Complex Analysis

Written by: George Cain (GA Tech)

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 singularities

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

Chapter Eleven – Argument Principle
11.1 Argument principle
11.2 Rouche’s Theorem


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Complex Analysis

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