Potential theory in the complex plane:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
1995
|
| Edition: | 1. publ. |
| Series: | London Mathematical Society student texts
28 |
| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | X, 232 S. |
| ISBN: | 0521461200 0521466547 |
Staff View
MARC
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Record in the Search Index
| _version_ | 1804125092921212928 |
|---|---|
| adam_text | Contents
Preface
ix
A Word about Notation
1
1
Harmonic Functions
3
1.1
Harmonic and Holomorphic Functions
............ 3
1.2
The Dirichlet Problem on the Disc
.............. 7
1.3
Positive Harmonic Functions
................. 13
Notes on Chapter
1.......................... 22
2
Sub harmonic Functions
25
2.1
Upper Semicontinuous Functions
............... 25
2.2
Subharmonic Functions
.................... 27
2.3
The Maximum Principle
.................... 29
2.4
Criteria for Subharmonicity
.................. 35
2.5
Integrability
........................... 39
2.6
Convexity
............................ 43
2.7
Smoothing
............................ 48
Notes on Chapter
2.......................... 51
3
Potential Theory
53
3.1
Potentials
............................ 53
3.2
Polar Sets
............................ 55
3.3
Equilibrium Measures
..................... 58
3.4
Upper Semicontinuous Regularization
............ 62
3.5
Minus-Infinity Sets
....................... 65
3.6
Removable Singularities
.................... 67
3.7
The Generalized Laplacian
................... 71
3.8
Thinness
............................. 78
Notes on Chapter
3.......................... 82
viii CONTENTS
4
The Dirichlet Problem
85
4.1
Solution of the Dirichlet Problem
............... 85
4.2
Criteria for Regularity
..................... 91
4.3
Harmonic Measure
....................... 96
4.4
Green s Functions
....................... 106
4.5
The
Poisson-
Jensen Formula
................. 116
Notes on Chapter
4.......................... 126
5
Capacity
127
5.1
Capacity as a Set Function
.................. 127
5.2
Computation of Capacity
................... 132
5.3
Estimation of Capacity
..................... 137
5.4
Criteria for Thinness
...................... 146
5.5
Transfinite
Diameter
...................... 152
Notes on Chapter
5.......................... 160
6
Applications
161
6.1
Interpolation of
/Лѕрасеѕ
................... 161
6.2
Homogeneous Polynomials
................... 166
6.3
Uniform Approximation
.................... 169
6.4
Banach Algebras
........................ 176
6.5
Complex Dynamics
....................... 190
Notes on Chapter
6.......................... 206
A Borei
Measures
209
A.I Supports
............................. 209
A.2 Regularity
............................ 210
A.3 Radon Measures
........................ 211
A.
4
Weak*-Convergence
...................... 216
Bibliography
219
Index
225
Glossary of Notation
231
Potential theory is the broad area of mathematical analysis encompassing
such topics as harmonic and subharmonic functions, the Dirichlet problem,
harmonic measure, Green s functions, potentials and capacity. This is an
introduction to the subject suitable for beginning graduate students,
concentrating on the important case of two dimensions. This permits a
simpler treatment than other books, yet is still sufficient for a wide range of
applications to complex analysis. These include
Picards
theorem, the
Phragmen-Lindelof principle, the Koebe one-quarter mapping theorem and a
sharp quantitative form of Runge s theorem. In addition there is a chapter on
connections with functional analysis and dynamical systems, which shows
how the theory can be applied to other parts of mathematics and gives a
flavour of some recent research.
Exercises are provided throughout, enabling the book to be used with
advanced courses on complex analysts or potential theory.
London Mathematical Society
Student Texts
Edited by
C
M.
SERIES, Mathematics Institute, University of Warwick,
Coventry CV4 7AL, United Kingdom
with assistance from
P. Diacoms {Harvard)
E Rees
(Edinburgh)
M
Roñan
{Birmingham)
M. Mtcallef
(
Warwick)
This
senes
consists of textbooks aimed at advanced undergraduates or
beginning graduate students. It will cover the whole range of pure
mathematics, as well as topics in applied mathematics and mathematical
physics that involve a substantial use of modern mathematical methods.
Topics not of a standard nature though of current interest are also covered by
volumes in this series.
|
| any_adam_object | 1 |
| author | Ransford, Thomas 1958- |
| author_GND | (DE-588)143777912 |
| author_facet | Ransford, Thomas 1958- |
| author_role | aut |
| author_sort | Ransford, Thomas 1958- |
| author_variant | t r tr |
| building | Verbundindex |
| bvnumber | BV010618526 |
| classification_rvk | SK 560 SK 780 |
| classification_tum | MAT 310f |
| ctrlnum | (OCoLC)490145830 (DE-599)BVBBV010618526 |
| 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 | 1. publ. |
| format | Book |
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| id | DE-604.BV010618526 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:56:04Z |
| institution | BVB |
| isbn | 0521461200 0521466547 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007083726 |
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| physical | X, 232 S. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | London Mathematical Society student texts |
| series2 | London Mathematical Society student texts |
| spelling | Ransford, Thomas 1958- Verfasser (DE-588)143777912 aut Potential theory in the complex plane Thomas Ransford 1. publ. Cambridge [u.a.] Cambridge Univ. Press 1995 X, 232 S. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society student texts 28 Hier auch später erschienene, unveränderte Nachdrucke Fonctions d'une variable complexe ram Potentiel, Théorie du ram Komplexe Ebene (DE-588)4481425-2 gnd rswk-swf Potenzialtheorie (DE-588)4046939-6 gnd rswk-swf Potenzialtheorie (DE-588)4046939-6 s Komplexe Ebene (DE-588)4481425-2 s DE-604 London Mathematical Society student texts 28 (DE-604)BV000841726 28 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007083726&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007083726&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ransford, Thomas 1958- Potential theory in the complex plane London Mathematical Society student texts Fonctions d'une variable complexe ram Potentiel, Théorie du ram Komplexe Ebene (DE-588)4481425-2 gnd Potenzialtheorie (DE-588)4046939-6 gnd |
| subject_GND | (DE-588)4481425-2 (DE-588)4046939-6 |
| title | Potential theory in the complex plane |
| title_auth | Potential theory in the complex plane |
| title_exact_search | Potential theory in the complex plane |
| title_full | Potential theory in the complex plane Thomas Ransford |
| title_fullStr | Potential theory in the complex plane Thomas Ransford |
| title_full_unstemmed | Potential theory in the complex plane Thomas Ransford |
| title_short | Potential theory in the complex plane |
| title_sort | potential theory in the complex plane |
| topic | Fonctions d'une variable complexe ram Potentiel, Théorie du ram Komplexe Ebene (DE-588)4481425-2 gnd Potenzialtheorie (DE-588)4046939-6 gnd |
| topic_facet | Fonctions d'une variable complexe Potentiel, Théorie du Komplexe Ebene Potenzialtheorie |
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