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Main Question or Discussion Point
- Author: Tristan Needham
- Title: Visual Complex Analysis
- Amazon Link: https://www.amazon.com/dp/0198534469/?tag=pfamazon01-20
- Prerequisities:
- Contents:
Table of Contents:
Code:
[LIST]
[*] Geometry and Complex Arithmetic
[LIST]
[*] Introduction
[LIST]
[*] Historical Sketch
[*] Bombelli's "Wild Thought"
[*] Some Terminology and Notation
[*] Practice
[*] Symbolic and Geometric Arithmetic
[/LIST]
[*] Euler's Formula
[LIST]
[*] Introduction
[*] Moving Particle Argument
[*] Power Series Argument
[*] Sine and Cosine in Terms of Euler's Formula
[/LIST]
[*] Some Applications
[LIST]
[*] Introduction
[*] Trigonometry
[*] Geometry
[*] Calculus
[*] Algebra
[*] Vectorial Operations
[/LIST]
[*] Transformations and Euclidean Geometry
[LIST]
[*] Geometry Through the Eyes of Felix Klein
[*] Classifying Motions
[*] Three Reflections Theorem
[*] Similarities and Complex Arithmetic
[*] Spatial Complex Numbers?
[/LIST]
[*] Exercises
[/LIST]
[*] Complex Functions as Transformations
[LIST]
[*] Introduction
[*] Polynomials
[LIST]
[*] Positive Integer Powers
[*] Cubics Revisited
[*] Cassinian Curves
[/LIST]
[*] Power Series
[LIST]
[*] 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
[/LIST]
[*] The Exponential Function
[LIST]
[*] Power Series Approach
[*] The Geometry of the Mapping
[*] Another Approach
[/LIST]
[*] Cosine and Sine
[LIST]
[*] Definitions and Identities
[*] Relation to Hyperbolic Functions
[*] The Geometry of the Mapping
[/LIST]
[*] Multifunctions
[LIST]
[*] Example: Fractional Powers
[*] Single-Valued Branches of a Multifunction
[*] Relevance to Power Series
[*] An Example with Two Branch Points
[/LIST]
[*] The Logarithm Function
[LIST]
[*] Inverse of the Exponential Function
[*] The Logarithmic Power Series
[*] General Powers
[/LIST]
[*] Averaging over Circles
[LIST]
[*] The Centroid
[*] Averaging over Regular Polygons
[*] Averaging over Circles
[/LIST]
[*] Exercises
[/LIST]
[*] Mobius Transformations and Inversion
[LIST]
[*] Introduction
[LIST]
[*] Definition of Mobius Transformations
[*] Connection with Einstein's Theory of Relativity
[*] Decomposition into Simple Transformations
[/LIST]
[*] Inversion
[LIST]
[*] Preliminary Definitions and Facts
[*] Preservation of Circles
[*] Construction Using Orthogonal Circles
[*] Preservation of Angles
[*] Preservation of Symmetry
[*] Inversion in a Sphere
[/LIST]
[*] Three Illustrative Applications of Inversion
[LIST]
[*] A Problem on Touching Circles
[*] Quadrilaterals with Orthogonal Diagonals
[*] Ptolemy's Theorem
[/LIST]
[*] The Riemann Sphere
[LIST]
[*] The Point at Infinity
[*] Stereographic Projection
[*] Transferring Complex Functions to the Sphere
[*] Behaviour of Functions at Infinity
[*] Stereographic Formulae
[/LIST]
[*] Mobius Transformations: Basic Results
[LIST]
[*] Preservation of Circles, Angles, and Symmetry
[*] Non-Uniqueness of the Coefficients
[*] The Group Property
[*] Fixed Points
[*] Fixed Points at Infinity
[*] The Cross-Ratio
[/LIST]
[*] Mobius Transformations as Matrices
[LIST]
[*] Evidence of a Link with Linear Algebra
[*] The Explanation: Homogeneous Coordinates
[*] Eigenvectors and Eigenvalues
[*] Rotations of the Sphere
[/LIST]
[*] Visualization and Classification
[LIST]
[*] The Main Idea
[*] Elliptic, Hyperbolic, and Loxodromic types
[*] Local Geometric Interpretation of the Multiplier
[*] Parabolic Transformations
[*] Computing the Multiplier
[*] Eigenvalue Interpretation of the Multiplier
[/LIST]
[*] Decomposition into 2 or 4 Reflections
[LIST]
[*] Introduction
[*] Elliptic Case
[*] Hyperbolic Case
[*] Parabolic Case
[*] Summary
[/LIST]
[*] Automorphisms of the Unit Disc
[LIST]
[*] Counting Degrees of Freedom
[*] Finding the Formula via the Symmetry Principle
[*] Interpreting the Formula Geometrically
[*] Introduction to Riemann's Mapping Theorem
[/LIST]
[*] Exercises
[/LIST]
[*] Differentiation: The Amplitwist Concept
[LIST]
[*] Introduction
[*] A Puzzling Phenomenon
[*] Local Description of Mappings in the Plane
[LIST]
[*] Introduction
[*] The Jacobian Matrix
[*] The Amplitwist Concept
[/LIST]
[*] The Complex Derivative as Amplitwist
[LIST]
[*] The Real Derivative Re-examined
[*] The Complex Derivative
[*] Analytic Functions
[*] A Brief Summary
[/LIST]
[*] Some Simple Examples
[*] Conformal = Analytic
[LIST]
[*] Introduction
[*] Conformality Throughout a Region
[*] Conformality and the Riemann Sphere
[/LIST]
[*] Critical Points
[LIST]
[*] Degrees of Crushing
[*] Breakdown of Conformality
[*] Branch Points
[/LIST]
[*] The Cauchy-Riemann Equations
[LIST]
[*] Introduction
[*] The Geometry of Linear Transformations
[*] The Cauchy-Riemann Equations
[/LIST]
[*] Exercises
[/LIST]
[*] Further Geometry of Differentiation
[LIST]
[*] Cauchy-Riemann Revealed
[LIST]
[*] Introduction
[*] The Cartesian Form
[*] The Polar Form
[/LIST]
[*] An Intimation of Rigidity
[*] Visual Differentiation of log(z)
[*] Rules of Differentiation
[LIST]
[*] Composition
[*] Inverse Functions
[*] Addition and Multiplication
[/LIST]
[*] Polynomials, Power Series, and Rational Functions
[LIST]
[*] Polynomials
[*] Power Series
[*] Rational Functions
[/LIST]
[*] Visual Differentiation of the Power Function
[*] Visual Differentiation of exp(z)
[*] Geometric Solution of E' = E 232
[*] An Application of Higher Derivatives: Curvature
[LIST]
[*] Introduction
[*] Analytic Transformation of Curvature
[*] Complex Curvature
[/LIST]
[*] Celestial Mechanics
[LIST]
[*] 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
[/LIST]
[*] Analytic Continuation
[LIST]
[*] Introduction
[*] Rigidity
[*] Uniqueness
[*] Preservation of Identities
[*] Analytic Continuation via Reflections
[/LIST]
[*] Exercises
[/LIST]
[*] Non-Euclidean Geometry
[LIST]
[*] Introduction
[LIST]
[*] 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
[/LIST]
[*] Spherical Geometry
[LIST]
[*] 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
[/LIST]
[*] Hyperbolic Geometry
[LIST]
[*] 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
[/LIST]
[*] Exercises
[/LIST]
[*] Winding Numbers and Topology
[LIST]
[*] Winding Number
[LIST]
[*] The Definition
[*] What does "inside" mean?
[*] Finding Winding Numbers Quickly
[/LIST]
[*] Hopf's Degree Theorem
[LIST]
[*] The Result
[*] Loops as Mappings of the Circle
[*] The Explanation
[/LIST]
[*] Polynomials and the Argument Principle
[*] A Topological Argument Principle
[LIST]
[*] Counting Preimages Algebraically
[*] Counting Preimages Geometrically
[*] Topological Characteristics of Analyticity
[*] A Topological Argument Principle
[*] Two Examples
[/LIST]
[*] Rouche's Theorem
[LIST]
[*] The Result
[*] The Fundamental Theorem of Algebra
[*] Brouwer's Fixed Point Theorem
[/LIST]
[*] Maxima and Minima
[LIST]
[*] Maximum-Modulus Theorem
[*] Related Results
[/LIST]
[*] The Schwarz-Pick Lemma
[LIST]
[*] Schwarz's Lemma
[*] Liouville's Theorem
[*] Pick's Result
[/LIST]
[*] The Generalized Argument Principle
[LIST]
[*] Rational Functions
[*] Poles and Essential Singularities
[*] The Explanation
[/LIST]
[*] Exercises
[/LIST]
[*] Complex Integration: Cauchy's Theorem
[LIST]
[*] Introduction
[*] The Real Integral
[LIST]
[*] The Riemann Sum
[*] The Trapezoidal Rule
[*] Geometric Estimation of Errors
[/LIST]
[*] The Complex Integral
[LIST]
[*] Complex Riemann Sums
[*] A Visual Technique
[*] A Useful Inequality
[*] Rules of Integration
[/LIST]
[*] Complex Inversion
[LIST]
[*] A Circular Arc
[*] General Loops
[*] Winding Number
[/LIST]
[*] Conjugation
[LIST]
[*] Introduction
[*] Area Interpretation
[*] General Loops
[/LIST]
[*] Power Functions
[LIST]
[*] Integration along a Circular Arc
[*] Complex Inversion as a Limiting Case
[*] General Contours and the Deformation Theorem
[*] A Further Extension of the Theorem
[*] Residues
[/LIST]
[*] The Exponential Mapping
[*] The Fundamental Theorem
[LIST]
[*] Introduction
[*] An Example
[*] The Fundamental Theorem
[*] The Integral as Antiderivative
[*] Logarithm as Integral
[/LIST]
[*] Parametric Evaluation
[*] Cauchy's Theorem
[LIST]
[*] Some Preliminaries
[*] The Explanation
[/LIST]
[*] The General Cauchy Theorem
[LIST]
[*] The Result
[*] The Explanation
[*] A Simpler Explanation
[/LIST]
[*] The General Formula of Contour Integration
[*] Exercises
[/LIST]
[*] Cauchy's Formula and Its Applications
[LIST]
[*] Cauchy's Formula
[LIST]
[*] Introduction
[*] First Explanation
[*] Gauss' Mean Value Theorem
[*] General Cauchy Formula
[/LIST]
[*] Infinite Differentiability and Taylor Series
[LIST]
[*] Infinite Differentiability
[*] Taylor Series
[/LIST]
[*] Calculus of Residues
[LIST]
[*] 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
[/LIST]
[*] Annular Laurent Series
[LIST]
[*] An Example
[*] Laurent's Theorem
[/LIST]
[*] Exercises
[/LIST]
[*] Vector Fields: Physics and Topology
[LIST]
[*] Vector Fields
[LIST]
[*] Complex Functions as Vector Fields
[*] Physical Vector Fields
[*] Flows and Force Fields
[*] Sources and Sinks
[/LIST]
[*] Winding Numbers and Vector Fields
[LIST]
[*] The Index of a Singular Point
[*] The Index According to Poincare
[*] The Index Theorem
[/LIST]
[*] Flows on Closed Surfaces
[LIST]
[*] Formulation of the Poincare-Hopf Theorem
[*] Defining the Index on a Surface
[*] An Explanation of the Poincare-Hopf Theorem
[/LIST]
[*] Exercises
[/LIST]
[*] Vector Fields and Complex Integration
[LIST]
[*] Flux and Work
[LIST]
[*] Flux
[*] Work
[*] Local Flux and Local Work
[*] Divergence and Curl in Geometric Form
[*] Divergence-Free and Curl-Free Vector Fields
[/LIST]
[*] Complex Integration in Terms of Vector Fields
[LIST]
[*] 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
[/LIST]
[*] The Complex Potential
[LIST]
[*] Introduction
[*] The Stream Function
[*] The Gradient Field
[*] The Potential Function
[*] The Complex Potential
[*] Examples
[/LIST]
[*] Exercises
[/LIST]
[*] Flows and Harmonic Functions
[LIST]
[*] Harmonic Duals
[LIST]
[*] Dual Flows
[*] Harmonic Duals
[/LIST]
[*] Conformal Invariance
[LIST]
[*] Conformal Invariance of Harmonicity
[*] Conformal Invariance of the Laplacian
[*] The Meaning of the Laplacian
[/LIST]
[*] A Powerful Computational Tool
[*] The Complex Curvature Revisited
[LIST]
[*] Some Geometry of Harmonic Equipotentials
[*] The Curvature of Harmonic Equipotentials
[*] Further Complex Curvature Calculations
[*] Further Geometry of the Complex Curvature
[/LIST]
[*] Flow Around an Obstacle
[LIST]
[*] Introduction
[*] An Example
[*] The Method of Images
[*] Mapping One Flow Onto Another
[/LIST]
[*] The Physics of Riemann's Mapping Theorem
[LIST]
[*] 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
[/LIST]
[*] Dirichlet's Problem
[LISt]
[*] Introduction
[*] Schwarz's Interpretation
[*] Dirichlet's Problem for the Disc
[*] The Interpretations of Neumann and Bocher
[*] Green's General Formula
[/LIST]
[*] Exercises
[/LIST]
[*] References
[*] Index
[/LIST]
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