[Append[#, First[#]]&[ (mp + 0.5 (# - mp))& /@ x[[1]]], 2, 1]}]}]] /@ Map[(* this deforms the faces *) (#/Sqrt[#.#] Sqrt[#.#]^α)&, Join @@ (* making a cube; every side has 6×6 polygons *) Apply[ParametricPlot3D[##, PlotPoints -> 7, DisplayFunction -> Identity][[1]]&, {#[[1]], Flatten[Append[{#[[2, 1]]}, {-1, 1}]], Flatten[Append[{#[[2, 2]]}, {-1, 1}]]}& /@ ({#, Cases[#, _Symbol]}& /@ Select[Flatten[Outer[List, {x, 1, -1}, {y, 1, -1}, {z, 1, -1}], 2], Length[Cases[#, _Symbol]] == 2&]), {1}], {-2}])}, Axes -> False, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}]], (* the values for the pure function parameter a *){-3, -1, 1, 2}], GraphicsSpacing -> -0.05]]

Images can also be created directly from graphics primitives—such as points, lines, and polygons—rather than as plots of functions. Here is a problem that was investigated already by Kepler. It involves the recursive subdivision of a regular pentagon according to the following visualized rule [385÷], [870÷]. (Here, we also implement the routines needed for the next two images.) The implementation itself is straightforward. For clarity, we do not enclose all pieces of the code in scoping constructs but rather use global variables like fac and MHJ. We discuss similar graphics in detail in Chapter 1 of the Graphics volume of the GuideBooks [1283÷].



33

startPentagon = Polygon[Table[{Cos[x], Sin[x]}, {x, Pi/2, -11/10 Pi, -2Pi/5}]]; (* makes a vector, perpendicular to vec *) perpendicular[vec_] := #/Sqrt[#.#]&[{vec[[2]], -vec[[1]]}] // N; (* for the pentagon-specific constants *) {fac, MHJ} = N[{1/(2 + 2 Sin[18 Degree]), Sin[72 Degree]}]; (* new points for making smaller pentagon *) threeNewPoints[{p1_, p2_}] := {p1 + fac (p2 - p1), p2 + fac (p1 - p2), (p1 + p2)/2 + MHJ fac Sqrt[(p2 - p1).(p2 - p1)] perpendicular[p2 - p1]}; sixNewPentagons[Polygon[l_]] := (* treating every side *) Module[{p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20}, (* the new points *) {p1, p2, p3, p4, p5} = l; {{p6, p7, p16}, {p8, p9, p17}, {p10, p11, p18}, {p12, p13, p19}, {p14, p15, p20}} = threeNewPoints /@ Partition[Append[l, First[l]], 2, 1]; (* the six new pentagons *) Polygon /@ {{p1, p6, p16, p20, p15}, {p7, p2, p8, p17, p16}, {p9, p3, p10, p18, p17}, {p11, p4, p12, p19, p18}, {p13, p5, p14, p20, p19}, {p16, p17, p18, p19, p20}}] Show[GraphicsArray[ {Graphics[startPentagon, AspectRatio -> Automatic], Graphics[sixNewPentagons[startPentagon], AspectRatio -> Automatic]}] /. Polygon[l_] :> Line[Append[l, First[l]]]]

If we repeat this subdivision four times, we get a figure consisting of 64 = 1296 pentagons in interesting positions. subdividedPentagons[0] = {startPentagon}; subdividedPentagons[k_] := subdividedPentagons[k] = Flatten[sixNewPentagons /@ subdividedPentagons[k - 1]] Show[Graphics[{Thickness[0.001], Line[Append[#, First[#]]]& @@@ subdividedPentagons[4]} // N] AspectRatio -> Automatic]

Now, we color the pentagons in each step with some color and stack them up. Show[Graphics[Table[{Hue[k/5], subdividedPentagons[k]} // N, {k, 0, 4}]], AspectRatio -> Automatic]

Here, we project Kepler’s recursive subdivision of a pentagon onto a sphere. toSphere[{x_, y_}] := Function[{ϕ, ϑ}, {Cos[ϕ] Sin[ϑ], Sin[ϕ] Sin[ϑ], Cos[ϑ]}][ ArcTan[x, y], Sqrt[x^2 + y^2] N[Pi]] (* a function that cuts a hole in a polygon *) makeHole[Polygon[l_], factor_] := Module[{mp = Mean[l], newl, nOld, nNew}, (* inner points *) newl = (mp + factor(# - mp))& /@ l; {nOld, nNew} = Partition[Append[#, First[#]], 2, 1]& /@ {l, newl}; {MapThread[Polygon[Join[#1, Reverse[#2]]]&, {nOld, nNew}]}]



34

Show[Graphics3D[{EdgeForm[], Thickness[0.001], {SurfaceColor[Hue[Random[]], Hue[Random[]], 3 Random[]], makeHole[#, 0.8]}& /@ Map[toSphere, subdividedPentagons[4], {3}]}], Boxed -> False]

Several 3D figures can be directly constructed from polygons. Here is a fractal sign post. (* normalize a vector *) normalize[a_List] = a/Sqrt[a.a]; (* make one elementary part of the sign post *) post[α_, dir_, ortho_, size_] := Module[{dir1, orthoh, ortho1, bi1, p1, p2, p3, p4, p5, p6, p7, p8, p9, s1 = 1, s2 = 0.3, s3 = 0.2, s4 = 1.2, h1, h2, h3, h4, h5}, (* direction the new sign will point to *) dir1 = normalize[dir]; (* first orthogonal direction *) ortho1 = normalize[normalize[ortho] + normalize[Cross[dir, ortho]]]; (* second orthogonal direction *) bi = normalize[Cross[dir1, ortho1]]; h1 = s2 size ortho1; h2 = s2 size bi; h3 = s3 size ortho1; h4 = s3 size bi; h5 = s1 size dir1; p1 = α + h1; p2 = α + h2; p3 = α - h1; p4 = α - h2; p5 = α + h3 + h5; p6 = α + h4 + h5; p7 = α - h3 + h5; p8 = α - h4 + h5; p9 = α + s4 size dir1; (* polygons forming the next generation *) Polygon /@ {{p1, p4, p8, p5}, {p4, p3, p7, p8}, {p3, p2, p6, p7}, {p2, p1, p5, p6}, {p5, p9, p8}, {p8, p7, p9}, {p6, p7, p9}, {p5, p6, p9}}] (* the start part *) postHierarchy[0] = {post[{0., 0., 0.}, {0., 0., 1.}, {1., 0., 0.}, 1 (* add new parts at the sides *) postHierarchy[i_] := postHierarchy[i] = (post @@ newData[#, 0.4î])& /@ Flatten[(Take[#, 4]& /@ postHierarchy[i - 1])]; (* iterate the process *) newData[poly_Polygon, size_] := Module[{f = poly[[1]], ortho, dir}, ortho = (f[[1]] + f[[2]])/2 - (f[[3]] + f[[4]])/2; p = (f[[3]] + f[[4]])/2 + 0.2 ortho; dir = -Cross[f[[1]] - f[[2]], f[[1]] - f[[4]]]; {p, dir, ortho, size}] Show[Graphics3D[{EdgeForm[Thickness[0.001]], SurfaceColor[Hue[0.11], Hue[0.10], 2], Table[postHierarchy[i], {i, 0, 4}]}], AspectRatio -> Automatic, Boxed -> False, PlotRange -> All]

With Mathematica’s symbolic, numerical, and graphical capabilities, much more complicated images with many more points and polygons can be created and displayed. However, this often requires some more CPU time and memory resources. Here is an example of such an image involving a flower made out of a dodecahedron. It consists of 6300 polygons.



35

Needs["Graphics`Polyhedra`"]; Module[{preCup, preBlossom, cup, blossom, allPolys, rotation, mat, rotMat, ]LJ = {0.324919, 0.324919, 0.180513}}, (* the elementary parts, made with ParametricPlot3D *) preCup = ParametricPlot3D[{Sin[ϑ]^2/3 Cos[ϕ], Sin[ϑ]^2/3 Sin[ϕ], ϑ}, {ϕ, -Pi, Pi}, {ϑ, 0, Pi/2}, PlotPoints -> {26, 8}, DisplayFunction -> Identity]; preBlossom = ParametricPlot3D[{(2 - 5/3 Sin[ϑ]) Cos[(Pi - ϑ)/(Pi/2) ϕ], (2 - 5/3 Sin[ϑ]) Sin[(Pi - ϑ)/(Pi/2) ϕ], Pi/2 + 2(ϑ - Pi/2)}, {ϕ, -Pi/5, Pi/5}, {ϑ, Pi/2, Pi}, PlotPoints -> {6, 15}, DisplayFunction -> Identity]; (* a rotation matrix *) mat = {{Cos[#], Sin[#], 0}, {-Sin[#], Cos[#], 0}, {0, 0, 1}}&[Pi/5.]; (* the cup *) cup = Map[]LJ (mat.#)&, preCup[[1]], {-2}]; (* one part of the blossom *) blossom[0] = Map[]LJ (mat.#)&, preBlossom[[1]], {-2}]; (* rotation matrices for other five subparts of one part *) Do[9[i] = {{ Cos[2Pi/5 i], Sin[2Pi/5 i], 0}, {-Sin[2Pi/5 i], Cos[2Pi/5 i], 0}, {0, 0, 1}} // N, {i, 4}]; (* the blossom *) Do[blossom[i] = Map[9[i].#&, blossom[0], {-2}], {i, 4}]; allPolys = Flatten[{cup, Table[blossom[i], {i, 0, 4}]}]; (* rotation matrices for other eleven parts *) With[{aMat = Table[H[k, l][i], {k, 3}, {l, 3}]}, (* rotation matrices for other faces of dodecahedron *) Do[rotation[i] = (aMat /. Solve[Flatten[Table[Thread[ aMat.Polyhedron[Dodecahedron][[1, 1, 1, j]] == Polyhedron[Dodecahedron][[1, i, 1, j]]], {j, 3}]], Flatten[aMat]])[[1]], {i, 12}]]; (* display cup and blossoms *) Show[Graphics3D[{EdgeForm[{Hue[0], Thickness[0.001]}], SurfaceColor[RGBColor[0, 0.8, 0.2], RGBColor[0.1, 0.9, 0.4], 1], Table[Map[rotation[i].#&, allPolys, {-2}], {i, 12}]}], Boxed -> False, PlotRange -> All, ViewPoint -> {2.1, -2.4, 2.3}]]

Mathematica has built-in functions for many kinds of graphics. The following picture shows a contour plot of the absolute value of the Gauss map z ö1 ê z - d1 ê zt over the complex z-plane. ContourPlot[Abs[1/(x + I y) - Floor[1/(x + I y)]], {x, -1.1, 1.1}, {y, -1.1, 1.1}, PlotPoints -> 400, ColorFunction -> Hue, ContourStyle -> {Thickness[0.001]}]

In the following, we use a sum of three Gauss maps to create an animation. Let 8z< denote the fractional part of z. We will animate a contour plot of the function f HzL =

1

Jz - 1N

a

+

Ke

1 2 i pí3

a

z - 1O

+

Ke

1 4 i pí3

a

z - 1O

as the parameter a varies from 1 ê 2 to 3.



36

fractionalPartContourPlot[α_, opts___] := Module[{r = 2.15, ring, color, cp}, (* cut out circular area *) ring = {GrayLevel[1], Polygon[Join[Table[3.05 {Cos[ϕ], Sin[ϕ]}, {ϕ, 0, 2Pi, 2Pi/200}], Reverse[Table[r {Cos[ϕ], Sin[ϕ]}, {ϕ, 0, 2Pi, 2Pi/200}]]]]}; (* coloring for the contour lines *) color[l_] := {Hue[α + 0.8 Sqrt[#.#&[Total[l]/2]]], Line[l]}; (* make the contour plot *) cp = ContourPlot[Evaluate[Sum[ Abs[FractionalPart[(Exp[I ϕ](zr + I zi) - 1)^-α]], {ϕ, 0, -4/3Pi, -2/3Pi}]], {zi, -r, r}, {zr, -r, r}, PlotPoints -> 301, PlotRange -> All, DisplayFunction -> Identity, Frame -> False, Epilog -> ring, ColorFunctionScaling -> False, Contours -> Table[ξ, {ξ, 0, 9/2, 3/10}], (* color contour zones alternatingly *) ColorFunction :> (Which[# == 0, RGBColor[1, 0, 0], # == 1, RGBColor[0, 0, 1]]&[ Mod[Ceiling[# 10], 2]]&)]; (* display the contour plot with re-colored contour lines *) Show[Graphics[cp] /. Line[l_] :> (color /@ Partition[l, 2, 1]), opts, DisplayFunction -> $DisplayFunction, PlotRange -> {{-r, r}, {-r, r}}]] (* show 3×3-array of graphics for various a *) Show[GraphicsArray[fractionalPartContourPlot[#, DisplayFunction -> Identity]& /@ #, GraphicsSpacing -> 0.2]]& /@ Partition[Table[α, {α, 1/2, 3, 5/2/8}], 3] Make Input

Show Animation

(* generate frames of the animation *) Do[fractionalPartContourPlot[α], {α, 1/2, 3, 5/2/75}];

Let us give a few more graphics examples. Here is an iterative construction of a fractal tree using n iteration levels.



37

FractalTree[n_] := With[{α = 0.65, β = 0.87, γ = 0.46, δ = 0.8}, Graphics[Polygon[Join[{{1.35, -0.2}, {1.1, 0}}, Map[{0.5, 0} + (* deform the pattern *) 1/(1 - 0.4 Cos[2 ArcTan @@ (# - {0.5, 0})]^2)* (# - {0.5, 0})&, Flatten[MapIndexed[ If[#2[[1]] == 1 || #2[[1]] == 5^n, #1, Drop[#1, -1]]&, {#[[1]], #[[1]] + γ(#[[4]] - #[[1]]) + δ(#[[2]] - #[[1]]), #[[4]]} & /@ (Function[p, Module[{mp}, mp = Mean[p]; (mp + β(# - mp))& /@ p]] /@ Nest[Flatten[(* just a "random" fancy form; many others are possible here *) Apply[{{#1, #5, #11, #6}, {#6, #2, #7, #12}, {#12, #11, #10, #9}, {#9, #7, #3, #8}, {#8, #10, #5, #4}}& @@ {#1, #2, #3, #4, #4 + α(#1 - #4), #1 + α(#2 - #1), #2 + α(#3 - #2), #3 + α(#4 - #3), #4 + (1 - α)(#1 + #3 - 2#4), (2α - 1)#1 + (1 - α)(#2 + #4), α(#1 + #3 - 2#4) + #4, (1 - α)#1 + α #3}&, #, {1}], 1]&, {{{1, 0}, {1, 1}, {0, 1}, {0, 0}}}, n]), {1}], 1]], {{-0.1, 0}, {-0.35, -0.2}}]], AspectRatio -> Automatic]]

These are the first three levels of growth. Show[GraphicsArray[Table[FractalTree[k], {k, 4}], GraphicsSpacing -> -0.05]]

The growth of this tree proceeds deterministically. We show the fifth level separately because of the fine details involved. Show[FractalTree[5]]

The next graphic is a fractal based on the iteration of the function z Ø 4 H1 + iL IH3 + i + 5 H1 + 2 iL z ê cL-1-Â M . We 2Â

display the number of iterations carried out until the condition †z§ > 100 is fulfilled as a function of the complex parameter c. DensityPlot[Function[c, (* iterate until |z| > 100 *) Module[{k = 1, z = 1.0 + 1.0 I, max = 100., maxk = 100}, While[k < maxk && Abs[z] < max, k++; z = (1/4 + I/4)((3 + I + (1/5 + 2I/5)* z/c)^(-1 - I))^(2I)]; k]][cx + I cy], {cx, -2, 2.25}, {cy, -3.4, 1.5}, ColorFunction -> (Hue[Pi #]&), Mesh -> False, ColorFunctionScaling -> False, FrameTicks -> None, (* use many points *) PlotPoints -> 600, Compiled -> True]

We now iterate a random subdivision of two triangles. The thickness of the edges of the triangles decreases with each iteration.



38

With[{level = 10}, Show[Graphics[Reverse[ MapIndexed[{Hue[#2[[1]]/9], Thickness[0.03/#2[[1]]], Line[Append[#, First[#]]]& /@ #1}&, NestList[Flatten[((* iteration of the subdivision *) Apply[Function[{d1, d2, d3}, (* divide longest side *) Which[# == 1, {{d1, #, d3}, {d2, #, d3}}&[ d1 + Random[Real, {0.25, 0.75}] (d2 - d1)], # == 2, {{d1, #, d2}, {d3, #, d2}}&[ d1 + Random[Real, {0.25, 0.75}] (d3 - d1)], # == 3, {{d2, #, d1}, {d3, #, d1}}&[ d2 + Random[Real, {0.25, 0.75}] (d3 - d2)]]&[ (* position of longest side *) (Position[#, Max[#]]&[#.#& /@ {#1 - #2, #1 - #3, #2 - #3}&[d1, d2, d3]])[[1, 1]]]], #, {1}]), 1]&, (* start triangles *) {{{0, -1}, {0, 1}, {3, -1}}, {{0, +1}, {3, 1}, {3, -1}}} // N, level], {1}]]], AspectRatio -> Automatic, PlotRange -> All]]

Lines can also be drawn in 3D space, as in this abstract branch. Module[{extend}, (* add some new hairs *) extend[x_, ω_] := Module[{c = N[Cos[ω]], s = N[Sin[ω]], vOld, vm, vPerp, v, α, β}, (* orthogonal directions *) {vOld, vm} = {x[[2]] - x[[1, 1]], x[[1, 2]] - x[[1, 1]]}; vPerp = #/Sqrt[#.#]&[vOld - vm vm.vOld]; v3 = #/Sqrt[#.#]&[Cross[vm, vPerp]]; {α, β} = x[[1]]; (* the new hairs *) Function[f, {{β, β + #/Sqrt[#.#]&[c vm + s vPerp + f s v3]}, α}] /@ {0, 1, -1}]; (* display iterated addition of hairs *) Show[Graphics3D[Rest[ MapIndexed[{Hue[(#2[[1]] - 2)/8], (* color and add various thickness *) Thickness[2^(-#2[[1]] - 3)], Line /@ #1}&, Map[First, (* iterate the process *) FoldList[Flatten[Function[x, extend[x, #2]] /@ #1, 1]&, {{{{0, 0, 0}, {0, 0, 1}}, {-Sin[28. Degree], 0, Cos[28. Degree]}}} // N, {30, 25, 20, 16, 11, 8, 5} Degree], {-3}]]]], PlotRange -> All, Boxed -> False]]

In the following construction, the edges of Platonic solids are taken and rotated continuously outward until they have the position of an edge again. (See [438÷] for a description of the resulting surfaces.) Needs["Graphics`Polyhedra`"]



39

RotatedSideWireFrame[platonicSolid: (Cube | Tetrahedron | Octahedron | Dodecahedron | Icosahedron), steps_Integer?(# > 2&), opts___] := Module[{l = Length[Faces[platonicSolid][[1]]] - 1, makeLines, combis, allLines, s = steps}, (* rotate edges outwards *) makeLines[points_] := Module[{l = Length[points]}, Join @@ Table[{(1 + t) points[[1]], (1 - (l - 2) (t - (i - 2)/(l - 2))) (1 + t) points[[i]] + (l - 2) (t - (i - 2)/(l - 2)) (1 + t) points[[i + 1]]}, {i, 2, l - 1}, {t, (i - 2)/(l - 2), (i - 1)/(l - 2), 1/(l - 2)/s}]] (* all possible combinations of points to rotate about *) combis = Join[Flatten[Table[RotateRight[#, i], {i, 0, l}]& /@ Faces[platonicSolid], 1], (* rotate in both directions *) Flatten[Table[RotateRight[#, i], {i, 0, l}]& /@ (Reverse /@ Faces[platonicSolid]), 1]]; (* all lines *) allLines = makeLines /@ Map[#/Sqrt[#.#]&[N[Vertices[platonicSolid][[#]]]]&, combis, {2}]; (* display all rotated lines *) Show[Graphics3D[{Thickness[0.001], MapIndexed[{Hue[(#2[[2]] - 1)/s 3/4], Line[#1]}&, allLines, {2}], MapIndexed[{Hue[(#2[[2]] - 1)/s 3/4], Line[#1]}&, Transpose[allLines, {1, 3, 2, 4}], {2}]}], opts, PlotRange -> {{-2, 2}, {-2, 2}, {-2, 2}}, Boxed -> False, ViewPoint -> {2, 2, 2}]] Show[GraphicsArray[ (* all five Platonic solids *) Apply[RotatedSideWireFrame[##, DisplayFunction -> Identity]&, {{Tetrahedron, 16}, {Octahedron, 15}, {Cube, 12}, {Dodecahedron, 8}, {Icosahedron, 10}}, {1}], GraphicsSpacing -> -0.25]]

Our next example involves an iterated construction using equilateral triangles. Each new magnified or shrunken triangle is attached to an old vertex. It is drawn in the plane formed by the normal to the old triangle and the line connecting the center of the old triangle to the vertex. (* the new triangles at the correct position *) newTriangle[x_, fac_] := Module[{mpo = Mean[x], mp2, mp3, dir1, dir2, poly2, poly3}, (* midpoint *) mp2 = x[[2]] + fac (x[[2]] - mpo); (* orthogonal directions *) dir1 = mpo - x[[2]]; dir2 = Cross[x[[2]] - x[[1]], x[[2]] - x[[3]]]; dir2 = #/Sqrt[#.#]&[dir2]; poly2 = Table[mp2 + fac (Cos[ϕ] dir1 + Sin[ϕ] dir2), {ϕ, 0, 2 2Pi/3, 2Pi/3}] // N; mp3 = x[[3]] + fac (x[[3]] - mpo); dir1 = mpo - x[[3]]; poly3 = Table[mp3 + fac Cos[ϕ] dir1 + fac Sin[ϕ] dir2, {ϕ, 0, 2 2Pi/3, 2Pi/3}] // N; {poly2, poly3}]; (* make three new polygons *) three[x_] := N[ {x, Map[({{-1, +Sqrt[3], 0}, {-Sqrt[3], -1, 0}, {0, 0, 1}}/2).#&, x, {-2}], Map[({{-1, -Sqrt[3], 0}, {+Sqrt[3], -1, 0}, {0, 0, 1}}/2).#&, x, {-2}]}



40

Here is a visualization of the first two steps of attaching new triangles. Show[Graphics3D[ Join[{{Hue[0], {Polygon[{{2, 0, 0}, {-1, Sqrt[3], 0}, {-1, -Sqrt[3], 0}}/2.]}}}, MapIndexed[{Hue[#2[[1]]/10], (* add color *) Polygon /@ Flatten[#1, 1]}&, Transpose[three[FoldList[ (* iterate the construction *) Flatten[Function[x, newTriangle[x, #2]] /@ #1, 1]&, {{{-2, 2Sqrt[3], 0}, {-5, 5Sqrt[3], -2Sqrt[3]}, {-5, 5Sqrt[3], 2Sqrt[3]}}/4 // N}, {1}]]]]]], PlotRange -> All, Lighting -> False, Boxed -> True, ViewPoint -> {3, 3, 3}]

Now, this process is repeated eight times. Show[Graphics3D[ Join[{{Hue[0], {Polygon[{{1, 0, 0}, {-1/2, 3^(1/2)/2, 0}, {-1/2, -3^(1/2)/2, 0}}]}}}, MapIndexed[{Hue[#2[[1]]/4], (* color the triangles *) Polygon /@ Flatten[#1, 1]}&, Transpose[three[FoldList[ (* iterate the construction *) Flatten[Function[x, newTriangle[x, #2]] /@ #1, 1]&, {{{-1/2, 1/2 Sqrt[3], 0 }, {-5/4, 5/4 Sqrt[3], -1/2 Sqrt[3]}, {-5/4, 5/4 Sqrt[3], +1/2 Sqrt[3]}} // N}, {1, 1, 1, 1, 1, 1, 1}]]]]]], PlotRange -> All, Lighting -> False, Boxed -> False, ViewPoint -> {3, 3, 3}]

Mathematica also includes functions to manipulate a graphic as a whole without explicitly manipulating, removing, or adding graphics primitives. The next image selects and shows only those triangles in the previous image whose centers have x-coordinates § 0. Show[Graphics3D[ {#[[1]], Select[#[[2]], (* the selection criteria *) (N[First[Total[#[[1]]]/3]] All, Lighting -> False, ViewPoint -> {3, 0, 1}, Boxed -> False]

In the next graphic, we manipulate directly the polygons of the above picture. Show[DeleteCases[ Show[%% /. (* invert *) Polygon[l_] :> Polygon[#/#.#& /@ l], (* split intersecting polygons *) PolygonIntersections -> False], _Line, Infinity] /. (* shrink resulting polygons *) Polygon[l_] :> With[{mp = Mean[l]}, {EdgeForm[], Polygon[(mp + 0.7(# - mp))& /@ l]}], PlotRange -> All, Boxed -> False, Lighting -> False, BoxRatios -> {1, 1, 1}]

3D graphics can be converted into 2D graphics and the resulting 2D polygons, lines, and points can be further manipulated within Mathematica. The following Christmas-themed input generates 413 random polyhedra, projects them into 2D, and places the resulting graphics on a grid in a random order. (* load polyhedra package *) Needs["Graphics`Polyhedra`"];



41

manyRandomPolyhedra[{Lx_, Ly_, δ_}] := Module[{randomPolyhedra, randomProjectedPolyhedra, randomPermutation, frame, d = Min[Lx, Ly]/20, ∂ = 0.5}, (* a random polyhedron *) randomPolyhedra[n_] := {SurfaceColor[Hue[Random[]], Hue[Random[]], 3 Random[]], (* iterate a random truncation/stellation *) Nest[(* random truncation or stellation *) If[Random[Integer] === 0, Truncate[#, Random[Real, {0.1, 0.4}]], Stellate[#, Random[Real, {1.3, 1.9}]]]&, (* randomly select a Platonic solid *) Polyhedron[{Tetrahedron, Hexahedron, Octahedron, Dodecahedron, Icosahedron}[[ Random[Integer, {1, 5}]]]], n][[1]]}; (* project into 2D *) randomProjectedPolyhedra[mp2D_] := Graphics[ Show[Graphics3D[randomPolyhedra[Random[Integer, {2, 3}]]], ViewPoint -> Table[Random[Real, {3/4, 4}], {3}], Boxed -> False, DisplayFunction -> Identity, PlotRange -> All, SphericalRegion -> True] /. (* color each face differently *) p_Polygon :> {SurfaceColor[Hue[Random[]], Hue[Random[]], 3 Random[]], p}] /. (* center approximately around origin and color lines *) (pl:(Polygon | Line))[l_] :> pl[(mp2D + # - {0.4, 0.4})& /@ l] /. Graphics[l_] :> Graphics[{Thickness[0.001], GrayLevel[Random[Real, {0.25, 1}]], l}]; (* random permutation of a list *) randomPermutation[l_] := Module[{b = l, n = Length[l]}, Do[(b[[{k, #}]] = b[[{#, k}]])&[ Random[Integer, {k, n}]], {k, n}]; b]; (* frame *) frame[{lx_, ly_}, d_] := With[{b = {{-1, -1}, {1, -1}, {1, 1}, {-1, 1}, {-1, -1}}}, Polygon[Join[{lx, ly} #& /@ b, Reverse[{lx + d, ly + d} #& /@ b]]]]; (* centers of projected polyhedron a grid; use random order of centers *) mps = randomPermutation[Flatten[Table[{x, y}, {x, -Lx, Lx, δ}, {y, -Ly + δ/2 Mod[x/δ, 2], Ly, δ}], 1]]; (* display graphic *) Show[{(* random projected polyhedra *) randomProjectedPolyhedra /@ mps, Graphics[{Hue[0], frame[{Lx, Ly}, d]}]}, AspectRatio -> Automatic, AspectRatio -> Automatic, PlotRange -> {(Lx + ∂ d) {-1, 1}, (Ly + ∂ d) {-1, 1}}]] SeedRandom[999]; manyRandomPolyhedra[{4, 3/2, 1/4}]

Because of its symbolic, numeric, pattern-matching, and graphical capabilities, constructing a variety of pictures is easy with Mathematica. Here are two further variations of a polyhedral flower.



42

Needs["Graphics`Polyhedra`"]; Needs["Graphics`Shapes`"]; With[{pp = 30}, Show[GraphicsArray[{Graphics3D[{ EdgeForm[{Hue[0.22], Thickness[0.001]}], SurfaceColor[Hue[0.3], Hue[0.45], 1.2], (* make polygons *) Map[MapThread[Polygon[Join[#1, Reverse[#2]]]&, #]&, Map[Partition[#, 2, 1]&, Map[ Partition[#, 2, 1]&, Transpose[Map[First, Table[ (* rotate faces outwards *) RotateShape[Map[Function[l, Module[{mp = Mean[l]}, mp + 0.6(1 - p^2) (# - mp)& /@ l]], Map[(p - 1)#&, Line[Append[#, First[#]]]& /@ First /@ Polyhedron[Dodecahedron][[1]], {-1}], {2}], p^2/2, -p^2/2, p^2/2], {p, -1, 1, 2/pp}], {2}]], {1}], {3}], {2}]}, Boxed -> False], (* form Graphics3D-object *) Graphics3D[{EdgeForm[{Hue[0.77], Thickness[0.001]}], SurfaceColor[Hue[0.22], Hue[0.85], 1.6], (* make polygons *) Map[MapThread[Polygon[Join[#1, Reverse[#2]]]&, #]&, Map[Partition[#, 2, 1]&, Map[ Partition[#, 2, 1]&, Transpose[Map[First, Table[ (* rotate faces outwards *) RotateShape[Map[Function[l, Module[{mp = Total[l]/3}, mp + 0.5(1 - p^2) (# - mp)& /@ l]], Map[(p - 1)#&, Line[Append[#, First[#]]]& /@ First /@ Polyhedron[Icosahedron][[1]], {-1}], {2}], p^3/2, Sin[Pi p]/2, p/4], {p, -1, 1, 2/pp}], {2}]], {1}], {3}], {2}]}, Boxed -> False]}]]]

It is possible to visualize real objects by using points, lines, and polygons directly in 3D space. Obtaining “realistic” images usually requires generating a large number of polygons. We could go on and display windmills, torsos, autos, starfish, cathedrals, castles, gears, the Eiffel tower [511÷], the Sagrada Familia, and so on. We will use more graphics in the next two subsections for various visualizations. Σ (* session summary *) TMGBs`PrintSessionSummary[]

à 1.2.3 Symbolic Calculations

D[f[x], x] differentiates f HxL once with respect to x. D[Sin[x], x]

Here is a slightly more complicated expression. f = Sin[Log[Tan[(ξ^2 + Exp[x])/(Cos[ξ^2 - 1] + Sqrt[ξ])]]]

The resulting manual differentiation is somewhat unpleasant. The result of differentiating this expression twice with respect to x is quite big, so we use Short to force Mathematica to show only a part. D[f, {ξ, 2}] // Short[#, 4]&

Here is a simple integral.



43

Integrate[Sin[x], x]

The following integral is tedious to find by hand. Integrate[ξ^3 Sin[ξ]^4, ξ]

By differentiating and simplifying, we get x 3 sinHxL4 again. D[%, ξ] Simplify[%]

Here is the definite integral Ÿ-¶ Ix4 + 4M ¶

-2

dx.

Integrate[1/(x^4 + 4)^2, {x, -Infinity, Infinity}]

We now consider a function that is complicated for integration. g = t^(2/3) Exp[-2 t] (t - 1)^(4/5)

It can be integrated analytically over the domain 1 to ¶. Integrate[g, {t, 1, Infinity}]

Because all of the special functions are numerically implemented for arbitrary complex arguments (in their domains) with arbitrary accuracy, we can also compute the numerical value with 50 digits. N[%, 50]

Here is the same integral calculated numerically to ten digits. NIntegrate[Evaluate[g], {t, 1, Infinity}, PrecisionGoal -> 10] // InputForm

Here, the function sinIx2 M is integrated five times.

Integrate[Sin[x^2], x, x, x, x, x]

Differentiating the result five times brings us back to sinIx2 M. D[%, x, x, x, x, x] // Simplify

The next input calculates the even Bernoulli numbers through the integral representation [580÷] m

B2 n = -H-1L 2

-H2 m+1L

‡

¶ -¶

d m-1 sech2 HxL d xm-1

2

.

Table[-(-1)^n 2^-(2n + 1) Integrate[D[Sech[x]^2, {x, n - 1}]^2, {x, -Infinity, Infinity}], {n, 10}]

The Bernoulli numbers are built-in functions of Mathematica. Table[BernoulliB[2n], {n, 10}]

Now let us consider a limit. The function e1êHx-1L has two different limit values, one from the left and from the right at the point x = 1. Limit[Exp[-1/(1 - x)], x -> 1, Direction -> +1] Limit[Exp[-1/(1 - x)], x -> 1, Direction -> -1]

We now solve a differential equation describing a damped oscillation x££ HtL + g x£ HtL + w2 xHtL = 0. DSolve[x''[t] + γ x'[t] + ω^2 x[t] == 0, x[t], t]



44

Suppose we want to approximate a function f HxL with the following properties by a polynomial in x: f H0L f £ H0L f H4L f £ H4L

= 1 = 2 = 8 = 45

f ≥ H4L =

0.

InterpolatingPolynomial[{{0, {1, 2}}, {4, {8, 45, 0}}}, x]

Here is the same polynomial in a simpler, but less practical, form. Simplify[%]

We check that it interpolates. {% /. {x -> 0}, D[%, x] /. {x -> 0}, % /. {x -> 4}, D[%, x] /. {x -> 4}, D[%, {x, 2}] /. {x -> 4}}

Here is the piecewise-continuous function pwHxL = qHxL qH1 - xL e5 x6 - 1u fracIx4 - 4M J¢ax2 + 2 x - 1q ¶N 3

6 1ê3

defined.

pw[x_] = If[0 < x < 1, Floor[(5 x^6 - 1)] FractionalPart[x^4 - 4]^3* Abs[Ceiling[x^2 + 2 x - 1]^6]^(1/3), 0]

Here is a canonical form of this function. PiecewiseExpand[pw[x]]

The next input calculates Ÿ-¶ pwHxL2 dx. ¶

Integrate[pw[x]^2, {x, -Infinity, Infinity}]

And here is a series expansion of this function at a point where a discontinuity occurs. Series[pw[x], {x, Sqrt[3] - 1, 1}]

Next, we solve a well known-differential equation of mathematical physics describing (among other things) the behavior of a quantum particle in a constant electric field. DSolve[ψ''[z] + e F z ψ[z] == ψ[z], ψ[z], z]

The Vandermonde matrix of the nth-order is easy to implement in the following way. VandermondMatrix[n_, x_] := Table[x[i]^j, {i, 0, n}, {j, 0, n}]

Here is the Vandermonde matrix of the third order. x[i] is a typical Mathematica equivalent for xi . MatrixForm[VandermondMatrix[3, x]]

Here is the value of its determinant. Det[VandermondMatrix[3, x]]

This product can also be written as a product. Factor[%]

The following function LUMatricesVandermonde implements the LU-decomposition of the nth-order Vandermonde matrix [1401÷].



45

LUMatricesVandermonde[n_, x_] := Module[{X = Table[x[k], {k, 0, n}], e, h, b, L, U}, (* recursive definitions for elementary and complete symmetric polynomials *) e[0, _] := 1; e[_, {}] := 0; h[0, _] := 1; h[_, {}] := 0; e[r_, l_] := Factor[e[r, Most[l]] + Last[l] e[r - 1, Most[l]]]; h[r_, l_] := Factor[h[r, Most[l]] + Last[l] h[r - 1, l]]; b[r_, y_] := Factor[Sum[(-1)^(r - k) e[r - k, Take[X, r]] y^k, {k, 0, r}]] (* lower and upper triangular matrices *) L = Table[If[i < j, 0, h[i - j, Take[X, j + 1]]], {i, 0, n}, {j, 0, n}]; U = Table[If[i > j, 0, b[i, X[[j + 1]]]], {i, 0, n}, {j, 0, n}]; (* return matrices *) {L, U}]

Here is the decomposition for the third order Vandermonde matrix. {L, U} = LUMatricesVandermonde[3, x]; {L, U} // (MatrixForm /@ #)&

Multiplying the two matrices recovers the original matrix. L.U // Expand

The last LU-decomposition is not unique. The next inputs use the function Solve to calculate the most general solution. (* general ansatz form for the matrices L and U *) With[{n = 3}, MatrixForm /@ {L = Table[If[i < j, 0, l[i, j]], {i, 0, n}, {j, 0, n}], U = Table[If[i > j, 0, u[i, j]], {i, 0, n}, {j, 0, n}]}] Solve[(* the decomposition identity that must hold *) L.U == VandermondMatrix[3, x], (* the 10 variables l[i, j] and the 10 variables u[i, j] *) Cases[{L, U}, _l | _u, Infinity]] // (Factor //@ #)&

Next, we calculate symbolically the eigenvalues of a 50 μ 50 Redheffer matrix. The matrix elements ai j are 1 if j = 1 or if i divides j, and 0 otherwise. Redheffer[d_] := Table[If[j == 1 || IntegerQ[j/i], 1, 0], {i, d}, {j, d}];

A Redheffer matrix of dimension n has n - elog2 HnLu - 1 eigenvalues 1 (see [1312÷] and [1313÷]). The remaining six (for n = 50) eigenvalues are the roots of an irreducible polynomial of degree 6. They are represented as Root-objects. Eigenvalues[Redheffer[50]]

The following example is a linear inhomogeneous system of equations with eight unknowns. (We show the equations in abbreviated form.) gls = Table[Sum[(i + j)^j x[i], {i, 8}] == j, {j, 8}]; Short /@ gls

We get its exact solution. Solve[gls, Table[x[i], {i, 8}]]

Here is a simple system of nonlinear equations and its solution. x2 + y2 = 1 x4 + y4 = 4 Solve[{x^2 + y^2 == 1, x^4 + y^4 == 4}, {x, y}]



46

The function Eliminate eliminates variables from a system of polynomial equations. Eliminate[{x^6 + y^6 == 6, x^8 + y^8 == 8}, {y}]

Mathematica can also solve higher order univariate polynomial equations. Solve[x^7 - a x + 3 == 0, x]

The result contained again the Root function. Root-objects are symbolic representations of the roots of polynomials. The first argument specifies the polynomial, and the second, the root number. (See Chapter 1 of the Symbolics volume of the GuideBooks [1285÷] for details.) Like any other function in Mathematica, they can be manipulated, for instance, differentiated. D[Root[-3 + a #1 - #1^7 &, 1], {a, 2}]

Here is the numerical value of the root for a given value of a to 50 digits. N[Root[-3 + a #1 - #1^7 &, 1] /. a -> 7, 50]

Backsubstitution shows that the equation gives zero to a good approximation. x^7 - a x + 3 /. a -> 7 /. x -> %

Here is a plot of the root; the parameter a varies between -10 and 10. Plot[Root[-3 + a #1 - #1^7 &, 1], {a, -10, 10}]

The following input solves a transcendental equation. Solve[Log[2 x] + Log[3 x] + Log[5 x] == 1/2, x]

The following example is a simple power series expansion up to the ninth order. Series[Sqrt[1 + x], {x, 0, 9}]

The next one is not so simple. It is not a Taylor series because logarithms appear. Series[x^x, {x, 0, 4}]

What is the first nonvanishing term in the series expansion of sinHtanHxLL - tanHsinHxLL [1235÷]? Series[Sin[Tan[x]] - Tan[Sin[x]], {x, 0, 9}]

Here is a Laurent series. Series[1/(Sin[x] - x - x^3/3 + x^5), {x, 0, 6}]

Here is a short program using l’Hôspital’s rule for determining the limit of

sinHtanHxLL-tanHsinHxLL arcsinHarctanHxLL-arctanHarcsinHxLL

as x Ø 0.

numerator = Sin[Tan[x]] - Tan[Sin[x]]; denominator = ArcSin[ArcTan[x]] - ArcTan[ArcSin[x]];

We differentiate the numerator and denominator until we get a determined quantity. lHospitalList = Table[D[numerator, {x, i}]/D[denominator, {x, i}], {i, 7}]; If[(* zero denominator? *) (Denominator[#] /. x -> 0) == 0, Indeterminate, # /. x -> 0]& /@ lHospitalList

Of course, Mathematica can also calculate this limit directly. (Mathematica can also compute limits in cases in which l’Hôspital’s rule is not applicable [586÷], [163÷], [1131÷].) Limit[(Sin[Tan[x]] - Tan[Sin[x]])/ (ArcSin[ArcTan[x]] - ArcTan[ArcSin[x]]), x -> 0]

Here is an exact value for the Gauss hypergeometric function with numeric arguments.



47

Hypergeometric2F1[3/2, 4, 1, z]

We can also evaluate a high-order Hermite polynomial. HermiteH[23, z]

The command FunctionExpand rewrites an expression using a simpler function than the original one. In the following, a trigonometric expression is converted to one involving square roots only. FunctionExpand[Sin[1/(2^3 3 5) Pi]]

Here is a similar example. FunctionExpand[Tan[Pi/32]]

The previous expression is an algebraic number. It is a root of the polynomial that is the first argument of the following Root-object. RootReduce[%]

Next, we find the prime factor decomposition of a relatively large number. FactorInteger[4951486756871515]

Here is an abbreviated list of all numbers dividing the number 4951486756871515. Divisors[4951486756871515] // Short[#, 6]&

This is the one-billionth prime number. Prime[10^9]

We now decompose a polynomial into smaller ones that, when plugged into each other, give again the starting polynomial. (This specific example was already decomposed by Vieta in 1594 [232÷].) Decompose[45 x - 3795 x^3 + 95634 x^5 - 1138500 x^7 + 7811375 x^9 34512075 x^11 + 105306075 x^13 - 232676280 x^15 + 384942375 x^17 - 488494125 x^19 + 483841800 x^21 378658800 x^23 + 236030652 x^25 - 117679100 x^27 + 46955700 x^29 - 14945040 x^31 + 3764565 x^33 740259 x^35 + 111150 x^37 - 12300 x^39 + 945 x^41 45 x^43 + x^45, x]

Sums can also be computed symbolically. Here are the first few partial sums for ⁄nk=1 k j . TableForm[Table[Sum[k^j, {k, n}], {j, 1, 8}]]

It is even possible to compute infinite sums analytically. Sum[1/k^6, {k, Infinity}]

Here are two more complicated sums. The summands and the result contain Riemann’s Zeta function. Sum[(-1)^n/n^2 Gamma[n]^2/Gamma[2n], {n, Infinity}] Sum[(Zeta[k] - 1) Exp[-k], {k, 2, Infinity}]

Here is a complicated finite sum. The result contains the Polygamma function and a sum of roots of a quartic polynomial. Sum[(k^2 - 1)/(k^4

+ 1), {k, 1, n}]

Mathematica’s functions Integrate, Sum, DSolve are very powerful and can integrate, sum, and solve differential equations of quite complicated functions. However, for efficiency, the solution is typically not automatically simplified. But Mathematica provides a variety of functions allowing us to rewrite results from functions like Integrate, Sum,



48

DSolve in various ways. For instance, here is a more explicit form of the last result (not containing the function RootSum any more). Normal[%] // Simplify

Using the function FullSimplify, we can further collapse the last result. % // FullSimplify

A closed form for the partial sum of the first n Taylor coefficients of sinHxL. Sum[(-1)^k/(2k + 1)! x^(2k + 1), {k, 0, n}] // FullSimplify

For a given value of n, we recover the first n Taylor coefficients of sinHxL. Series[% /. n -> 12, {x, 0, 12}]

Here is a complicated finite sum that can be expressed in polylogarithmic and Lerch functions: ‚ k -1ê2 Hk wL n

x

iw

m

k

k=1

x2 i p g k .

M[{m_, ω_, γ_}, n_, x_] := Sum[k^(-1/2) (k ω)^(I ω) (x/k)^m x^(I 2Pi γ k), {k, n}]; M[{m, ω, γ}, n, x] // PowerExpand // TraditionalForm

For larger n the last sums shows a complicated, hierarchical behavior [537÷]. Here is an example for the parameter values m = 0.2, w = 7, g = 1.007, and n = 100. Plot[Evaluate[Im[M[{0.2, 7, 1.007}, 100, x]]], {x, 0, 1}, PlotPoints -> 1000, Frame -> True, Axes -> False]

The following sum calculates the interaction energy of a point charge at position z0 between two flat, parallel, perfectly conducting walls of distance a using mirror charges [1192÷]. Sum[1/(a n) - 1/(2n a - 2 z0) - 1/(2 (n - 1) a + 2 z0), {n, Infinity}] // Normal // Simplify

A Green’s function approach to the same problem yields the following integral and, of course, evaluates to the same result [1192÷]. Integrate[Cosh[k a]/Sinh[k a] - 1 - Cosh[k(a - 2 z0)]/Sinh[k a], {k, 0, Infinity}, Assumptions -> a > 0 && z0 > 0 && z0/a < 1]

A series expansion of the energy around z0 = 0 or z0 = a yields the force on the point charge. {Series[%, {z0, 0, 6}], Series[%, {z0, a, 6}]} // FullSimplify

Next, we use Mathematica to prove a neat identity discovered by Ramanujan:

3

cos

2p 9

+

3

cos

4p 9

-

3

p

cos

9

=

3

3

3

2

9

-3 .

Cos[2Pi/9]^(1/3) + Cos[4Pi/9]^(1/3) (Cos[1Pi/9])^(1/3) - (3 9^(1/3)/2 - 3)^(1/3)

The last identity contains algebraic and trigonometric expressions. For algorithmic treatments, algebraic expressions are always preferable. In algebraic form, the identity has the following form. Together[TrigToExp[%]]



49

The function RootReduce canonicalizes algebraic expressions. The identity can be simplified to 0. RootReduce[%]

Here is more challenging example: A three-line proof of Legendre’s celebrated identity for complete elliptic integrals [442÷] p EHmL KH1 - mL - KHmL KH1 - mL + EH1 - mL KHmL = . 2 The integral ‡ ‡ 0

p

p

2

2

0

1 - m sin2 HxL - H1 - mL sin2 HyL

1 - m sin HxL 2

1 - H1 - mL sin HyL

dx d y

2

is the left-hand side of Legendre’s identity. integrand[x_, y_, m_] := (1 - m Sin[x]^2 - (1 - m) Sin[y]^2)/ Sqrt[1 - m Sin[x]^2]/Sqrt[1 - (1 - m) Sin[y]^2] Integrate[integrand[x, y, m], {y, 0, Pi/2}, {x, 0, Pi/2}, GenerateConditions -> False] // FunctionExpand // Together

Differentiation shows that the last expression is independent of m. D[%, m] // Together

This means EHmL KH1 - mL - KHmL KH1 - mL + EH1 - mL KHmL equals a constant, and evaluating the above integrand for m = 0 shows that the constant is p ê 2.

Integrate[integrand[x, y, 0], {x, 0, Pi/2}, {y, 0, Pi/2}]

A powerful command for algebraic computations is GroebnerBasis. Given a set of polynomials, the function GroebnerBasis can transform this set into triangular form, so that a numerical solution is easily possible. Groeb nerBasis can also be used to eliminate certain variables from a set of polynomials. In the following example, we are looking for an equation connecting the area A of a triangle with the radius of its circumscribed circle, with radius R and the edge lengths l12 , l13 , and l23 . Clear[x1, y1, x2, y2, x3, y3, X, Y, R, A]; GroebnerBasis[{(* all equations of the problem *) (* defining equations for the circumscribed circle *) (X - x1)^2 + (Y - y1)^2 - R^2, (X - x2)^2 + (Y - y2)^2 - R^2, (X - x3)^2 + (Y - y3)^2 - R^2, (* defining equations for the length of the edges *) (x2 - x1)^2 + (y2 - y1)^2 - l12^2, (x3 - x2)^2 + (y3 - y2)^2 - l32^2, (x1 - x3)^2 + (y1 - y3)^2 - l13^2, (* defining equations for area *) (1/2(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3))^2 - A^2}, (* the variables to keep *) {R, l12, l23, l13, A}, (* the variables to eliminate *) {x1, y1, x2, y2, x3, y3, X, Y}]

The last polynomial in the result means that the relation we were looking for is A = l12 l23 l13 ê H4 RL. The first polynomial in the result expresses the area in the edge lengths only. Let us use GroebnerBasis [288÷] again to solve a slightly more complicated example: the area of the medial parallelogram [28÷] of a tetrahedron expressed through the edge length of the tetrahedron [1404÷], [65÷]. We start with a generic tetrahedron. From this tetrahedron, we remove two nonincident edges. The midpoints of the remaining



50

four edges form a parallelogram, the medial parallelogram. We want to express the area of this parallelogram through the six lengths of the edges of the original tetrahedron. Below is a sketch of the tetrahedron. The two red-colored edges P1 P2 and P3 P4 are the removed edges. P4

P3 P1 P2

The next input calculates the formula we are looking for. Module[{(* coordinates of the four vertices *) p1 = {0, 0, 0}, p2 = {p2x, 0, 0}, p3 = {p3x, p3y, 0}, p4 = {p4x, p4y, p4z}, p13, p14, p23, p24}, (* coordinates of the midpoints of the edges *) p13 = (p1 + p3)/2; p14 = (p1 - p4)/2; p23 = (p2 + p3)/2; p24 = (p2 + p4)/2; GroebnerBasis[ {(* edge lengths expressed through coordinates of vertices *) l12^2 - (p1 - p2).(p1 - p2), l13^2 - (p1 - p3).(p1 - p3), l23^2 - (p2 - p3).(p2 - p3), l14^2 - (p1 - p4).(p1 - p4), l24^2 - (p2 - p4).(p2 - p4), l34^2 - (p3 - p4).(p3 - p4), (* medial parallelogram area A expressed through coordinates of vertices *) A^2 - Cross[p14 - p13, p23 - p13].Cross[p14 - p13, p23 - p13]}, (* the variables to keep *) {l12, l13, l14, l23, l24, l34, A}, (* the variables to eliminate *) {p2x, p3x, p3y, p4x, p4y, p4z}, MonomialOrder -> EliminationOrder]]

In the last subsection, we made use of the Polyhedra` package. Mathematica comes with a wide set of standard packages carrying out various numerical, graphical, and symbolic operations not built into the Mathematica kernel. Let us make use of the package AlgebraÌnequalitySolve` for doing some symbolic calculations. The package implements the function InequalitySolve. Needs["AlgebraÌnequalitySolve`"]

As the function name indicates, InequalitySolve “solves” inequalities. Solving an inequality here means describing the solution sets in a canonicalized manner. The canonicalized form is a hierarchical description of the allowed intervals for the variables. ?InequalitySolve

Next, we “solve” the inequality I-16 x6 + 24 x4 - 9 x2 - 4 y4 + 4 y2 M - 1 ê 8 < 0. 2

[x_, y_] = (24x^4 - 9x^2 - 16x^6 + 4y^2 - 4y^4)^2 - 1/8; iSol = InequalitySolve[[x, y] < 0, {x, y}];

Because the result iSol is quite large and its structure is not immediately recognizable, we do not display the result. It has 25 independent parts. iSol // Length



51

Here is the first part. First[iSol]

è è It is of the form x1 < x < x2 Ï Iy1 HxL < y < y2 HxL Í y1 HxL < y < y2 HxLM. This form is the canonicalized description of one

region where the above inequality holds. The regions are areas or lines extending along the y-direction over a fixed xinterval. (For a more detailed description, see Section 1.2.3 of the Symbolics volume [1285÷] of the GuideBooks.) Similar to the above Solve example, when “solving” inequalities, one often ends up with Root-objects. The x1 , x2 are è è algebraic numbers and y1 HxL, y2 HxL, y1 HxL, and y2 HxL are algebraic functions of x1 and x2 , which means they are inverse functions of polynomials that generically cannot be inverted using elementary functions. It is straightforward to visualize the canonicalized regions where the inequality holds. We just form polygons by è è traversing y1 HxL from x1 to x2 and going back along y2 HxL from x2 to x1 and similarly for y1 HxL, y2 HxL. The little function makePolygon forms a polygon from a logical combination of inequalities. makePolygon[Inequality[x1_, Less, x, Less, x2_] && Inequality[y1_, Less, y, Less, y2_], plotpoints:pp_Integer] := With[{(* avoid endpoints *) ∂ = 10.^-12}, Polygon[Join[ (* bottom and top boundaries *) Table[{x, y1}, {x, x1 + ∂, x2 - ∂, (x2 - x1 - 2∂)/pp}], Table[{x, y2}, {x, x2 - ∂, x1 + ∂, (x1 - x2 + 2∂)/pp}]]]]

iSol contains 41 independent 2D regions. Here, we show them; each one has a randomly assigned color. (The regions described by the inequality are “thickened” versions of the Lissajous curve 8x, y< = 8sinH2 JL, cosH3 JL ((a && #)& /@ b))]}], (* the Lissajous curve *) ParametricPlot[{Sin[2ϑ], Cos[3ϑ]}, {ϑ, 0, 2Pi}, PlotRange -> All, PlotPoints -> 200, DisplayFunction -> Identity, PlotStyle -> {{GrayLevel[0.5], Thickness[0.01]}}]}, AspectRatio -> Automatic, Frame -> True, PlotRange -> {{-1.2, 1.2}, {-1.2, 1.2}}]

The next input finds the smallest value of R, such that all points 8x, h< with †8x, h R, the value of the left-hand side of the above inequality is positive (meaning the maximal spatial extension of the set defined by the inequality from the origin). ForAll[{ξ, η}, {ξ, η, R} ∈ Reals && Norm[{ξ, η}] > R, [ξ, η] > 0] // (* write in quantifier-free form *) Resolve

The resulting value of R is a root of a polynomial of degree 15 with integer coefficients. Its numerical value is 1.38143…. N[%]

The next input proves for positive a, b, c the so-called Nesbitt inequality a ê Hb + cL + b ê Ha + cL + c ê Ha + bL ¥ 3 ê 2 [1244÷]. We specify the inequality using the forall quantifier. ForAll[{a, b, c}, Element[{a, b, c}, Reals] && a > 0 && b > 0 && c > 0, a/(b + c) + b/(a + c) + c/(a + b) >= 3/2] // Resolve



52

In case the constant 3 ê 2 were not known in advance, one could easily determine it, either through quantifier elimination or minimization. ForAll[{a, b, c}, Element[{a, b, c, R}, Reals] && a > 0 && b > 0 && c > 0, a/(b + c) + b/(a + c) + c/(a + b) >= R] // Resolve Minimize[a/(b + c) + b/(a + c) + c/(a + b), a > 0 && b > 0 && c > 0, {a, b, c}]

While being inherently of algebraic nature, functions like Resultant and GroebnerBasis can often be fruitfully used to deal with analysis problems (as we will do repeatedly in the GuideBooks). Here we use them to derive nonlinear polynomial differential equations for the function @HzL = tanHlnHzLL. Differentiating @HzL repeatedly shows powers of the secHlnHzLL and @HzL. Table[Derivative[k][@][z] - D[Tan[Log[z]], {z, k}], {k, 0, 3}] // Together // Numerator

Eliminating secHlnHzLLn and tanHlnHzLLm yields polynomial differential equations such as z @≥ HzL = @£ HzL H2 @HzL - 1L in z, @HzL, @£ HzL, @≥ HzL and maybe higher derivatives of @HzL.

GroebnerBasis[%, {}, {Tan[Log[z]], Sec[Log[z]]}, MonomialOrder -> EliminationOrder] // Factor

Taking two such differential equations yields a z-free, nonlinear, third-order differential equation for @HzL = tanHlnHzLL. Resultant[%[[1, -1]], %[[2, -1]], z] // Simplify

Substituting tanHlnHzLL for @HzL in the last differential equations gives zero. % /. {@[z] :> Tan[Log[z]], Derivative[k_][@][z] :> D[Tan[Log[z]], {z, k}]} // Simplify

Next, we examine a self-defined rule. The function x p sinHxq L lnHxr L cannot be integrated by Mathematica with respect to x (it is not possible to express this integral in named special functions). Integrate[x^p Tan[x^q] Log[x^r], {x, 0, Pi}]

However, we can create a new symbol XtoPowerαTimesSinOfXtoPowerβTimesLogOfXtoPowerγ[p, q, r] for this integral. Unprotect[Integrate]; Integrate[x_^α_. Tan[x_^β_.] Log[x_^γ_.], {x_, 0, Pi}] := XtoPowerαTimesTanOfXtoPowerβTimesLogOfXtoPowerγ[α, β, γ]; Protect[Integrate];

Mathematica can use this rule when it is possible. Integrate[zÎ Tan[z^23] Log[z], {z, 0, Pi}]

Mathematica is good at matching patterns. For example, we can extract all elements from a list that are the product of x with any factor, including the not explicitly written factor 1. Cases[{3, 2 + 7 I, 6 x, I x, u x, x, a x, u}, Optional[_] x]

Here is an umbral example [380÷]. When one interprets the even powers ,k in the expanded form of H, - iLn = 0 as indexed numbers ,k , then the ,k are just the absolute values of the Euler numbers †Ek § [439÷], [479÷]. Here is an

example for n = 12. This is the expanded form. Expand[(, - Sqrt[-1])^12]



53

Using patterns and replacements is straightforward to go from the monomials ,k to the indexed quantities ,k . % /. ,^k_. :> Subscript[,, k]

This checks the above statement about the Euler numbers. % /. Subscript[,, k_] :> Abs[EulerE[k]]

The next input tests if the first four digits of p appear somewhere within the first 50000 digits of the decimal representation of 17-1000 . (It turns out that within the periodic part of the decimal expansion of 17-1000 , the first 1230 digits of p appear many times [1251÷], [1252÷]; almost all real numbers are lexicons [226÷], [584÷].) MatchQ[First[RealDigits[N[1/17^1000, 50000]]], {___, 3, 1, 4, 1, ___}]

The first four digits of p appear also in the (integer) digits of 171000 . MatchQ[IntegerDigits[17^1000], {___, 3, 1, 4, 1, ___}]

Mathematica can simplify expressions when it knows properties of the variables. In the next input, it is assumed that p is an odd prime. Simplify[Sin[p^2 Pi] + (-1)^p, Element[p, Primes] && p > 2]

The following expression does not automatically “simplify” to x + 1. Sqrt[x^2 + 2 x + 1]

Actually, such a transformation would be mathematically wrong for many complex numbers. {Sqrt[x^2 + 2 x + 1], x + 1} /. x -> -3 + 2I

Under the additional assumption that x is a positive real number, Mathematica can simplify

x2 + 2 x + 1 to x + 1.

Simplify[Sqrt[x^2 + 2 x + 1], Element[x, Reals] && x > 0]

Many more functions in Mathematica perform symbolic mathematics. The Numerics [1284÷] and Symbolics [1285÷] volumes of the GuideBooks discuss many more details. Mathematica can carry out complicated and never-before-carried out calculations in various mathematical topics with great ease. The following short code, for instance, searches for a number whose digits of its decimal expansion digits agree with the terms of its (nonsimple) continued fraction expansion. Make Input

(* difference between decimal expansion and continued fraction *) δ[l_] := N[Abs[FromDigits[{l, 1}, 10] Fold[#2[[2]]/(#2[[1]] + #1)&, l[[-2]]/l[[-1]], Partition[Reverse[Drop[l, -2]], 2]]]]; (* recursively add digit pair and keep a set of best lists *) Nest[First /@ Take[#, Min[43, Length[#]]]&[Sort[{#, δ[#]}& /@ Flatten[Flatten[Table[Join[#, {i, j}], {i, 0, 9}, {j, 9}], 1]& /@ #, 1], (#1[[2]] < #2[[2]])&]]&, {{0}}, 72][[1]]

After running, the code above returns the following result.



54

{0, 7, 9, 6, 7, 8, 9, 9,

2, 1, 2, 2, 7, 3, 2, 6,

7, 7, 7, 9, 4, 6, 5, 7,

3, 1, 3, 1, 1, 2, 8, 6,

9, 4, 7, 9, 8, 5, 9, 9,

4, 5, 2, 1, 1, 1, 4, 2,

4, 9, 5, 7, 9, 6, 9, 8,

1, 1, 1, 3, 6, 1, 8, 1,

9, 5, 8, 8, 9, 5, 9, 9,

5, 2, 7, 3, 4, 4, 1, 4,

7, 7, 7, 7, 6, 8, 5, 5,

3, 2, 2, 5, 1, 5, 1, 3,

9, 4, 9, 5, 9, 9, 7, 5,

2, 2, 8, 2, 1, 3, 2, 1,

7, 8, 8, 8, 7, 6, 7, 6,

1, 5, 1, 1, 3, 4, 3, 3,

6, 9, 9, 7, 8, 7, 9, 8,

1, 1, 8, 1, 2, 1, 1, 1, 6};

The next input forms the continued fraction corresponding to the last list. With[{b = C /@ %}, DeleteCases[Hold[0 + #]& @@ {Fold[#2[[2]]/(#2[[1]] + #1)&, b[[-2]]/b[[-1]], Partition[Reverse[Drop[b, -2]], 2]]}, C, Infinity, Heads -> True]] // InputForm

Collapsing the last expression into a fraction and then calculating a high-precision approximation of this fraction yields a decimal number, showing that the first 100 digits agree with the continued fraction terms. ReleaseHold[%] N[%, 100]

Here is a short way to show the agreement of the first 100 digits using Mathematica. RealDigits[%%, 10, 100, 0][[1]] == Take[%%%%, 100]

The code above can easily be adapted to calculate numbers with many identical decimal and continued fraction digits, for the case of a simple continued fraction, and to deal with the case for a base different from 10. Mathematica also allows larger mathematical formulas and algorithms to be entered in a direct way. As a small example, let us implement the calculation of the series of the conformal map w = f HzL after Szegö’s method (see [346÷], [509÷], [1106÷], [702÷], and [1180÷]), which maps a square in the z-plane onto the unit disk in the w-plane. The approximation of w = f HzL of order n is given by: h jk = HnL

H

1 l

j ê k ‡ z z ds C

= h jk

dn = det H

GHnL HxL =

j, k = 0, 1, …, n

HnL

h jk ,

j = 0, 1, …, n, k = 0, 1, …, n - 1

x j,

j = 0, 1, …, n, k = n

ln HxL = det GHnL HxL ln HxL

pn HxL =

p0 HxL = 1

dn-1 dn

kn Ha, bL = ‚ pi HaL pi HbL n

wn HzL =

i=0

p

4 kn H0, 0L

2 ‡ kn H0, xL dx z

0

Here, l is the length of the boundary of the square, and the integration has to be carried out along the boundary of the

square. The pn HxL form orthogonal polynomials. HHnL and GHnL are square matrices of dimension n with elements h j,k ,



55

and g j,k respectively. Here, the above-described method is implemented. ord determines the order in z. ConformalMapSquareToUnitDisk[ord_, z_] := Module[{h, H, G, d, l, p, k, t, a, b, λ, integrand, edgeList = {-1 + I, 1 + I, 1 - I, -1 - I}}, lineSegments = Partition[Append[edgeList, First[edgeList]], 2, 1]; (* edge length *) λ = Total[Abs[#[[2]] - #[[1]]]& /@ lineSegments]; (* the h-integrals *) integrand[j_, k_] = Plus @@ ((Abs[#[[2]] - #[[1]]]* (#[[1]] + t (#[[2]] - #[[1]]))^j*((#[[1]] + t (#[[2]] - #[[1]]))^k /. c_Complex :> Conjugate[c]))& /@ lineSegments); (* scalar product *) h[j_, k_] := h[j, k] = 1/λ Integrate[integrand[j, k], {t, 0, 1}]; (* Hankel-Hadamard-Gram determinants *) H[n_] := Array[h, {n + 1, n + 1}, 0]; d[n_] := d[n] = Det[H[n]]; G[n_, ξ_] := Array[If[#2 < n, h[#1, #2], ξ^#1]&, {n + 1, n + 1}, 0]; l[n_, ξ_] := l[n, ξ] = Det[G[n, ξ]]; (* Szegö polynomials *) p[0, ξ_] = 1; p[n_, ξ_] := p[n, x] = l[n, ξ]/Sqrt[d[n] d[n - 1]]; (* Szegö kernel *) k[a_, b_] = Sum[p[i, a] p[i, b], {i, 0, ord}]; Cancel[Pi/(4 k[0, 0]) Expand[Integrate[k[0, ξ]^2, {ξ, 0, z}]]]]

Here is an example (for ord = 8, the constant term deviates about 0.09 % from its exact value). ConformalMapSquareToUnitDisk[8, z]

Using Mathematica’s graphics capabilities, we can easily visualize the conformal map generated by the last function. The left picture shows a mesh in the square with the corners -1 + i, 1 + i, 1 - i, -1 - i, and the right picture shows the mesh after mapping; the unit disk is shown underlying in gray. With[{pp = 15}, Module[{points, opts}, (* points forming the grid *) points = Table[N[x + I y], {x, -1, 1, 1/pp}, {y, -1, 1, 1/pp}]; (* common graphics options *) opts[label_] := Sequence[AspectRatio -> Automatic, PlotLabel -> label, PlotRange -> {{-1.2, 1.2}, {-1.2, 1.2}}]; Show[GraphicsArray[{ (* the original square *) Graphics[{Thickness[0.001], Line /@ #, Line /@ Transpose[#]}&[ Map[{Re[#], Im[#]}&, points, {-1}]], opts["z-plane"]], (* the mapped square *) Graphics[{{GrayLevel[3/4], Disk[{0, 0}, 1]}, Thickness[0.001], Line /@ #, Line /@ Transpose[#]}&[ Map[{Re[#], Im[#]}&, Map[Function[z, Evaluate[N[%]]], points, {-1}], {-1}]], opts["w-plane"]]}]]]]

Mathematica has most of the special functions of mathematical physics (see Chapter 3 of the Symbolics volume [1285÷] of the GuideBooks) [865÷]. Using elliptic functions, it is possible to find an exact formula for a conformal map from a rectangle to the unit disk [1013÷]. The next graphic visualizes the exact map. To avoid repeating the last input, we modify the last input in a programmatic way and then evaluate the new code.



56

Module[{wExact, k = InverseEllipticNomeQ[Exp[-2. Pi]], K}, K = EllipticK[k]; (* the exact conformal map *) wExact[z_] := (1 + I JacobiSN[K (z + I), k])/ (1 - I JacobiSN[K (z + I), k]); (* or another map in elliptic functions: {g2, g3} = WeierstrassInvariants[{1., I}]; wExact[z_] := (1 - I WeierstrassP[(z + (1 + I))/2, {g2, g3}])/ (1 + I WeierstrassP[(z + (1 + I))/2, {g2, g3}]); *) (* to obtain above symmetry, apply in addition: makeAboveSymmetry[z_] := (3 - Sqrt[2] + I + (2 Sqrt[2] + 1 + (Sqrt[2] + 1) I) z)/ (2 Sqrt[2] + 1 + (Sqrt[2] + 1) I + (3 - Sqrt[2] + I) z); *) (* reuse the above input *) Last[DownValues[In][[-2]] /. (* make changes to last input *) HoldPattern[%] -> (* makeAboveSymmetry @ *) wExact[z]]]

The typesetting capabilities of Mathematica allow mathematical formulas and algorithms to be entered in a still more direct way. ConformalMapSquareToUnitDiskSF @ω_Integer ? Positive, z_D := ModuleB8h, H, G, d, l, p, k, t, a, b, ρ,

* = 8−1 + I, 1 + I, 1 − I, −1 − I 1}; Show[(* use different initial conditions of the form {x0, 0} *) Table[(* solve equations of motion *) nsol = NDSolve[Join[odesN, {x[0] == x0, y[0] == 0}], {x, y}, {t, 0, 2T/x0}, MaxSteps -> 10000]; (* plot the path *) ParametricPlot[Evaluate[{x[t], y[t]} /. nsol], {t, 0, 2T/x0}, Axes -> False, DisplayFunction -> Identity, PlotStyle -> {{Thickness[0.001], Hue[x0/2.6]}}], {x0, 0.1, 1.9, 0.1}], DisplayFunction -> $DisplayFunction, Frame -> True, FrameTicks -> False, FrameStyle -> {Thickness[0.02]}]]

The penultimate example of this subsection deals with a slightly more complicated example: the lines of magnetic induction (which in a 2D cylindrical geometry are also the lines of constant vector potential) of a cylindrical magnet with an air gap. The z component Az of the vector potential A (a is the inner radius, b is the outer radius of the magnet, 2 p - 2 a is the slit width, and the slit is pointing into the -x direction) is given by the following sums [1220÷]. We use

Mathematica’s typesetting capabilities for this example.

A@r_, θ_, 8a_, b_, α_, Evaluate êê@ WhichB WithB:θW = n2 ∞ ∞ b a n b n θW − ‚ θW, r > b, α LogB F + ‚ b r r n=1 n=1 ∞ b a b > r > a, α LogB F + ‚ r r n=1

a > r,

∞ b r α LogB F + ‚ a a n=1

n

θW − ‚ ∞

n=1 n

θW − ‚ ∞

n=1

r

n

θW,

b r b

n

θWFF

As the result shows, Mathematica was able to sum all three of the above symbolic infinite sums in closed form. The normal component of the field is everywhere differentiable.



58

Plot@Evaluate@A@3 ê 2, θ, 81, 2, 7 ê 8 π

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