| Foreword | p. vii |
| Introduction | p. xv |
| Preliminaries to Complex Analysis | p. 1 |
| Complex numbers and the complex plane | p. 1 |
| Basic properties | p. 1 |
| Convergence | p. 5 |
| Sets in the complex plane | p. 5 |
| Functions on the complex plane | p. 8 |
| Continuous functions | p. 8 |
| Holomorphic functions | p. 8 |
| Power series | p. 14 |
| Integration along curves | p. 18 |
| Exercises | p. 24 |
| Cauchy's Theorem and Its Applications | p. 32 |
| Goursat's theorem | p. 34 |
| Local existence of primitives and Cauchy's theorem in a disc | p. 37 |
| Evaluation of some integrals | p. 41 |
| Cauchy's integral formulas | p. 45 |
| Further applications | p. 53 |
| Morera's theorem | p. 53 |
| Sequences of holomorphic functions | p. 53 |
| Holomorphic functions defined in terms of integrals | p. 55 |
| Schwarz reflection principle | p. 57 |
| Runge's approximation theorem | p. 60 |
| Exercises | p. 64 |
| Problems | p. 67 |
| Meromorphic Functions and the Logarithm | p. 71 |
| Zeros and poles | p. 72 |
| The residue formula | p. 76 |
| Examples | p. 77 |
| Singularities and meromorphic functions | p. 83 |
| The argument principle and applications | p. 89 |
| Homotopies and simply connected domains | p. 93 |
| The complex logarithm | p. 97 |
| Fourier series and harmonic functions | p. 101 |
| Exercises | p. 103 |
| Problems | p. 108 |
| The Fourier Transform | p. 111 |
| The class F | p. 113 |
| Action of the Fourier transform on F | p. 114 |
| Paley-Wiener theorem | p. 121 |
| Exercises | p. 127 |
| Problems | p. 131 |
| Entire Functions | p. 134 |
| Jensen's formula | p. 135 |
| Functions of finite order | p. 138 |
| Infinite products | p. 140 |
| Generalities | p. 140 |
| Example: the product formula for the sine function | p. 142 |
| Weierstrass infinite products | p. 145 |
| Hadamard's factorization theorem | p. 147 |
| Exercises | p. 153 |
| Problems | p. 156 |
| The Gamma and Zeta Functions | p. 159 |
| The gamma function | p. 160 |
| Analytic continuation | p. 161 |
| Further properties of T | p. 163 |
| The zeta function | p. 168 |
| Functional equation and analytic continuation | p. 168 |
| Exercises | p. 174 |
| Problems | p. 179 |
| The Zeta Function and Prime Number Theorem | p. 181 |
| Zeros of the zeta function | p. 182 |
| Estimates for 1/s(s) | p. 187 |
| Reduction to the functions v and v1 | p. 188 |
| Proof of the asymptotics for v1 | p. 194 |
| Note on interchanging double sums | p. 197 |
| Exercises | p. 199 |
| Problems | p. 203 |
| Conformal Mappings | p. 205 |
| Conformal equivalence and examples | p. 206 |
| The disc and upper half-plane | p. 208 |
| Further examples | p. 209 |
| The Dirichlet problem in a strip | p. 212 |
| The Schwarz lemma; automorphisms of the disc and upper half-plane | p. 218 |
| Automorphisms of the disc | p. 219 |
| Automorphisms of the upper half-plane | p. 221 |
| The Riemann mapping theorem | p. 224 |
| Necessary conditions and statement of the theorem | p. 224 |
| Montel's theorem | p. 225 |
| Proof of the Riemann mapping theorem | p. 228 |
| Conformal mappings onto polygons | p. 231 |
| Some examples | p. 231 |
| The Schwarz-Christoffel integral | p. 235 |
| Boundary behavior | p. 238 |
| The mapping formula | p. 241 |
| Return to elliptic integrals | p. 245 |
| Exercises | p. 248 |
| Problems | p. 254 |
| An Introduction to Elliptic Functions | p. 261 |
| Elliptic functions | p. 262 |
| Liouville's theorems | p. 264 |
| The Weierstrass p function | p. 266 |
| The modular character of elliptic functions and Eisenstein series | p. 273 |
| Eisenstein series | p. 273 |
| Eisenstein series and divisor functions | p. 276 |
| Exercises | p. 278 |
| Problems | p. 281 |
| Applications of Theta Functions | p. 283 |
| Product formula for the Jacobi theta function | p. 284 |
| Further transformation laws | p. 289 |
| Generating functions | p. 293 |
| The theorems about sums of squares | p. 296 |
| The two-squares theorem | p. 297 |
| The four-squares theorem | p. 304 |
| Exercises | p. 309 |
| Problems | p. 314 |
| Asymptotics | p. 318 |
| Bessel functions | p. 319 |
| Laplace's method; Stirling's formula | p. 323 |
| The Airy function | p. 328 |
| The partition function | p. 334 |
| Problems | p. 341 |
| Simple Connectivity and Jord | |
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