Complex analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York u.a.
Springer
1993
|
| Ausgabe: | 3. ed. |
| Schriftenreihe: | Graduate texts in mathematics
103 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 458 S. graph. Darst. |
| ISBN: | 0387978860 3540978860 |
Internformat
MARC
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| 245 | 1 | 0 | |a Complex analysis |c Serge Lang |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a New York u.a. |b Springer |c 1993 | |
| 300 | |a XIV, 458 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Graduate texts in mathematics |v 103 | |
| 650 | 4 | |a Analyse mathématique | |
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| 650 | 7 | |a Fonctions d'une variable complexe |2 ram | |
| 650 | 4 | |a Functions of complex variables | |
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Datensatz im Suchindex
| _version_ | 1804120388921196544 |
|---|---|
| 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
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 116
§6. Existence of Global Primitives. Definition of the Logarithm 119
§7. The Local Cauchy Formula 126
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 218
§5. Fractional Linear Transformations 227
CONTENTS xiii
CHAPTER VIII
Harmonic Functions 237
§1. Definition 237
Application: Perpendicularity 241
Application: Flow Lines 242
§2. Examples 247
§3. Basic Properties of Harmonic Functions 254
§4. The Poisson Formula 264
§5. Construction of Harmonic Functions 267
PART TWO
Geometric Function Theory 277
CHAPTER IX
Schwarz Reflection 279
§1. Schwarz Reflection (by Complex Conjugation) 279
§2. Reflection Across Analytic Arcs 283
§3. Application of Schwarz Reflection 287
CHAPTER X
The Riemann Mapping Theorem 291
§1. Statement of the Theorem 291
§2. Compact Sets in Function Spaces 293
§3. Proof of the Riemann Mapping Theorem 296
§4. Behavior at the Boundary 299
CHAPTER XI
Analytic Continuation Along Curves 307
§1. Continuation Along a Curve 307
§2. The Dilogarithm 315
§3. Application to Picard s Theorem 319
PART THREE
Various Analytic Topics 321
CHAPTER XII
Applications of the Maximum Modulus Principle and Jensen s Formula 323
§1. Jensen s Formula 324
§2. The Picard Borel Theorem 330
§3. Bounds by the Real Part, Borel Caratheodory Theorem 338
§4. The Use of Three Circles and the Effect of Small Derivatives 340
Hermite Interpolation Formula 342
§5. Entire Functions with Rational Values 344
§6. The Phragmen Lindelof and Hadamard Theorems 349
xiv CONTENTS
CHAPTER XIII
Entire and Meromorphic Functions 356
§1. Infinite Products 356
§2. Weierstrass Products 360
§3. Functions of Finite Order 366
§4. Meromorphic Functions, Mittag Leffler Theorem 371
CHAPTER XIV
Elliptic Functions 374
§1. The Liouville Theorems 374
§2. The Weierstrass Function 378
§3. The Addition Theorem 383
§4. The Sigma and Zeta Functions 386
CHAPTER XV
The Gamma and Zeta Functions 391
§1. The Differentiation Lemma 392
§2. The Gamma Function 396
Weierstrass Product 396
The Mellin Transform 401
Proof of Stirling s Formula 406
§3. The Lerch Formula 412
§4. Zeta Functions 415
CHAPTER XVI
The Prime Number Theorem 422
§1. Basic Analytic Properties of the Zeta Function 423
§2. The Main Lemma and its Application 428
§3. Proof of the Main Lemma 431
Appendix 435
§1. Summation by Parts and Non Absolute Convergence 435
§2. Difference Equations 437
§3. Analytic Differential Equations 441
§4. Fixed Points of a Fractional Linear Transformation 445
§5. Cauchy s Formula for C°° Functions 447
Bibliography 454
Index 455
|
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| classification_tum | MAT 300f |
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| dewey-full | 515/.9 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.9 |
| dewey-search | 515/.9 |
| dewey-sort | 3515 19 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 3. ed. |
| format | Book |
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| id | DE-604.BV006159144 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:41:18Z |
| institution | BVB |
| isbn | 0387978860 3540978860 |
| language | English |
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| physical | XIV, 458 S. graph. Darst. |
| publishDate | 1993 |
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| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Lang, Serge 1927-2005 Verfasser (DE-588)119305119 aut Complex analysis Serge Lang 3. ed. New York u.a. Springer 1993 XIV, 458 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 103 Analyse mathématique Analyse mathématique ram Fonctions d'une variable complexe Fonctions d'une variable complexe ram Functions of complex variables Mathematical analysis Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Funktionentheorie (DE-588)4018935-1 s DE-604 Funktionalanalysis (DE-588)4018916-8 s 1\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=003896413&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lang, Serge 1927-2005 Complex analysis Graduate texts in mathematics Analyse mathématique Analyse mathématique ram Fonctions d'une variable complexe Fonctions d'une variable complexe ram Functions of complex variables Mathematical analysis 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 |
| 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 | Analyse mathématique Analyse mathématique ram Fonctions d'une variable complexe Fonctions d'une variable complexe ram Functions of complex variables Mathematical analysis Funktionalanalysis (DE-588)4018916-8 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Analyse mathématique Fonctions d'une variable complexe Functions of complex variables Mathematical analysis Funktionalanalysis Funktionentheorie Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003896413&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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