vi Contents

Chapter 4. Power Series Expansions 67

1. Geometric series 67

2. The radius of convergence 70

3. Generating functions 73

4. Fibonacci numbers 74

5. An application of power series 77

6. Rationality 79

Chapter 5. Complex Diﬀerentiation 83

1. Deﬁnitions of complex analytic function 83

2. Complex diﬀerentiation 85

3. The Cauchy-Riemann equations 86

4. Orthogonal trajectories and harmonic functions 89

5. A glimpse at harmonic functions 90

6. What is a diﬀerential form? 96

Chapter 6. Complex Integration 99

1. Complex-valued functions 99

2. Line integrals 101

3. Goursat’s proof 109

4. The Cauchy integral formula 111

5. A return to the deﬁnition of complex analytic function 116

Chapter 7. Applications of Complex Integration 119

1. Singularities and residues 119

2. Evaluating real integrals using complex variables methods 121

3. Fourier transforms 128

4. The Gamma function 131

Chapter 8. Additional Topics 137

1. The minimum-maximum theorem 137

2. The fundamental theorem of algebra 138

3. Winding numbers, zeroes, and poles 141

4. Pythagorean triples 147

5. Elementary mappings 150

6. Quaternions 153

7. Higher-dimensional complex analysis 155

Further reading 158

Bibliography 159

Index 161