Applied Complex Variable

Part I. Analytic Function Theory
Chapter 1. The Complex Number Plane
1.1 Introduction
1.2 Complex Numbers
1.3 The Complex Plane
1.4 Point Sets in the Plane
1.5 Stereographic Projection. The Extended Complex Plane
1.6 Curves and Regions
Chapter 2. Functions of a Complex Variable
2.1 Functions and Limits
2.2 Differentiability and Analyticity
2.3 The Cauchy-Riemann Conditions
2.4 Linear Fractional Transformations
2.5 Transcendental functions
2.6 Riemann Surfaces
Chapter 3. Integration in the Complex Plane
3.1 Line Integrals
3.2 The Definite Integral
3.3 Cauchy's Theorem
3.4 Implications of Cauchy's Theorem
3.5 Functions Defined by Integration
3.6 Cauchy Formulas
3.7 Maximum Modulus Principle
Chapter 4. Sequences and Series
4.1 Sequences of Complex Numbers
4.2 Sequences of Complex Functions
4.3 Infinite Series
4.4 Power Series
4.5 Analytic Continuation
4.6 Laurent Series
4.7 Double Series
4.8 Infinite Products
4.9 Improper Integrals
4.10 The Gamma Function
Chapter 5. Residue Calculus
5.1 The Residue Theorem
5.2 Evaluation of Real Integrals
5.3 The Principle of the Argument
5.4 Meromorphic Functions
5.5 Entire Functions
Part II. Applications of Analytic Function Theory
Chapter 6. Potential Theory
6.1 Laplace's Equation in Physics
6.2 The Dirichlet Problem
6.3 Green's Functions
6.4 Conformal Mapping
6.5 The Schwarz-Christoffel Transformation
6.6 Flows with Sources and Sinks
6.7 Volume and Surface Distributions
6.8 Singular Integral Equations
Chapter 7. Ordinary Differential Equations
7.1 Separation of Variables
7.2 Existence and Uniqueness Theorems
7.3 Solution of a Linear Second-Order Differential Equation Near an Ordinary Point
7.4 Solution of a Linear Second-Order Differential Equation Near a Regular Singular Point
7.5 Bessel Functions
7.6 Legendre Functions
7.7 Sturm-Liouville Problems
7.8 Fredholm Integral Equations
Chapter 8. Fourier Transforms
8.1 Fourier Series
8.2 The Fourier Integral Theorem
8.3 The Complex Fourier Transform
8.4 Properties of the Fourier Transform
8.5 The Solution of Ordinary Differential Equations
8.6 The Solution of Partial Differential Equations
8.7 The Solution of Integral Equations
Chapter 9. Laplace Transforms
9.1 From Fourier to Laplace Transform
9.2 Properties of the Laplace Transform
9.3 Inversion of Laplace Transforms
9.4 The Solution of Ordinary Differential Equations
9.5 Stability
9.6 The Solution of Partial Differential Equations
9.7 The Solution of Integral Equations
Chapter 10. Asymptotic Expansions
10.1 Introduction and Definitions
10.2 Operations on Asymptotic Expansions
10.3 Asymptotic Expansion of Integrals
10.4 Asymptotic Solutions of Ordinary Differential Equations
References; Index