Nine Introductions in Complex Analysis - Revised Edition

Nine Introductions in Complex Analysis - Revised Edition

1st Edition - September 6, 2007

Write a review

  • Author: Sanford Segal
  • eBook ISBN: 9780080550763

Purchase options

Purchase options
DRM-free (EPub, PDF, Mobi)
Sales tax will be calculated at check-out

Institutional Subscription

Free Global Shipping
No minimum order


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

Table of Contents

  • 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


    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



Product details

  • No. of pages: 500
  • Language: English
  • Copyright: © Elsevier Science 2007
  • Published: September 6, 2007
  • Imprint: Elsevier Science
  • eBook ISBN: 9780080550763

About the Author

Sanford Segal

Affiliations and Expertise

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

Ratings and Reviews

Write a review

There are currently no reviews for "Nine Introductions in Complex Analysis - Revised Edition"