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
Previous Page Next Page