Complex analysis: in the spirit of Lipman Bers
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| Previous Title: | Gilman, Jane P. Complex analysis |
|---|---|
| Main Authors: | , , |
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
[2013]
|
| Edition: | Second edition |
| Series: | Graduate texts in mathematics
245 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xviii, 306 Seiten Illustrationen, Diagramme |
| ISBN: | 9781441973221 9781489999085 |
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MARC
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| 245 | 1 | 0 | |a Complex analysis |b in the spirit of Lipman Bers |c Rubí E. Rodríguez ; Irwin Kra ; Jane P. Gilman |
| 250 | |a Second edition | ||
| 264 | 1 | |a New York, NY |b Springer |c [2013] | |
| 264 | 4 | |c © 2013 | |
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Record in the Search Index
| _version_ | 1804149741928316928 |
|---|---|
| adam_text | Contents
The Fundamental Theorem in Complex Function Theory
............ 1
1.1
Some Motivation
..................................................... 1
1.1.1
Where Do Series
Converge
............................... 1
1.1.2
A Problem on Partitions
.................................... 2
1.1.3
Evaluation of Definite Real Integrals
...................... 3
1.2
The Fundamental Theorem of Complex Function Theory
......... 3
1.3
The Plan for the Proof
................................................ 5
1.4
Outline of Text
........................................................ 6
1.5
Appendix: Historical Notes by Ranjan Roy
......................... 6
Foundations
................................................................. 15
2.1
Introduction and Preliminaries
....................................... 15
2.1.1
Properties of Conjugation
.................................. 17
2.1.2
Properties of Absolute Value
............................... 18
2.1.3
Linear Representation of
С
................................ 18
2.1.4
Additional Properties of Absolute Value
.................. 18
2.1.5
Lines, Circles, and Half Planes
............................ 19
2.1.6
Polar Coordinates
........................................... 21
2.1.7
Coordinates on
С
........................................... 22
2.2
More Preliminaries that Rely on Topology, Metrics,
and Sequences
........................................................ 23
2.3
Differentiability and Holomorphic Mappings
...................... 28
2.3.1
Convention
.................................................. 30
2.3.2
The Cauchy-Riemann (CR) Equations
.................... 30
Power Series
................................................................. 39
3.1
Complex Power Series
............................................... 40
3.1.1
Properties of Limits Superior and
Inf
enor
................ 45
3.1.2
The Radius of Convergence
................................ 46
3.2
More on Power Series
................................................ 47
xiv Contents
3.3
The Exponential Function, the Logarithm Function,
and Some Complex Trigonometric Functions
...................... 53
3.3.1
The Exponential Function
.................................. 53
3.3.2
The Complex Trigonometric Functions
................... 56
3.3.3
The Definition of
π
and the Logarithm Function
......... 57
3.4
An Identity Principle
................................................. 62
3.5
Zeros and Poles
....................................................... 67
4
The Cauchy Theory
:
A Fundamental Theorem
......................... 81
4.1
Line Integrals and Differential Forms
............................... 82
4.1.1
Reparameterization.
......................................... 84
4.1.2
Subdivision of Interval
..................................... 84
4.1.3
The Line Integral
........................................... 85
4.2
The Precise Difference Between Closed and Exact Forms
......... 88
4.2.1
Caution
...................................................... 94
4.2.2
Existence and Uniqueness
.................................. 94
4.3
Integration of Closed Forms and the Winding Number
............ 95
4.4
Homotopy and Simple Connectivity
................................ 97
4.5
More on the Winding Number
....................................... 100
4.6
Cauchy Theory: Initial Version
...................................... 103
4.7
Appendix I: The Exterior Differential Calculus
.................... 106
4.8
Appendix II: An Alternative Approach to the Cauchy Theory
..... 107
4.8.1
Integration of Functions
.................................... 108
4.8.2
The Key Theorem
.......................................... 109
5
The Cauchy Theory: Key Consequences
................................. 119
5.1
Consequences of the Cauchy Theory
................................ 119
5.2
Cycles and Homology
............................................... 126
5.3
Jordan Curves
........................................................ 129
5.4
The Mean Value Property
............................................ 131
5.5
Appendix: Cauchy s Integral Formula for Smooth Functions
...... 134
6
Cauchy Theory: Local Behavior and Singularities
of Holomorphic Functions
................................................. 139
6.1
Functions Holomorphic on An Anmilus
............................ 139
6.2
Isolated Singularities
................................................. 143
6.3
Residues
.............................................................. 147
6.4
Zeros and Poles of Meromorphic Functions
........................ 149
6.5
Local Properties of Holomorphic Maps
............................. 153
6.6
Evaluation of Definite Integrals
..................................... 156
6.7
Appendix: Cauchy Principal Value
.................................. 162
7
Sequences and Series of Holomorphic Functions
....................... 171
7.1
Consequences of Uniform Convergence
............................ 171
7.2
A Metric on C(D)
.................................................... 174
7.2.1
Properties of
d
.............................................. 175
7.2.2
Properties of
ρ
.............................................. 176
Contents xv
7.3
The Cotangent Function
............................................. 179
7.4
Compact Sets in
H(ű)
.............................................. 183
7.5
Runge s Theorem
..................................................... 187
7.5.1
Preliminaries for the Proof of Runge s Theorem
.......... 189
7.5.2
Proof of Runge s Theorem
................................. 190
7.5.3
Two Major Lemmas
........................................ 191
1
7.5.4
Approximating
------................................. 193
z
-
с
8
Conforma!
Equivalence and Hyperbolic Geometry
.................... 199
8.1
Fractional Linear
(Möbius)
Transformations
....................... 200
8.1.1
Fixed Points of
Möbius
Transformations
.................. 202
8.1.2
Cross Ratios
................................................. 202
8.2
Aut(D) for
D
=
C, C, B, and H2
................................... 205
8.3
The Riemann Mapping Theorem
.................................... 207
8.4
Hyperbolic Geometry
................................................ 211
8.4.1
The
Poincaré
Metric
........................................ 212
8.4.2
Upper Half-plane Model
................................... 214
8.4.3
Unit Disc Model
............................................ 218
8.4.4
Contractions and the Schwarz s Lemma
................... 219
8.5
Finite Blaschke Products
............................................. 221
9
Harmonic Functions
........................................................ 229
9.1
Harmonic Functions and the Laplacian
............................. 230
9.2
Integral Representation of Harmonic Functions
.................... 232
9.3
The Dinchlet Problem
............................................... 235
9.3.1
Geometric Interpretation of the
Poisson
Formula
......... 237
9.3.2
Founer Series Interpretation of the
Poisson
Formula
..... 239
9.3.3
Classical Reformulation of the
Poisson
Formula
......... 240
9.4
The Mean Value Property: A Characterization of Harmomcity
.... 243
9.5
The Reflection Principle
............................................. 244
9.6
Subharmonic Functions
.............................................. 245
9.7
Perron Families
....................................................... 249
9.8
The Dinchlet Problem (Revisited)
.................................. 251
9.9
Green s Function and
RMT
Revisited
............................... 256
10
Zeros of Holomorphic Functions
......................................... 267
10.1
Infinite Products
...................................................... 268
10.2
Holomorphic Functions with Prescribed Zeros
..................... 272
10.3
The Ring
H(ű)
....................................................... 276
10.4
Euler s F-Function
................................................... 280
10.4.1
Basic Properties
............................................. 280
10.4.2
Estimates for
Г (г) ..........................................
284
10.4.3
The Formulae for the Function
............................. 287
10.5
Divisors and the Field of Meromorphic Functions
................. 289
10.6
Infinite Blaschke Products
........................................... 290
xvi Contents
Bibliographical Notes
............................................................ 297
References
......................................................................... 299
Index
............................................................................... 301
|
| any_adam_object | 1 |
| author | Rodríguez, Rubí E. 1953- Kra, Irwin 1937- Gilman, Jane P. 1945- |
| author_GND | (DE-588)123785119X (DE-588)134052498 (DE-588)134012801 |
| author_facet | Rodríguez, Rubí E. 1953- Kra, Irwin 1937- Gilman, Jane P. 1945- |
| author_role | aut aut aut |
| author_sort | Rodríguez, Rubí E. 1953- |
| author_variant | r e r re rer i k ik j p g jp jpg |
| building | Verbundindex |
| bvnumber | BV040630983 |
| callnumber-first | Q - Science |
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| callnumber-search | QA331.7 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 700 |
| ctrlnum | (OCoLC)823252582 (DE-599)BVBBV040630983 |
| 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 | Second edition |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV040630983 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:27:51Z |
| institution | BVB |
| isbn | 9781441973221 9781489999085 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025458249 |
| oclc_num | 823252582 |
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| owner_facet | DE-706 DE-384 DE-11 DE-20 DE-188 DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-83 |
| physical | xviii, 306 Seiten Illustrationen, Diagramme |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Rodríguez, Rubí E. 1953- Verfasser (DE-588)123785119X aut Complex analysis in the spirit of Lipman Bers Rubí E. Rodríguez ; Irwin Kra ; Jane P. Gilman Second edition New York, NY Springer [2013] © 2013 xviii, 306 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 245 Functions of complex variables Mathematical analysis Funktion Mathematik (DE-588)4071510-3 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Mehrere komplexe Variable (DE-588)4169285-8 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 Gilman, Jane P. 1945- Verfasser (DE-588)134012801 aut Erscheint auch als Online-Ausgabe 978-1-4419-7323-8 Fortsetzung von Gilman, Jane P. Complex analysis in the spirit of Lipman Bers 2007 978-0-387-74714-9 Graduate texts in mathematics 245 (DE-604)BV000000067 245 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025458249&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rodríguez, Rubí E. 1953- Kra, Irwin 1937- Gilman, Jane P. 1945- Complex analysis in the spirit of Lipman Bers Graduate texts in mathematics Functions of complex variables Mathematical analysis Funktion Mathematik (DE-588)4071510-3 gnd Funktionentheorie (DE-588)4018935-1 gnd Mehrere komplexe Variable (DE-588)4169285-8 gnd |
| subject_GND | (DE-588)4071510-3 (DE-588)4018935-1 (DE-588)4169285-8 (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_full | Complex analysis in the spirit of Lipman Bers Rubí E. Rodríguez ; Irwin Kra ; Jane P. Gilman |
| title_fullStr | Complex analysis in the spirit of Lipman Bers Rubí E. Rodríguez ; Irwin Kra ; Jane P. Gilman |
| title_full_unstemmed | Complex analysis in the spirit of Lipman Bers Rubí E. Rodríguez ; Irwin Kra ; Jane P. Gilman |
| title_old | Gilman, Jane P. 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 Funktion Mathematik (DE-588)4071510-3 gnd Funktionentheorie (DE-588)4018935-1 gnd Mehrere komplexe Variable (DE-588)4169285-8 gnd |
| topic_facet | Functions of complex variables Mathematical analysis Funktion Mathematik Funktionentheorie Mehrere komplexe Variable Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025458249&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT rodriguezrubie complexanalysisinthespiritoflipmanbers AT krairwin complexanalysisinthespiritoflipmanbers AT gilmanjanep complexanalysisinthespiritoflipmanbers |