Function Theory of One Complex Variable: Third Edition (Graduate Studies in Mathematics, 40)

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 and exercises 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 $H^p$ 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. Cover 1 Title: Function Theory ofOne Complex Variable, Third Edition 2 Editorial Board & Copyright 3 2006 by the American Mathematical Society 3 ISBN 0-8218-3962-4 3 Dedicated To Paige and Randi 4 Contents 6 Preface to the Third Edition 12 Preface to the Second Edition 14 Preface to the First Edition 16 Acknowledgments 18 Chapter 1: Fundamental Concepts 20 1.1. Elementary Properties of the Complex Numbers 20 1.2. Further Properties of the Complex Numbers 22 1.3. Complex Polynomials 29 1.4. Holomorphic Functions, the Cauchy-Riemann Equations, and Harmonic Functions 33 1.5. Real and Holomorphic Antiderivatives 36 Exercises 39 Chapter 2: Complex Line Integrals 48 2.1. Real and Complex Line Integrals 48 2.2. Complex Differentiability and Conformality 53 2.3. Antiderivatives Revisited 59 2.4. The Cauchy Integral Formula and the Cauchy Integral Theorem 62 2.5. The Cauchy Integral Formula: Some Examples 69 2.6. An Introduction to the Cauchy Integral Theorem and the Cauchy Integral Formula for More General Curves 72 Exercises 79 Chapter 3: Applications of the Cauchy Integral 88 3.1. Differentiability Properties of Holomorphic Functions 88 3.2. Complex Power Series 93 3.3. The Power Series Expansion for a Holomorphic Function 100 3.4. The Cauchy Estimates and Liouville's Theorem 103 3.5. Uniform Limits of Holomorphic Functions 107 3.6. The Zeros of a Holomorphic Function 109 Exercises 113 Chapter 4: Meromorphic Functions and Residues 124 4.1. The Behavior of a Holomorphic Function Near an Isolated Singularity 124 4.2. Expansion around Singular Points 128 4.3. Existence of Laurent Expansions 132 4.4. Examples of Laurent Expansions 138 4.5. The Calculus of Residues 141 4.6. Applications of the Calculus of Residues to the Calculation of Definite Integrals and Sums 147 4.7. Meromorphic Functions and Singularities at Infinity 156 Exercises 164 Chapter 5: The Zeros of a Holomorphic Function 176 5.1. Counting Zeros and Poles 176 5.2. The Local Geometry of Holomorphic Functions 181 5.3. Further Results on the Zeros of Holomorphic Functions 185 5.4. The Maximum Modulus Principle 188 5.5. The Schwarz Lemma 190 Exercises 193 Chapter 6: Holomorphic Functions as Geometric Mappings 198 6.1. Biholomorphic Mappings of the Complex Plane to Itself 199 6.2. Biholomorphic Mappings of the Unit Disc to Itself 201 6.3. Linear Fractional 'fransformations 203 6.4. The Riemann Mapping Theorem: Statement and Idea of Proof 208 6.5. Normal Families 211 6.6. Holomorphically Simply Connected Domains 215 6.7. The Proof of the Analytic Form of the Riemann Mapping Theorem 217 Exercises 221 Chapter 7: Harmonic Functions 226 7.1 Basic Properties of Harmonic Functions 227 7.2. The Maximum Principle and the Mean Value Property 229 7.3. The Poisson Integral Formula 231 7.4. Regularity of Harmonic Functions 237 7.5. The Schwarz Reflection Principle 239 7.6. Harnack's Principle 243 7.7. The Dirichlet Problem and Subharmonic Functions 245 7.8. The Perron Method and the Solution of the Dirichlet Problem 255 7.9. Conformal Mappings of Annuli 259 Exercises 262 Chapter 8: Infinite Series and Products 274 8.1. Basic Concepts Concerning Infinite Sums and Products 274 8.2. The Weierstrass Factorization Theorem 282 8.3. The Theorems of Weierstrass and Mittag-Leffler: Interpolation Problems 285 Exercises 293 Chapter 9: Applications of Infinite Sums and Products 298 9.1. Jensen's Formula and an Introduction to Blaschke Products 298 9.2. The Hadamard Gap Theorem 304 9.3. Entire Functions of Finite Order 307 Exercises 315 Chapter 10: Analytic Continuation 318 10.1. Definition of an Analytic Function Element 318 10.2. Analytic Continuation along a Curve 323 10.3. The Monodromy Theorem 326 10.4. The Idea of a Riemann Surface 329 10.5. The Elliptic Modular Function and Picard's Theorem 333 Introductory Remarks 333 The Action of the Modular Group 335 The Modular Function 339 The Little Picard Theorem 340 10.6. Elliptic Functions 342 Exercises 349 Chapter 11: Topology 354 11.1. Multiply Connected Domains 354 11.2. The Cauchy Integral Formula for Multiply Connected Domains 357 11.3. Holomorphic Simple Connectivity and Topological Simple Connectivity 362 11.4. Simple Connectivity and Connectedness of the Complement 363 11.5. Multiply Connected Domains Revisited 368 Exercises 371 Chapter 12: Rational Approximation Theory 382 12.1. Runge's Theorem 382 12.2. Mergelyan's Theorem 388 12.3. Some Remarks about Analytic Capacity 397 Exercises 400 Chapter 13: Special Classes of Holomorphic Functions 404 13.1. Schlicht FUnctions and the Bieberbach Conjecture 405 13.2. Continuity to the Boundary of Conformal Mappings 411 13.4. Boundary Behavior of Functions in Hardy Classes 425 Exercises 431 Chapter 14: Hilbert Spaces of Holomorphic Functions, the Bergman Kernel, and Biholomorphic Mappings 434 14.1. The Geometry of Hilbert Space 434 14.2. Orthonormal Systems in Hilbert Space 445 14.3. The Bergman Kernel 450 14.4. Bell's Condition R 457 14.5. Smoothness to the Boundary of Conformal Mappings 462 Exercises 465 Chapter 15: Special Functions 468 15.1. The Gamma and Beta Functions 468 15.2. The Riemann Zeta Function 476 Exercises 486 Chapter 16: The Prime Number Theorem 490 16.0. Introduction 490 16.1. Complex Analysis and the Prime Number Theorem 492 16.2. Precise Connections to Complex Analysis 497 16.3. Proof of the Integral Theorem 502 Exercises 504 APPENDIX A: Real Analysis 506 Commuting a Limit with an Integral [RUDl, p. 51, Theorem7.16] 506 Commuting a Limit with a Sum [RUDl, p. 149 506 Commuting a Sum with an Integral [RUD1, p. 152, Corollary to Theorem 7.16] 507 Differentiation under the Integral Sign [RUD1, p. 151, Theorem 7.16] 507 Uniformly Cauchy Sequences [RUD1, p. 147, Theorem 7.8]: 507 The Weierstrass M-Test [RUD1, p. 148, Theorem 7.10]: 507 Existence of Piecewise Linear Paths 508 Convergence of C1 Functions [RUD1, p. 152, Theorem 7.17]: 508 The Ascoli-Arzela Theorem [RUD1, p. 158, Theorem 7.25]: 508 The Tietze Extension Theorem [RUD2, p. 385, Theorem 20.4]: 508 Change of Variables Theorem in Two Variables [RUD1, p. 252, Theorem 10.9: 509 Jensen's Inequality (from measure theory) [RUD2, p. 61, Theorem 3.3] 509 Green's Theorem [RUD1, p. 282, Theorem 10.45]: 509 Fubini's Theorem [RUD2, Theorem 7.8]: 510 Switching the Order of Summation [RUDl, Theorem 8.3]: 510 A Lebesgue Monotone Convergence Theorem [RUD2, Theorem 10.28]: 511 APPENDIX B: The Statement and Proof of Goursat 's Theorem 512 References 516 Index 520 Titles in This Series 524 "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 and exercises 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 H[superscript p] 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 a modern introduction to the subject, reflecting the authors' expertise both as mathematicians and as expositors."--BOOK JACKET