A Friendly Approach To Complex Analysis

by Amol Sasane, Sara Maad Sasane

Overview

The book constitutes a basic, concise, yet rigorous course in complex analysis, for students who have studied calculus in one and several variables, but have not previously been exposed to complex analysis. The textbook should be particularly useful and relevant for undergraduate students in joint programmes with mathematics, as well as engineering students. The aim of the book is to cover the bare bones of the subject with minimal prerequisites. The core content of the book is the three main pillars of complex analysis: the Cauchy-Riemann equations, the Cauchy Integral Theorem, and Taylor and Laurent series expansions.Each section contains several problems, which are not purely drill exercises, but are rather meant to reinforce the fundamental concepts. Detailed solutions to all the exercises appear at the end of the book, making the book ideal also for self-study. There are many figures illustrating the text.

Product Details

ISBN-13:	9789814578981
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	02/19/2014
Pages:	288
Product dimensions:	5.80(w) x 9.00(h) x 0.90(d)

Table of Contents

Preface v

1 Complex numbers and their geometry 1

1.1 The field of complex numbers 1

1.2 Geometric representation of complex numbers 5

1.3 Topology of C 11

1.3.1 Metric on C 12

1.3.2 Open discs, open sets, closed sets and compact sets 13

1.3.3 Convergence and continuity 14

1.3.4 Domains 15

1.4 The exponential function and kith 17

1.4.1 The exponential exp z 18

1.4.2 Trigonometric functions 22

1.4.3 Logarithm function 23

1.5 Notes 27

2 Complex differentiability 29

2.1 Complex differentiability 30

2.2 Cauchy-Riemann equations 36

2.3 Geometric meaning of the complex derivative 49

2.4 The d-bar operator 56

2.5 Notes 58

3 Cauchy Integral Theorem and consequences 59

3.1 Definition of the contour integral 59

3.1.1 An important integral 67

3.2 Properties of contour integration 69

3.3 Fundamental Theorem of Contour Integration 73

3.4 The Cauchy Integral Theorem 77

3.4.1 Special case: simply connected domains 82

3.4.2 What happens with nonholomorphic functions? 85

3.5 Existence of a primitive 88

3.6 The Cauchy Integral Formula 91

3.7 Holomorphic functions are infinitely differentiable 96

3.8 Liouville's Theorem; Fundamental Theorem of Algebra 99

3.9 Morera's Theorem: converse to Cauchy's Integral Theorem 101

3.10 Notes 103

4 Taylor and Laurent series 105

4.1 Series 106

4.2 Power series 107

4.2.1 Power series and their region of convergence 107

4.2.2 Power series are holomorphic 113

4.3 Taylor series 117

4.4 Classification of zeros 122

4.5 The Identity Theorem 126

4.6 The Maximum Modulus Theorem 129

4.7 Laurent series 131

4.8 Classification of singularities 141

4.8.1 Wild behaviour near essential singularities 151

4.9 Residue Theorem 153

4.10 Notes 162

5 Harmonic functions 163

5.1 What is a harmonic function? 163

5.2 What is the link between harmonic functions and holomorphic functions? 165

5.3 Consequences of the two way traffic: holomorphic ↔ harmonic 170

5.3.1 Harmonic functions are smooth 171

5.3.2 Mean value property 172

5.3.3 Maximum Principle 172

5.4 Uniqueness of solution for the Dirichlet problem 173

5.5 Notes 176

Solutions 177

Bibliography 269

Index 271