Verfügbarkeit: Complex analysis

Complex analysis:

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Bibliographische Detailangaben
1. Verfasser:	Lang, Serge 1927-2005 (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	New York [u.a.] Springer 1999
Ausgabe:	4. ed.
Schriftenreihe:	Graduate texts in mathematics 103
Schlagworte:	Análisis matemático Funciones de variable compleja Funktionalanalysis Funktionentheorie Einführung Aufgabensammlung
Online-Zugang:	Inhaltsverzeichnis Klappentext
Beschreibung:	XIV, 485 S. Ill., graph. Darst.
ISBN:	0387985921

Internformat

MARC


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Datensatz im Suchindex

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adam_text	Contents Foreword v Prerequisites ix PART ONE Basic Theory 1 CHAPTER I Complex Numbers and Functions 3 §1. Definition 3 §2. Polar Form 8 §3. Complex Valued Functions 12 §4. Limits and Compact Sets 17 Compact Sets 21 §5. Complex Differentiability 27 §6. The Cauchy-Riemann Equations 31 §7. Angles Under Holomorphic Maps 33 CHAPTER II Power Series 37 §1. Formal Power Series 37 §2. Convergent Power Series 47 §3. Relations Between Formal and Convergent Series 60 Sums and Products 60 Quotients 64 Composition of Series 66 §4. Analytic Functions 68 §5. Differentiation of Power Series 72 xi Xii CONTENTS §6. The Inverse and Open Mapping Theorems 76 §7. The Local Maximum Modulus Principle 83 CHAPTER III Cauchy s Theorem, First Part 86 §1. Holomorphic Functions on Connected Sets 86 Appendix: Connectedness 92 §2. Integrals Over Paths 94 §3. Local Primitive for a Holomorphic Function 104 §4. Another Description of the Integral Along a Path 110 §5. The Homotopy Form of Cauchy s Theorem 115 §6. Existence of Global Primitives. Definition of the Logarithm 119 §7. The Local Cauchy Formula 125 CHAPTER IV Winding Numbers and Cauchy s Theorem 133 §1. The Winding Number 134 §2. The Global Cauchy Theorem 138 Dixon s Proof of Theorem 2.5 (Cauchy s Formula) 147 §3. Artin s Proof 149 CHAPTER V Applications of Cauchy s Integral Formula 156 §1. Uniform Limits of Analytic Functions 156 §2. Laurent Series 161 §3. Isolated Singularities 165 Removable Singularities 165 Poles 166 Essential Singularities 168 CHAPTER VI Calculus of Residues 173 §1. The Residue Formula 173 Residues of Differentials 184 §2. Evaluation of Definite Integrals 191 Fourier Transforms 194 Trigonometric Integrals 197 Mellin Transforms 199 CHAPTER VII Conformal Mappings 208 §1. Schwarz Lemma 210 §2. Analytic Automorphisms of the Disc 212 §3. The Upper Half Plane 215 §4. Other Examples 220 §5. Fractional Linear Transformations 231 CONTENTS xiii CHAPTER VIII Harmonic Functions 241 §1. Definition 241 Application: Perpendicularity 246 Application: Flow Lines 248 §2. Examples 252 §3. Basic Properties of Harmonic Functions 259 §4. The Poisson Formula 271 The Poisson Integral as a Convolution 273 §5. Construction of Harmonic Functions 276 §6. Appendix. Differentiating Under the Integral Sign 286 PART TWO Geometric Function Theory 291 CHAPTER IX Schwarz Reflection 293 §1. Schwarz Reflection (by Complex Conjugation) 293 §2. Reflection Across Analytic Arcs 297 §3. Application of Schwarz Reflection 303 CHAPTER X The Riemann Mapping Theorem 306 §1. Statement of the Theorem 306 §2. Compact Sets in Function Spaces 308 §3. Proof of the Riemann Mapping Theorem 311 §4. Behavior at the Boundary 314 CHAPTER XI Analytic Continuation Along Curves 322 §1. Continuation Along a Curve 322 §2. The Dilogarithm 331 §3. Application to Picard s Theorem 335 PART THREE Various Analytic Topics 337 CHAPTER XII Applications of the Maximum Modulus Principle and Jensen s Formula 339 §1. Jensen s Formula 340 §2. The Picard-Borel Theorem 346 §3. Bounds by the Real Part, Borel-Caratheodory Theorem 354 §4. The Use of Three Circles and the Effect of Small Derivatives 356 Hermite Interpolation Formula 358 §5. Entire Functions with Rational Values 360 §6. The Phragmen-Lindelof and Hadamard Theorems 365 xiv CONTENTS CHAPTER XIII Entire and Meromorphic Functions 372 §1. Infinite Products 372 §2. Weierstrass Products 376 §3. Functions of Finite Order 382 §4. Meromorphic Functions, Mittag-Leffler Theorem 387 CHAPTER XIV Elliptic Functions 391 §1. The Liouville Theorems 391 §2. The Weierstrass Function 395 §3. The Addition Theorem 400 §4. The Sigma and Zeta Functions 403 CHAPTER XV The Gamma and Zeta Functions 408 §1. The Differentiation Lemma 409 §2. The Gamma Function 413 Weierstrass Product 413 The Gauss Multiplication Formula (Distribution Relation) 416 The (Other) Gauss Formula 418 The Mellin Transform 420 The Stirling Formula 422 Proof of Stirling s Formula 424 §3. The Lerch Formula 431 §4. Zeta Functions 433 CHAPTER XVI The Prime Number Theorem 440 §1. Basic Analytic Properties of the Zeta Function 441 §2. The Main Lemma and its Application 446 §3. Proof of the Main Lemma 449 Appendix 453 §1. Summation by Parts and Non-Absolute Convergence 453 §2. Difference Equations 457 §3. Analytic Differential Equations 461 §4. Fixed Points of a Fractional Linear Transformation 465 §5. Cauchy s Formula for C Functions 467 §6. Cauchy s Theorem for Locally Integrable Vector Fields 472 Bibliography 479 Index 481 Contents Foreword .......................................... v Prerequisites ....................................... ix PART ONE Basic Theory ...................................... 1 CHAPTER I Complex Numbers and Functions .......................... 3 §1. Definition ........................................ 3 §2. Polar Form ...................................... 8 §3. Complex Valued Functions ............................ 12 §4. Limits and Compact Sets ............................. 17 Compact Sets .................................... 21 §5. Complex Differentiability .............................. 27 §6. The Cauchy—Riemann Equations ........................ 31 §7. Angles Under Holomorphic Maps ........................ 33 CHAPTER II Power Series ....................................... 37 §t. Formal Power Series ................................ 37 §2. Convergent Power Series .............................. 47 §3. Relations Between Formal and Convergent Series ............. 60 Sums and Products ................................ 60 Quotients ...................................... 64 Composition of Series .............................. 66 §4. Analytic Functions .................................. 68 §5. Differentiation of Power Series .......................... 72 ХП CONTENTS §6. The Inverse and Open Mapping Theorems .................. 76 §7. The Local Maximum Modulus Principle ................... 83 CHAPTER 111 Cauchy s Theorem, First Part ............................ 86 §1. Holom orphie Functions on Connected Sets .................. 86 Appendix: Connectedness ............................ 92 §2. Integrals Over Paths ........,....................... 94 §3. Local Primitive for a Holomorphic Function ................. 104 §4. Another Description of the Integral Along a Path ............. 110 §5. The Homotopy Form of Cauchy s Theorem ................. 115 §6. Existence of Global Primitives. Definition of the Logarithm ....... 119 §7. The Local Cauchy Formula ............................ 125 CHAPTER IV Winding Numbers and Cauchy s Theorem .................... 133 §1. The Winding Number ................................ 134 §2. The Global Cauchy Theorem ........................... 138 Dixon s Proof of Theorem 2-5 (Cauchy s Formula) ........... 147 §3. Artin s Proof ...................................... 149 CHAPTER V Applications of Cauchy s Integral Formula ................... . 156 §1. Uniform Limits of Analytic Functions ..................... 156 §2. Laurent Series ..................................... 161 §3. Isolated Singularities ................................ 165 Removable Singularities ............................. 165 Poles ......................................... 166 Essential Singularities .............................. 168 CHAPTER VI î Calculus of Residues .................................. 173 §1. The Residue Formula ................................ 173 ł Residues of Differentials ............................. 184 §2. Evaluation of Definite Integrals ......................... 191 Fourier Transforms ................................ 194 i Trigonometric Integrals ............................. 197 Mellin Transforms ................................ 199 CHAPTER VII Conformai Mappings .................................. 208 §1. Schwarz Lemma ................................... 210 §2. Analytic Automorphisms of the Disc ...................... 212 §3. The Upper Half Plane ............................... 215 §4. Other Examples ................................... 220 §5. Fractional Linear Transformations ....................... 231 CONTENTS xiii CHAPTER VIII , Harmonic Functions ................................... 241 §1. Definition ........................................ 241, Application: Perpendicularity ......................... 246¡ Application: Flow Lines ............................. 248- §2. Examples ........................................ 252> §3. Basic Properties of Harmonic Functions .................... 259 §4. The Poisson Formula ................................ 271 The Poisson Integral as a Convolution .................... 273 §5. Construction of Harmonic Functions ...................... 276 §6. Appendix. Differentiating Under the Integral Sign .............. 286 PART TWO Geometric Function Theory ............................ 291 CHAPTER IX Schwarz Reflection ................................... 293 §1. Schwarz Reflection (by Complex Conjugation) ................ 293 §2. Reflection Across Analytic Arcs ........................, 297 §3. Application of Schwarz Reflection ........................ 303 CHAPTER X The Rlemann Mapping Theorem .......................... 306 §1. Statement of the Theorem ............................. 306 §2. Compact Sets in Function Spaces ........................ 308 §3. Proof of the Riemann Mapping Theorem ,.................. 311 §4. Behavior at the Boundary ............................. 314 CHAPTER XI Analytic Continuation Along Curves ........................ 322 §1. Continuation Along a Curve ........................... 322 §2. The Dilogarithm ................................... 331 §3. Application to Picard s Theorem ......................... 335 PART THREE Various Analytic Topics ............................... 337 CHAPTER XII Applications of the Maximum Modulus Principle and Jensen s Formula 339 §1. Jensen s Formula ................................... 340 §2, The Picard-Borei Theorem ............................ 346 §3. Bounds by the Real Part, Borel-Carathéodory Theorem ......... 354 §4. The Use of Three Circles and the Effect of Small Derivatives ...... 356 Hermite Interpolation Formula ........................ 358 §5. Entire Functions with Rational Values ..................... 360 §6. The Phragmen— Lindelof and Hadamard Theorems ............. 365 í XIV CONTENTS j і ; ţ CHAPTER ХШ Entire and Meromorphic Functions ......................... 372 §1. Infinite Products ................................... 372 §2. Weierstrass Products ................................ 376 §3. Functions of Finite Order ............................. 382 §4. Meromorphic Functions, Mittag- Leffler Theorem ............. 387 CHAPTER XIV Elliptic Functions ..................................... 391 §1. The Liouville Theorems .............................. 391 §2. The Weierstrass Function ............................. 395 §3. The Addition Theorem ............................... 400 §4. The Sigma and Zeta Functions .......................... 403 CHAPTER XV The Gamma and Zeta Functions .......................... 408 §1. The Differentiation Lemma ............................ 409 §2. The Gamma Function ............................... 413 Weierstrass Product ............................... 413 The Gauss Multiplication Formula (Distribution Relation) ....... 416 The (Other) Gauss Formula .......................... 418 The Mellin Transform .............................. 420 The Stirling Formula .............................. 422 Proof of Stirling s Formula ........................... 424 §3. The Lerch Formula ................................. 431 §4. Zeta Functions .................................... 433 CHAPTER XVI The Prime Number Theorem ............................. 440 §1. Basic Analytic Properties of the Zeta Function ............... 441 §2. The Main Lemma and its Application ..................... 446 §3. Proof of the Main Lemma ............................. 449 Appendix .......................................... 453 §1. Summation by Parts and Non-Absolute Convergence ........... 453 §2. Difference Equations ................................ 457 §3. Analytic Differential Equations .......................... 461 §4. Fixed Points of a Fractional Linear Transformation ............ 465 §5. Cauchy s Formula for C00 Functions ...................... 467 §6. Cauchy s Theorem for Locally Integrable Vector Fields .......... 472 §7. More on Cauchy-Riemann ............................. 477 Bibliography ......................................... 479 Index .............................................. 481
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spelling	Lang, Serge 1927-2005 Verfasser (DE-588)119305119 aut Complex analysis Serge Lang 4. ed. New York [u.a.] Springer 1999 XIV, 485 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 103 Análisis matemático Funciones de variable compleja Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content 2\p (DE-588)4143389-0 Aufgabensammlung gnd-content Funktionentheorie (DE-588)4018935-1 s DE-604 Funktionalanalysis (DE-588)4018916-8 s 3\p DE-604 Graduate texts in mathematics 103 (DE-604)BV000000067 103 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008418594&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008418594&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk
spellingShingle	Lang, Serge 1927-2005 Complex analysis Graduate texts in mathematics Análisis matemático Funciones de variable compleja Funktionalanalysis (DE-588)4018916-8 gnd Funktionentheorie (DE-588)4018935-1 gnd
subject_GND	(DE-588)4018916-8 (DE-588)4018935-1 (DE-588)4151278-9 (DE-588)4143389-0
title	Complex analysis
title_auth	Complex analysis
title_exact_search	Complex analysis
title_full	Complex analysis Serge Lang
title_fullStr	Complex analysis Serge Lang
title_full_unstemmed	Complex analysis Serge Lang
title_short	Complex analysis
title_sort	complex analysis
topic	Análisis matemático Funciones de variable compleja Funktionalanalysis (DE-588)4018916-8 gnd Funktionentheorie (DE-588)4018935-1 gnd
topic_facet	Análisis matemático Funciones de variable compleja Funktionalanalysis Funktionentheorie Einführung Aufgabensammlung
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work_keys_str_mv	AT langserge complexanalysis

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