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

3.1 Generalities

3.2 Example:the product formula for the sine function

4 Weierstrass infinite products

5 Hadamard's factorization theorem

6 Exercises

7 Problems

Chapter 6．The Gamma and Zeta Functions

1 The gamma function

1.1 Analytic continuation

1.2 Further properties of г

2 The zeta function

2.1 Functional equation and analytic continuation

3 Exercises

4 Problems

Chapter 7．The Zeta Function and Prime Number The-orem

1 Zeros of the zeta function

1.1 Estimates for 1/ζ(s)

2 Reduction to the functionsψandψ1

2.1 Proof of the asymptotics forψ1

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

3.2 The four-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