Visual Complex Analysis

cover
titlepage
copyright
dedication
Foreword
Preface to the 25th Anniversary Edition
Preface
Acknowledgements
Contents
1 Geometry and Complex Arithmetic
1.1 Introduction
1.1.1 Historical Sketch
1.1.2 Bombelli's ``Wild Thought''
1.1.3 Some Terminology and Notation
1.1.4 Practice
1.1.5 Equivalence of Symbolic and Geometric Arithmetic
1.2 Euler's Formula
1.2.1 Introduction
1.2.2 Moving Particle Argument
1.2.3 Power Series Argument
1.2.4 Sine and Cosine in Terms of Euler's Formula
1.3 Some Applications
1.3.1 Introduction
1.3.2 Trigonometry
1.3.3 Geometry
1.3.4 Calculus
1.3.5 Algebra
1.3.6 Vectorial Operations
1.4 Transformations and Euclidean Geometry*
1.4.1 Geometry Through the Eyes of Felix Klein
1.4.2 Classifying Motions
1.4.3 Three Reflections Theorem
1.4.4 Similarities and Complex Arithmetic
1.4.5 Spatial Complex Numbers?
1.5 Exercises
2 Complex Functions as Transformations
2.1 Introduction
2.2 Polynomials
2.2.1 Positive Integer Powers
2.2.2 Cubics Revisited*
2.2.3 Cassinian Curves*
2.3 Power Series
2.3.1 The Mystery of Real Power Series
2.3.2 The Disc of Convergence
2.3.3 Approximating a Power Series with a Polynomial
2.3.4 Uniqueness
2.3.5 Manipulating Power Series
2.3.6 Finding the Radius of Convergence
2.3.7 Fourier Series*
2.4 The Exponential Function
2.4.1 Power Series Approach
2.4.2 The Geometry of the Mapping
2.4.3 Another Approach
2.5 Cosine and Sine
2.5.1 Definitions and Identities
2.5.2 Relation to Hyperbolic Functions
2.5.3 The Geometry of the Mapping
2.6 Multifunctions
2.6.1 Example: Fractional Powers
2.6.2 Single-Valued Branches of a Multifunction
2.6.3 Relevance to Power Series
2.6.4 An Example with Two Branch Points
2.7 The Logarithm Function
2.7.1 Inverse of the Exponential Function
2.7.2 The Logarithmic Power Series
2.7.3 General Powers
2.8 Averaging over Circles*
2.8.1 The Centroid
2.8.2 Averaging over Regular Polygons
2.8.3 Averaging over Circles
2.9 Exercises
3 Möbius Transformations and Inversion
3.1 Introduction
3.1.1 Definition and Significance of Möbius Transformations
3.1.2 The Connection with Einstein's Theory of Relativity*
3.1.3 Decomposition into Simple Transformations
3.2 Inversion
3.2.1 Preliminary Definitions and Facts
3.2.2 Preservation of Circles
3.2.3 Constructing Inverse Points Using Orthogonal Circles
3.2.4 Preservation of Angles
3.2.5 Preservation of Symmetry
3.2.6 Inversion in a Sphere
3.3 Three Illustrative Applications of Inversion
3.3.1 A Problem on Touching Circles
3.3.2 A Curious Property of Quadrilaterals with Orthogonal Diagonals
3.3.3 Ptolemy's Theorem
3.4 The Riemann Sphere
3.4.1 The Point at Infinity
3.4.2 Stereographic Projection
3.4.3 Transferring Complex Functions to the Sphere
3.4.4 Behaviour of Functions at Infinity
3.4.5 Stereographic Formulae*
3.5 Möbius Transformations: Basic Results
3.5.1 Preservation of Circles, Angles, and Symmetry
3.5.2 Non-Uniqueness of the Coefficients
3.5.3 The Group Property
3.5.4 Fixed Points
3.5.5 Fixed Points at Infinity
3.5.6 The Cross-Ratio
3.6 Möbius Transformations as Matrices*
3.6.1 Empirical Evidence of a Link with Linear Algebra
3.6.2 The Explanation: Homogeneous Coordinates
3.6.3 Eigenvectors and Eigenvalues*
3.6.4 Rotations of the Sphere as Möbius Transformations*
3.7 Visualization and Classification*
3.7.1 The Main Idea
3.7.2 Elliptic, Hyperbolic, and Loxodromic Transformations
3.7.3 Local Geometric Interpretation of the Multiplier
3.7.4 Parabolic Transformations
3.7.5 Computing the Multiplier*
3.7.6 Eigenvalue Interpretation of the Multiplier*
3.8 Decomposition into 2 or 4 Reflections*
3.8.1 Introduction
3.8.2 Elliptic Case
3.8.3 Hyperbolic Case
3.8.4 Parabolic Case
3.8.5 Summary
3.9 Automorphisms of the Unit Disc*
3.9.1 Counting Degrees of Freedom
3.9.2 Finding the Formula via the Symmetry Principle
3.9.3 Interpreting the Simplest Formula Geometrically*
3.9.4 Introduction to Riemann's Mapping Theorem
3.10 Exercises
4 Differentiation: The Amplitwist Concept
4.1 Introduction
4.2 A Puzzling Phenomenon
4.3 Local Description of Mappings in the Plane
4.3.1 Introduction
4.3.2 The Jacobian Matrix
4.3.3 The Amplitwist Concept
4.4 The Complex Derivative as Amplitwist
4.4.1 The Real Derivative Re-examined
4.4.2 The Complex Derivative
4.4.3 Analytic Functions
4.4.4 A Brief Summary
4.5 Some Simple Examples
4.6 Conformal = Analytic
4.6.1 Introduction
4.6.2 Conformality Throughout a Region
4.6.3 Conformality and the Riemann Sphere
4.7 Critical Points
4.7.1 Degrees of Crushing
4.7.2 Breakdown of Conformality
4.7.3 Branch Points
4.8 The Cauchy–Riemann Equations
4.8.1 Introduction
4.8.2 The Geometry of Linear Transformations
4.8.3 The Cauchy–Riemann Equations
4.9 Exercises
5 Further Geometry of Differentiation
5.1 Cauchy–Riemann Revealed
5.1.1 Introduction
5.1.2 The Cartesian Form
5.1.3 The Polar Form
5.2 An Intimation of Rigidity
5.3 Visual Differentiation of log(z)
5.4 Rules of Differentiation
5.4.1 Composition
5.4.2 Inverse Functions
5.4.3 Addition and Multiplication
5.5 Polynomials, Power Series, and Rational Functions
5.5.1 Polynomials
5.5.2 Power Series
5.5.3 Rational Functions
5.6 Visual Differentiation of the Power Function
5.7 Visual Differentiation of exp(z)
5.8 Geometric Solution of E' = E
5.9 An Application of Higher Derivatives: Curvature*
5.9.1 Introduction
5.9.2 Analytic Transformation of Curvature
5.9.3 Complex Curvature
5.10 Celestial Mechanics*
5.10.1 Central Force Fields
5.10.2 Two Kinds of Elliptical Orbit
5.10.3 Changing the First into the Second
5.10.4 The Geometry of Force
5.10.5 An Explanation
5.10.6 The Kasner–Arnol'd Theorem
5.11 Analytic Continuation*
5.11.1 Introduction
5.11.2 Rigidity
5.11.3 Uniqueness
5.11.4 Preservation of Identities
5.11.5 Analytic Continuation via Reflections
5.12 Exercises
6 Non-Euclidean Geometry*
6.1 Introduction
6.1.1 The Parallel Axiom
6.1.2 Some Facts from Non-Euclidean Geometry
6.1.3 Geometry on a Curved Surface
6.1.4 Intrinsic versus Extrinsic Geometry
6.1.5 Gaussian Curvature
6.1.6 Surfaces of Constant Curvature
6.1.7 The Connection with Möbius Transformations
6.2 Spherical Geometry
6.2.1 The Angular Excess of a Spherical Triangle
6.2.2 Motions of the Sphere: Spatial Rotations and Reflections
6.2.3 A Conformal Map of the Sphere
6.2.4 Spatial Rotations as Möbius Transformations
6.2.5 Spatial Rotations and Quaternions
6.3 Hyperbolic Geometry
6.3.1 The Tractrix and the Pseudosphere
6.3.2 The Constant Negative Curvature of the Pseudosphere*
6.3.3 A Conformal Map of the Pseudosphere
6.3.4 Beltrami's Hyperbolic Plane
6.3.5 Hyperbolic Lines and Reflections
6.3.6 The Bolyai–Lobachevsky Formula*
6.3.7 The Three Types of Direct Motion
6.3.8 Decomposing an Arbitrary Direct Motion into Two Reflections
6.3.9 The Angular Excess of a Hyperbolic Triangle
6.3.10 The Beltrami–Poincaré Disc
6.3.11 Motions of the Beltrami–Poincaré Disc
6.3.12 The Hemisphere Model and Hyperbolic Space
6.4 Exercises
7 Winding Numbers and Topology
7.1 Winding Number
7.1.1 The Definition
7.1.2 What Does ``Inside'' Mean?
7.1.3 Finding Winding Numbers Quickly
7.2 Hopf's Degree Theorem
7.2.1 The Result
7.2.2 Loops as Mappings of the Circle*
7.2.3 The Explanation*
7.3 Polynomials and the Argument Principle
7.4 A Topological Argument Principle*
7.4.1 Counting Preimages Algebraically
7.4.2 Counting Preimages Geometrically
7.4.3 What's Topologically Special About Analytic Functions?
7.4.4 A Topological Argument Principle
7.4.5 Two Examples
7.5 Rouché's Theorem
7.5.1 The Result
7.5.2 The Fundamental Theorem of Algebra
7.5.3 Brouwer's Fixed Point Theorem*
7.6 Maxima and Minima
7.6.1 Maximum-Modulus Theorem
7.6.2 Related Results
7.7 The Schwarz–Pick Lemma*
7.7.1 Schwarz's Lemma
7.7.2 Liouville's Theorem
7.7.3 Pick's Result
7.8 The Generalized Argument Principle
7.8.1 Rational Functions
7.8.2 Poles and Essential Singularities
7.8.3 The Explanation*
7.9 Exercises
8 Complex Integration: Cauchy's Theorem
8.1 Introduction
8.2 The Real Integral
8.2.1 The Riemann Sum
8.2.2 The Trapezoidal Rule
8.2.3 Geometric Estimation of Errors
8.3 The Complex Integral
8.3.1 Complex Riemann Sums
8.3.2 A Visual Technique
8.3.3 A Useful Inequality
8.3.4 Rules of Integration
8.4 Complex Inversion
8.4.1 A Circular Arc
8.4.2 General Loops
8.4.3 Winding Number
8.5 Conjugation
8.5.1 Introduction
8.5.2 Area Interpretation
8.5.3 General Loops
8.6 Power Functions
8.6.1 Integration along a Circular Arc
8.6.2 Complex Inversion as a Limiting Case*
8.6.3 General Contours and the Deformation Theorem
8.6.4 A Further Extension of the Theorem
8.6.5 Residues
8.7 The Exponential Mapping
8.8 The Fundamental Theorem
8.8.1 Introduction
8.8.2 An Example
8.8.3 The Fundamental Theorem
8.8.4 The Integral as Antiderivative
8.8.5 Logarithm as Integral
8.9 Parametric Evaluation
8.10 Cauchy's Theorem
8.10.1 Some Preliminaries
8.10.2 The Explanation
8.11 The General Cauchy Theorem
8.11.1 The Result
8.11.2 The Explanation
8.11.3 A Simpler Explanation
8.12 The General Formula of Contour Integration
8.13 Exercises
9 Cauchy's Formula and Its Applications
9.1 Cauchy's Formula
9.1.1 Introduction
9.1.2 First Explanation
9.1.3 Gauss's Mean Value Theorem
9.1.4 A Second Explanation and the General Cauchy Formula
9.2 Infinite Differentiability and Taylor Series
9.2.1 Infinite Differentiability
9.2.2 Taylor Series
9.3 Calculus of Residues
9.3.1 Laurent Series Centred at a Pole
9.3.2 A Formula for Calculating Residues
9.3.3 Application to Real Integrals
9.3.4 Calculating Residues using Taylor Series
9.3.5 Application to Summation of Series
9.4 Annular Laurent Series
9.4.1 An Example
9.4.2 Laurent's Theorem
9.5 Exercises
10 Vector Fields: Physics and Topology
10.1 Vector Fields
10.1.1 Complex Functions as Vector Fields
10.1.2 Physical Vector Fields
10.1.3 Flows and Force Fields
10.1.4 Sources and Sinks
10.2 Winding Numbers and Vector Fields*
10.2.1 The Index of a Singular Point
10.2.2 The Index According to Poincaré
10.2.3 The Index Theorem
10.3 Flows on Closed Surfaces*
10.3.1 Formulation of the Poincaré–Hopf Theorem
10.3.2 Defining the Index on a Surface
10.3.3 An Explanation of the Poincaré–Hopf Theorem
10.4 Exercises
11 Vector Fields and Complex Integration
11.1 Flux and Work
11.1.1 Flux
11.1.2 Work
11.1.3 Local Flux and Local Work
11.1.4 Divergence and Curl in Geometric Form*
11.1.5 Divergence-Free and Curl-Free Vector Fields
11.2 Complex Integration in Terms of Vector Fields
11.2.1 The Pólya Vector Field
11.2.2 Cauchy's Theorem
11.2.3 Example: Area as Flux
11.2.4 Example: Winding Number as Flux
11.2.5 Local Behaviour of Vector Fields*
11.2.6 Cauchy's Formula
11.2.7 Positive Powers
11.2.8 Negative Powers and Multipoles
11.2.9 Multipoles at Infinity
11.2.10 Laurent's Series as a Multipole Expansion
11.3 The Complex Potential
11.3.1 Introduction
11.3.2 The Stream Function
11.3.3 The Gradient Field
11.3.4 The Potential Function
11.3.5 The Complex Potential
11.3.6 Examples
11.4 Exercises
12 Flows and Harmonic Functions
12.1 Harmonic Duals
12.1.1 Dual Flows
12.1.2 Harmonic Duals
12.2 Conformal Invariance
12.2.1 Conformal Invariance of Harmonicity
12.2.2 Conformal Invariance of the Laplacian
12.2.3 The Meaning of the Laplacian
12.3 A Powerful Computational Tool
12.4 The Complex Curvature Revisited*
12.4.1 Some Geometry of Harmonic Equipotentials
12.4.2 The Curvature of Harmonic Equipotentials
12.4.3 Further Complex Curvature Calculations
12.4.4 Further Geometry of the Complex Curvature
12.5 Flow Around an Obstacle
12.5.1 Introduction
12.5.2 An Example
12.5.3 The Method of Images
12.5.4 Mapping One Flow Onto Another
12.6 The Physics of Riemann's Mapping Theorem
12.6.1 Introduction
12.6.2 Exterior Mappings and Flows Round Obstacles
12.6.3 Interior Mappings and Dipoles
12.6.4 Interior Mappings, Vortices, and Sources
12.6.5 An Example: Automorphisms of the Disc
12.6.6 Green's Function
12.7 Dirichlet's Problem
12.7.1 Introduction
12.7.2 Schwarz's Interpretation
12.7.3 Dirichlet's Problem for the Disc
12.7.4 The Interpretations of Neumann and Bôcher
12.7.5 Green's General Formula
12.8 Exercises
Bibliography
Index