Complex analysis 2: Riemann surfaces, several complex variables, Abelian Functions, higher modular functions
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English German |
| Published: |
Berlin [u.a.]
Springer
2011
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Bildet Ergänzung zu: Complex Analysis / Eberhard Freitag |
| Physical Description: | XIII, 506 Seiten graph. Darst. |
| ISBN: | 9783642205538 3642205534 |
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| adam_text |
IMAGE 1
CONTENTS
CHAPTER I. RIEMANN SURFACES 1
0 BASIC TOPOLOGICAL NOTIONS 3
1 THE NOTION OF A RIEMANN SURFACE 13
2 THE ANALYTISCHES GEBILDE 27
3 THE RIEMANN SURFACE OF AN ALGEBRAIC FUNCTION 35
APPENDIX A. A SPECIAL CASE OF COVERING THEORY 46
APPENDIX B. A THEOREM OF IMPLICIT FUNCTIONS 51
CHAPTER II. HARMONIC FUNCTIONS ON RIEMANN SURFACES 54
1 THE POISSON INTEGRAL FORMULA 56
2 STABILITY OF HARMONIC FUNCTIONS ON TAKING LIMITS 60
3 THE BOUNDARY VALUE PROBLEM FOR DISKS 63
4 THE FORMULATION OF THE BOUNDARY VALUE PROBLEM ON RIEMANN SURFACES AND
THE UNIQUENESS OF THE SOLUTION 68
5 SOLUTION OF THE BOUNDARY VALUE PROBLEM BY MEANS OF THE SCHWARZ
ALTERNATING METHOD 74
6 THE NORMALIZED SOLUTION OF THE EXTERNAL SPACE PROBLEM 80
APPENDIX. COUNTABILITY OF RIEMANN SURFACES 88
7 CONSTRUCTION OF HARMONIC FUNCTIONS WITH PRESCRIBED SINGULARITIES: THE
BORDERED CASE 91
8 CONSTRUCTION OF HARMONIC FUNCTIONS WITH A LOGARITHMIC SINGULARITY: THE
GREEN'S FUNCTION 95
9 CONSTRUCTION OF HARMONIC FUNCTIONS WITH A PRESCRIBED SINGULARITY: THE
CASE OF A POSITIVE BOUNDARY 99
10 A LEMMA OF NEVANLINNA 103
11 CONSTRUCTION OF HARMONIC FUNCTIONS WITH A PRESCRIBED SINGULARITY: THE
CASE OF A ZERO BOUNDARY 113
12 THE MOST IMPORTANT CASES OF THE EXISTENCE THEOREMS 118
13 APPENDIX TO CHAPTER II. STOKES'S THEOREM 120
CHAPTER III. UNIFORMIZATION 141
1 THE UNIFORMIZATION THEOREM 143
2 A ROUGH CLASSIFICATION OF RIEMANN SURFACES 152
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1010702688
DIGITALISIERT DURCH
IMAGE 2
VIII CONTENTS
3 PICARD'S THEOREMS 159
4 APPENDIX A. THE FUNDAMENTAL GROUP 163
5 APPENDIX B. THE UNIVERSAL COVERING 170
6 APPENDIX C. THE MONODROMY THEOREM 180
CHAPTER IV. COMPACT RIEMANN SURFACES 184
1 MEROMORPHIC DIFFERENTIALS 184
2 COMPACT RIEMANN SURFACES AND ALGEBRAIC FUNCTIONS 192
3 THE TRIANGULATION OF A COMPACT RIEMANN SURFACE 204
APPENDIX. THE RIEMANN-HURWITZ RAMIFICATION FORMULA 208 4 COMBINATORIAL
SCHEMES 211
5 GLUING OF BOUNDARY EDGES 218
6 THE NORMAL FORM OF COMPACT RIEMANN SURFACES 222
7 DIFFERENTIALS OF THE FIRST KIND 231
APPENDIX. THE POLYHEDRON THEOREM 237
8 SOME PERIOD RELATIONS 238
APPENDIX. PIECEWISE SMOOTHNESS 243
9 THE RIEMANN -ROCH THEOREM 245
10 MORE PERIOD RELATIONS 254
11 ABEL'S THEOREM 261
12 THE JACOBI INVERSION PROBLEM 270
APPENDIX. CONTINUITY OF ROOTS 278
APPENDICES TO CHAPTER IV 279
13 MULTICANONICAL FORMS 279
14 DIMENSIONS OF VECTOR SPACES OF MODULAR FORMS 285
15 DIMENSIONS OF VECTOR SPACES OF MODULAR FORMS WITH MULTIPLIER SYSTEMS
295
CHAPTER V. ANALYTIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES 300
1 ELEMENTARY PROPERTIES OF ANALYTIC FUNCTIONS OF SEVERAL VARIABLES 300
2 POWER SERIES IN SEVERAL VARIABLES 302
3 ANALYTIC MAPS 311
4 THE WEIERSTRASS PREPARATION THEOREM 317
5 REPRESENTATION OF MEROMORPHIC FUNCTIONS AS QUOTIENTS OF ANALYTIC
FUNCTIONS 326
UE ALTERNATING DIFFERENTIAL FORMS 337
IMAGE 3
CONTENTS IX
CHAPTER V I. A B E L I AN FUNCTIONS 347
1 LATTICES AND TORI 347
2 HODGE THEORY OF THE REAL TORUS 352
3 HODGE THEORY OF A COMPLEX TORUS 355
4 AUTOMORPHY SUMMANDS 357
5 QUASI-HERMITIAN FORMS ON LATTICES 364
6 RIEMANNIAN FORMS 372
7 CANONICAL LATTICE BASES 378
8 THETA SERIES (CONSTRUCTION OF THE SPACES [Q, I, E}) 384
APPENDIX. COMPLEX FOURIER SERIES 392
9 GRADED RINGS OF THETA SERIES 393
10 A NONDEGENERATENESS THEOREM 396
11 THE FIELD OF ABELIAN FUNCTIONS 403
12 POLARIZED ABELIAN MANIFOLDS 408
13 THE LIMITS OF CLASSICAL COMPLEX ANALYSIS 412
CHAPTER VII. MODULAR FORMS OF SEVERAL VARIABLES 427
1 SIEGEL'S MODULAR GROUP 427
2 THE NOTION OF A MODULAR FORM OF DEGREE N 431
3 KOECHER'S PRINCIPLE 435
4 SPECIALIZATION OF MODULAR FORMS 438
5 GENERATORS FOR SOME MODULAR GROUPS 441
6 COMPUTATION OF SOME INDICES 449
7 THETA SERIES 452
8 GROUP-THEORETIC CONSIDERATIONS 459
9 IGUSA'S CONGRUENCE SUBGROUPS 462
10 THE FUNDAMENTAL DOMAIN OF THE MODULAR GROUP OF DEGREE TWO 466 11 THE
ZEROS OF THE THETA SERIES OF DEGREE TWO 469
12 A RING OF MODULAR FORMS 475
CHAPTER VIII. APPENDIX: ALGEBRAIC TOOLS 483
1 DIVISIBILITY 483
2 FACTORIAL RINGS (UFD RINGS) 485
3 THE DISCRIMINANT 488
4 ALGEBRAIC FUNCTION FIELDS 490
REFERENCES 494
INDEX 499 |
| any_adam_object | 1 |
| author | Freitag, Eberhard 1942- |
| author_GND | (DE-588)142950017 |
| author_facet | Freitag, Eberhard 1942- |
| author_role | aut |
| author_sort | Freitag, Eberhard 1942- |
| author_variant | e f ef |
| building | Verbundindex |
| bvnumber | BV039158437 |
| classification_rvk | SK 750 SK 700 |
| classification_tum | MAT 300 |
| ctrlnum | (OCoLC)725042306 (DE-599)DNB1010702688 |
| discipline | Mathematik |
| format | Book |
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| genre_facet | Lehrbuch |
| id | DE-604.BV039158437 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T11:15:41Z |
| institution | BVB |
| isbn | 9783642205538 3642205534 |
| language | English German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024176013 |
| oclc_num | 725042306 |
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| physical | XIII, 506 Seiten graph. Darst. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| series2 | Universitext |
| spelling | Freitag, Eberhard 1942- Verfasser (DE-588)142950017 aut Funktionentheorie Complex analysis 2 Riemann surfaces, several complex variables, Abelian Functions, higher modular functions Eberhard Freitag Berlin [u.a.] Springer 2011 XIII, 506 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Bildet Ergänzung zu: Complex Analysis / Eberhard Freitag Riemannsche Fläche (DE-588)4049991-1 gnd rswk-swf Funktionstheorie (DE-588)4301988-2 gnd rswk-swf Modulform (DE-588)4128299-1 gnd rswk-swf Abelsche Funktion (DE-588)4140987-5 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Riemannsche Fläche (DE-588)4049991-1 s Abelsche Funktion (DE-588)4140987-5 s Modulform (DE-588)4128299-1 s DE-604 Funktionentheorie (DE-588)4018935-1 s 2\p DE-604 Funktionstheorie (DE-588)4301988-2 s 3\p DE-604 Erscheint auch als Online-Ausgabe 978-3-642-20554-5 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3705615&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024176013&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Freitag, Eberhard 1942- Complex analysis 2 Riemann surfaces, several complex variables, Abelian Functions, higher modular functions Riemannsche Fläche (DE-588)4049991-1 gnd Funktionstheorie (DE-588)4301988-2 gnd Modulform (DE-588)4128299-1 gnd Abelsche Funktion (DE-588)4140987-5 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| subject_GND | (DE-588)4049991-1 (DE-588)4301988-2 (DE-588)4128299-1 (DE-588)4140987-5 (DE-588)4018935-1 (DE-588)4123623-3 |
| title | Complex analysis 2 Riemann surfaces, several complex variables, Abelian Functions, higher modular functions |
| title_alt | Funktionentheorie |
| title_auth | Complex analysis 2 Riemann surfaces, several complex variables, Abelian Functions, higher modular functions |
| title_exact_search | Complex analysis 2 Riemann surfaces, several complex variables, Abelian Functions, higher modular functions |
| title_full | Complex analysis 2 Riemann surfaces, several complex variables, Abelian Functions, higher modular functions Eberhard Freitag |
| title_fullStr | Complex analysis 2 Riemann surfaces, several complex variables, Abelian Functions, higher modular functions Eberhard Freitag |
| title_full_unstemmed | Complex analysis 2 Riemann surfaces, several complex variables, Abelian Functions, higher modular functions Eberhard Freitag |
| title_short | Complex analysis 2 |
| title_sort | complex analysis 2 riemann surfaces several complex variables abelian functions higher modular functions |
| title_sub | Riemann surfaces, several complex variables, Abelian Functions, higher modular functions |
| topic | Riemannsche Fläche (DE-588)4049991-1 gnd Funktionstheorie (DE-588)4301988-2 gnd Modulform (DE-588)4128299-1 gnd Abelsche Funktion (DE-588)4140987-5 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Riemannsche Fläche Funktionstheorie Modulform Abelsche Funktion Funktionentheorie Lehrbuch |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3705615&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024176013&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT freitageberhard funktionentheorie AT freitageberhard complexanalysis2riemannsurfacesseveralcomplexvariablesabelianfunctionshighermodularfunctions |