This textbook has been utilized at SUNY Binghamton, California State University Dominguez Hills, Drexel University as well as Georgia Tech where the author is a professor. George Cain wrote his introductory course book on *Complex Analysis* in 2001. He has been teaching at Georgia Institute of Technology since 1965.

## Table of Contents for *Complex Analysis*

- 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 singularitiesApplications of the Residue Theorem to Real Integrals-Supplementary Material by Pawel Hitczenko

Chapter Eleven – Argument Principle

11.1 Argument principle

11.2 Rouche’s TheoremView this Free Online Material at the source:

Complex Analysis