Complex Analysis: The Geometric Viewpoint

"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.