**The Carus Mathematical Monographs**

Volume: 23;
2004;
219 pp;
Hardcover

MSC: Primary 30;
Secondary 32

**Print ISBN: 978-0-88385-035-0
Product Code: CAR/23**

List Price: $56.00

AMS Member Price: $42.00

MAA Member Price: $42.00

**Electronic ISBN: 978-0-88385-968-1
Product Code: CAR/23.E**

List Price: $56.00

AMS Member Price: $42.00

MAA Member Price: $42.00

# Complex Analysis: The Geometric Viewpoint: Second Edition

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*Steven G. Krantz*

MAA Press: An Imprint of the American Mathematical Society

Recipient 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 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. 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.

#### Reviews & Endorsements

A first-rate 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 function-theoretic background is needed and no knowledge whatsoever of differential geometry is assumed.

-- Mathematical Reviews

# Table of Contents

## Complex Analysis: The Geometric Viewpoint: Second Edition

- Cover cover11
- Title iii4
- Preface to the Second Edition xi12
- Preface to the First Edition xiii14
- Contents xv16
- 0. Principal Ideas of Classical Function Theory 120
- 1. Basic Notions of Differential Geometry 2948
- Ch. 2. Curvature and Applications 6786
- Ch. 3. Some New Invariant Metrics 89108
- Ch. 4. Introduction to the Bergman Theory 137156
- Ch. 5. A Glimpse of Several Complex Variables 161180
- Epilogue 189208
- Appendix on the Structure Equations and Curvature 191210
- Table of Symbols 205224
- References 209228
- Index 213232