Cover image for Complex Analysis : The Geometric Viewpoint.
Complex Analysis : The Geometric Viewpoint.
Title:
Complex Analysis : The Geometric Viewpoint.
Author:
Krantz, Steven G.
ISBN:
9780883859681
Personal Author:
Edition:
2nd ed.
Physical Description:
1 online resource (234 pages)
Series:
Carus ; v.23

Carus
Contents:
Cover -- Title -- Preface to the Second Edition -- Preface to the First Edition -- Contents -- 0. Principal Ideas of Classical Function Theory -- 1. A Glimpse of Complex Analysis -- 2. The Maximum Principle, the Schwarz Lemma, and Applications -- 3. Normal Families and the Riemann Mapping Theorem -- 4. Isolated Singularities and the Theorems of Picard -- 1. Basic Notions of Differential Geometry -- 0. Introductory Remarks -- 1. Riemannian Metrics and the Concept of Length -- 2. Calculus in the Complex Domain -- 3. Isometries -- 4. The Poincaré Metric -- 5. The Schwarz Lemma -- 6. A Detour into Non-Euclidean Geometry -- Ch. 2. Curvature and Applications -- 1. Curvature and the Schwarz Lemma Revisited -- 2. Liouville's Theorem and Other Applications -- 3. Normal Families and the Spherical Metric -- 4. A Generalization of Montel's Theorem and the Great Picard Theorem -- Ch. 3. Some New Invariant Metrics -- 0. Introductory Remarks -- 1. The Carathéodory Metric -- 2. The Kobayashi Metric -- 3. Completeness of the Carathéodory and Kobayashi Metrics -- 4. An Application of Completeness: Automorphisms -- 5. Hyperbolicity and Curvature -- Ch. 4. Introduction to the Bergman Theory -- 0. Introductory Remarks -- 1. Bergman Basics -- 2. Invariance Properties of the Bergman Kernel -- 3. Calculation of the Bergman Kernel -- 3.1 Construction of the Bergman Kernel for the Disc by Conformal Invariance -- 3.2 Construction of the Bergman Kernel by Means of an Orthonormal System -- 3.3 The Bergman Kernel by Way of Differential Equations -- krantz_chap5 -- 4. About the Bergman Metric -- 5. More on the Bergman Metric -- 6. Application to Conformal Mapping -- Ch. 5. A Glimpse of Several Complex Variables -- 0. Functions of Several Complex Variables -- 1. Basic Concepts -- 2. The Automorphism Groups of the Ball and Bidisc.

3. Invariant Metrics and the Inequivalence of the Ball and the Bidisc -- Epilogue -- Appendix on the Structure Equations and Curvature -- 1. Introduction -- 2. Expressing Curvature Intrinsically -- Table of Symbols -- References -- Index.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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