Nine Introductions in Complex Analysis - Revised Edition

Nine Introductions in Complex Analysis - Revised Edition, 1st Edition

Nine Introductions in Complex Analysis - Revised Edition, 1st Edition,Sanford Segal,ISBN9780444518316


Elsevier Science




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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


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.


This book is primarily intended for graduate students in mathematics

Sanford Segal

Affiliations and Expertise

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

Nine Introductions in Complex Analysis - Revised Edition, 1st Edition


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’ ?- and s-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 ?(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



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