Nine Introductions in Complex Analysis - Revised Edition - 1st Edition - ISBN: 9780444518316, 9780080550763

Nine Introductions in Complex Analysis - Revised Edition, Volume 208

1st Edition

Authors: Sanford Segal
eBook ISBN: 9780080550763
Hardcover ISBN: 9780444518316
Imprint: Elsevier Science
Published Date: 6th September 2007
Page Count: 500
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Table of Contents


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


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




The book addresses many topics not usually in "second course in complex analysis" texts. It also contains multiple proofs of several central results, and it has a minor historical perspective.

Key Features

  • Proof of Bieberbach conjecture (after DeBranges)
  • Material on asymptotic values
  • Material on Natural Boundaries
  • First four chapters are comprehensive introduction to entire and metomorphic functions
  • First chapter (Riemann Mapping Theorem) takes up where "first courses" usually leave off


This book is primarily intended for graduate students in mathematics


No. of pages:
© Elsevier Science 2008
Elsevier Science
eBook ISBN:
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About the Authors

Sanford Segal Author

Affiliations and Expertise

University of Rochester, NY, U.S.A.