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eBook ISBN:  9780883859681 
Product Code:  CAR/23.E 
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Hardcover ISBN:  9780883850350 
eBook: ISBN:  9780883859681 
Product Code:  CAR/23.B 
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AMS Member Price:  $90.00 $69.38 
Hardcover ISBN:  9780883850350 
Product Code:  CAR/23 
List Price:  $65.00 
MAA Member Price:  $48.75 
AMS Member Price:  $48.75 
eBook ISBN:  9780883859681 
Product Code:  CAR/23.E 
List Price:  $55.00 
MAA Member Price:  $41.25 
AMS Member Price:  $41.25 
Hardcover ISBN:  9780883850350 
eBook ISBN:  9780883859681 
Product Code:  CAR/23.B 
List Price:  $120.00 $92.50 
MAA Member Price:  $90.00 $69.38 
AMS Member Price:  $90.00 $69.38 

Book DetailsThe Carus Mathematical MonographsVolume: 23; 2004; 219 ppMSC: Primary 30; Secondary 32Recipient of the Mathematical Association of America's Beckenbach Book Prize in 1994!
In this second edition of a Carus Monograph Classic, Steven Krantz develops material on classical nonEuclidean 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 rethinking of the original successful volume. This 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.
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 a) the Caratheodory and Kobayashi metrics, b) the Bergman kernel and metric, and c) boundary continuation of conformal maps. There is also an introduction to the theory of several complex variables. Poincaré's celebrated theorem about the biholomorphic inequivalence of the ball and polydisc is discussed and proved.

Table of Contents

Chapters

Chapter 0. Principal Ideas of Classical Function Theory

Chapter 1. Basic Notions of Differential Geometry

Chapter 2. Curvature and Applications

Chapter 3. Some New Invariant Metrics

Chapter 4. Introduction to the Bergman Theory

Chapter 5. A Glimpse of Several Complex Variables


Additional Material

Reviews

A firstrate book, which can be used either as a text or reference.
Choice 
In five very nicely written chapters this book gives an introduction to the approach to function theory via Riemannian geometry. Very little functiontheoretic background is needed and no knowledge whatsoever of differential geometry is assumed.
Mathematical Reviews


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In this second edition of a Carus Monograph Classic, Steven Krantz develops material on classical nonEuclidean 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 rethinking of the original successful volume. This 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.
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 a) the Caratheodory and Kobayashi metrics, b) the Bergman kernel and metric, and c) boundary continuation of conformal maps. There is also an introduction to the theory of several complex variables. Poincaré's celebrated theorem about the biholomorphic inequivalence of the ball and polydisc is discussed and proved.

Chapters

Chapter 0. Principal Ideas of Classical Function Theory

Chapter 1. Basic Notions of Differential Geometry

Chapter 2. Curvature and Applications

Chapter 3. Some New Invariant Metrics

Chapter 4. Introduction to the Bergman Theory

Chapter 5. A Glimpse of Several Complex Variables

A firstrate book, which can be used either as a text or reference.
Choice 
In five very nicely written chapters this book gives an introduction to the approach to function theory via Riemannian geometry. Very little functiontheoretic background is needed and no knowledge whatsoever of differential geometry is assumed.
Mathematical Reviews