Complex analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
1985
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Graduate texts in mathematics
103 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 367 S. graph. Darst. |
| ISBN: | 0387960856 3540960856 |
Internformat
MARC
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| 245 | 1 | 0 | |a Complex analysis |c Serge Lang |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1985 | |
| 300 | |a XIV, 367 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 103 | |
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Datensatz im Suchindex
| _version_ | 1804116643638411264 |
|---|---|
| adam_text | Contents
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 28
§6. The Cauchy Riemann Equations 32
§7. Angles Under Holomorphic Maps 34
CHAPTER II
Power Series 38
§1. Formal Power Series 38
§2. Convergent Power Series 49
§3. Relations Between Formal and Convergent Series 62
Sums and Products 62
Quotients 66
Composition of Series 67
§4. Analytic Functions 69
§5. The Inverse and Open Mapping Theorems 72
§6. The Local Maximum Modulus Principle 79
§7. Differentiation of Power Series 82
Xii CONTENTS
CHAPTER III
Cauchy s Theorem, First Part 87
§1. Holomorphic Functions on Connected Sets 87
Appendix: Connectedness 93
§2. Integrals Over Paths 94
§3. Local Primitive for a Holomorphic Function 103
§4. Another Description of the Integral Along a Path 109
§5. The Homotopy Form of Cauchy s Theorem 115
§6. Existence of Global Primitives. Definition of the Logarithm 118
CHAPTER IV
Cauchy s Theorem, Second Part 123
§1. The Winding Number 124
§2. Statement of Cauchy s Theorem 128
§3. Artin s Proof 137
CHAPTER V
Applications of Cauchy s Integral Formula 144
§1. Cauchy s Integral Formula on a Disc 144
§2. Laurent Series 151
§3. Isolated Singularities 155
Removable Singularities 155
Poles 156
Essential Singularities 158
§4. Dixon s Proof of Cauchy s Theorem 162
CHAPTER VI
Calculus of Residues 165
§1. The Residue Formula 165
§2. Evaluation of Definite Integrals 180
Fourier Transforms 182
Trigonometric Integrals 185
Mellin Transforms 187
CHAPTER VII
Conformal Mappings 196
§1. Schwarz Lemma 198
§2. Analytic Automorphisms of the Disc 200
§3. The Upper Half Plane 203
§4. Other Examples 206
§5. Fractional Linear Transformations 215
CONTENTS xiii
CHAPTER VIII
Harmonic Functions 224
§1. Definition 224
Application: Perpendicularity 228
Application: Flow Lines 230
§2. Examples 234
§3. Basic Properties of Harmonic Functions 241
§4. Construction of Harmonic Functions 244
§5. The Poisson Representation 249
PART TWO
Various Analytic Topics 253
CHAPTER IX
Applications of the Maximum Modulus Principle 255
§1. The Effect of Zeros, Jensen Schwarz Lemma 255
§2. The Effect of Small Derivatives 260
Hermite Interpolation Formula 261
§3. Entire Functions with Rational Values 262
§4. The Phragmen Lindelof and Hadamard Theorems 268
§5. Bounds by the Real Part, Borel Carath odory Theorem 273
CHAPTER X
Entire and Meromorphic Functions 276
§1. Infinite Products 276
§2. Weierstrass Products 280
§3. Functions of Finite Order 286
§4. Meromorphic Functions, Mittag Leffler Theorem 290
CHAPTER XI
Elliptic Functions 292
§1. The Liouville Theorems 292
§2. The Weierstrass Function 295
§3. The Addition Theorem 299
§4. The Sigma and Zeta Functions 302
CHAPTER XII
Differentiating Under an Integral 307
§1. The Differentiation Lemma 308
§2. The Gamma Function 311
Proof of Stirling s Formula 316
Xiv CONTENTS
CHAPTER XIII
Analytic Continuation 324
§1. Schwarz Reflection 324
§2. Continuation Along a Path 330
CHAPTER XIV
The Riemann Mapping Theorem 340
§1. Statement and Application to Picard s Theorem 340
§2. Compact Sets in Function Spaces 344
§3. Proof of the Riemann Mapping Theorem 347
§4. Behavior at the Boundary 351
Appendix
Cauchy s Formula for C00 Functions 359
Index 365
|
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| classification_rvk | SK 700 |
| classification_tum | MAT 300f |
| ctrlnum | (OCoLC)720960729 (DE-599)BVBBV002188163 |
| 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 | 3515.9 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| genre_facet | Einführung |
| id | DE-604.BV002188163 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T15:41:46Z |
| institution | BVB |
| isbn | 0387960856 3540960856 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001437325 |
| oclc_num | 720960729 |
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| physical | XIV, 367 S. graph. Darst. |
| publishDate | 1985 |
| publishDateSearch | 1985 |
| publishDateSort | 1985 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Lang, Serge 1927-2005 Verfasser (DE-588)119305119 aut Complex analysis Serge Lang 2. ed. New York [u.a.] Springer 1985 XIV, 367 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 103 Analyse (wiskunde) gtt Analyse mathématique Complexe variabelen gtt Fonctions d'une variable complexe 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 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=001437325&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lang, Serge 1927-2005 Complex analysis Graduate texts in mathematics Analyse (wiskunde) gtt Analyse mathématique Complexe variabelen gtt Fonctions d'une variable complexe 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 (wiskunde) gtt Analyse mathématique Complexe variabelen gtt Fonctions d'une variable complexe Functions of complex variables Mathematical analysis Funktionalanalysis (DE-588)4018916-8 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Analyse (wiskunde) Analyse mathématique Complexe variabelen Fonctions d'une variable complexe Functions of complex variables Mathematical analysis Funktionalanalysis Funktionentheorie Einführung |
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