**AMS Chelsea Publishing**

Volume: 368;
1965;
317 pp;
Hardcover

MSC: Primary 32;

Print ISBN: 978-0-8218-2165-7

Product Code: CHEL/368.H

List Price: $50.00

Individual Member Price: $45.00

**Electronic ISBN: 978-1-4704-1576-1
Product Code: CHEL/368.H.E**

List Price: $50.00

Individual Member Price: $40.00

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#### Supplemental Materials

# Analytic Functions of Several Complex Variables

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*Robert C. Gunning; Hugo Rossi*

The theory of analytic functions of several complex variables enjoyed a
period of remarkable development in the middle part of the twentieth
century. After initial successes by Poincaré and others in the late 19th
and early 20th centuries, the theory encountered obstacles that prevented
it from growing quickly into an analogue of the theory for functions of
one complex variable. Beginning in the 1930s, initially through the work
of Oka, then H. Cartan, and continuing with the work of Grauert, Remmert,
and others, new tools were introduced into the theory of several complex
variables that resolved many of the open problems and fundamentally
changed the landscape of the subject. These tools included a central role
for sheaf theory and increased uses of topology and algebra. The book by
Gunning and Rossi was the first of the modern era of the theory of several
complex variables, which is distinguished by the use of these methods.

The intention of Gunning and Rossi's book is to provide an extensive
introduction to the Oka-Cartan theory and some of its applications, and to
the general theory of analytic spaces. Fundamental concepts and techniques
are discussed as early as possible. The first chapter covers material
suitable for a one-semester graduate course, presenting many of the
central problems and techniques, often in special cases. The later
chapters give more detailed expositions of sheaf theory for analytic
functions and the theory of complex analytic spaces.

Since its original publication, this book has become a classic resource
for the modern approach to functions of several complex variables and the
theory of analytic spaces.

Further information about this book, including updates, can be found at
the following URL:
www.ams.org/bookpages/chel-368.

#### Table of Contents

# Table of Contents

## Analytic Functions of Several Complex Variables

- Cover Cover11
- Title page i2
- Dedication iii4
- Preface to the AMS/Chelsea edition v6
- Preface vii8
- Contents xi12
- Analytic functions of several complex variables xiii14
- Holomorphic functions 116
- Local rings of holomorphic functions 6580
- Varieties 93108
- Analytic sheaves 118133
- Analytic spaces 147162
- Cohomology theory 172187
- Stein spaces, geometric theory 208223
- Stein spaces, sheaf theory 234249
- Pseudoconvexity 260275
- Appendix A. Partitions of unity 288303
- Appendix B. The theorem of Schwartz on Frechet spaces 290305
- References 296311
- Bibliography 297312
- Index 313328
- Back Cover Back Cover1338

#### Readership

Graduate students and research mathematicians interested in several complex variables.

#### Reviews

...it is a pure pleasure to read: the prose is crystal clear and anything but prolix. ... They are happy to get into relatively elementary material in some detail, and sophisticated stuff. And the i's are dotted and the t's are crossed. It's a wonderful book!

-- MAA Reviews

This book is an excellent survey of the present state of the modern theory of several complex variables. Because of the style of presentation, the wide scope and precise treatment of the material, it is destined to become a classic.

-- Mathematical Reviews