# Entire Functions of Several Complex Variables

• Pierre Lelong
• Lawrence Gruman
Book

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 282)

1. Front Matter
Pages I-XI
2. Pierre Lelong, Lawrence Gruman
Pages 1-29
3. Pierre Lelong, Lawrence Gruman
Pages 30-58
4. Pierre Lelong, Lawrence Gruman
Pages 59-94
5. Pierre Lelong, Lawrence Gruman
Pages 95-115
6. Pierre Lelong, Lawrence Gruman
Pages 116-154
7. Pierre Lelong, Lawrence Gruman
Pages 155-166
8. Pierre Lelong, Lawrence Gruman
Pages 167-176
9. Pierre Lelong, Lawrence Gruman
Pages 177-200
10. Pierre Lelong, Lawrence Gruman
Pages 201-229
11. Back Matter
Pages 230-272

### 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 transcen­ dence, 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 asymp­ totic 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 spaces. However, for applications it is of utmost importance to develop precise and explicit representations of the solutions.

### Keywords

Area Complex analysis Frechet Spaces Functions Lelong Schwarz lemma Variables analytic number theory approximation boundary element method equation function information number theory presentation

#### Authors and affiliations

• Pierre Lelong
• 1
• Lawrence Gruman
• 2
1. 1.Université Paris VIParis Cedex 05France
2. 2.UER de mathématiquesC.N.R.S., Université de ProvenceMarseilleFrance

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-70344-7
• Copyright Information Springer-Verlag Berlin Heidelberg 1986
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-70346-1
• Online ISBN 978-3-642-70344-7
• Series Print ISSN 0072-7830