A course in complex analysis and Riemann surfaces:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Math. Soc.
2014
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| Schriftenreihe: | Graduate studies in mathematics
154 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 384 S. graph. Darst. |
| ISBN: | 9780821898475 |
Internformat
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Datensatz im Suchindex
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| adam_text | Titel: A course in complex analysis and Riemann surfaces
Autor: Schlag, Wilhelm
Jahr: 2014
Contents Preface vii Acknowledgments xv Chapter 1. From i to 2 : the basics of complex analysis 1 §1.1. The field of complex numbers 1 §1.2. Holomorphic, analytic, and conformal 4 §1.3. The Ricmann sphere 9 §1.4. Möbius transformations 11 §1.5. The hyperbolic plane and the Poincare disk 15 §1.6. Complex integration, Cauchy theorems 18 §1.7. Applications of Cauchy’s theorems 23 §1.8. Harmonic functions 33 §1.9. Problems 36 Chapter 2. From 2 to the Riemann mapping theorem: some finer points of basic complex analysis 41 §2.1. The winding number 41 §2.2. The global form of Cauchy’s theorem 45 §2.3. Isolated singularities and residues 47 §2.4. Analytic continuation 56 §2.5. Convergence and normal families 60 §2.6. The Mittag-Leffler and Weierstrass theorems 63 §2.7. The Riemann mapping theorem 69 §2.8. Runge’s theorem and simple connectivity 74 iii
IV Contents §2.9. Problems 79 Chapter 3. Harmonic functions 85 §3.1. The Poisson kernel 85 §3.2. The Poisson kernel from the probabilistic point of view 91 §3.3. Hardy classes of harmonic: functions 95 §3.4. Almost, everywhere convergence to the boundary data 100 §3.5. Hardy spaces of analytic functions 105 §3.6. Ricsz theorems 109 §3.7. Entire functions of finite order 111 §3.8. A gallery of conformal plots 117 §3.9. Problems 122 Chapter 4. Riemann surfaces: definitions, examples, basic properties 129 §4.1. The 1 basic definitions 129 §4.2. Examples and constructions of Riemann surfaces 131 §4.3. Functions on Riemann surfaces 143 §4.4. Degree and genus 146 §4.5. Riemann surfaces as quotients 148 §4.6. Elliptic functions 151 §4.7. Covering the plane with two or more points removed 160 §4.8. Groups of Möbius transforms 164 §4.9. Problems 174 Chapter 5. Analytic continuation, covering surfaces, and algebraic functions 179 §5.1. Analytic continuation 179 §5.2. The unramified Riemann surface of an analytic germ 185 §5.3. The ramified Riemann surface of an analytic germ 189 §5.4. Algebraic germs and functions 192 §5.5. Algebraic equations generated by compact surfaces 199 §5.6. Some compact surfaces and their associated polynomials 206 §5.7. ODEs with merornorphic, coefficients 211 §5.8. Problems 221 Chapter 6. Differential forms on Riemann surfaces 225 §6.1. Holomorphic and merornorphic differentials 225 §6.2. Integrating differentials and residues 227
Contents v §6.3. The Hodge-* operator and harmonic differentials 230 §6.4. Statement and examples of the Hodge decomposition 236 §6.5. Weyl’s lemma and the Hodge decomposition 244 §6.6. Existence of nonconstant meromorphic functions 250 §6.7. Examples of meromorphic functions and differentials 258 §6.8. Problems 266 Chapter 7. The Theorems of Riemann-Roch, Abel, and Jacobi 269 §7.1. Homology bases and holomorphic differentials 269 §7.2. Periods and bilinear relations 273 §7.3. Divisors 280 §7.4. The Riemann-Roch theorem 285 §7.5. Applications and general divisors 289 §7.6. Applications to algebraic curves 292 §7.7. The theorems of Abel and Jacobi 295 §7.8. Problems 303 Chapter 8. Uniformization 305 §8.1. Green functions and Riemann mapping 306 §8.2. Perron families 310 §8.3. Solution of Dirichlet’s problem 314 §8.4. Green’s functions on Riemann surfaces 317 §8.5. Uniformization for simply-connected surfaces 326 §8.6. Uniformization of non-simply-connected surfaces 335 §8.7. Fuchsian groups 338 §8.8. Problems 349 Appendix A. Review of some basic background material 353 §A.l. Geometry and topology 353 §A.2. Algebra 363 §A.3. Analysis 365 Bibliography 371 Index 377
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| isbn | 9780821898475 |
| language | English |
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| publishDate | 2014 |
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| spelling | Schlag, Wilhelm 1969- Verfasser (DE-588)1022462636 aut A course in complex analysis and Riemann surfaces Wilhelm Schlag Providence, RI American Math. Soc. 2014 XV, 384 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 154 Riemannsche Fläche (DE-588)4049991-1 gnd rswk-swf Komplexe Funktion (DE-588)4217733-9 gnd rswk-swf Riemannsche Fläche (DE-588)4049991-1 s Komplexe Funktion (DE-588)4217733-9 s DE-604 Graduate studies in mathematics 154 (DE-604)BV009739289 154 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027480194&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schlag, Wilhelm 1969- A course in complex analysis and Riemann surfaces Graduate studies in mathematics Riemannsche Fläche (DE-588)4049991-1 gnd Komplexe Funktion (DE-588)4217733-9 gnd |
| subject_GND | (DE-588)4049991-1 (DE-588)4217733-9 |
| title | A course in complex analysis and Riemann surfaces |
| title_auth | A course in complex analysis and Riemann surfaces |
| title_exact_search | A course in complex analysis and Riemann surfaces |
| title_full | A course in complex analysis and Riemann surfaces Wilhelm Schlag |
| title_fullStr | A course in complex analysis and Riemann surfaces Wilhelm Schlag |
| title_full_unstemmed | A course in complex analysis and Riemann surfaces Wilhelm Schlag |
| title_short | A course in complex analysis and Riemann surfaces |
| title_sort | a course in complex analysis and riemann surfaces |
| topic | Riemannsche Fläche (DE-588)4049991-1 gnd Komplexe Funktion (DE-588)4217733-9 gnd |
| topic_facet | Riemannsche Fläche Komplexe Funktion |
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