Analysis Visual Complex Analysis by Tristan Needham

 Geometry and Complex Arithmetic
 Introduction
 Historical Sketch
 Bombelli's "Wild Thought"
 Some Terminology and Notation
 Practice
 Symbolic and Geometric Arithmetic

 Euler's Formula
 Introduction
 Moving Particle Argument
 Power Series Argument
 Sine and Cosine in Terms of Euler's Formula

 Some Applications
 Introduction
 Trigonometry
 Geometry
 Calculus
 Algebra
 Vectorial Operations

 Transformations and Euclidean Geometry
 Geometry Through the Eyes of Felix Klein
 Classifying Motions
 Three Reflections Theorem
 Similarities and Complex Arithmetic
 Spatial Complex Numbers?

 Exercises

 Complex Functions as Transformations
 Introduction
 Polynomials
 Positive Integer Powers
 Cubics Revisited
 Cassinian Curves

 Power Series
 The Mystery of Real Power Series
 The Disc of Convergence
 Approximating a Power Series with a Polynomial
 Uniqueness
 Manipulating Power Series
 Finding the Radius of Convergence
 Fourier Series

 The Exponential Function
 Power Series Approach
 The Geometry of the Mapping
 Another Approach

 Cosine and Sine
 Definitions and Identities
 Relation to Hyperbolic Functions
 The Geometry of the Mapping

 Multifunctions
 Example: Fractional Powers
 Single-Valued Branches of a Multifunction
 Relevance to Power Series
 An Example with Two Branch Points

 The Logarithm Function
 Inverse of the Exponential Function
 The Logarithmic Power Series
 General Powers

 Averaging over Circles
 The Centroid
 Averaging over Regular Polygons
 Averaging over Circles

 Exercises

 Mobius Transformations and Inversion
 Introduction
 Definition of Mobius Transformations
 Connection with Einstein's Theory of Relativity
 Decomposition into Simple Transformations

 Inversion
 Preliminary Definitions and Facts
 Preservation of Circles
 Construction Using Orthogonal Circles
 Preservation of Angles
 Preservation of Symmetry
 Inversion in a Sphere

 Three Illustrative Applications of Inversion
 A Problem on Touching Circles
 Quadrilaterals with Orthogonal Diagonals
 Ptolemy's Theorem

 The Riemann Sphere
 The Point at Infinity
 Stereographic Projection
 Transferring Complex Functions to the Sphere
 Behaviour of Functions at Infinity
 Stereographic Formulae

 Mobius Transformations: Basic Results
 Preservation of Circles, Angles, and Symmetry
 Non-Uniqueness of the Coefficients
 The Group Property
 Fixed Points
 Fixed Points at Infinity
 The Cross-Ratio

 Mobius Transformations as Matrices
 Evidence of a Link with Linear Algebra
 The Explanation: Homogeneous Coordinates
 Eigenvectors and Eigenvalues
 Rotations of the Sphere

 Visualization and Classification
 The Main Idea
 Elliptic, Hyperbolic, and Loxodromic types
 Local Geometric Interpretation of the Multiplier
 Parabolic Transformations
 Computing the Multiplier
 Eigenvalue Interpretation of the Multiplier

 Decomposition into 2 or 4 Reflections
 Introduction
 Elliptic Case
 Hyperbolic Case
 Parabolic Case
 Summary

 Automorphisms of the Unit Disc
 Counting Degrees of Freedom
 Finding the Formula via the Symmetry Principle
 Interpreting the Formula Geometrically
 Introduction to Riemann's Mapping Theorem

 Exercises

 Differentiation: The Amplitwist Concept
 Introduction
 A Puzzling Phenomenon
 Local Description of Mappings in the Plane
 Introduction
 The Jacobian Matrix
 The Amplitwist Concept

 The Complex Derivative as Amplitwist
 The Real Derivative Re-examined
 The Complex Derivative
 Analytic Functions
 A Brief Summary

 Some Simple Examples
 Conformal = Analytic
 Introduction
 Conformality Throughout a Region
 Conformality and the Riemann Sphere

 Critical Points
 Degrees of Crushing
 Breakdown of Conformality
 Branch Points

 The Cauchy-Riemann Equations
 Introduction
 The Geometry of Linear Transformations
 The Cauchy-Riemann Equations

 Exercises

 Further Geometry of Differentiation
 Cauchy-Riemann Revealed
 Introduction
 The Cartesian Form
 The Polar Form

 An Intimation of Rigidity
 Visual Differentiation of log(z)
 Rules of Differentiation
 Composition
 Inverse Functions
 Addition and Multiplication

 Polynomials, Power Series, and Rational Functions
 Polynomials
 Power Series
 Rational Functions

 Visual Differentiation of the Power Function
 Visual Differentiation of exp(z)
 Geometric Solution of E' = E 232
 An Application of Higher Derivatives: Curvature
 Introduction
 Analytic Transformation of Curvature
 Complex Curvature

 Celestial Mechanics
 Central Force Fields
 Two Kinds of Elliptical Orbit
 Changing the First into the Second
 The Geometry of Force
 An Explanation
 The Kasner-Arnol'd Theorem

 Analytic Continuation
 Introduction
 Rigidity
 Uniqueness
 Preservation of Identities
 Analytic Continuation via Reflections

 Exercises

 Non-Euclidean Geometry
 Introduction
 The Parallel Axiom
 Some Facts from Non-Euclidean Geometry
 Geometry on a Curved Surface
 Intrinsic versus Extrinsic Geometry
 Gaussian Curvature
 Surfaces of Constant Curvature
 The Connection with Mobius Transformations

 Spherical Geometry
 The Angular Excess of a Spherical Triangle
 Motions of the Sphere
 A Conformal Map of the Sphere
 Spatial Rotations as Mobius Transformations
 Spatial Rotations and Quaternions

 Hyperbolic Geometry
 The Tractrix and the Pseudosphere
 The Constant Curvature of the Pseudosphere
 A Conformal Map of the Pseudosphere
 Beltrami's Hyperbolic Plane
 Hyperbolic Lines and Reflections
 The Bolyai-Lobachevsky Formula
 The Three Types of Direct Motion
 Decomposition into Two Reflections
 The Angular Excess of a Hyperbolic Triangle
 The Poincare Disc
 Motions of the Poincare Disc
 The Hemisphere Model and Hyperbolic Space

 Exercises

 Winding Numbers and Topology
 Winding Number
 The Definition
 What does "inside" mean?
 Finding Winding Numbers Quickly

 Hopf's Degree Theorem
 The Result
 Loops as Mappings of the Circle
 The Explanation

 Polynomials and the Argument Principle
 A Topological Argument Principle
 Counting Preimages Algebraically
 Counting Preimages Geometrically
 Topological Characteristics of Analyticity
 A Topological Argument Principle
 Two Examples

 Rouche's Theorem
 The Result
 The Fundamental Theorem of Algebra
 Brouwer's Fixed Point Theorem

 Maxima and Minima
 Maximum-Modulus Theorem
 Related Results

 The Schwarz-Pick Lemma
 Schwarz's Lemma
 Liouville's Theorem
 Pick's Result

 The Generalized Argument Principle
 Rational Functions
 Poles and Essential Singularities
 The Explanation

 Exercises

 Complex Integration: Cauchy's Theorem
 Introduction
 The Real Integral
 The Riemann Sum
 The Trapezoidal Rule
 Geometric Estimation of Errors

 The Complex Integral
 Complex Riemann Sums
 A Visual Technique
 A Useful Inequality
 Rules of Integration

 Complex Inversion
 A Circular Arc
 General Loops
 Winding Number

 Conjugation
 Introduction
 Area Interpretation
 General Loops

 Power Functions
 Integration along a Circular Arc
 Complex Inversion as a Limiting Case
 General Contours and the Deformation Theorem
 A Further Extension of the Theorem
 Residues

 The Exponential Mapping
 The Fundamental Theorem
 Introduction
 An Example
 The Fundamental Theorem
 The Integral as Antiderivative
 Logarithm as Integral

 Parametric Evaluation
 Cauchy's Theorem
 Some Preliminaries
 The Explanation

 The General Cauchy Theorem
 The Result
 The Explanation
 A Simpler Explanation

 The General Formula of Contour Integration
 Exercises

 Cauchy's Formula and Its Applications
 Cauchy's Formula
 Introduction
 First Explanation
 Gauss' Mean Value Theorem
 General Cauchy Formula

 Infinite Differentiability and Taylor Series
 Infinite Differentiability
 Taylor Series

 Calculus of Residues
 Laurent Series Centred at a Pole
 A Formula for Calculating Residues
 Application to Real Integrals
 Calculating Residues using Taylor Series
 Application to Summation of Series

 Annular Laurent Series
 An Example
 Laurent's Theorem

 Exercises

 Vector Fields: Physics and Topology
 Vector Fields
 Complex Functions as Vector Fields
 Physical Vector Fields
 Flows and Force Fields
 Sources and Sinks

 Winding Numbers and Vector Fields
 The Index of a Singular Point
 The Index According to Poincare
 The Index Theorem

 Flows on Closed Surfaces
 Formulation of the Poincare-Hopf Theorem
 Defining the Index on a Surface
 An Explanation of the Poincare-Hopf Theorem

 Exercises

 Vector Fields and Complex Integration
 Flux and Work
 Flux
 Work
 Local Flux and Local Work
 Divergence and Curl in Geometric Form
 Divergence-Free and Curl-Free Vector Fields

 Complex Integration in Terms of Vector Fields
 The Polya Vector Field
 Cauchy's Theorem
 Example: Area as Flux
 Example: Winding Number as Flux
 Local Behaviour of Vector Fields
 Cauchy's Formula
 Positive Powers
 Negative Powers and Multipoles
 Multipoles at Infinity
 Laurent's Series as a Multipole Expansion

 The Complex Potential
 Introduction
 The Stream Function
 The Gradient Field
 The Potential Function
 The Complex Potential
 Examples

 Exercises

 Flows and Harmonic Functions
 Harmonic Duals
 Dual Flows
 Harmonic Duals

 Conformal Invariance
 Conformal Invariance of Harmonicity
 Conformal Invariance of the Laplacian
 The Meaning of the Laplacian

 A Powerful Computational Tool
 The Complex Curvature Revisited
 Some Geometry of Harmonic Equipotentials
 The Curvature of Harmonic Equipotentials
 Further Complex Curvature Calculations
 Further Geometry of the Complex Curvature

 Flow Around an Obstacle
 Introduction
 An Example
 The Method of Images
 Mapping One Flow Onto Another

 The Physics of Riemann's Mapping Theorem
 Introduction
 Exterior Mappings and Flows Round Obstacles
 Interior Mappings and Dipoles
 Interior Mappings, Vortices, and Sources
 An Example: Automorphisms of the Disc
 Green's Function

 Dirichlet's Problem 
 Introduction
 Schwarz's Interpretation
 Dirichlet's Problem for the Disc
 The Interpretations of Neumann and Bocher
 Green's General Formula

 Exercises

 References
 Index