An Introductory textbook on Complex Analysis.
The book has been divided into two parts. The first part deals with the fundamentals of analytic function theory and should be studied sequentially before going on to the second part which deals with the applications of analytic function theory. The second part introduces five of the major applications, more or less independently of one another. In other words, once the first half of the book is mastered, one can pick any one of the five applications to study in detail. This makes it possible for the book to be used for a one-semester course, where one or more applications can be introduced, depending on the time available, or for a two-semester course, in which virtually all the applications are studied.
Although a fairly brisk pace is maintained throughout, I have assumed no advanced calculus background. A well-motivated student with a modern calculus course behind him should be able to study this material with profit. This should make it possible to use the book for a junior- or senior-level advanced engineering mathematics course or for an undergraduate
introduction to complex variables for mathematics majors.
Author(s): John W. Dettman
Publisher: Dover Publications
Year: 2012
Language: English
Pages: 666
Tags: Complex variables, Complex Analysis
Title Page
Copyright Page
Preface
Contents
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
