Nine Introductions in Complex Analysis - Revised Edition

Foreword

A Note on Notational Conventions

Chapter 1: Conformal Mapping and the Riemann Mapping Theorem

1.1 Introduction

1.2 Linear fractional transformations

1.3 Univalent Functions

1.4 Normal Families

1.5 The Riemann Mapping Theorem

Chapter 2: Picard’s Theorems

2.1 Introduction

2.2 The Bloch-Landau Approach

2.3 The Elliptic Modular Function

2.4 Introduction

2.5 The Constants of Bloch and Landau

Chapter 3: An Introduction to Entire Functions

3.1 Growth, Order, and Zeros

3.2 Growth, Coefficients, and Type

3.3 The Phragmén-Lindelöf Indicator

3.4 Composition of entire functions

Chapter 4: Introduction to Meromorphic Functions

4.1 Nevanlinna’s Characteristic and its Elementary Properties

4.2 Nevanlinna’s Second Fundamental Theorem

4.3 Nevanlinna’s Second Fundamental Theorem: Some Applications

Chapter 5: Asymptotic Values

5.1 Julia’s Theorem

5.2 The Denjoy-Carleman-Ahlfors Theorem

Chapter 6: Natural Boundaries

6.1 Natural Boundaries—Some Examples

6.2 The Hadamard Gap Theorem and Over-convergence

6.3 The Hadamard Multiplication Theorem

6.4 The Fabry Gap Theorem

6.5 The Pólya-Carlson Theorem

Chapter 7: The Bieberbach Conjecture

7.1 Elementary Area and Distortion Theorems

7.2 Some Coefficient Theorems

Chapter 8: Elliptic Functions

8.1 Elementary properties

8.2 Weierstrass’ -function

8.3 Weierstrass’ ζ- and σ-functions

8.4 Jacobi’s Elliptic Functions

8.5 Theta Functions

8.6 Modular functions

Chapter 9: Introduction to the Riemann Zeta-Function

9.1 Prime Numbers and ζ(s)

9.2 Ordinary Dirichlet Series

9.3 The Functional Equation, the Prime Number Theorem, and De La Vallée-Poussin’s Estimate

9.4 The Riemann Hypothesis

Appendix

1 The Area Theorem

2 The Borel-Carathéodory Lemma

3 The Schwarz Reflection Principle

4 A Special Case of the Osgood-Carathéodory Theorem

5 Farey Series

6 The Hadamard Three Circles Theorem

7 The Poisson Integral Formula

8 Bernoulli Numbers

9 The Poisson Summation Formula

10 The Fourier Integral Theorem

11 Carathéodory Convergence

Bibliography

Index