Visual complex analysis / Tristan Needham.

Format:

Book

Author/Creator:

Needham, Tristan, author.

Language:

English

Subjects (All):

Mathematics.

Functions of complex variables.

Physical Description:

1 online resource (720 pages)

Place of Publication:

New York, New York : Oxford University Press, [2023]

Summary:

This new 25th anniversary edition of Visual Complex Analysis introduces this powerful method combining complex numbers with ordinary calculus, and includes new introductory content and brand-new captions that fully explain the geometrical reasoning.

Contents:

cover

titlepage

copyright

dedication

Foreword

Preface to the 25th Anniversary Edition

Preface

Acknowledgements

Contents

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

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

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 Reflections*

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

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

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

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.

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 Rouché'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

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.

Notes:

Description based on print version record.

Other Format:

Print version: Needham, Tristan Visual Complex Analysis

ISBN:

0-19-196494-8

0-19-269545-2