Entire Functions of Several Complex Variables
I - Entire functions of several complex variables constitute an important and original chapter in complex analysis. The study is often motivated by certain applications to specific problems in other areas of mathematics: partial differential equations via
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Editors M.Artin S.S.Chem 1.M.Frohlich E.Heinz H. Hironaka F. Hirzebruch L. Hormander S.MacLane W.Magnus C.C.Moore 1.K.Moser M.Nagata W.Schmidt D.S.Scott Ya.G.Sinai 1. Tits B.L. van der Waerden M. Waldschmidt S.Watanabe Managing Editors M. Berger B. Eckmann S.R.S. Varadhan
Pierre Lelong Lawrence Gruman
Entire Functions of Several Complex Variables
Springer-Verlag Berlin Heidelberg NewYork Tokyo
Professor Dr. Pierre Lelong Universite Paris VI 4, Place Jussieu, Tour 45-46 75230 Paris Cedex 05 France Dr. Lawrence Gruman UER de mathematiques Universite de Provence 3 place Victor Hugo 13331 Marseille France
Mathematics Subject Classification (1980): 32A15
ISBN-13: 978-3-642-70346-1 e-ISBN-13: 978-3-642-70344-7 DOl: 10.1007/978-3-642-70344-7
Library of Congress Cataloging in Publication Data Lelong, Pierre. Entire Functions of several complex variables. (Grundlehren der mathematischen WissenschaFten; 282) Bibliography: p. Includes index. \. Functions, Entire. 2. Functions of several complex variables. I. Gruman. Lawrence, 1942-. II. Title. III. Series. QA353.E5L44 1986 515.914 85-25028 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, spefifically those of translation, reprinting, re-use of illustrations, broadcasting reproduction by photocopying machine or similar means, and storage in data banks. Under ~ 54 of the German Copyright Law where copies are made for other than private use a fee is payable to "Verwertungsgesellschaft Wort'", Munich.
© Springer-Verlag Berlin Heidelberg 1986 Softcover reprint of the hardcover I 5t edition 1986 2141/3140-543210
Introduction
I - Entire functions of several complex variables constitute an important and original chapter in complex analysis. The study is often motivated by certain applications to specific problems in other areas of mathematics: partial differential equations via the Fourier-Laplace transformation and convolution operators, analytic number theory and problems of transcendence, or approximation theory, just to name a few. What is important for these applications is to find solutions which satisfy certain growth conditions. The specific problem defines inherently a growth scale, and one seeks a solution of the problem which satisfies certain growth conditions on this scale, and sometimes solutions of minimal asymptotic growth or optimal solutions in some sense. For one complex variable the study of solutions with growth conditions forms the core of the classical theory of entire functions and, historically, the relationship between the number of zeros of an entire function f(z) of one complex variable and the growth of If I (or equivalently log If I) was the first example of a systematic study of growth conditions in a general setting. Problems with growth conditions on the solutions demand much more precise information than existence theorems. The correspondence between two scales of growth can be interpreted often as a correspondence between families of bounded sets in certain Frechet space
