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9780821839621

Function Theory of One Complex Variable

by ;
  • ISBN13:

    9780821839621

  • ISBN10:

    0821839624

  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 2006-04-30
  • Publisher: Amer Mathematical Society

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Summary

Complex analysis is one of the most central subjects in mathematics. It is compelling and rich in its own right, but it is also remarkably useful in a wide variety of other mathematical subjects, both pure and applied. This book is different from others in that it treats complex variables as a direct development from multivariable real calculus. As each new idea is introduced, it is related to the corresponding idea from real analysis and calculus. The text is rich with examples andexercises that illustrate this point. The authors have systematically separated the analysis from the topology, as can be seen in their proof of the Cauchy theorem. The book concludes with several chapters on special topics, including full treatments of special functions, the prime number theorem,and the Bergman kernel. The authors also treat $Hp$ spaces and Painleve's theorem on smoothness to the boundary for conformal maps. This book is a text for a first-year graduate course in complex analysis. It is an engaging and modern introduction to the subject, reflecting the authors' expertise both as mathematicians and as expositors.

Table of Contents

Preface to the Third Edition xiii
Preface to the Second Edition xv
Preface to the First Edition xvii
Acknowledgments xix
Chapter 1. Fundamental Concepts 1(28)
§1.1. Elementary Properties of the Complex Numbers
1(2)
§1.2. Further Properties of the Complex Numbers
3(7)
§1.3. Complex Polynomials
10(4)
§1.4. Holomorphic Functions, the Cauchy-Riemann Equations, and Harmonic Functions
14(3)
§1.5. Real and Holomorphic Antiderivatives
17(3)
Exercises
20(9)
Chapter 2. Complex Line Integrals 29(40)
§2.1. Real and Complex Line Integrals
29(5)
§2.2. Complex Differentiability and Conformality
34(6)
§2.3. Antiderivatives Revisited
40(3)
§2.4. The Cauchy Integral Formula and the Cauchy Integral Theorem
43(7)
§2.5. The Cauchy Integral Formula: Some Examples
50(3)
§2.6. An Introduction to the Cauchy Integral Theorem and the Cauchy Integral Formula for More General Curves
53(7)
Exercises
60(9)
Chapter 3. Applications of the Cauchy Integral 69(36)
§3.1. Differentiability Properties of Holomorphic Functions
69(5)
§3.2. Complex Power Series
74(7)
§3.3. The Power Series Expansion for a Holomorphic Function
81(3)
§3.4. The Cauchy Estimates and Liouville's Theorem
84(4)
§3.5. Uniform Limits of Holomorphic Functions
88(2)
§3.6. The Zeros of a Holomorphic Function
90(4)
Exercises
94(11)
Chapter 4. Meromorphic Functions and Residues 105(52)
§4.1. The Behavior of a Holomorphic Function Near an Isolated Singularity
105(4)
§4.2. Expansion around Singular Points
109(4)
§4.3. Existence of Laurent Expansions
113(6)
§4.4. Examples of Laurent Expansions
119(3)
§4.5. The Calculus of Residues
122(6)
§4.6. Applications of the Calculus of Residues to the Calculation of Definite Integrals and Sums
128(9)
§4.7. Meromorphic Functions and Singularities at Infinity
137(8)
Exercises
145(12)
Chapter 5. The Zeros of a Holomorphic Function 157(22)
§5.1. Counting Zeros and Poles
157(5)
§5.2. The Local Geometry of Holomorphic Functions
162(4)
§5.3. Further Results on the Zeros of Holomorphic Functions
166(3)
§5.4. The Maximum Modulus Principle
169(2)
§5.5. The Schwarz Lemma
171(3)
Exercises
174(5)
Chapter 6. Holomorphic Functions as Geometric Mappings 179(28)
§6.1. Biholomorphic Mappings of the Complex Plane to Itself
180(2)
§6.2. Biholomorphic Mappings of the Unit Disc to Itself
182(2)
§6.3. Linear Fractional Transformations
184(5)
§6.4. The Riemann Mapping Theorem: Statement and Idea of Proof
189(3)
§6.5. Normal Families
192(4)
§6.6. Holomorphically Simply Connected Domains
196(2)
§6.7. The Proof of the Analytic Form of the Riemann Mapping Theorem
198(4)
Exercises
202(5)
Chapter 7. Harmonic Functions 207(48)
§7.1. Basic Properties of Harmonic Functions
208(2)
§7.2. The Maximum Principle and the Mean Value Property
210(2)
§7.3. The Poisson Integral Formula
212(6)
§7.4. Regularity of Harmonic Functions
218(2)
§7.5. The Schwarz Reflection Principle
220(4)
§7.6. Harnack's Principle
224(2)
§7.7. The Dirichlet Problem and Subharmonic Functions
226(10)
§7.8. The Perron Method and the Solution of the Dirichlet Problem
236(4)
§7.9. Conformal Mappings of Annuli
240(3)
Exercises
243(12)
Chapter 8. Infinite Series and Products 255(24)
§8.1. Basic Concepts Concerning Infinite Sums and Products
255(8)
§8.2. The Weierstrass Factorization Theorem
263(3)
§8.3. The Theorems of Weierstrass and Mittag-Leffler: Interpolation Problems
266(8)
Exercises
274(5)
Chapter 9. Applications of Infinite Sums and Products 279(20)
§9.1. Jensen's Formula and an Introduction to Blaschke Products
279(6)
§9.2. The Hadamard Gap Theorem
285(3)
§9.3. Entire Functions of Finite Order
288(8)
Exercises
296(3)
Chapter 10. Analytic Continuation 299(36)
§10.1. Definition of an Analytic Function Element
299(5)
§10.2. Analytic Continuation along a Curve
304(3)
§10.3. The Monodromy Theorem
307(3)
§10.4. The Idea of a Riemann Surface
310(4)
§10.5. The Elliptic Modular Function and Picard's Theorem
314(9)
§10.6. Elliptic Functions
323(7)
Exercises
330(5)
Chapter 11. Topology 335(28)
§11.1. Multiply Connected Domains
335(3)
§11.2. The Cauchy Integral Formula for Multiply Connected Domains
338(5)
§11.3. Holomorphic Simple Connectivity and Topological Simple Connectivity
343(1)
§11.4. Simple Connectivity and Connectedness of the Complement
344(5)
§11.5. Multiply Connected Domains Revisited
349(3)
Exercises
352(11)
Chapter 12. Rational Approximation Theory 363(22)
§12.1. Runge's Theorem
363(6)
§12.2. Mergelyan's Theorem
369(9)
§12.3. Some Remarks about Analytic Capacity
378(3)
Exercises
381(4)
Chapter 13. Special Classes of Holomorphic Functions 385(30)
§13.1. Schlicht Functions and the Bieberbach Conjecture
386(6)
§13.2. Continuity to the Boundary of Conformal Mappings
392(9)
§13.3. Hardy Spaces
401(5)
§13.4. Boundary Behavior of Functions in Hardy Classes [An Optional Section for Those Who Know Elementary Measure Theory]
406(6)
Exercises
412(3)
Chapter 14. Hilbert Spaces of Holomorphic Functions, the Bergman Kernel, and Biholomorphic Mappings 415(34)
§14.1. The Geometry of Hilbert Space
415(11)
§14.2. Orthonormal Systems in Hilbert Space
426(5)
§14.3. The Bergman Kernel
431(7)
§14.4. Bell's Condition R
438(5)
§14.5. Smoothness to the Boundary of Conformal Mappings
443(3)
Exercises
446(3)
Chapter 15. Special Functions 449(22)
§15.1. The Gamma and Beta Functions
449(8)
§15.2. The Riemann Zeta Function
457(10)
Exercises
467(4)
Chapter 16. The Prime Number Theorem 471(16)
§16.0. Introduction
471(2)
§16.1. Complex Analysis and the Prime Number Theorem
473(5)
§16.2. Precise Connections to Complex Analysis
478(5)
§16.3. Proof of the Integral Theorem
483(2)
Exercises
485(2)
APPENDIX A: Real Analysis 487(6)
APPENDIX B: The Statement and Proof of Goursat's Theorem 493(4)
References 497(4)
Index 501

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