Geometric function theory: explorations in complex analysis
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston, Mass. [u.a.]
Birkhäuser
2006
|
| Series: | Cornerstones
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Auch als Internetausgabe |
| Physical Description: | XIII, 314 Seiten graph. Darst. |
| ISBN: | 0817643397 0817644407 9780817643393 |
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MARC
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| 100 | 1 | |a Krantz, Steven G. |d 1951- |e Verfasser |0 (DE-588)130535907 |4 aut | |
| 245 | 1 | 0 | |a Geometric function theory |b explorations in complex analysis |c Steven G. Krantz |
| 264 | 1 | |a Boston, Mass. [u.a.] |b Birkhäuser |c 2006 | |
| 300 | |a XIII, 314 Seiten |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Cornerstones | |
| 500 | |a Auch als Internetausgabe | ||
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| 650 | 7 | |a Functietheorie |2 gtt | |
| 650 | 4 | |a Geometric function theory | |
| 650 | 4 | |a Functions of complex variables | |
| 650 | 0 | 7 | |a Geometrische Funktionentheorie |0 (DE-588)4156711-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Funktionentheorie |0 (DE-588)4018935-1 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1804133540824088576 |
|---|---|
| adam_text | Contents
Preface
........................................................ xi
Part I Classical Function Theory
Overview
...................................................... 3
1
Invariant Geometry
........................................ 5
1.1
Conformality and
Invariance
.............................. 5
1.2
Bergman s Construction
.................................. 9
1.3
Calculation of the Bergman Kernel for the Disk
............. 15
1.3.1
Construction of the Bergman Kernel for the Disk by
Conformai Invariance..............................
16
1.3.2
Construction of the Bergman Kernel by means of an
Orthonormal
System
............................... 17
1.3.3
Construction of the Bergman Kernel by way of
Differential Equations
.............................. 18
1.4
A New Application
...................................... 21
1.5
An Application to Mapping Theory
........................ 24
Problems for Study and Exploration
............................ 26
2
Variations on the Theme of the
Schwarz
Lemma
........... 29
2.1
Introduction
............................................ 29
2.2
Other Versions of Schwarz s Lemma
....................... 32
2.3
A Geometric View of the
Schwarz
Lemma
.................. 33
2.3.1
Geometric Ideas
................................... 33
2.3.2
Calculus in the Complex Domain
.................... 35
2.3.3
Isometries
........................................ 37
2.3.4
The
Poincaré
Metric
............................... 39
2.3.5
The
Schwarz
Lemma
............................... 47
2.4
Ahlfors s Version of the
Schwarz
Lemma
.................... 50
2.5
Liouville
and Picard
Theorems
............................ 54
2.6
The
Schwarz
Lemma at the Boundary
...................... 58
Problems for Study and Exploration
............................ 63
Normal Families
........................................... 65
3.1
Introduction
............................................ 65
3.2
Topologies on the Space of Holomorphic Functions
........... 66
3.3
Normal Families in Their Natural Context
.................. 67
3.4
Advanced Results on Normal Families
...................... 73
3.5
Robinson s Heuristic Principle
............................. 78
Problems for Study and Exploration
............................ 80
The Riemann Mapping Theorem and Its Generalizations
.. 83
4.1
The Riemann Mapping Theorem by way of the Green s
Function
............................................... 84
4.2
Canonical Representations for Multiply Connected Regions
... 85
4.3
Review of Some Topological Ideas
......................... 86
4.3.1
Cycles and Periods
................................ 86
4.3.2
Harmonic Functions
............................... 88
4.4
Uniformization of Multiply Connected Domains
............. 90
4.5
The Ahlfors Map
........................................ 94
4.6
The Uniformization Theorem
.............................104
Problems for Study and Exploration
............................106
Boundary Regularity of
Conformai
Maps
..................109
5.1
Continuity to the Boundary
...............................110
5.2
Preliminary Facts about Boundary Smoothness
.............118
5.3
Smoothness of
Conformai
Mappings
.......................127
Problems for Study and Exploration
............................131
The Boundary Behavior of Holomorphic Functions
........135
6.0
Introductory Remarks
....................................135
6.1
Review of the Classical Theory of Hp Spaces on the disk
.....136
6.2
The
Lindelof
Principle
...................................146
Problems for Study and Exploration
............................152
Part II Real and Harmonic Analysis
Overview
......................................................157
7
The Cauchy—Riemann Equations
..........................159
7.1
Introduction
............................................160
7.2
Solution of the Inhomogeneous Cauchy-Riemann Equations.
. . 161
7.3
Development and Application of the
В
Equation
.............163
Problems for Study and Exploration
............................166
8
The Green s Function and the
Poisson
Kernel
.............169
8.1
The Laplacian and Its Fundamental Solution
................169
8.2
The Green s Function and Consequences
...................176
8.3
Calculation of the
Poisson
Kernel
..........................178
Problems for Study and Exploration
............................182
9
Harmonic Measure
........................................185
9.1
Introductory Remarks
....................................185
9.2
The Idea of Harmonic Measure
............................188
9.3
Some Examples
.........................................190
9.4
Hadamard s Three Circles Theorem
........................194
9.5
A Discussion of Interpolation of Linear Operators
...........197
9.6
The F. and M. Riesz Theorem
............................200
Problems for Study and Exploration
............................203
10
Conjugate Functions and the Hubert Transform
...........205
10.0
Introduction
............................................205
10.1
Discovering the Hubert Transform
.........................206
10.2
The Modified Hubert Transform
...........................207
10.3
The Hubert Transform and Fourier Series
..................209
10.4
Proof of Theorem
10.3.8..................................219
Problems for Study and Exploration
............................222
11
Wolff s Proof of the Corona Theorem
......................225
11.1
Introductory Remarks
....................................226
11.2
The Banach Algebra
Я°°
.................................228
11.3
Statement of the Corona Theorem
.........................228
11.4
Carleson
Measures
.......................................229
11.5
A Key Technical Lemma
.................................234
11.6
Proof of the Corona Theorem
.............................240
11.7
Proof of Lemma
11.6.1...................................245
Problems for Study and Exploration
............................250
Part III Algebraic Topics
Overview
......................................................255
12
Automorphism Groups of Domains in the Plane
...........257
12.1
Introductory Concepts
...................................257
12.2
Noncompact Automorphism Groups
.......................259
12.3
The Dimension of the Automorphism Group
................265
12.4
The Iwasawa Decomposition
..............................268
12.5
General Properties of Holomorphic Maps
...................271
Problems for Study and Exploration
............................278
13 Cousin Problems,
Cohomology,
and Sheaves
...............281
13.1
The Cousin Problems
....................................282
13.2
A Few Words About Sheaves
.............................287
Problems for Study and Exploration
............................301
Bibliography
...................................................303
Index
..........................................................307
|
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.9 |
| dewey-search | 515/.9 |
| dewey-sort | 3515 19 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T20:10:21Z |
| institution | BVB |
| isbn | 0817643397 0817644407 9780817643393 |
| language | English |
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| spelling | Krantz, Steven G. 1951- Verfasser (DE-588)130535907 aut Geometric function theory explorations in complex analysis Steven G. Krantz Boston, Mass. [u.a.] Birkhäuser 2006 XIII, 314 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cornerstones Auch als Internetausgabe Analytische meetkunde gtt Functietheorie gtt Geometric function theory Functions of complex variables Geometrische Funktionentheorie (DE-588)4156711-0 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 s DE-604 Geometrische Funktionentheorie (DE-588)4156711-0 s Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013315371&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Krantz, Steven G. 1951- Geometric function theory explorations in complex analysis Analytische meetkunde gtt Functietheorie gtt Geometric function theory Functions of complex variables Geometrische Funktionentheorie (DE-588)4156711-0 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| subject_GND | (DE-588)4156711-0 (DE-588)4018935-1 |
| title | Geometric function theory explorations in complex analysis |
| title_auth | Geometric function theory explorations in complex analysis |
| title_exact_search | Geometric function theory explorations in complex analysis |
| title_full | Geometric function theory explorations in complex analysis Steven G. Krantz |
| title_fullStr | Geometric function theory explorations in complex analysis Steven G. Krantz |
| title_full_unstemmed | Geometric function theory explorations in complex analysis Steven G. Krantz |
| title_short | Geometric function theory |
| title_sort | geometric function theory explorations in complex analysis |
| title_sub | explorations in complex analysis |
| topic | Analytische meetkunde gtt Functietheorie gtt Geometric function theory Functions of complex variables Geometrische Funktionentheorie (DE-588)4156711-0 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Analytische meetkunde Functietheorie Geometric function theory Functions of complex variables Geometrische Funktionentheorie Funktionentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013315371&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT krantzsteveng geometricfunctiontheoryexplorationsincomplexanalysis |