**Translations of Mathematical Monographs**

2001;
366 pp;
Hardcover

MSC: Primary 32;

Print ISBN: 978-0-8218-0816-0

Product Code: MMONO/193

List Price: $135.00

AMS Member Price: $108.00

MAA Member Price: $121.50

**Electronic ISBN: 978-1-4704-4607-9
Product Code: MMONO/193.E**

List Price: $135.00

AMS Member Price: $108.00

MAA Member Price: $121.50

# Function Theory in Several Complex Variables

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*Toshio Nishino*

“Kiyoshi Oka, at the beginning of his research, regarded the collection of problems which he encountered in the study of domains of holomorphy as large mountains which separate today and tomorrow. Thus, he believed that there could be no essential progress in analysis without climbing over these mountains … this book is a worthwhile initial step for the reader in order to understand the mathematical world which was created by Kiyoshi Oka.”

—from the Preface

This book explains results in the theory of functions of several complex
variables which were mostly established from the late nineteenth century
through the middle of the twentieth century. In the work, the author introduces
the mathematical world created by his advisor, Kiyoshi Oka.

In this volume, Oka's work is divided into two parts. The first is the study of
analytic functions in univalent domains in \({\mathbf C}^n\). Here Oka proved that
three concepts are equivalent: domains of holomorphy, holomorphically convex
domains, and pseudoconvex domains; and moreover that the Poincaré problem,
the Cousin problems, and the Runge problem, when stated properly, can be solved
in domains of holomorphy satisfying the appropriate conditions.
The second part of Oka's work established a method for the study of analytic
functions defined in a ramified domain over \({\mathbf C}^n\) in which the branch
points are considered as interior points of the domain. Here analytic functions
in an analytic space are treated, which is a slight generalization of a
ramified domain over \({\mathbf C}^n\).

In writing the book, the author's goal was to bring to readers a real
understanding of Oka's original papers. This volume is an English translation
of the original Japanese edition, published by the University of Tokyo Press
(Japan). It would make a suitable course text for advanced graduate level
introductions to several complex variables.

#### Readership

Graduate students and research mathematicians interested in several complex variables.

#### Reviews & Endorsements

This book has its own point of view, and it is one that is not well represented in the more modern books. So it is a welcome addition to the literature … I conclude by noting that the translation is a particularly mellifluous one. The book is a pleasure to read.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Function Theory in Several Complex Variables

- Cover Cover11
- Title page v6
- Contents vii8
- Preface ix10
- Preface to the English edition xiii14
- Fundamental theory 116
- Holomorphic functions and domains of holomorphy 318
- Implicit functions and analytic sets 3752
- The Poincaré, Cousin, and Runge problems 7388
- Pseudoconvex domains and pseudoconcave sets 105120
- Holomorphic mappings 147162
- Theory of analytic spaces 165180
- Ramified domains 167182
- Analytic sets and holomorphic functions 209224
- Analytic spaces 267282
- Normal pseudoconvex spaces 321336
- Bibliography 359374
- Index 363378
- Back Cover Back Cover1383