## Complex Analysis

by

### George Cain

This is a textbook for an introductory course in complex analysis. It has been used for our undergraduate complex analysis course here at Georgia Tech and at a few other places that I know of.

I owe a special debt of gratitude to Professor Matthias Beck who used the book in his class at SUNY Binghamton and found many errors and made many good suggestions for changes and additions to the book. I thank him very much.

Many thanks also to Professor Serban Raianu of California State University Dominguez Hills whose many helpful suggestions have considerably improved the book.

I am also grateful to Professor Pawel Hitczenko of Drexel University, who prepared the nice supplement to Chapter 10 on applications of the Residue Theorem to real integration.

Chapter One - Complex Numbers
1.1 Introduction
1.2 Geometry
1.3 Polar coordinates

Chapter Two - Complex Functions
2.1 Functions of a real variable
2.2 Functions of a complex variable
2.3 Derivatives

Chapter Three - Elementary Functions
3.1 Introduction
3.2 The exponential function
3.3 Trigonometric functions
3.4 Logarithms and complex exponents

Chapter Four - Integration
4.1 Introduction
4.2 Evaluating integrals
4.3 Antiderivative

Chapter Five - Cauchy's Theorem
5.1 Homotopy
5.2 Cauchy's Theorem

Chapter Six - More Integration
6.1 Cauchy's Integral Formula
6.2 Functions defined by integrals
6.3 Liouville's Theorem
6.4 Maximum moduli

Chapter Seven - Harmonic Functions
7.1 The Laplace equation
7.2 Harmonic functions
7.3 Poisson's integral formula

Chapter Eight - Series
8.1 Sequences
8.2 Series
8.3 Power series
8.4 Integration of power series
8.5 Differentiation of power series

Chapter Nine - Taylor and Laurent Series
9.1 Taylor series
9.2 Laurent series

Chapter Ten - Poles, Residues, and All That
10.1 Residues
10.2 Poles and other singularities

Chapter Eleven - Argument Principle
11.1 Argument principle
11.2 Rouche's Theorem

1 May 2009