Visual Complex Analysis: 25th Anniversary Edition

Copyright
Foreword
Preface to the 25th Anniversary Edition
Preface
	A Parable
	Computers
	The Book\'s Newtonian Genesis
	Reading This Book
	Teaching from this Book
	Omissions and Apologies
Acknowledgements
Contents
Chapter 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
Chapter 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
Chapter 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 Reflexions*
		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
Chapter 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
Chapter 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
Chapter 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
Chapter 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 Rouche\'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
Chapter 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
Chapter 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
Chapter 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 Poincare
		10.2.3 The Index Theorem
	10.3 Flows on Closed Surfaces*
		10.3.1 Formulation of the Poincare-Hopf Theorem
		10.3.2 Defining the Index on a Surface
		10.3.3 An Explanation of the Poincare-Hopf Theorem
	10.4 Exercises
Chapter 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
Chapter 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
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Index
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	FG
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	KLM
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	OP
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	VWYZ