Table of Contents
Chapter 1: Complex Numbers
1 The Algebra of Complex Numbers
1.1 Arithmetic Operations
1.2 Square Roots
1.3 Justification
1.4 Conjugation, Absolute Value
1.5 Inequalities2 The Geometric Representation of Complex Numbers
2.1 Geometric Addition and Multiplication
2.2 The Binomial Equation
2.3 Analytic Geometry
2.4 The Spherical Representation
Chapter 2: Complex Functions1 Introduction to the Concept of Analytic Function
1.1 Limits and Continuity
1.2 Analytic Functions
1.3 Polynomials
1.4 Rational Functions2 Elementary Theory of Power Series
2.1 Sequences
2.2 Series
2.3 Uniform Coverages
2.4 Power Series
2.5 Abel's Limit Theorem3 The Exponential and Trigonometric Functions
3.1 The Exponential
3.2 The Trigonometric Functions
3.3 The Periodicity
3.4 The Logarithm
Chapter 3: Analytic Functions as Mappings1 Elementary Point Set Topology
1.1 Sets and Elements
1.2 Metric Spaces
1.3 Connectedness
1.4 Compactness
1.5 Continuous Functions
1.6 Topological Spaces2 Conformality
2.1 Arcs and Closed Curves
2.2 Analytic Functions in Regions
2.3 Conformal Mapping
2.4 Length and Area3 Linear Transformations
3.1 The Linear Group
3.2 The Cross Ratio
3.3 Symmetry
3.4 Oriented Circles
3.5 Families of Circles4 Elementary Conformal Mappings
4.1 The Use of Level Curves
4.2 A Survey of Elementary Mappings
4.3 Elementary Riemann Surfaces
Chapter 4: Complex Integration1 Fundamental Theorems
1.1 Line Integrals
1.2 Rectifiable Arcs
1.3 Line Integrals as Functions of Arcs
1.4 Cauchy's Theorem for a Rectangle
1.5 Cauchy's Theorem in a Disk2 Cauchy's Integral Formula
2.1 The Index of a Point with Respect to a Closed Curve
2.2 The Integral Formula
2.3 Higher Derivatives3 Local Properties of Analytical Functions
3.1 Removable Singularities. Taylor's Theorem
3.2 Zeros and Poles
3.3 The Local Mapping
3.4 The Maximum Principle4 The General Form of Cauchy's Theorem
4.1 Chains and Cycles
4.2 Simple Connectivity
4.3 Homology
4.4 The General Statement of Cauchy's Theorem
4.5 Proof of Cauchy's Theorem
4.6 Locally Exact Differentials
4.7 Multiply Connected Regions5 The Calculus of Residues
5.1 The Residue Theorem
5.2 The Argument Principle
5.3 Evaluation of Definite Integrals6 Harmonic Functions
6.1 Definition and Basic Properties
6.2 The Mean-value Property
6.3 Poisson's Formula
6.4 Schwarz's Theorem
6.5 The Reflection Principle
Chapter 5: Series and Product Developments1 Power Series Expansions
1.1 Wierstrass's Theorem
1.2 The Taylor Series
1.3 The Laurent Series2 Partial Fractions and Factorization
2.1 Partial Fractions
2.2 Infinite Products
2.3 Canonical Products
2.4 The Gamma Function
2.5 Stirling's Formula3 Entire Functions
3.1 Jensen's Formula
3.2 Hadamard's Theorem4 The Riemann Zeta Function
4.1 The Product Development
4.2 Extension of ζ(s) to the Whole Plane
4.3 The Functional Equation
4.4 The Zeros of the Zeta Function5 Normal Families
5.1 Equicontinuity
5.2 Normality and Compactness
5.3 Arzela's Theorem
5.4 Families of Analytic Functions
5.5 The Classical Definition
Chapter 6: Conformal Mapping, Dirichlet's Problem1 The Riemann Mapping Theorem
1.1 Statement and Proof
1.2 Boundary Behavior
1.3 Use of the Reflection Principle
1.4 Analytic Arcs2 Conformal Mapping of Polygons
2.1 The Behavior at an Angle
2.2 The Schwarz-Christoffel Formula
2.3 Mapping on a Rectangle
2.4 The Triangle Functions of Schwarz3 A Closer Look at Harmonic Functions
3.1 Functions with Mean-value Property
3.2 Harnack's Principle4 The Dirichlet Problem
4.1 Subharmonic Functions
4.2 Solution of Dirichlet's Problem5 Canonical Mappings of Multiply Connected Regions
5.1 Harmonic Measures
5.2 Green's Function
5.3 Parallel Slit Regions
Chapter 7: Elliptic Functions1 Simply Periodic Functions
1.1 Representation by Exponentials
1.2 The Fourier Development
1.3 Functions of Finite Order2 Doubly Periodic Functions
2.1 The Period Module
2.2 Unimodular Transformations
2.3 The Canonical Basis
2.4 General Properties of Elliptic Functions3 The Weierstrass Theory
3.1 The Weierstrass p-function
3.2 The Functions ζ(z) and σ(z)
3.3 The Differential Equation
3.4 The Modular Function λ(r)
3.5 The Conformal Mapping by λ(r)
Chapter 8: Global Analytic Functions1 Analytic Continuation
1.1 The Weierstrass Theory
1.2 Germs and Sheaves
1.3 Sections and Riemann Surfaces
1.4 Analytic Continuations along Arcs
1.5 Homotopic Curves
1.6 The Monodromy Theorem
1.7 Branch Points2 Algebraic Functions
2.1 The Resultant of Two Polynomials
2.2 Definition and Properties of Algebraic Functions
2.3 Behavior at the Critical Points3 Picard's Theorem
3.1 Lacunary Values4 Linear Differential Equations
4.1 Ordinary Points
4.2 Regular Singular Points
4.3 Solutions at Infinity
4.4 The Hypergeometric Differential Equation
4.5 Riemann's Point of View
Index