Foreword viiIntroduction xvChapter 1. Preliminaries to Complex Analysis 11 Complex numbers and the complex plane 11.1 Basic properties 11.2 Convergence 51.3 Sets in the complex plane 52 Functions on the complex plane 82.1 Continuous functions 82.2 Holomorphic functions 82.3 Power series 143 Integration along curves 184 Exercises 24Chapter 2. Cauchy's Theorem and Its Applications 321 Goursat's theorem 342 Local existence of primitives and Cauchy's theorem in a disc 373 Evaluation of some integrals 414 Cauchy's integral formulas 455 Further applications 535.1 Morera's theorem 535.2 Sequences of holomorphic functions 535.3 Holomorphic functions defined in terms of integrals 555.4 Schwarz reflection principle 575.5 Runge's approximation theorem 606 Exercises 647 Problems 67Chapter 3. Meromorphic Functions and the Logarithm 711 Zeros and poles 722 The residue formula 762.1 Examples 773 Singularities and meromorphic functions 834 The argument principle and applications 895 Homotopies and simply connected domains 936 The complex logarithm 977 Fourier series and harmonic functions 1018 Exercises 1039 Problems 108Chapter 4. The Fourier Transform 1111 The class F 1132 Action of the Fourier transform on F 1143 Paley-Wiener theorem 1214 Exercises 1275 Problems 131Chapter 5. Entire Functions 1341 Jensen's formula 1352 Functions of finite order 1383 Infinite products 1403.1 Generalities 1403.2 Example: the product formula for the sine function 1424 Weierstrass infinite products 1455 Hadamard's factorization theorem 1476 Exercises 1537 Problems 156Chapter 6. The Gamma and Zeta Functions 1591 The gamma function 1601.1 Analytic continuation 1611.2 Further properties of T 1632 The zeta function 1682.1 Functional equation and analytic continuation 1683 Exercises 1744 Problems 179Chapter 7. The Zeta Function and Prime Number Theorem 1811 Zeros of the zeta function 1821.1 Estimates for 1/s(s) 1872 Reduction to the functions v and v1 1882.1 Proof of the asymptotics for v1 194Note on interchanging double sums 1973 Exercises 1994 Problems 203Chapter 8. Conformal Mappings 2051 Conformal equivalence and examples 2061.1 The disc and upper half-plane 2081.2 Further examples 2091.3 The Dirichlet problem in a strip 2122 The Schwarz lemma; automorphisms of the disc and upper half-plane 2182.1 Automorphisms of the disc 2192.2 Automorphisms of the upper half-plane 2213 The Riemann mapping theorem 2243.1 Necessary conditions and statement of the theorem 2243.2 Montel's theorem 2253.3 Proof of the Riemann mapping theorem 2284 Conformal mappings onto polygons 2314.1 Some examples 2314.2 The Schwarz-Christoffel integral 2354.3 Boundary behavior 2384.4 The mapping formula 2414.5 Return to elliptic integrals 2455 Exercises 2486 Problems 254Chapter 9. An Introduction to Elliptic Functions 2611 Elliptic functions 2621.1 Liouville's theorems 2641.2 The Weierstrass p function 2662 The modular character of elliptic functions and Eisenstein series 2732.1 Eisenstein series 2732.2 Eisenstein series and divisor functions 2763 Exercises 2784 Problems 281Chapter 10. Applications of Theta Functions 2831 Product formula for the Jacobi theta function 2841.1 Further transformation laws 2892 Generating functions 2933 The theorems about sums of squares 2963.1 The two-squares theorem 2973.2 The four-squares theorem 3044 Exercises 3095 Problems 314Appendix A: Asymptotics 3181 Bessel functions 3192 Laplace's method; Stirling's formula 3233 The Airy function 3284 The partition function 3345 Problems 341Appendix B: Simple Connectivity and Jordan Curve Theorem 3441 Equivalent formulations of simple connectivity 3452 The Jordan curve theorem 3512.1 Proof of a general form of Cauchy's theorem 361Notes and References 365Bibliography 369Symbol Glossary 373Index 375