Concise complex analysis:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2007
|
| Edition: | Rev. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 237 S. graph. Darst. |
| ISBN: | 9789812706935 9812706933 |
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MARC
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| 245 | 1 | 0 | |a Concise complex analysis |c Sheng Gong ; Youhong Gong |
| 250 | |a Rev. ed. | ||
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 2007 | |
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Record in the Search Index
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| adam_text | Titel: Concise complex analysis
Autor: Gong, Sheng
Jahr: 2007
Contents
Preface to the Revised Edition vii
Preface to the First Edition ix
Foreword xi
1. Calculus 1
1.1 A Brief Review of Calculus.................. 1
1.2 The Field of Complex Numbers, The Extended Complex
Plane and Its Spherical Representation............ 8
1.3 Derivatives of Complex Functions............... 11
1.4 Complex Integration...................... 17
1.5 Elementary Functions..................... 19
1.6 Complex Series......................... 26
Exercise I............................... 29
2. Cauchy Integral Theorem and Cauchy Integral Formula 39
2.1 Cauchy-Green Formula (Pompeiu Formula)......... 39
2.2 Cauchy-Goursat Theorem................... 44
2.3 Taylor Series and Liouville Theorem............. 52
2.4 Some Results about the Zeros of Holomorphic Functions . . 59
2.5 Maximum Modulus Principle, Schwarz Lemma and Group
of Holomorphic Automorphisms................ 64
2.6 Integral Representation of Holomorphic Functions ..... 69
Exercise II.............................. 75
Appendix I Partition of Unity.................. 82
xviii Concise Complex Analysis
3. Theory of Series of Weierstrass 85
3.1 Laurent Series.......................... 85
3.2 Isolated Singularity....................... 90
3.3 Entire Functions and Meromorphic Functions........ 93
3.4 Weierstrass Factorization Theorem, Mittag-Leffler Theorem
and Interpolation Theorem.................. 97
3.5 Residue Theorem........................ 106
3.6 Analytic Continuation..................... 113
Exercise III.............................. 117
4. Riemann Mapping Theorem 123
4.1 Conformal Mapping...................... 123
4.2 Normal Family......................... 128
4.3 Riemann Mapping Theorem.................. 131
4.4 Symmetry Principle...................... 134
4.5 Some Examples of Riemann Surface............. 136
4.6 Schwarz-Christoffel Formula.................. 138
Exercise IV.............................. 141
Appendix II Riemann Surface.................. 143
5. Differential Geometry and Picard Theorem 145
5.1 Metric and Curvature..................... 145
5.2 Ahlfors-Schwarz Lemma.................... 151
5.3 The Generalization of Liouville Theorem and Value Distri-
bution .............................. 153
5.4 The Little Picard Theorem.................. 154
5.5 The Generalization of Normal Family ............ 156
5.6 The Great Picard Theorem.................. 159
Exercise V.............................. 162
Appendix III Curvature ..................... 163
6. A First Taste of Function Theory of Several Complex Variables 169
6.1 Introduction........................... 169
6.2 Cartan Theorem........................ 172
6.3 Groups of Holomorphic Automorphisms of The Unit Ball
and The Bidisc......................... 174
6.4 Poincare Theorem....................... 179
6.5 Hartogs Theorem........................ 181
Contents xix
7. Elliptic Functions 185
7.1 The Concept of Elliptic Functions .............. 185
7.2 The Weierstrass Theory.................... 191
7.3 The Jacobi Elliptic Functions................. 197
7.4 The Modular Function..................... 200
8. The Riemann ^-Function and The Prime Number Theo-
rem 207
8.1 The Gamma Function..................... 207
8.2 The Riemann C-function.................... 211
8.3 The Prime Number Theorem................. 218
8.4 The Proof of The Prime Number Theorem.......... 222
Bibliography 231
Index 235
|
| adam_txt |
Titel: Concise complex analysis
Autor: Gong, Sheng
Jahr: 2007
Contents
Preface to the Revised Edition vii
Preface to the First Edition ix
Foreword xi
1. Calculus 1
1.1 A Brief Review of Calculus. 1
1.2 The Field of Complex Numbers, The Extended Complex
Plane and Its Spherical Representation. 8
1.3 Derivatives of Complex Functions. 11
1.4 Complex Integration. 17
1.5 Elementary Functions. 19
1.6 Complex Series. 26
Exercise I. 29
2. Cauchy Integral Theorem and Cauchy Integral Formula 39
2.1 Cauchy-Green Formula (Pompeiu Formula). 39
2.2 Cauchy-Goursat Theorem. 44
2.3 Taylor Series and Liouville Theorem. 52
2.4 Some Results about the Zeros of Holomorphic Functions . . 59
2.5 Maximum Modulus Principle, Schwarz Lemma and Group
of Holomorphic Automorphisms. 64
2.6 Integral Representation of Holomorphic Functions . 69
Exercise II. 75
Appendix I Partition of Unity. 82
xviii Concise Complex Analysis
3. Theory of Series of Weierstrass 85
3.1 Laurent Series. 85
3.2 Isolated Singularity. 90
3.3 Entire Functions and Meromorphic Functions. 93
3.4 Weierstrass Factorization Theorem, Mittag-Leffler Theorem
and Interpolation Theorem. 97
3.5 Residue Theorem. 106
3.6 Analytic Continuation. 113
Exercise III. 117
4. Riemann Mapping Theorem 123
4.1 Conformal Mapping. 123
4.2 Normal Family. 128
4.3 Riemann Mapping Theorem. 131
4.4 Symmetry Principle. 134
4.5 Some Examples of Riemann Surface. 136
4.6 Schwarz-Christoffel Formula. 138
Exercise IV. 141
Appendix II Riemann Surface. 143
5. Differential Geometry and Picard Theorem 145
5.1 Metric and Curvature. 145
5.2 Ahlfors-Schwarz Lemma. 151
5.3 The Generalization of Liouville Theorem and Value Distri-
bution . 153
5.4 The Little Picard Theorem. 154
5.5 The Generalization of Normal Family . 156
5.6 The Great Picard Theorem. 159
Exercise V. 162
Appendix III Curvature . 163
6. A First Taste of Function Theory of Several Complex Variables 169
6.1 Introduction. 169
6.2 Cartan Theorem. 172
6.3 Groups of Holomorphic Automorphisms of The Unit Ball
and The Bidisc. 174
6.4 Poincare Theorem. 179
6.5 Hartogs Theorem. 181
Contents xix
7. Elliptic Functions 185
7.1 The Concept of Elliptic Functions . 185
7.2 The Weierstrass Theory. 191
7.3 The Jacobi Elliptic Functions. 197
7.4 The Modular Function. 200
8. The Riemann ^-Function and The Prime Number Theo-
rem 207
8.1 The Gamma Function. 207
8.2 The Riemann C-function. 211
8.3 The Prime Number Theorem. 218
8.4 The Proof of The Prime Number Theorem. 222
Bibliography 231
Index 235 |
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| edition | Rev. ed. |
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| index_date | 2024-07-02T22:17:02Z |
| indexdate | 2024-07-09T21:22:31Z |
| institution | BVB |
| isbn | 9789812706935 9812706933 |
| language | English |
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| spelling | Gong, Sheng Verfasser aut Concise complex analysis Sheng Gong ; Youhong Gong Rev. ed. Singapore [u.a.] World Scientific 2007 XIX, 237 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Calculus Functions of complex variables Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Funktionentheorie (DE-588)4018935-1 s DE-604 Gong, Youhong Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016777543&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gong, Sheng Gong, Youhong Concise complex analysis Calculus Functions of complex variables Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd |
| subject_GND | (DE-588)4018935-1 (DE-588)4123623-3 |
| title | Concise complex analysis |
| title_auth | Concise complex analysis |
| title_exact_search | Concise complex analysis |
| title_exact_search_txtP | Concise complex analysis |
| title_full | Concise complex analysis Sheng Gong ; Youhong Gong |
| title_fullStr | Concise complex analysis Sheng Gong ; Youhong Gong |
| title_full_unstemmed | Concise complex analysis Sheng Gong ; Youhong Gong |
| title_short | Concise complex analysis |
| title_sort | concise complex analysis |
| topic | Calculus Functions of complex variables Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Calculus Functions of complex variables Mathematical analysis Funktionentheorie Lehrbuch |
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