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Mathematics
Visual Complex Analysis
Chapter 1
I Introduction
II Euler's Formula
III Some Applications
IV Transformations and Euclidian Geometry
V Exercises chapter 1
Errata for chapter 1
Chapter 2
I Introduction
II Polynomials
III Power Series
IV The Exponential Function
V Cosine and Sine
VI Multifunctions
VII The Logarithm Function
VIII Averaging over Circles
IX Exercises chapter 2
Errata for chapter 2
Chapter 3
I Introduction
II Inversion
III Three Illustrative Applications of Inversion
IV The Riemann Sphere
V Möbius Transformations: Basic Results
VI Möbius Transformations as Matrices*
VII Visualization and Classification*
VIII Decomposition into 2 or 4 Reflections*
IX Automorphisms of the Unit Disc*
X Exercises chapter 3
Errata for chapter 3
Chapter 4
I Introduction
II A Puzzling Phenomenon
III Local Description of Mappings in the Plane
IV The Complex Derivative as Amplitwist
V Some Simple Examples
VI Conformal = Analytic
VII Critical Points
VIII The Cauchy-Riemann Equations
IX Exercises chapter 4
Errata for chapter 4
Chapter 5
I Cauchy-Riemann Revealed
II An Intimation of Rigidity
III Visual differentiation of log(z)
IV Rules of Differentiation
V Polynomials, Power Series, and Rational Functions
VI Visual Differentiation of the Power Function
VII Visual Differentiation of exp(z)
VIII Geometric Solution of E'=E
IX An Application of Higher Derivatives: Curvature*
X Celestial Mechanics*
XI Analytic Continuation*
XII Exercises chapter 5
Errata for chapter 5
Chapter 6
I Introduction
II Spherical Geometry
III Hyperbolic Geometry
1 The Tractrix and the Pseudosphere
2 The constant curvature of the Pseudosphere*
3 A Conformal Map of the Pseudosphere
4 Beltrami's Hyperbolic Plane
5 Hyperbolic Lines and Reflections
6 The Bolyai-Lobachevsky Formula*
7 The Three Types of Direct Motion
8 Decomposition into Two Reflections
9 The Angular Excess of a Hyperbolic Triangle
10 The Poincaré Disc
11 Motions of the Poincaré Disc
12 The Hemisphere model and Hyperbolic Space
IV Exercises chapter 6
Errata for chapter 6
Chapter 7
I Winding Number
II Hopf''s Degree Theorem
III Polynomials and the Argument Principle
IV A Topological Argument Principle
V Rouché's Theorem
VI Maxima and Minima
VII The Schwarz-Pick Lemma*
VIII The Generalized argument Principle
IX Exercises
Errata for Chapter 7
Chapter 8
I Introduction
II The Real Integral
III The Complex Integral
IV Complex Inversion
V Conjugation
VI Power Functions
VII The Exponential Mapping
VIII The Fundamental Theorem
IX Parametric Evaluation
X Cauchy's Theorem
XI The General Cauchy Theorem
XII The General Formula of Contour Integration
XIII Exercises chapter 8
Errata for chapter 8
Chapter 9
I Cauchy's Formula
II Infinite Differentiability and Taylor Series
III Calculus of Residues
IV Annular Laurent Series
V Exercises for chapter 9
Errata for chapter 9
Chapter 10
I Vector Fields
II Winding Numbers and Vector Fields*
III Flows on Closed Surfaces*
Exercises for Chapter 10
Errata for Chapter 10
Chapter 11
I Flux and Work
II Complex Integration in Terms of Vector Fields
III The Complex Potential
IV Exercises for Chapter 11
Errata for Chapter 11
Chapter 12
I Harmonic Duals
II Conformal Invariance
III A Powerful Computational Tool
IV The Complex Curvature Revisited*
V Flow around an Obstacle
VI The Physics of Riemann's Mapping Theorem
VII Dirichlet's Problem
VIII Exercises
Errata for Chapter 12
Mathematics and its History
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