Nine Introductions in Complex Analysis - Revised Edition

Nine Introductions in Complex Analysis - Revised Edition

1st Edition - September 6, 2007

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  • Author: Sanford Segal
  • eBook ISBN: 9780080550763

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Description

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

Readership

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

    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

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.

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