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Bibliographic Details
Main Authors:	Greene, Robert Everist 1943- (Author), Krantz, Steven G. 1951- (Author)
Format:	Book
Language:	English
Published:	Providence, RI American Mathematical Society 2002
Edition:	2. ed.
Series:	Graduate studies in mathematics 40
Subjects:	ANÁLISE MATEMÁTICA FUNÇÕES DE VÁRIAS VARIÁVEIS COMPLEXAS Fonctions d'une variable complexe Functions of complex variables Funktion > Mathematik Komplexe Variable Funktionentheorie
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009890284&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Item Description:	Includes bibliographical references and index
Physical Description:	xix, 502 p. ill. : 26 cm
ISBN:	082182905X

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Record in the Search Index

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adam_text	Contents Preface to the Second Edition xv Preface to the First Edition xvii Acknowledgments xix Chapter 1. Fundamental Concepts 1 §1.1. Elementary Properties of the Complex Numbers 1 §1.2. Further Properties of the Complex Numbers 3 §1.3. Complex Polynomials 10 §1.4. Holomorphic Functions, the Cauchy Riemann Equations, and Harmonic Functions 14 §1.5. Real and Holomorphic Antiderivatives 17 Exercises 20 Chapter 2. Complex Line Integrals 29 §2.1. Real and Complex Line Integrals 29 §2.2. Complex Differentiability and Conformality 34 §2.3. Antiderivatives Revisited 40 §2.4. The Cauchy Integral Formula and the Cauchy Integral Theorem 43 §2.5. The Cauchy Integral Formula: Some Examples 50 ix x Contents §2.6. An Introduction to the Cauchy Integral Theorem and the Cauchy Integral Formula for More General Curves 53 Exercises 60 Chapter 3. Applications of the Cauchy Integral 69 §3.1. Differentiability Properties of Holomorphic Functions 69 §3.2. Complex Power Series 74 §3.3. The Power Series Expansion for a Holomorphic Function 81 §3.4. The Cauchy Estimates and Liouville s Theorem 85 §3.5. Uniform Limits of Holomorphic Functions 88 §3.6. The Zeros of a Holomorphic Function 90 Exercises 94 Chapter 4. Meromorphic Functions and Residues 105 §4.1. The Behavior of a Holomorphic Function Near an Isolated Singularity 105 §4.2. Expansion Around Singular Points 109 §4.3. Existence of Laurent Expansions 113 §4.4. Examples of Laurent Expansions 119 §4.5. The Calculus of Residues 122 §4.6. Applications of the Calculus of Residues to the Calculation of Definite Integrals and Sums 128 §4.7. Meromorphic Functions and Singularities at Infinity 137 Exercises 145 Chapter 5. The Zeros of a Holomorphic Function 157 §5.1. Counting Zeros and Poles 157 §5.2. The Local Geometry of Holomorphic Functions 162 §5.3. Further Results on the Zeros of Holomorphic Functions 166 §5.4. The Maximum Modulus Principle 169 §5.5. The Schwarz Lemma 171 Exercises 174 Contents xi Chapter 6. Holomorphic Functions as Geometric Mappings 179 §6.1. Biholomorphic Mappings of the Complex Plane to Itself 180 §6.2. Biholomorphic Mappings of the Unit Disc to Itself 182 §6.3. Linear Fractional Transformations 184 §6.4. The Riemann Mapping Theorem: Statement and Idea of Proof 189 §6.5. Normal Families 192 §6.6. Holomorphically Simply Connected Domains 196 §6.7. The Proof of the Analytic Form of the Riemann Mapping Theorem 198 Exercises 202 Chapter 7. Harmonic Functions 207 §7.1. Basic Properties of Harmonic Functions 208 §7.2. The Maximum Principle and the Mean Value Property 210 §7.3. The Poisson Integral Formula 212 §7.4. Regularity of Harmonic Functions 218 §7.5. The Schwarz Reflection Principle 220 §7.6. Harnack s Principle 224 §7.7. The Dirichlet Problem and Subharmonic Functions 227 §7.8. The Perron Method and the Solution of the Dirichlet Problem 236 §7.9. Conformal Mappings of Annuli 240 Exercises 243 Chapter 8. Infinite Series and Products 255 §8.1. Basic Concepts Concerning Infinite Sums and Products 255 §8.2. The Weierstrass Factorization Theorem 263 §8.3. The Theorems of Weierstrass and Mittag Leffler: Interpolation Problems 266 Exercises 275 Chapter 9. Applications of Infinite Sums and Products 279 xii Contents §9.1. Jensen s Formula and an Introduction to Blaschke Products 279 §9.2. The Hadamard Gap Theorem 285 §9.3. Entire Functions of Finite Order 288 Exercises 296 Chapter 10. Analytic Continuation 299 §10.1. Definition of an Analytic Function Element 299 §10.2. Analytic Continuation Along a Curve 305 §10.3. The Monodromy Theorem 307 §10.4. The Idea of a Riemann Surface 310 §10.5. The Elliptic Modular Function and Picard s Theorem 314 §10.6. Elliptic Functions 323 Exercises 330 Chapter 11. Topology 335 §11.1. Multiply Connected Domains 335 §11.2. The Cauchy Integral Formula for Multiply Connected Domains 338 §11.3. Holomorphic Simple Connectivity and Topological Simple Connectivity 343 §11.4. Simple Connectivity and Connectedness of the Complement 344 §11.5. Multiply Connected Domains Revisited 349 Exercises 352 Chapter 12. Rational Approximation Theory 361 §12.1. Runge s Theorem 361 §12.2. Mergelyan s Theorem 367 §12.3. Some Remarks about Analytic Capacity 376 Exercises 379 Chapter 13. Special Classes of Holomorphic Functions 383 §13.1. Schlicht Functions and the Bieberbach Conjecture 384 Contents xiii §13.2. Continuity to the Boundary of Conformal Mappings 390 §13.3. Hardy Spaces 399 §13.4. Boundary Behavior of Functions in Hardy Classes [An Optional Section for Those Who Know Elementary Measure Theory] 404 Exercises 410 Chapter 14. Hilbert Spaces of Holomorphic Functions, the Bergman Kernel, and Biholomorphic Mappings 413 §14.1. The Geometry of Hilbert Space 413 §14.2. Orthonormal Systems in Hilbert Space 424 §14.3. The Bergman Kernel 429 §14.4. Bell s Condition R 435 §14.5. Smoothness to the Boundary of Conformal Mappings 441 Exercises 444 Chapter 15. Special Functions 447 §15.1. The Gamma and Beta Functions 447 §15.2. The Riemann Zeta Function 455 Exercises 465 Chapter 16. The Prime Number Theorem 469 §16.0. Introduction 469 §16.1. Complex Analysis and the Prime Number Theorem 471 §16.2. Precise Connections to Complex Analysis 476 §16.3. Proof of the Integral Theorem 481 Exercises 482 APPENDIX A: Real Analysis 485 APPENDIX B: The Statement and Proof of Goursafs Theorem 491 References 495 Index 499
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physical	xix, 502 p. ill. : 26 cm
publishDate	2002
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publisher	American Mathematical Society
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series	Graduate studies in mathematics
series2	Graduate studies in mathematics
spellingShingle	Greene, Robert Everist 1943- Krantz, Steven G. 1951- Function theory of one complex variable Graduate studies in mathematics ANÁLISE MATEMÁTICA larpcal FUNÇÕES DE VÁRIAS VARIÁVEIS COMPLEXAS larpcal Fonctions d'une variable complexe Functions of complex variables Funktion Mathematik (DE-588)4071510-3 gnd Komplexe Variable (DE-588)4164905-9 gnd Funktionentheorie (DE-588)4018935-1 gnd
subject_GND	(DE-588)4071510-3 (DE-588)4164905-9 (DE-588)4018935-1
title	Function theory of one complex variable
title_auth	Function theory of one complex variable
title_exact_search	Function theory of one complex variable
title_full	Function theory of one complex variable Robert E. Greene and Steven G. Krantz
title_fullStr	Function theory of one complex variable Robert E. Greene and Steven G. Krantz
title_full_unstemmed	Function theory of one complex variable Robert E. Greene and Steven G. Krantz
title_short	Function theory of one complex variable
title_sort	function theory of one complex variable
topic	ANÁLISE MATEMÁTICA larpcal FUNÇÕES DE VÁRIAS VARIÁVEIS COMPLEXAS larpcal Fonctions d'une variable complexe Functions of complex variables Funktion Mathematik (DE-588)4071510-3 gnd Komplexe Variable (DE-588)4164905-9 gnd Funktionentheorie (DE-588)4018935-1 gnd
topic_facet	ANÁLISE MATEMÁTICA FUNÇÕES DE VÁRIAS VARIÁVEIS COMPLEXAS Fonctions d'une variable complexe Functions of complex variables Funktion Mathematik Komplexe Variable Funktionentheorie
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009890284&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV009739289
work_keys_str_mv	AT greeneroberteverist functiontheoryofonecomplexvariable AT krantzsteveng functiontheoryofonecomplexvariable

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