Complex Analysis

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Contents

Preface vii

1. Complex Numbers 1

1.1 Introduction 1

1.2 Complex Numbers 1

1.3 Graphical Representation of a

Complex Number 6

1.4 Vector Form of Complex Numbers 7

1.5 Absolute Value and Conjugate 8

1.6 Triangle Inequality 11

1.7 Polar Form of a Complex Number 15

1.8 Exponential Form of a Complex

Number 21

1.9 De Moivre’s Theorem 26

1.10 Roots of Complex Numbers 29

1.11 Stereographic Projection 33

1.12 Regions in the Complex Plane 35

Summary 40

2. Analytic Functions 41

2.1 Introduction 41

2.2 Functions of a Complex Variable 41

2.3 Limit 45

2.4 Continuity 56

2.5 Differentiability 61

2.6 Analytic Functions 68

2.7 Cauchy–Riemann ...