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Foreword vii Introduction xv Chapter 1. Preliminaries to ComplexAnalysis 1 1 Complex numbers and the complex plane 1 1.1 Basic properties 11.2 Convergence 5 1.3 Sets in the complex plane 5 2 Functions on thecomplex plane 8 2.1 Continuous functions 8 2.2 Holomorphic functions 8 2.3Power series 14 3 Integration along curves 18 4 Exercises 24 Chapter 2.Cauchy's Theorem and Its Applications 32 1 Goursat's theorem 34 2 Localexistence of primitives and Cauchy's theorem in a disc 37 3 Evaluation ofsome integrals 41 4 Cauchy's integral formulas 45 5 Further applications 535.1 Morera's theorem 53 5.2 Sequences of holomorphic functions 53 5.3Holomorphic functions defined in terms of integrals 55 5.4 Schwarzreflection principle 57 5.5 Runge's approximation theorem 60 6 Exercises 647 Problems 67 Chapter 3. Meromorphic Functions and the Logarithm 71 1Zeros and poles 72 2 The residue formula 76 2.1 Examples 77 3Singularities and meromorphic functions 83 4 The argument principle andapplications 89 5 Homotopies and simply connected domains 93 6 The complexlogarithm 97 7 Fourier series and harmonic functions 101 8 Exercises 103 9Problems 108 Chapter 4. The Fourier Transform 111 1 The class F 113 2Action of the Fourier transform on F 114 3 Paley-Wiener theorem 121 4Exercises 127 5 Problems 131 Chapter 5. Entire Functions 134 1 Jensen'sformula 135 2 Functions of finite order 138 3 Infinite products 140 3.1Generalities 140 3.2 Example: the product formula for the sine function 1424 Weierstrass infinite products 145 5 Hadamard's factorization theorem 1476 Exercises 153 7 Problems 156 Chapter 6. The Gamma and Zeta Functions 1591 The gamma function 160 1.1 Analytic continuation 161 1.2 Furtherproperties of T 163 2 The zeta function 168 2.1 Functional equation andanalytic continuation 168 3 Exercises 174 4 Problems 179 Chapter 7. TheZeta Function and Prime Number Theorem 181 1 Zeros of the zeta function 1821.1 Estimates for 1/s(s) 187 2 Reduction to the functions v and v1 188 2.1Proof of the asymptotics for v1 194 Note on interchanging double sums 197 3Exercises 199 4 Problems 203 Chapter 8. Conformal Mappings 205 1 Conformalequivalence and examples 206 1.1 The disc and upper half-plane 208 1.2Further examples 209 1.3 The Dirichlet problem in a strip 212 2 The Schwarzlemma; automorphisms of the disc and upper half-plane 218 2.1 Automorphismsof the disc 219 2.2 Automorphisms of the upper half-plane 221 3 The Riemannmapping theorem 224 3.1 Necessary conditions and statement of the theorem224 3.2 Montel's theorem 225 3.3 Proof of the Riemann mapping theorem 2284 Conformal mappings onto polygons 231 4.1 Some examples 231 4.2 TheSchwarz-Christoffel integral 235 4.3 Boundary behavior 238 4.4 The mappingformula 241 4.5 Return to elliptic integrals 245 5 Exercises 248 6Problems 254 Chapter 9. An Introduction to Elliptic Functions 261 1Elliptic functions 262 1.1 Liouville's theorems 264 1.2 The Weierstrass pfunction 266 2 The modular character of elliptic functions and Eisensteinseries 273 2.1 Eisenstein series 273 2.2 Eisenstein series and divisorfunctions 276 3 Exercises 278 4 Problems 281 Chapter 10. Applications ofTheta Functions 283 1 Product formula for the Jacobi theta function 284 1.1Further transformation laws 289 2 Generating functions 293 3 The theoremsabout sums of squares 296 3.1 The two-squares theorem 297 3.2 Thefour-squares theorem 304 4 Exercises 309 5 Problems 314 Appendix A:Asymptotics 318 1 Bessel functions 319 2 Laplace's method; Stirling'sformula 323 3 The Airy function 328 4 The partition function 334 5Problems 341 Appendix B: Simple Connectivity and Jordan Curve Theorem 344 1Equivalent formulations of simple connectivity 345 2 The Jordan curvetheorem 351 2.1 Proof of a general form of Cauchy's theorem 361 Notes andReferences 365 Bibliography 369 Symbol Glossary 373 Index 375