Nine Introductions in Complex Analysis - Revised Edition
- 1st Edition, Volume 208 - September 6, 2007
- Imprint: Elsevier Science
- Author: Sanford L. Segal
- Language: English
- Hardback ISBN:9 7 8 - 0 - 4 4 4 - 5 1 8 3 1 - 6
- Paperback ISBN:9 7 8 - 0 - 4 4 4 - 6 0 2 8 3 - 1
- eBook ISBN:9 7 8 - 0 - 0 8 - 0 5 5 0 7 6 - 3
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… Read more
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Request a sales quoteThe 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.
- 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
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
- Edition: 1
- Volume: 208
- Published: September 6, 2007
- No. of pages (eBook): 500
- Imprint: Elsevier Science
- Language: English
- Hardback ISBN: 9780444518316
- Paperback ISBN: 9780444602831
- eBook ISBN: 9780080550763
SS
Sanford L. Segal
Affiliations and expertise
University of Rochester, NY, U.S.A.Read Nine Introductions in Complex Analysis - Revised Edition on ScienceDirect