Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions
Chapter I. Riemann Surfaces -- Chapter II. Harmonic Functions on Riemann Surfaces -- Chapter III. Uniformization -- Chapter IV. Compact Riemann Surfaces -- Appendices to Chapter IV -- Chapter V. Analytic Functions of Several Complex Variables -- Chapter V. Analytic Functions of Several Complex Varia...
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| Main Author: | |
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| Format: | Book/Monograph |
| Language: | English |
| Published: |
Berlin, Heidelberg
Springer-Verlag Berlin Heidelberg
2011
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| Series: | Universitext
SpringerLink Bücher |
| DOI: | 10.1007/978-3-642-20554-5 |
| Subjects: | |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1007/978-3-642-20554-5 Resolving-System, lizenzpflichtig, Volltext: http://dx.doi.org/10.1007/978-3-642-20554-5 Cover: https://swbplus.bsz-bw.de/bsz34711136xcov.jpg Verlag, Zentralblatt MATH, Inhaltstext: https://zbmath.org/?q=an:1234.30001 
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| Author Notes: | by Eberhard Freitag |
Table of Contents:
- ""Complex Analysis II""; ""Contents""; ""Introduction""; ""I. Riemann Surfaces""; ""0. Basic Topological Notions""; ""I Metric Spaces and Their Topology""; ""II The Induced Topology""; ""III The Quotient Topology""; ""IV The Product Topology""; ""Derived Topological Notions""; ""Universal Properties of the Constructed Topologies""; ""The Induced Topology""; ""The Quotient Topology""; ""The Product Topology""; ""Topological Mappings""; ""Some Properties of Topological Spaces""; ""Locally Compact Spaces and Proper Mappings""; ""Convergent Sequences""; ""Connectedness""; ""Arc Components""
- ""Exercises for Sect. I.0""""1. The Notion of a Riemann Surface""; ""Some Simple Permanence Properties""; ""Simple Examples of Riemann Surfaces""; ""The Complex Plane as a Riemann Surface""; ""The Riemann Sphere as Riemann Surface""; ""The Torus as a Riemann Surface""; ""The Maximal Atlas of a Riemann Surface""; ""Some Elementary Properties of Riemann Surfaces""; ""Exercises for Sect. I.1""; ""2. The Analytisches Gebilde""; ""The Analytisches Gebilde as a Riemann Surface""; ""Curves on the Analytisches Gebilde""; ""The Analytisches Gebilde of an Analytic Function""; ""A Simple Example""
- ""Exercises for Sect. I.2""""3. The Riemann Surface of an Algebraic Function""; ""Alternative Construction of the Riemann Surface by Means of the Analytisches Gebilde (Sketch)""; ""Appendix A. A Special Case of Covering Theory""; ""Appendix B. A Theorem of Implicit Functions""; ""Exercises for Sect. I.3""; ""II. Harmonic Functions on Riemann Surfaces""; ""1. The Poisson Integral Formula""; ""Poisson Integral Formula""; ""The Maximum Principle""; ""Exercises for Sect. II.1""; ""2. Stability of Harmonic Functions on Taking Limits""; ""The Harnack Inequality (A. Harnack, 1887)""
- ""Harnack�s Principle""""3. The Boundary Value Problem for Disks""; ""Additional remark.""; ""Exercises for Sect. II.3""; ""4. The Formulation of the Boundary Value Problem on Riemann Surfaces and the Uniqueness of the Solution""; ""Variant of the Maximum Principle""; ""Exercises for Sect. II.4""; ""5. Solution of the Boundary Value Problem by Means of the Schwarz Alternating Method""; ""The Exceptional Points of the Boundary Value Problem""; ""Exercises for Sect. II.5""; ""6. The Normalized Solution of the External Space Problem""
- ""Construction of the Solution of the Normalized External Space Problem""""Exercises for Sect. II.6""; ""Appendix to 6. Countability of Riemann Surfaces""; ""Metrization of X""; ""Construction of a Countable Dense Subset""; ""Existence of a Harmonic Function""; ""7. Construction of Harmonic Functions with Prescribed Singularities: The Bordered Case""; ""Exercises for Sect. II.7""; ""8. Construction of Harmonic Functions with a Logarithmic singularity: The Green�s Function""; ""Green�s Function""; ""Exercises for Sect. II.8""
- ""9. Construction of Harmonic Functions with a Prescribed Singularity: The Case of a Positive Boundary""