Harmonic function theory:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York ; Inc
Springer
2004
|
| Edition: | Second edition |
| Series: | Graduate texts in mathematics
137 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 259 Seiten Illustrationen |
| ISBN: | 0387952187 9781441929112 9780387952185 |
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MARC
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| 100 | 1 | |a Axler, Sheldon Jay |d 1949- |0 (DE-588)113221231 |4 aut | |
| 245 | 1 | 0 | |a Harmonic function theory |c Sheldon Axler ; Paul Bourdon ; Wade Ramey |
| 250 | |a Second edition | ||
| 264 | 1 | |a New York ; Inc |b Springer |c 2004 | |
| 300 | |a XI, 259 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 137 | |
| 650 | 0 | 7 | |a Harmonische Funktion |0 (DE-588)4159122-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Harmonische Funktion |0 (DE-588)4159122-7 |D s |
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| 700 | 1 | |a Bourdon, Paul S. |0 (DE-588)17417229X |4 aut | |
| 700 | 1 | |a Ramey, Wade |4 aut | |
| 830 | 0 | |a Graduate texts in mathematics |v 137 |w (DE-604)BV000000067 |9 137 | |
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Record in the Search Index
| _version_ | 1805073738165649408 |
|---|---|
| adam_text |
Contents
Preface
ix
Acknowledgments
xi
Chapter
1
Basic Properties of Harmonic Functions
1
Definitions and Examples
. 1
Invariance
Properties
. 2
The Mean-Value Property
. 4
The Maximum Principle
. 7
The
Poisson
Kernel for the Ball
. 9
The Dirichlet Problem for the Ball
. 12
Converse of the Mean-Value Property
. 17
Real
Analytici
ty and Homogeneous Expansions
. 19
Origin of the Term "Harmonic"
. 25
Exercises
. 26
Chapter
2
Bounded Harmonic Functions
31
Liouville's Theorem
. 31
Isolated Singularities
. 32
Cauchy's Estimates
. 33
Normal Families
. 35
Maximum Principles
. 36
Limits Along Rays
. 38
Bounded Harmonic Functions on the Ball
. 40
Exercises
. 42
Chapter
3
Positive Harmonic Functions
45
Liouville's Theorem
. 45
Harnack's Inequality and Harnack's Principle
. 47
Isolated Singularities
. 50
Positive Harmonic Functions on the Ball
. 55
Exercises
. 56
Chapter
4
The Kelvin Transform
59
Inversion in the Unit Sphere
. 59
Motivation and Definition
. 61
The Kelvin Transform Preserves Harmonic Functions
. 62
Harmonicity
at Infinity
. 63
The Exterior Dirichlet Problem
. 66
Symmetry and the
Schwarz
Reflection Principle
. 67
Exercises
. 71
Chapter
5
Harmonic Polynomials
73
Polynomial Decompositions
. 74
Spherical Harmonic Decomposition of I2
(5) . 78
Inner Product of Spherical Harmonics
. 82
Spherical Harmonics Via Differentiation
. 85
Explicit Bases of 9fm(Rn) and tfm{S)
. 92
Zonal Harmonics
. 94
The
Poisson
Kernel Revisited
. 97
A Geometric Characterization of Zonal Harmonics
. 100
An Explicit Formula for Zonal Harmonics
. 104
Exercises
. 106
Chapter
6
Harmonic Hardy Spaces 111
Poisson
Integrals of Measures
.
Ill
Weak* Convergence
. 115
The Spaces h?(B)
. 117
The Hilbert Space h2(B)
. 121
The
Schwarz
Lemma
. 123
The Fatou Theorem
. 128
Exercises
. 138
Chapter
7
Harmonic Functions on Half-Spaces
143
The
Poisson
Kernel for the Upper Half-Space
.144
The Dirichlet Problem for the Upper Half-Space
.146
The Harmonic Hardy Spaces hv(H)
. 151
From the Ball to the Upper Half-Space, and Back
.153
Positive Harmonic Functions on the Upper Half-Space
.156
Nontangential Limits
.160
The Local Fatou Theorem
.161
Exercises
. 167
Chapter
8
Harmonic Bergman Spaces
171
Reproducing Kernels
. 172
The Reproducing Kernel of the Ball
.176
Examples in b"(5)
.181
The Reproducing Kernel of the Upper Half-Space
.185
Exercises
. 188
Chapter
9
The Decomposition Theorem
191
The Fundamental Solution of the Laplacian
.191
Decomposition of Harmonic Functions
.193
Bôcher's
Theorem Revisited
.197
Removable Sets for Bounded Harmonic Functions
.200
The Logarithmic Conjugation Theorem
.203
Exercises
.206
Chapter
10
Annular Regions
209
Laurent Series
.209
Isolated Singularities
.210
The Residue Theorem
.213
The
Poisson
Kernel for Annular Regions
.215
Exercises
.219
Chapter
11
The Dirichlet Problem and Boundary Behavior
223
The Dirichlet Problem
. 223
Subharmonic Functions
.224
viii
_Contents
The Perron Construction
.226
Barrier Functions and Geometric Criteria for Solvability
. 227
Nonextendability Results
.233
Exercises
.236
Appendix A
Volume, Surface Area, and Integration on Spheres
239
Volume of the Ball and Surface Area of the Sphere
.239
Slice Integration on Spheres
. 241
Exercises
.244
Appendix
В
Harmonic Function Theory and
Mathematica
247
References
249
Symbol Index 2S1
Index
255 |
| any_adam_object | 1 |
| author | Axler, Sheldon Jay 1949- Bourdon, Paul S. Ramey, Wade |
| author_GND | (DE-588)113221231 (DE-588)17417229X |
| author_facet | Axler, Sheldon Jay 1949- Bourdon, Paul S. Ramey, Wade |
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| bvnumber | BV036089950 |
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| ctrlnum | (OCoLC)634167240 (DE-599)BVBBV036089950 |
| discipline | Mathematik |
| edition | Second edition |
| format | Book |
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| id | DE-604.BV036089950 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T05:14:23Z |
| institution | BVB |
| isbn | 0387952187 9781441929112 9780387952185 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018980568 |
| oclc_num | 634167240 |
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| owner | DE-739 DE-83 DE-634 |
| owner_facet | DE-739 DE-83 DE-634 |
| physical | XI, 259 Seiten Illustrationen |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Axler, Sheldon Jay 1949- (DE-588)113221231 aut Harmonic function theory Sheldon Axler ; Paul Bourdon ; Wade Ramey Second edition New York ; Inc Springer 2004 XI, 259 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 137 Harmonische Funktion (DE-588)4159122-7 gnd rswk-swf Harmonische Funktion (DE-588)4159122-7 s DE-604 Bourdon, Paul S. (DE-588)17417229X aut Ramey, Wade aut Graduate texts in mathematics 137 (DE-604)BV000000067 137 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018980568&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Axler, Sheldon Jay 1949- Bourdon, Paul S. Ramey, Wade Harmonic function theory Graduate texts in mathematics Harmonische Funktion (DE-588)4159122-7 gnd |
| subject_GND | (DE-588)4159122-7 |
| title | Harmonic function theory |
| title_auth | Harmonic function theory |
| title_exact_search | Harmonic function theory |
| title_full | Harmonic function theory Sheldon Axler ; Paul Bourdon ; Wade Ramey |
| title_fullStr | Harmonic function theory Sheldon Axler ; Paul Bourdon ; Wade Ramey |
| title_full_unstemmed | Harmonic function theory Sheldon Axler ; Paul Bourdon ; Wade Ramey |
| title_short | Harmonic function theory |
| title_sort | harmonic function theory |
| topic | Harmonische Funktion (DE-588)4159122-7 gnd |
| topic_facet | Harmonische Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018980568&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
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