Complex Analysis

CONTENTS
Foreword
Introduction
CHAPTER 1. Preliminaries to Complex Analysis
1 Complex numbers and the complex plane, 1.1 Basic properties, 1.2 Convergence, 1.3 Sets in the complex plane, 2 Functions on the complex plane, 2.1 Continuous functions, 2.2 Holomorphic functions
2.3 Power series, 3 Integration along curves, 4 Exercises,
CHAPTER 2. Cauchy’s Theorem and Its Applications, 1 Goursat’s theorem, 2 Local existence of primitives and Cauchy’s theorem in a disc, 3 Evaluation of some integrals, 4 Cauchy’s integral formulas, 5 Further applications, 5.1 Morera’s theorem, 5.2 Sequences of holomorphic functions, 5.3 Holomorphic functions defined in terms of integrals, 5.4 Schwarz reflection principle, 5.5 Runge’s approximation theorem, 6 Exercises, 7 Problems,
CHAPTER 3. Meromorphic Functions and the Logarithm, 1 Zeros and poles, 2 The residue formula, 2.1 Examples, 3 Singularities and meromorphic functions, 4 The argument principle and applications, 5 Homotopies and simply connected domains, 6 The complex logarithm, 7 Fourier series and harmonic functions, 8 Exercises, 9 Problems,
CHAPTER 4. The Fourier Transform, 1 The class,
2 Action of the Fourier transform on, 3 Paley-Wiener theorem, 4 Exercises, 5. Problems,
CHAPTER 5. Entire Functions, 1 Jensen’s formula, 2 Functions of finite order, 3. Infinite products, 4 Problems
CHAPTER 6. The Gamma and Zeta Functions, 1 The gamma function, 1.1 Analytic continuation, 1.2 Further properties of I, ,2 The zeta function, 2.1 Functional equation and analytic continuation,
3 Exercises,
CHAPTER 7. The Zeta Function and Prime Number The orem, 1 Zeros of the zeta function, 1.1 Estimates for 1/6(s), 2 Reduction to the functions and V1, 2.1 Proof of the asymptotics for V1, Note on interchanging double sums, 3 Exercises, 4 Problems
CHAPTER 8. Conformal Mappings, 1 Conformal equivalence and examples,1.1 The disc and upper half-plane, 1.2 Further examples, 1.3 The Dirichlet problem in a strip, 2 The Schwarz lemma; automorphisms of the disc and upper half-plane, 2.1 Automorphisms of the disc, 2.2 Automorphisms of the upper half-plane, 3 The Riemann mapping theorem, 3.1 Necessary conditions and statement of the theorem, 3.2 Montel’s theorem, 3.3 Proof of the Riemann mapping theorem, 4 Conformal mappings onto polygons, 4.1 Some examples, 4.2 The Schwarz-Christoffel integral, 4.3 Boundary behavior, 4.4 The mapping formula, 4.5 Return to elliptic integrals, 5 Exercises, 6 Problems
CHAPTER 9. An Introduction to Elliptic Functions, 1 Elliptic functions, 1.1 Liouville’s theorems,
1.2 The Weierstrass function, 2 The modular character of elliptic functions and Eisenstein series
2.1 Eisenstein series, 2.2 Eisenstein series and divisor functions, 3 Exercises, 4 Problems
CHAPTER 10. Applications of Theta Functions, 1 Product formula for the Jacobi theta function
1.1 Further transformation laws, 2 Generating functions,3 The theorems about sums of squares,
3.1 The two-squares theorem, 4 Exercises, 5 Problems
Appendix A: Asymptotics
1 Bessel functions, 2 Laplace’s method; Stirling’s formula, 3 The Airy function, 4 The partition function,5 Problems
Appendix B: Simple Connectivity and Jordan Curve Theorem, 1 Equivalent formulations of simple connectivity, 2 The Jordan curve theorem, 2.1 Proof of a general form of Cauchy’s theorem
Notes and References
Bibliography
Symbol Glossary
Index