Basic Complex Analysis - 3rd edition

Preface

1. Analytic Functions

Introduction to Complex Numbers
Properties of Complex Numbers
Some Elementary Functions
Continuous Functions
Analytic Functions
Differentiation of the Elementary Functions
Review Exercises for Chapter 1

2. Cauchy's Theorem

Contour Integrals
Supplement: Riemann Sums
Cauchy's Theorem: Intuitive Version
Cauchy's Theorem: Precise Version
Supplement A: Integrals Along
Continuous Curves
Supplement B: Relationship of Cauchy's Theorem to the Jordan Curve Theorem
Cauchy's Integral Formula
Maximum Modulus Theorem and Harmonic Function
Review Exercises for Chapter 2

3. Series Representation of Analytic Functions

Convergent Series of Analytic Functions
Power Series and Taylor's Theorem
Laurent's Series and Classification of Singularities
Review Exercises for Chapter 3

4. Calculus of Residues

Calculation of Residues
The Residue Theorem
Evaluation of Definite Integrals
Evaluation of Infinite Series and
Partial-Fraction Expansions
Review Exercises for Chapter 4

5. Conformal Mappings

Basic Theory of Conformal Mappings
Fractional Linear and Schwarz-Christoffel Transformations
Applications of Conformal Mapping to Laplace's Equation, Heat
Conduction, Electrostatics, and Hydrodynamics
Review Exercises for Chapter 5

6. Further Development of the Theory

Analytic Continuation and Elementary Riemann Surfaces
Rouché's Theorem and the Principle of the Argument
Mapping Properties of Analytic
Functions
Supplement A: Normal Families and The Riemann Mapping Theorem
Supplement B: The Dynamics of Complex Analytic Mappings
Review Exercises for Chapter 6

7. Asymptotic Methods

Infinite Products and the Gamma Function
Asymptotic Expansions and the Method of Steepest Descent
Supplement: Bounded Variation and the Proof of the Stationary Phase Formula
Stirling's Formula and Bessel
Functions
Review Exercises for Chapter 7

8. The Laplace Transform and Applications

Basic Properties of LaPlace Transforms
The Complex Inversion Formula
Application of Laplace Transforms to Ordinary Differential Equations
Supplement: The Fourier Transform and the Wave Equation
Review Exercises for Chapter 8

Index