Contents

Preface vii

Chapter 1. The Complex Numbers 1

1.1. Definition and Simple Properties 1

1.2. Convergence in C 8

1.3. The Exponential Function 14

1.4. Polar Form for Complex Numbers 19

Chapter 2. Analytic Functions 27

2.1. Continuous Functions 27

2.2. The Complex Derivative 34

2.3. Contour Integrals 41

2.4. Properties of Contour Integrals 47

2.5. Cauchy’s Integral Theorem for a Triangle 53

2.6. Cauchy’s Theorem for a Convex Set 60

2.7. Properties of the Index Function 66

Chapter 3. Power Series Expansions 75

3.1. Uniform Convergence 75

3.2. Power Series Expansions 82

3.3. Liouville’s Theorem 89

3.4. Zeroes and Singularities 95

3.5. The Maximum Modulus Principle 102

Chapter 4. The General Cauchy Theorems 109

4.1. Chains and Cycles 110

4.2. Cauchy’s Theorems 118

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