9780070006577

A standard source of information of functions of one complex variable, this text has retained its wide popularity in this field by being consistently rigorous without becoming needlessly concerned with advanced or overspecialized material. Difficult points have been clarified, the book has been reviewed for accuracy, and notations and terminology have been modernized. Chapter 2, Complex Functions, features a brief section on the change of length and area under conformal mapping, and much of Chapter 8, Global-Analytic Functions, has been rewritten in order to introduce readers to the terminology of germs and sheaves while still emphasizing that classical concepts are the backbone of the theory. Chapter 4, Complex Integration, now includes a new and simpler proof of the general form of Cauchy's theorem. There is a short section on the Riemann zeta function, showing the use of residues in a more exciting situation than in the computation of definite integrals.

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 Inequalities

2 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 Functions

1 Introduction to the Concept of Analytic Function

1.1 Limits and Continuity

1.2 Analytic Functions

1.3 Polynomials

1.4 Rational Functions

2 Elementary Theory of Power Series

2.1 Sequences

2.2 Series

2.3 Uniform Coverages

2.4 Power Series

2.5 Abel's Limit Theorem

3 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 Mappings

1 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 Spaces

2 Conformality

2.1 Arcs and Closed Curves

2.2 Analytic Functions in Regions

2.3 Conformal Mapping

2.4 Length and Area

3 Linear Transformations

3.1 The Linear Group

3.2 The Cross Ratio

3.3 Symmetry

3.4 Oriented Circles

3.5 Families of Circles

4 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 Integration

1 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 Disk

2 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 Derivatives

3 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 Principle

4 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 Regions

5 The Calculus of Residues

5.1 The Residue Theorem

5.2 The Argument Principle

5.3 Evaluation of Definite Integrals

6 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 Developments

1 Power Series Expansions

1.1 Wierstrass's Theorem

1.2 The Taylor Series

1.3 The Laurent Series

2 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 Formula

3 Entire Functions

3.1 Jensen's Formula

3.2 Hadamard's Theorem

4 The Riemann Zeta Function

4.1 The Product Development

4.2 Extension of ç(s) to the Whole Plane

4.3 The Functional Equation

4.4 The Zeros of the Zeta Function

5 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 Problem

1 The Riemann Mapping Theorem

1.1 Statement and Proof

1.2 Boundary Behavior

1.3 Use of the Reflection Principle

1.4 Analytic Arcs

2 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 Schwarz

3 A Closer Look at Harmonic Functions

3.1 Functions with Mean-value Property

3.2 Harnack's Principle

4 The Dirichlet Problem

4.1 Subharmonic Functions

4.2 Solution of Dirichlet's Problem

5 Canonical Mappings of Multiply Connected Regions

5.1 Harmonic Measures

5.2 Green's Function

5.3 Parallel Slit Regions

Chapter 7: Elliptic Functions

1 Simply Periodic Functions

1.1 Representation by Exponentials

1.2 The Fourier Development

1.3 Functions of Finite Order

2 Doubly Periodic Functions

2.1 The Period Module

2.2 Unimodular Transformations

2.3 The Canonical Basis

2.4 General Properties of Elliptic Functions

3 The Weierstrass Theory

3.1 The Weierstrass p-function

3.2 The Functions ç(z) and ó(z)

3.3 The Differential Equation

3.4 The Modular Function µ(r)

3.5 The Conformal Mapping by µ(r)

Chapter 8: Global Analytic Functions

1 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 Points

2 Algebraic Functions

2.1 The Resultant of Two Polynomials

2.2 Definition and Properties of Algebraic Functions

2.3 Behavior at the Critical Points

3 Picard's Theorem

3.1 Lacunary Values

4 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

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