Harmonic function theory:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
2001
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Graduate texts in mathematics
137 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 259 S. graph. Darst. : 25 cm |
| ISBN: | 0387952187 |
Internformat
MARC
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| 100 | 1 | |a Axler, Sheldon Jay |d 1949- |e Verfasser |0 (DE-588)113221231 |4 aut | |
| 245 | 1 | 0 | |a Harmonic function theory |c Sheldon Axler ; Paul Bourdon ; Wade Ramey |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2001 | |
| 300 | |a XI, 259 S. |b graph. Darst. : 25 cm | ||
| 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 | 7 | |a Complexe variabelen |2 gtt | |
| 650 | 4 | |a Fonctions harmoniques | |
| 650 | 7 | |a Functies (wiskunde) |2 gtt | |
| 650 | 4 | |a Harmonic functions | |
| 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 |e Verfasser |4 aut | |
| 700 | 1 | |a Ramey, Wade |e Verfasser |4 aut | |
| 830 | 0 | |a Graduate texts in mathematics |v 137 |w (DE-604)BV000000067 |9 137 | |
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Datensatz im Suchindex
| _version_ | 1804128491449352192 |
|---|---|
| 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 Analyticity 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
v
vi Contents
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 L2 (5) 78
Inner Product of Spherical Harmonics 82
Spherical Harmonics Via Differentiation 85
Explicit Bases of 5fm(RM) and Jfm(5) 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
Contents vjj
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 hp(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 bp(B) 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
Bocher 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
vin 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 B
Harmonic Function Theory and Mathematica 247
References 249
Symbol Index 251
Index 255
|
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
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| id | DE-604.BV013678621 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:50:05Z |
| institution | BVB |
| isbn | 0387952187 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009346201 |
| oclc_num | 45315455 |
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| physical | XI, 259 S. graph. Darst. : 25 cm |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Axler, Sheldon Jay 1949- Verfasser (DE-588)113221231 aut Harmonic function theory Sheldon Axler ; Paul Bourdon ; Wade Ramey 2. ed. New York [u.a.] Springer 2001 XI, 259 S. graph. Darst. : 25 cm txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 137 Complexe variabelen gtt Fonctions harmoniques Functies (wiskunde) gtt Harmonic functions Harmonische Funktion (DE-588)4159122-7 gnd rswk-swf Harmonische Funktion (DE-588)4159122-7 s DE-604 Bourdon, Paul Verfasser aut Ramey, Wade Verfasser aut Graduate texts in mathematics 137 (DE-604)BV000000067 137 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009346201&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Axler, Sheldon Jay 1949- Bourdon, Paul Ramey, Wade Harmonic function theory Graduate texts in mathematics Complexe variabelen gtt Fonctions harmoniques Functies (wiskunde) gtt Harmonic functions 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 | Complexe variabelen gtt Fonctions harmoniques Functies (wiskunde) gtt Harmonic functions Harmonische Funktion (DE-588)4159122-7 gnd |
| topic_facet | Complexe variabelen Fonctions harmoniques Functies (wiskunde) Harmonic functions Harmonische Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009346201&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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