Complex analysis: an introduction to the theory of analytic functions of one complex variable
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York ; St. Louis ; San Francisco ; Toronto ; London ; Sydney
McGraw-Hill Book Company
1966
|
| Edition: | Second edition |
| Series: | International Series in Pure and Applied Mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiii, 317 Seiten |
| ISBN: | 0070006563 |
Staff View
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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, Professor of Mathematics, Harvard University |
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Record in the Search Index
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| adam_text | CONTENTS
Preface vii
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
/. 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 34
2.2. Series 35
2.3. Uniform Convergence 36
2.4. Power Series 38
2.5. Abel s Limit Theorem 42
X CONTENTS
3. The Exponential and Trigonometric Functions 43
3.1. The Exponential 43
3.2. The Trigonometric Functions 44
3.3. The Periodicity 45
3.4. The Logarithm 46
CHAPTER 3: ANALYTIC FUNCTIONS AS MAPPINGS 43
1. 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 64
1.6. Topological Spaces 67
2. Conformality 68
2.1. Arcs and Closed Curves 68
2.2. Analytic Functions in Regions 69
2.3. Conformal Mapping 73
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
/. 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 Circular Disk 112
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
CONTENTS Xl
3. Local Properties of Analytic 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 139
4.3. Exact Differentials in Simply Connected Regions 141
4.4. Multiply Connected Regions 144
5. The Calculus of Residues 147
5.1. The Residue Theorem 147
5.2. The Argument Principle 151
5.3. Evaluation of Definite Integrals 153
6. Harmonic Functions 160
6.1. Definition and Basic Properties 160
6.2. The Mean value Property 163
6.3. Poisson s Formula 165
6.4. Schwarz s Theorem 167
6.5. The Reflection Principle 170
CHAPTER 5: SERIES AND PRODUCT DEVELOPMENTS 173
1. Power Series Expansions 173
1.1. Weierstrass s Theorem 173
1.2. The Taylor Series 177
1.3. The Laurent Series 182
2. Partial Fractions and Factorization 185
2.1. Partial Fractions 185
2.2. Infinite Products 189
2.3. Canonical Products 192
2.4. The Gamma Function 196
2.5. Stirling s Formula 199
3. Entire Functions 205
3.1. Jensen s Formula 205
3.2. Hadamard s Theorem 206
4. Normal Families 210
4.1. Equicontinuity 210
4.2. Normality and Compactness 211
4.3. Arzela s Theorem 214
4.4. Families of Analytic Functions 215
4.5. The Classical Definition 217
Xlf CONTENTS
CHAPTER 6: CONFORMAL MAPPING. DIRICHLET S
PROBLEM 221
1. The Riemann Mapping Theorem 221
1.1. Statement and Proof 221
1.2. Boundary Behavior 224
1.3. Use of the Reflection Principle 225
1.4. Analytic Arcs 226
2. Conformal Mapping of Polygons 227
2.1. The Behavior at an Angle 227
2.2. The Schwarz Christoffel Formula 228
2.3. Mapping on a Rectangle 230
2.4. The Triangle Functions of Schwarz 233
3. A Closer Look at Harmonic Functions 233
3.1. Functions with the Mean value Property 234
3.2. Harnack s Principle 235
4. The Dirichlet Problem 237
4.1. Subharmonic Functions 237
4.2. Solution of Dirichlet s Problem 240
5. Canonical Mappings of Multiply Connected Regions 243
5.1. Harmonic Measures 244
5.2. Green s Function 249
5.3. Parallel Slit Regions 251
CHAPTER 7: ELLIPTIC FUNCTIONS 255
1. Simply Periodic Functions 255
1.1. Representation by Exponentials 255
1.2. The Fourier Development 256
1.3. Functions of Finite Order 256
2. Doubly Periodic Functions 257
2.1. The Period Module 257
2.2. Unimodular Transformations 258
2.3. The Canonical Basis 260
2.4. General Properties of Elliptic Functions 262
3. The Weierstrass Theory 264
3.1. The Weierstrass ^ function 264
3.2. The Functions f (t) and a{z) 265
3.3. The Differential Equation 267
3.4. The Modular Function X(t) 269
3.5. The Conformal Mapping by X(t) 271
CONTENTS Xiil
CHAPTER 8: GLOBAL ANALYTIC FUNCTIONS 275
1. Analytic Continuation 275
1.1. General Analytic Functions 275
1.2. The Riemann Surface of a Function 277
1.3. Analytic Continuation along Arcs 278
1.4. Homotopic Curves 281
1.5. The Monodromy Theorem 285
1.6. Branch Points 287
2. Algebraic Functions 291
2.1. The Resultant of Two Polynomials 291
2.2. Definition and Properties of Algebraic Functions 292
2.3. Behavior at the Critical Points 294
3. Picard s Theorem 297
3.1. Lacunary Values 297
4. Linear Differential Equations 299
4.1. Ordinary Points 300
4.2. Regular Singular Points 302
4.3. Solutions at Infinity 304
4.4. The Hypergeometric Differential Equation 305
4.5. Riemann s Point of View 309
Index 313
|
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| isbn | 0070006563 |
| language | English |
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| publishDateSearch | 1966 |
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| series2 | International Series in Pure and Applied Mathematics |
| 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, Professor of Mathematics, Harvard University Second edition New York ; St. Louis ; San Francisco ; Toronto ; London ; Sydney McGraw-Hill Book Company 1966 xiii, 317 Seiten txt rdacontent n rdamedia nc rdacarrier International Series in Pure and Applied Mathematics Komplexe Variable (DE-588)4164905-9 gnd rswk-swf Funktion Mathematik (DE-588)4071510-3 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 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 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001496082&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 | Ahlfors, Lars Valerian 1907-1996 Complex analysis an introduction to the theory of analytic functions of one complex variable Komplexe Variable (DE-588)4164905-9 gnd Funktion Mathematik (DE-588)4071510-3 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| subject_GND | (DE-588)4164905-9 (DE-588)4071510-3 (DE-588)4018935-1 (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, Professor of Mathematics, Harvard University |
| title_fullStr | Complex analysis an introduction to the theory of analytic functions of one complex variable Lars V. Ahlfors, Professor of Mathematics, Harvard University |
| title_full_unstemmed | Complex analysis an introduction to the theory of analytic functions of one complex variable Lars V. Ahlfors, Professor of Mathematics, Harvard University |
| 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 | Komplexe Variable (DE-588)4164905-9 gnd Funktion Mathematik (DE-588)4071510-3 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Komplexe Variable Funktion Mathematik Funktionentheorie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001496082&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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