CHAPTER 0 Introduction and Orientation

0.1 Overview

0.1.1 Content Summaries

0.1.2 Relation of the Four Volumes

0.1.3 Chapter Structure

0.1.4 Code Presentation Style

0.2 Requirements

0.2.1 Hardware and Software

0.2.2 Reader Prerequisites

0.3 What theGuideBooks Are and What They Are Not

0.3.1 Doing Computer Mathematics

0.3.2 Programming Paradigms

0.4 Exercises and Solutions

0.4.1 Exercises

0.4.2 Solutions

0.5 The Books Versus the Electronic Components

0.5.1 Working with the Notebook

0.5.2 Reproducibility of the Results

0.5.3 Earlier Versions of the Notebooks

0.6 Style and Design Elements

0.6.1 Text and Code Formatting

0.6.2 References

0.6.3 Variable Scoping, Input Numbering, and Warning Messages

0.6.4 Graphics

0.6.5 Notations and Symbols

0.6.6 Units

0.6.7 Cover Graphics

0.7 Production History

0.8 Four General Suggestions

References

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 Remarks

1.2.1 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 γ
• Numerical Root-Finding
• Roots of Polynomials
• Jensen Disks
• De Rham's Function
• Logistic Map
• Built-in Pseudo-Compiler
• 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

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'Ho^spital'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

• Computing Wishes and Proposals
• Computer Algebra Systems

Solutions

• 100 Proposals for Problems to Tackle
• Sources of Interesting and Challenging Problems
• ISSAC Challenge Problems
• 100$-100-Digit Challenge

References

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
• High-Precision 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
• pi
• Autoevaluations of Trigonometric Functions
• Base of the Natural Logarithm
• Golden Ratio
• Euler's Constant γ
• 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 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 (z^4)^1/4
• Branch Cuts of sqrt(z+1/z)sqrt(z-1/z)
• Riemann Surface of arctan(tan(z/2)/2)
• Repeated Mappings of Singularities

References

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

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 λ-Calculus

• λ-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

Overview

Exercises

• Predicting Results of Inputs
• Nice Polynomial Expansions
• Laguerre Polynomials
• Puzzles
• Unexpected Outputs
• Power Tower
• Cayley Multiplication

Solutions

• Matching Unevaluated Arguments
• Equality of Pure Functions
• Invalid Patterns
• Counting Function Applications

References

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

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

Overview

Exercises

• Frequently Seen Messages
• Unevaluated Arguments
• Predicting Results of Inputs
• Analyzing Context Changes
• Evaluated versus Unevaluated Expressions

Solutions

• 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 Remarks

5.1 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

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

Overview

Exercises

• 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

Solutions

• 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

References

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

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

Overview

Exercises

• Benford's Rule
• Timing Comparisons for List Operations
• Sum-Free 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

Solutions

• 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

References

Graphics Volume

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
• Influence of Each Options
• 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 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
• Po'lya'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 Remarks

1.5.1 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 L-Systems
• Space Filling Curves
• Filled Gosper Curve
• L-Systems with Branching
• L-Systems that Model Plants
• Random L-Systems

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

Overview

Exercises

• Game of Life
• Langton's Ant
• Brillouin Zones
• Maxwell-Helmholtz Color Triangle
• Conformal Maps
• Cornet Isogons
• Jarni'k 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 Sin-Curves
• 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 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 Remarks

2.3.1 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

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
• Body-Centered Lattice
• Face-Centered Lattice

Overview

Exercises

• 3D Surface Sampler
• Warped, Twisted, and Interlocked Tori
• Dodecahedra Iteratively Reflected on its Faces
• Snail
• Trinomial Theorem Visualization
• Ball Blending Method
• Loop Subdivision
• sqrt(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

Overview

Exercises

• 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

Solutions

• Visualizing Saddle Points
• Outer Products
• Repeatedly Mirrored Matrix
• Halley Map
• Generating Random Functions
• Weierstrass P 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

Numerics Volume

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 Remarks

1.1.1 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 pi
• 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

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 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

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 Remarks

1.12.1 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

Overview

Exercises

• 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
• Ising-Model 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
• Auto-Compiling 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

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
• Fibonacci-Binomial Theorem
• Discretized Cat Map

Overview

Exercises

• 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 \[Tau] 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
• Faá di Bruno Formula
• Symbolic Tables

References

Symbolics Volume

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 Remarks

1.2.1 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
• Finite-Dimensional 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 One-Dimensional Wave Equation
• Schrödinger Equation with a Time-dependent 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

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 and Difference Equations

1.7.0 Remarks

1.7.1 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
• δ-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 Remarks

1.10.1 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 cos(2pi/257) á la Gauss

• The Morning of March 29 in 1796
• Gauss Periods
• Primitive Roots
• Splitting and Combining Periods
• Thousands of Square Roots
• cos(2pi/65537)
• Fermat Primes

1.10.3 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

Overview

Exercises

• Heron's formula
• Tetrahedron Volume
• Apollonius Circles
• Proving Trigonometric Identities
• Icosahedron Inequalities
• Two-Point 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

• 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 nD
• 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

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
• PT-Invariant Oscillators

Overview

Exercises

• 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

Solutions

• 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

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
• PT-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 Half-Plane

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

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

Overview

Exercises

• 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 P 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 pi 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 η ODE
• Darboux-Halphen System
• Ramanujan Identities for \[CurlyPhi] and λ Functions
• Generating Identities in Gamma Functions
• Modular Equations for Dedekind η Function

Solutions

• 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 z^z
• Weierstrass P Function over the Riemann Sphere
• Using Gröbner Bases to Derive ODEs
• Riemann Surface of Inverse Weierstrass P 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

References

A.0 Remarks

A.1 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

B.0 Remarks

B.1 Notebooks and Cells as Expressions

B.1.1 The Structure of Notebook

B.1.2 The Appearance of Cells

B.1.3 Stylesheets

B.1.4 Selected Cell Options

B.2 Front End Functions and Operations

B.2.1 Navigating and Manipulating Notebook

B.2.2 Performing Menu and Keyboard Operations Programmatically

B.3 Typesetting and Boxes

B.3.1 Two-Dimensional Formatting

B.3.2 Tweaking Formula Appearances

B.3.3 Creating Typesetting Rules

B.4 Buttons, Hyperlinks, and Palettes

B.4.1 General Buttons

B.4.2 Hyperlinks

B.4.3 Palettes

B.5 Dynamic Boxes

B.5.1 Automatic Numbering

B.5.2 Displaying Values Automatically

B.6 Special Notebooks

B.6.1 Help Browser Notebooks

B.6.2 The Message Notebook

B.7 MathLink-Related Operations

B.8 Three Applications

B.8.0 Remarks

B.8.1 Analyzing the Notebook Version of The Mathematica Book

B.8.2 Incorporating theGuideBooks into the Help Browser

B.8.3 Evaluating a CompleteGuideBooks Chapter Programmatically

References

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-h/(2pi) 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 pi-Related Continued Fraction
• Plots of the Argument of Cyclotomic Polynomials

W.12 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