Complex Analysis: The Geometric ViewpointCambridge University Press, 2004 - 219 pages "The second edition of a classic Carus Monograph and a winner of the Beckenbach Book Prize! In this second edition of a Carus Monograph Classic, Steven Krantz develops material on classical non-Euclidean geometry. He shows how it can be developed in a natural way from the invariant geometry of the complex disc. He also introduces the Bergman kernel and metric and provides profound applications, some of them never having appeared before in print. In general, the new edition represents a considerable polishing and re-thinking of the original successful volume. Complex Analysis: The Geometric Viewpoint is the first and only book to describe the context, the background, the details, and the applications of Ahlfors's celebrated ideas about curvature, the Schwarz lemma, and applications in complex analysis. This new edition represents a considerable polishing and re-thinking of the original successful volume. It develops material on classical non-Euclidean geometry of the complex disc. Beginning from scratch, and requiring only a minimal background in complex variable theory, this book takes the reader up to ideas that are currently active areas of study. Such areas include the Caratheodory and Kobayashi metrics, the Bergman kernel and metric, and boundary continuation of conformal maps. There is also an introduction to the theory of several complex variables. Poincare's celebrated theorem about the biholomorphic inequivalence of the ball and polydisc is discussed and proved. Krantz is a leading researcher in complex analysis and a well-known mathematical expositor. His style is light and inviting, making this book accessible while also authoritative and precise. Complex Analysis: The Geometric Viewpoint will appeal and delight anyone interested in complex analysis."--Publisher's description. |
Table des matières
Acknowledgments | ix |
Preface to the Second Edition | xi |
Preface to the First Edition | xiii |
Principal Ideas of Classical Function Theory | 1 |
2 The Maximum Principle the Schwarz Lemma and Applications | 11 |
3 Normal Families and the Riemann Mapping Theorem | 16 |
4 Isolated Singularities and the Theorems of Picard | 22 |
Basic Notions of Differential Geometry | 29 |
3 Completeness of the Caratheodory and Kobayashi Metrics | 103 |
Automorphisms | 121 |
5 Hyperbolicity and Curvature | 133 |
Introduction to the Bergman Theory | 137 |
1 Bergman Basics | 138 |
2 Invariance Properties of the Bergman Kernel | 140 |
3 Calculation of the Bergman Kernel | 143 |
4 About the Bergman Metric | 151 |
1 Riemannian Metrics and the Concept of Length | 30 |
2 Calculus in the Complex Domain | 38 |
3 Isometrics | 42 |
4 The Poincare Metric | 45 |
5 The Schwarz Lemma | 55 |
6 A Detour into NonEuclidean Geometry | 58 |
Curvature and Applications | 67 |
2 Liouvilles Theorem and Other Applications | 73 |
3 Normal Families and the Spherical Metric | 79 |
4 A Generalization of Montels Theorem and the Great Picard Theorem | 86 |
Some New Invariant Metrics | 89 |
1 The Caratheodory Metric | 90 |
2 The Kobayashi Metric | 93 |
5 More on the Bergman Metric | 155 |
6 Application to Conformal Mapping | 156 |
A Glimpse of Several Complex Variables | 161 |
1 Basic Concepts | 164 |
2 The Automorphism Groups of the Ball and Bidisc | 170 |
3 Invariant Metrics and the Inequivalence of the Ball and the Bidisc | 180 |
Appendix | 191 |
3 Curvature Calculations on Planar Domains | 200 |
Symbols | 205 |
| 209 | |
| 213 | |
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Expressions et termes fréquents
analysis analytic apply assertion assume Aut(U automorphism ball Bergman kernel boundary bounded calculate called Carathéodory metric Cauchy Chapter circle classical closed compact complex variables conclude conformal mapping connected consider constant construction contains continuously differentiable converges Corollary course curvature curve define definition derivative discussion distance domain elements equal equations equicontinuity equivalent Euclidean Example exercise existence expression fact Figure fixed follows formula geometry given gives hence holomorphic function integral invariant isometry Kobayashi metric least length Let F limit linear mathematics neighborhood normal family Notice notion obtain plane Poincaré metric positive Proof Proposition prove Remark result Riemann rotation satisfies Schwarz lemma sequence shows singularity space subsequence suppose tangent theorem theory transformation unit disc values vector write zero