Complex analysis: in the spirit of Lipman Bers
Gespeichert in:
| Späterer Titel: | Rodríguez, Rubí E. Complex analysis |
|---|---|
| Hauptverfasser: | , , |
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer
2007
|
| Schriftenreihe: | Graduate Texts in Mathematics
245 |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XIII, 220 Seiten Illustrationen, Diagramme 235 mm x 155 mm |
| ISBN: | 9780387747149 0387747141 9781441925671 |
Internformat
MARC
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| 245 | 1 | 0 | |a Complex analysis |b in the spirit of Lipman Bers |c Jane P. Gilman, Irwin Kra, Rubí E. Rodríguez |
| 264 | 1 | |a New York, NY |b Springer |c 2007 | |
| 300 | |a XIII, 220 Seiten |b Illustrationen, Diagramme |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Graduate Texts in Mathematics |v 245 | |
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| 650 | 4 | |a Mathematical analysis | |
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| 785 | 0 | 0 | |i Fortgesetzt durch |a Rodríguez, Rubí E. |t Complex analysis |c in the spirit of Lipman Bers |b Second edition |d 2013 |z 978-1-4419-7322-1 |
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Datensatz im Suchindex
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| adam_text |
Contents
Preface
vii
Standard
Notation
and Commonly Used Symbols
ix
Chapter
1.
The Fundamental Theorem in Complex Function
Theory
1
1.1.
Some motivation
1
1.2.
The Fundamental Theorem
3
1.3.
The plan for the proof
4
1.4.
Outline of text
5
Chapter
2.
Foundations
7
2.1.
Introduction and preliminaries
7
2.2.
Differentiability and holomorphic mappings
14
Exercises
18
Chapter
3.
Power Series
23
3.1.
Complex power series
24
3.2.
More on power series
32
3.3.
The exponential function, the logarithm function,
and some complex trigonometric functions
36
3.4.
An identity principle
42
3.5.
Zeros and poles
47
Exercises
52
Chapter
4.
The Cauchy Theory-A Fundamental Theorem
59
4.1.
Line integrals and differential forms
60
4.2.
The precise difference between closed and exact forms
65
4.3.
Integration of closed forms and the winding number
70
4.4.
Homotopy and simple connectivity
72
4.5.
Winding number
75
4.6.
Cauchy Theory: initial version
78
Exercises
80
Chapter
5.
The Cauchy Theory-Key Consequences
83
5.1.
Consequences of the Cauchy Theory
83
xii CONTENTS
5.2.
Cycles and homology
89
5.3.
Jordan curves
90
5.4.
The Mean Value Property
93
5.5.
On elegance and conciseness
96
5.6.
Appendix: Cauchy's integral formula for smooth functions
96
Exercises
97
Chapter
6.
Cauchy Theory: Local Behavior and Singularities
of Holomorphic Functions
101
6.1.
Functions holomorphic on an annulus
101
6.2.
Isolated singularities
103
6.3.
Zeros and poles of meromorphic functions
106
6.4.
Local properties of holomorphic maps
110
6.5.
Evaluation of definite integrals
113
Exercises
117
Chapter
7.
Sequences and Series of Holomorphic Functions
123
7.1.
Consequences of uniform convergence on compact sets
123
7.2.
A metric on C(D)
126
7.3.
The cotangent function
130
7.4.
Compact sets in H(£>)
134
7.5.
Approximation theorems and Runge's theorem
138
Exercises
146
Chapter
8.
Conformai
Equivalence
147
8.1.
Fractional linear
(Möbius)
transformations
148
8.2.
Aut(JD) for
D
=
C, C, B, and H2
152
8.3.
The Riemann Mapping Theorem
154
8.4.
Hyperbolic geometry
158
8.5.
Finite Blaschke products
167
Exercises
169
Chapter
9.
Harmonic Functions
173
9.1.
Harmonic functions and the Laplacian
173
9.2.
Integral representation of harmonic functions
176
9.3.
The Dirichlet problem
179
9.4.
The Mean Value Property: a characterization
186
9.5.
The reflection principle
186
Exercises
188
Chapter
10.
Zeros of Holomorphic Functions
191
10.1.
Infinite products
191
10.2.
Holomorphic functions with prescribed zeros
195
10.3.
Euler
's
Г
-ŕunction
199
CONTENTS xiii
10.4.
The field of meromorphic functions
207
10.5.
Infinite Blaschke products
209
Exercises
209
BIBLIOGRAPHICAL NOTES
213
Bibliography
215
Index
217 |
| adam_txt |
Contents
Preface
vii
Standard
Notation
and Commonly Used Symbols
ix
Chapter
1.
The Fundamental Theorem in Complex Function
Theory
1
1.1.
Some motivation
1
1.2.
The Fundamental Theorem
3
1.3.
The plan for the proof
4
1.4.
Outline of text
5
Chapter
2.
Foundations
7
2.1.
Introduction and preliminaries
7
2.2.
Differentiability and holomorphic mappings
14
Exercises
18
Chapter
3.
Power Series
23
3.1.
Complex power series
24
3.2.
More on power series
32
3.3.
The exponential function, the logarithm function,
and some complex trigonometric functions
36
3.4.
An identity principle
42
3.5.
Zeros and poles
47
Exercises
52
Chapter
4.
The Cauchy Theory-A Fundamental Theorem
59
4.1.
Line integrals and differential forms
60
4.2.
The precise difference between closed and exact forms
65
4.3.
Integration of closed forms and the winding number
70
4.4.
Homotopy and simple connectivity
72
4.5.
Winding number
75
4.6.
Cauchy Theory: initial version
78
Exercises
80
Chapter
5.
The Cauchy Theory-Key Consequences
83
5.1.
Consequences of the Cauchy Theory
83
xii CONTENTS
5.2.
Cycles and homology
89
5.3.
Jordan curves
90
5.4.
The Mean Value Property
93
5.5.
On elegance and conciseness
96
5.6.
Appendix: Cauchy's integral formula for smooth functions
96
Exercises
97
Chapter
6.
Cauchy Theory: Local Behavior and Singularities
of Holomorphic Functions
101
6.1.
Functions holomorphic on an annulus
101
6.2.
Isolated singularities
103
6.3.
Zeros and poles of meromorphic functions
106
6.4.
Local properties of holomorphic maps
110
6.5.
Evaluation of definite integrals
113
Exercises
117
Chapter
7.
Sequences and Series of Holomorphic Functions
123
7.1.
Consequences of uniform convergence on compact sets
123
7.2.
A metric on C(D)
126
7.3.
The cotangent function
130
7.4.
Compact sets in H(£>)
134
7.5.
Approximation theorems and Runge's theorem
138
Exercises
146
Chapter
8.
Conformai
Equivalence
147
8.1.
Fractional linear
(Möbius)
transformations
148
8.2.
Aut(JD) for
D
=
C, C, B, and H2
152
8.3.
The Riemann Mapping Theorem
154
8.4.
Hyperbolic geometry
158
8.5.
Finite Blaschke products
167
Exercises
169
Chapter
9.
Harmonic Functions
173
9.1.
Harmonic functions and the Laplacian
173
9.2.
Integral representation of harmonic functions
176
9.3.
The Dirichlet problem
179
9.4.
The Mean Value Property: a characterization
186
9.5.
The reflection principle
186
Exercises
188
Chapter
10.
Zeros of Holomorphic Functions
191
10.1.
Infinite products
191
10.2.
Holomorphic functions with prescribed zeros
195
10.3.
Euler
's
Г
-ŕunction
199
CONTENTS xiii
10.4.
The field of meromorphic functions
207
10.5.
Infinite Blaschke products
209
Exercises
209
BIBLIOGRAPHICAL NOTES
213
Bibliography
215
Index
217 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Gilman, Jane P. 1945- Kra, Irwin 1937- Rodríguez, Rubí E. 1953- |
| author_GND | (DE-588)134012801 (DE-588)134052498 (DE-588)123785119X |
| author_facet | Gilman, Jane P. 1945- Kra, Irwin 1937- Rodríguez, Rubí E. 1953- |
| author_role | aut aut aut |
| author_sort | Gilman, Jane P. 1945- |
| author_variant | j p g jp jpg i k ik r e r re rer |
| building | Verbundindex |
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| callnumber-first | Q - Science |
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| classification_rvk | SK 700 |
| ctrlnum | (OCoLC)255687055 (DE-599)DNB985166665 |
| dewey-full | 515.9 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.9 |
| dewey-search | 515.9 |
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| discipline | Mathematik |
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| genre_facet | Lehrbuch |
| id | DE-604.BV023071232 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:33:08Z |
| indexdate | 2024-07-20T09:29:38Z |
| institution | BVB |
| isbn | 9780387747149 0387747141 9781441925671 |
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| physical | XIII, 220 Seiten Illustrationen, Diagramme 235 mm x 155 mm |
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| publisher | Springer |
| record_format | marc |
| series | Graduate Texts in Mathematics |
| series2 | Graduate Texts in Mathematics |
| spelling | Gilman, Jane P. 1945- Verfasser (DE-588)134012801 aut Complex analysis in the spirit of Lipman Bers Jane P. Gilman, Irwin Kra, Rubí E. Rodríguez New York, NY Springer 2007 XIII, 220 Seiten Illustrationen, Diagramme 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Graduate Texts in Mathematics 245 Functions of complex variables Mathematical analysis Mehrere komplexe Variable (DE-588)4169285-8 gnd rswk-swf Funktion Mathematik (DE-588)4071510-3 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Funktionentheorie (DE-588)4018935-1 s DE-604 Funktion Mathematik (DE-588)4071510-3 s Mehrere komplexe Variable (DE-588)4169285-8 s Kra, Irwin 1937- Verfasser (DE-588)134052498 aut Rodríguez, Rubí E. 1953- Verfasser (DE-588)123785119X aut Erscheint auch als Online-Ausgabe 978-0-387-74715-6 Fortgesetzt durch Rodríguez, Rubí E. Complex analysis in the spirit of Lipman Bers Second edition 2013 978-1-4419-7322-1 Graduate Texts in Mathematics 245 (DE-604)BV000000067 245 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2990409&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016274377&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gilman, Jane P. 1945- Kra, Irwin 1937- Rodríguez, Rubí E. 1953- Complex analysis in the spirit of Lipman Bers Graduate Texts in Mathematics Functions of complex variables Mathematical analysis Mehrere komplexe Variable (DE-588)4169285-8 gnd Funktion Mathematik (DE-588)4071510-3 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| subject_GND | (DE-588)4169285-8 (DE-588)4071510-3 (DE-588)4018935-1 (DE-588)4123623-3 |
| title | Complex analysis in the spirit of Lipman Bers |
| title_auth | Complex analysis in the spirit of Lipman Bers |
| title_exact_search | Complex analysis in the spirit of Lipman Bers |
| title_exact_search_txtP | Complex analysis in the spirit of Lipman Bers |
| title_full | Complex analysis in the spirit of Lipman Bers Jane P. Gilman, Irwin Kra, Rubí E. Rodríguez |
| title_fullStr | Complex analysis in the spirit of Lipman Bers Jane P. Gilman, Irwin Kra, Rubí E. Rodríguez |
| title_full_unstemmed | Complex analysis in the spirit of Lipman Bers Jane P. Gilman, Irwin Kra, Rubí E. Rodríguez |
| title_new | Rodríguez, Rubí E. Complex analysis |
| title_short | Complex analysis |
| title_sort | complex analysis in the spirit of lipman bers |
| title_sub | in the spirit of Lipman Bers |
| topic | Functions of complex variables Mathematical analysis Mehrere komplexe Variable (DE-588)4169285-8 gnd Funktion Mathematik (DE-588)4071510-3 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Functions of complex variables Mathematical analysis Mehrere komplexe Variable Funktion Mathematik Funktionentheorie Lehrbuch |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2990409&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016274377&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT gilmanjanep complexanalysisinthespiritoflipmanbers AT krairwin complexanalysisinthespiritoflipmanbers AT rodriguezrubie complexanalysisinthespiritoflipmanbers |