Complex analysis

Contents
Preface to the Second Edition
Preface to the First Edition
0 The Origins of Complex Analysis, and Its Challenge to Intuition
	0.1 The Origins of Complex Numbers
	0.2 The Origins of Complex Analysis
	0.3 The Puzzle
	0.4 Is Mathematics Discovered or Invented?
	0.5 Overview of the Book
1 Algebra of the Complex Plane
	1.1 Construction of the Complex Numbers
	1.2 The x +iy Notation
	1.3 A Geometric Interpretation
	1.4 Real and Imaginary Parts
	1.5 The Modulus
	1.6 The Complex Conjugate
	1.7 Polar Coordinates
	1.8 The Complex Numbers Cannot be Ordered
	1.9 Exercises
2 Topology of the Complex Plane
	2.1 Open and Closed Sets
	2.2 Limits of Functions
	2.3 Continuity
	2.4 Paths
		2.4.1 Standard Paths
		2.4.2 Visualising Paths
		2.4.3 The Image of a Path
	2.5 Change of Parameter
		2.5.1 Preserving Direction
	2.6 Subpaths and Sums of Paths
	2.7 The Paving Lemma
2.8 Connectedness
2.9 Space-filling Curves
2.10 Exercises
3 Power Series
	3.1 Sequences
	3.2 Series
	3.3 Power Series
	3.4 Manipulating Power Series
	3.5 Products of Series
	3.6 Exercises
4 Differentiation
	4.1 Basic Results
	4.2 The Cauchy–Riemann Equations
	4.3 Connected Sets and Differentiability
	4.4 Hybrid Functions
	4.5 Power Series
	4.6 A Glimpse Into the Future
		4.6.1 Real Functions Differentiable Only Finitely Many Times
		4.6.2 Bad Behaviour of Real Taylor Series
		4.6.3 The Blancmange function
		4.6.4 Complex Analysis is Better Behaved
	4.7 Exercises
5 The Exponential Function
	5.1 The Exponential Function
	5.2 Real Exponentials and Logarithms
	5.3 Trigonometric Functions
	5.4 An Analytic Definition of π
	5.5 The Behaviour of Real Trigonometric Functions
	5.6 Dynamic Explanation of Euler’s Formula
	5.7 Complex Exponential and Trigonometric Functions are Periodic
	5.8 Other Trigonometric Functions
	5.9 Hyperbolic Functions
	5.10 Exercises
6 Integration
	6.1 The Real Case
	6.2 Complex Integration Along a Smooth Path
	6.3 The Length of a Path
		6.3.1 Integral Formula for the Length of Smooth Paths and Contours
	6.4 If You Took the Short Cut
	6.5 Further Properties of Lengths
6.5.1 Lengths of More General Paths
6.6 Regular Paths and Curves
	6.6.1 Parametrisation by Arc Length
6.7 Regular and Singular Points
6.8 Contour Integration
	6.8.1 Definition of Contour Integral
6.9 The Fundamental Theorem of Contour Integration
6.10 An Integral that Depends on the Path
6.11 The Gamma Function
	6.11.1 Known Properties of the Gamma Function
6.12 The Estimation Lemma
6.13 Consequences of the Fundamental Theorem
6.14 Exercises
7 Angles, Logarithms, and the Winding Number
	7.1 Radian Measure of Angles
	7.2 The Argument of a Complex Number
	7.3 The Complex Logarithm
	7.4 The Winding Number
	7.5 The Winding Number as an Integral
	7.6 The Winding Number Round an Arbitrary Point
	7.7 Components of the Complement of a Path
	7.8 Computing the Winding Number by Eye
	7.9 Exercises
8 Cauchy’s Theorem
	8.1 The Cauchy Theorem for a Triangle
	8.2 Existence of an Antiderivative in a Star Domain
	8.3 An Example – the Logarithm
	8.4 Local Existence of an Antiderivative
	8.5 Cauchy’s Theorem
	8.6 Applications of Cauchy’s Theorem
		8.6.1 Cuts and Jordan Contours
	8.7 Simply Connected Domains
	8.8 Exercises
9 Homotopy Versions of Cauchy’s Theorem
	9.1 Informal Description of Homotopy
	9.2 Integration Along Arbitrary Paths
	9.3 The Cauchy Theorem for a Boundary
	9.4 Formal Definition of Homotopy
	9.5 Fixed End Point Homotopy
	9.6 Closed Path Homotopy
	9.7 Converse to Cauchy’s Theorem
9.8 The Cauchy Theorems Compared
9.9 Exercises
10 Taylor Series
	10.1 Cauchy Integral Formula
	10.2 Taylor Series
	10.3 Morera’s Theorem
	10.4 Cauchy’s Estimate
	10.5 Zeros
	10.6 Extension Functions
	10.7 Local Maxima and Minima
	10.8 The Maximum Modulus Theorem
	10.9 Exercises
11 Laurent Series
	11.1 Series Involving Negative Powers
	11.2 Isolated Singularities
	11.3 Behaviour Near an Isolated Singularity
	11.4 The Extended Complex Plane, or Riemann Sphere
	11.5 Behaviour of a Differentiable Function at Infinity
	11.6 Meromorphic Functions
	11.7 Exercises
12 Residues
	12.1 Cauchy’s Residue Theorem
	12.2 Calculating Residues
	12.3 Evaluation of Definite Integrals
	12.4 Summation of Series
	12.5 Counting Zeros
	12.6 Exercises
13 Conformal Transformations
	13.1 Measurement of Angles
		13.1.1 Real Numbers Modulo 2π
		13.1.2 Geometry of R/2π
		13.1.3 Operations on Angles
		13.1.4 The Argument Modulo 2π
	13.2 Conformal Transformations
	13.3 Critical Points
	13.4 Möbius Maps
		13.4.1 Möbius Maps Preserve Circles
		13.4.2 Classification of Möbius Maps
13.4.3 Extension of Möbius Maps to the Riemann Sphere
13.5 Potential Theory
13.5.1 Laplace’s Equation
13.5.2 Design of Aerofoils
13.6 Exercises
14 Analytic Continuation
	14.1 The Limitations of Power Series
	14.2 Comparing Power Series
	14.3 Analytic Continuation
		14.3.1 Direct Analytic Continuation
		14.3.2 Indirect Analytic Continuation
		14.3.3 Complete Analytic Functions
	14.4 Multiform Functions
		14.4.1 The Logarithm as a Multiform Function
		14.4.2 Singularities
	14.5 Riemann Surfaces
		14.5.1 Riemann Surface for the Logarithm
		14.5.2 Riemann Surface for the Square Root
		14.5.3 Constructing a General Riemann Surface by Gluing
	14.6 Complex Powers
	14.7 Conformal Maps Using Multiform Functions
	14.8 Contour Integration of Multiform Functions
	14.9 Exercises
15 Infinitesimals in Real and Complex Analysis
	15.1 Infinitesimals
	15.2 The Relationship Between Real and Complex Analysis
		15.2.1 Critical Points
	15.3 Interpreting Power Series Tending to Zero as Infinitesimals
	15.4 Real Infinitesimals as Variable Points on a Number Line
	15.5 Infinitesimals as Elements of an Ordered Field
	15.6 Structure Theorem for any Ordered Extension Field of R
	15.7 Visualising Infinitesimals as Points on a Number Line
	15.8 Complex Infinitesimals
	15.9 Non-standard Analysis and Hyperreals
	15.10 Outline of the Construction of Hyperreal Numbers
	15.11 Hypercomplex Numbers
	15.12 The Evolution of Meaning in Real and Complex Analysis
		15.12.1 A Brief History
		15.12.2 Non-standard Analysis in Mathematics Education
		15.12.3 Human Visual Senses
		15.12.4 Computer Graphics
		15.12.5 Summary
	15.13 Exercises
16 Homology Version of Cauchy’s Theorem
	16.0.1 Outline of Chapter
	16.0.2 Group-theoretic Interpretation
	16.1 Chains
	16.2 Cycles
		16.2.1 Sums and Formal Sums of Paths
	16.3 Boundaries
	16.4 Homology
	16.5 Proof of Cauchy’s Theorem, Homology Version
		16.5.1 Grid of Rectangles
		16.5.2 Proof of Theorem 16.2
		16.5.3 Rerouting Segments
		16.5.4 Resumption of Proof of Theorem 16.2
	16.6 Cauchy’s Residue Theorem, Homology Version
	16.7 Exercises
17 The Road Goes Ever On
	17.1 The Riemann Hypothesis
	17.2 Modular Functions
	17.3 Several Complex Variables
	17.4 Complex Manifolds
	17.5 Complex Dynamics
	17.6 Epilogue
References
Index