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 of SUNY Binghamton who used the book in his class at Binghamton and found many errors and made many good suggestions for changes and additions to the book. I thank him very much. I have corrected the errors and made some changes.
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.
The notes are available as Adobe Acrobat documents. If you do not have an Adobe Acrobat Reader, you may download a copy, free of charge, from Adobe.
Title page and Table of Contents
Table of Contents
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 Antiderivatives
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
Applications of the Residue Theorem to Real IntegralsSupplementary Material by Pawel Hitczenko
Chapter Eleven  Argument Principle
11.1 Argument principle
11.2 Rouche's Theorem
