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Complex Analysis 2 : Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions / by Eberhard Freitag
Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Universitext | SpringerLink BücherPublisher: Berlin, Heidelberg : Springer-Verlag Berlin Heidelberg, 2011Description: Online-Ressource (XIII, 506p. 51 illus, digital)ISBN:- 9783642205545
- 515.94
- QA331.7
- QA331
Contents:
Summary: 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 Variable -- Chapter VI. Abelian Functions -- Chapter VII. Modular Forms of Several Variables -- Chapter VIII. Appendix: Algebraic Tools -- References -- Index.Summary: The book provides a complete presentation of complex analysis, starting with the theory of Riemann surfaces, including uniformization theory and a detailed treatment of the theory of compact Riemann surfaces, the Riemann-Roch theorem, Abel's theorem and Jacobi's inversion theorem. This motivates a short introduction into the theory of several complex variables, followed by the theory of Abelian functions up to the theta theorem. The last part of the book provides an introduction into the theory of higher modular functions.PPN: PPN: 1650970560Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""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""
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