Function Theory of One Complex Variable

by Greene, Robert Everist; Krantz, Steven G.

• ISBN13: 9780821839621
• ISBN10: 0821839624
• Edition: 3rd
• Format: Hardcover
• Copyright: 2006-04-30
• Publisher: Amer Mathematical Society

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