Complex analysis: an introduction to the theory of analytic functions of one complex variable
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY [u.a.]
McGraw-Hill
1979
|
| Ausgabe: | 3. ed. |
| Schriftenreihe: | International series in pure and applied mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 331 S. graph. Darst. |
| ISBN: | 0070006571 |
Internformat
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| 245 | 1 | 0 | |a Complex analysis |b an introduction to the theory of analytic functions of one complex variable |c Lars V. Ahlfors |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a New York, NY [u.a.] |b McGraw-Hill |c 1979 | |
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Datensatz im Suchindex
| _version_ | 1804116594690883584 |
|---|---|
| adam_text | Contents
Preface xiii
CHAPTER 1 COMPLEX NUMBERS 1
1 The Algebra of Complex Numbers 1
1.1 Arithmetic Operations 1
1.2 Square Roots 3
1.3 Justification 4
1.4 Conjugation, Absolute Value 6
1.5 Inequalities 9
2 The Geometric Representation of Complex Numbers 12
2.1 Geometric Addition and Multiplication 12
2.2 The Binomial Equation 15
2.3 Analytic Geometry 17
2.4 The Spherical Representation 18
CHAPTER 2 COMPLEX FUNCTIONS 21
1 Introduction to the Concept of Analytic Function 21
1.1 Limits and Continuity 22
1.2 Analytic Functions 24
1.3 Polynomials 28
1.4 Rational Functions 30
2 Elementary Theory of Power Series 33
2.1 Sequences 33
2.2 Series 35
vil
vill CONTENTS
2.3 Uniform Convergence 35
2.4 Power Series 38
2.5 Abel s Limit Theorem 41
3 The Exponential and Trigonometric Functions 42
3.1 The Exponential 42
3.2 The Trigonometric Functions 43
3.3 The Periodicity. 44
3.4 The Logarithm 46
CHAPTER 3 ANALYTIC FUNCTIONS AS MAPPINGS 49
J Elementary Point Set Topology 50
1.1 Sets and Elements 50
1.2 Metric Spaces 51
1.3 Connectedness 54
1.4 Compactness 59
1.5 Continuous Functions 63
1.6 Topologieal Spaces 66
2 Conformality 67
2.1 Arcs and Closed Curves 67
2.2 Analytic Functions in Regions 69
2.3 Conformal Mapping 73
2.4 Length and Area 75
3 Linear Transformations 76
3.1 The Linear Group 76
3.2 The Cross Ratio 78
3.3 Symmetry 80
3.4 Oriented Circles 83
3.5 Families of Circles 84
4 Elementary Conformal Mappings 89
4.1 The Use of Level Curves 89
4.2 A Survey of Elementary Mappings 93
4.3 Elementary Riemann Surfaces 97
CHAPTER 4 COMPLEX INTEGRATION 101
1 Fundamental Theorems 101
1.1 Line Integrals 101
1.2 Rectifiable Arcs 104
1.3 Line Integrals as Functions of Arcs 105
1.4 Cauchy s Theorem for a Rectangle 109
1.5 Cauchy s Theorem in a Disk 112
CONTENTS ix
2 Cauchy s Integral Formula 114
2.1 The Index of a Point with Respect to a Closed Curve 114
2.2 The Integral Formula 118
2.3 Higher Derivatives 120
3 Local Properties of Analytical Functions 124
3.1 Removable Singularities. Taylor s Theorem 124
3.2 Zeros and Poles 126
3.3 The Local Mapping 130
3.4 The Maximum Principle 133
4 The General Form of Cauchy s Theorem 137
4.1 Chains and Cycles 137
4.2 Simple Connectivity 138
4.3 Homology 141
4.4 The General Statement of Cauchy s Theorem 141
4.5 Proof of Cauchy s Theorem 142
4.6 Locally Exact Differentials 144
4.7 Multiply Connected Regions 146
5 The Calculus of Residues 148
5.1 The Residue Theorem 148
5.2 The Argument Principle 152
5.3 Evaluation of Definite Integrals 154
6 Harmonic Functions 162
6.1 Definition and Basic Properties 162
6.2 The Mean value Property 165
6.3 Poisson s Formula 166
6.4 Schwarz s Theorem 168
6.5 The Reflection Principle 172
CHAPTER 5 SERIES AND PRODUCT DEVELOPMENTS 175
1 Power Series Expansions 175
1.1 Weierstrass s Theorem 175
1.2 The Taylor Series 179
1.3 The Laurent Series 184
2 Partial Fractions and Factorization 187
2.1 Partial Fractions 187
2.2 Infinite Products 191
2.3 Canonical Products 193
2.4 The Gamma Function 198
2.5 Stirling s Formula 201
X CONTENTS
3 Entire Functions 206
3.1 Jensen s Formula 207
3.2 Hadamard s Theorem 208
4 The Riemann Zeta Function 212
4.1 The Product Development 213
4.2 Extension of f(s) to the Whole Plane 214
4.3 The Functional Equation 216
4.4 The Zeros of the Zeta Function 218
5 Normal Families 219
5.1 Equicontinuity 219
5.2 Normality and Compactness 220
5.3 Arzela s Theorem 222
5.4 Families of Analytic Functions 223
5.5 The Classical Definition 225
CHAPTER 6 CONFORMAL MAPPING. DIRICHLET S
PROBLEM 229
1 The Riemann Mapping Theorem 229
1.1 Statement and Proof 229
1.2 Boundary Behavior 232
1.3 Use of the Reflection Principle 233
1.4 Analytic Arcs 234
2 Conformal Mapping of Polygons 235
2.1 The Behavior at an Angle 235
2.2 The Schwarz Christoffel Formula 236
2.3 Mapping on a Rectangle 238
2.4 The Triangle Functions of Schwarz 241
3 A Closer Look at Harmonic Functions 241
3.1 Functions with the Mean value Property 242
3.2 Harnack s Principle 243
4 The Dirichlet Problem 245
4.1 Subharmonic Functions 245
4.2 Solution of Dirichlet s Problem 248
5 Canonical Mappings of Multiply Connected Regions 251
5.1 Harmonic Measures 252
5.2 Green s Function 257
5.3 Parallel Slit Regions 259
CONTENTS xl
CHAPTER 7 ELLIPTIC FUNCTIONS 263
1 Simply Periodic Functions 263
1.1 Representation by Exponentials 263
1.2 The Fourier Development 264
1.3 Functions of Finite Order 264
2 Doubly Periodic Functions 265
2.1 The Period Module 265
2.2 Unimodular Transformations 266
2.3 The Canonical Basis 268
2.4 General Properties of Elliptic Functions 270
3 The Weierstrass Theory 272
3.1 The Weierstrass p function 272
3.2 The Functions £(a) and r(z) 273
3.3 The Differential Equation 275
3.4 The Modular Function X(t) 277
3.5 The Conformal Mapping by X(t) 279
CHAPTER 8 GLOBAL ANALYTIC FUNCTIONS 283
1 Analytic Continuation 283
1.1 The Weierstrass Theory 283
1.2 Germs and Sheaves 284
1.3 Sections and Riemann Surfaces 287
1.4 Analytic Continuations along Arcs 289
1.5 Homotopic Curves 291
1.6 The Monodromy Theorem 295
1.7 Branch Points 297
2 Algebraic Functions 300
2.1 The Resultant of Two Polynomials 300
2.2 Definition and Properties of Algebraic Functions 301
2.3 Behavior at the Critical Points 304
3 Picard s Theorem 306
3.1 Lacunary Values 307
4 Linear Differential Equations 308
4.1 Ordinary Points 309
4.2 Regular Singular Points 311
4.3 Solutions at Infinity 313
4.4 The Hypergeometric Differential Equation 315
4.5 Riemann s Point of View 318
Index 323
|
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| discipline | Mathematik |
| edition | 3. ed. |
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| spelling | Ahlfors, Lars Valerian 1907-1996 Verfasser (DE-588)104541164 aut Complex analysis an introduction to the theory of analytic functions of one complex variable Lars V. Ahlfors 3. ed. New York, NY [u.a.] McGraw-Hill 1979 XIV, 331 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier International series in pure and applied mathematics Funciones analíticas Funciones de variables complejas Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Komplexe Variable (DE-588)4164905-9 gnd rswk-swf Funktion Mathematik (DE-588)4071510-3 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Funktionentheorie (DE-588)4018935-1 s DE-604 Komplexe Variable (DE-588)4164905-9 s Funktion Mathematik (DE-588)4071510-3 s 2\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001403798&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 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Ahlfors, Lars Valerian 1907-1996 Complex analysis an introduction to the theory of analytic functions of one complex variable Funciones analíticas Funciones de variables complejas Funktionentheorie (DE-588)4018935-1 gnd Komplexe Variable (DE-588)4164905-9 gnd Funktion Mathematik (DE-588)4071510-3 gnd |
| subject_GND | (DE-588)4018935-1 (DE-588)4164905-9 (DE-588)4071510-3 (DE-588)4123623-3 |
| title | Complex analysis an introduction to the theory of analytic functions of one complex variable |
| title_auth | Complex analysis an introduction to the theory of analytic functions of one complex variable |
| title_exact_search | Complex analysis an introduction to the theory of analytic functions of one complex variable |
| title_full | Complex analysis an introduction to the theory of analytic functions of one complex variable Lars V. Ahlfors |
| title_fullStr | Complex analysis an introduction to the theory of analytic functions of one complex variable Lars V. Ahlfors |
| title_full_unstemmed | Complex analysis an introduction to the theory of analytic functions of one complex variable Lars V. Ahlfors |
| title_short | Complex analysis |
| title_sort | complex analysis an introduction to the theory of analytic functions of one complex variable |
| title_sub | an introduction to the theory of analytic functions of one complex variable |
| topic | Funciones analíticas Funciones de variables complejas Funktionentheorie (DE-588)4018935-1 gnd Komplexe Variable (DE-588)4164905-9 gnd Funktion Mathematik (DE-588)4071510-3 gnd |
| topic_facet | Funciones analíticas Funciones de variables complejas Funktionentheorie Komplexe Variable Funktion Mathematik Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001403798&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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