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## Visual Complex Analysis by Tristan Needham

Code:
 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
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
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

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 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

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 I bought it and did not like it. For one, it's not visual. Complex Analysis in my opinion is very visual and they did not in my opinion capture this visual component of the subject well at all. Sorry I can't give examples, I looked on my bookshelf and seem not to have it anymore.
 This is one of my favourite mathematics books. It has a strong "style" to it that some people may not like, and it is probably not the best book to use for a first course. However, as a supplemental text or a second course, it would be awesome. I deeply love Complex Analysis because of this book.

## Visual Complex Analysis by Tristan Needham

I haven't worked through the entirety of the text, but the section on Mobius transformations and their relationship linear projective transformations is honestly the most insightful and clear that I've ever read. The book is worth the price for that chapter alone.