vi Contents
4.3. Laurent Series 124
4.4. The Residue Theorem 131
4.5. Rouch´ e’s Theorem and Inverse Functions 137
4.6. Homotopy 141
Chapter 5. Residue Theory 153
5.1. Computing Residues 153
5.2. Evaluating Integrals Using Residues 159
5.3. Fourier Transforms 166
5.4. The Laplace and Mellin Transforms 175
5.5. Summing Infinite Series 179
Chapter 6. Conformal Mappings 185
6.1. Definition and Examples 185
6.2. The Riemann Sphere 191
6.3. Linear Fractional Transformations 197
6.4. The Riemann Mapping Theorem 202
6.5. The Poisson Integral 207
6.6. The Dirichlet Problem 213
Chapter 7. Analytic Continuation and the Picard Theorems 221
7.1. The Schwarz Reflection Principle 222
7.2. Continuation Along a Curve 226
7.3. Analytic Covering Maps 232
7.4. The Picard Theorems 236
Chapter 8. Infinite Products 245
8.1. Convergence of Infinite Products 245
8.2. Weierstrass Products 251
8.3. Entire Functions of Finite Order 257
8.4. Hadamard’s Factorization Theorem 262
Chapter 9. The Gamma and Zeta Functions 269
9.1. Euler’s Gamma Function 270
9.2. The Riemann Zeta Function 275
9.3. Properties of ζ 281
9.4. The Riemann Hypothesis and Prime Numbers 287
9.5. A Proof of the Prime Number Theorem 291
Bibliography 299
Index 301
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