Theory of Stein Spaces

  • Hans Grauert
  • Reinhold Remmert

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 236)

Table of contents

  1. Front Matter
    Pages I-XXI
  2. Hans Grauert, Reinhold Remmert
    Pages 1-27
  3. Hans Grauert, Reinhold Remmert
    Pages 28-44
  4. Hans Grauert, Reinhold Remmert
    Pages 45-55
  5. Hans Grauert, Reinhold Remmert
    Pages 56-82
  6. Hans Grauert, Reinhold Remmert
    Pages 83-99
  7. Hans Grauert, Reinhold Remmert
    Pages 100-124
  8. Hans Grauert, Reinhold Remmert
    Pages 125-185
  9. Hans Grauert, Reinhold Remmert
    Pages 186-203
  10. Hans Grauert, Reinhold Remmert
    Pages 204-239
  11. Back Matter
    Pages 240-252

About this book

Introduction

1. The classical theorem of Mittag-Leffler was generalized to the case of several complex variables by Cousin in 1895. In its one variable version this says that, if one prescribes the principal parts of a merom orphic function on a domain in the complex plane e, then there exists a meromorphic function defined on that domain having exactly those principal parts. Cousin and subsequent authors could only prove the analogous theorem in several variables for certain types of domains (e. g. product domains where each factor is a domain in the complex plane). In fact it turned out that this problem can not be solved on an arbitrary domain in em, m ~ 2. The best known example for this is a "notched" bicylinder in 2 2 e . This is obtained by removing the set { (z , z ) E e 11 z I ~ !, I z 1 ~ !}, from 1 2 1 2 2 the unit bicylinder, ~ :={(z , z ) E e llz1 < 1, lz1 < 1}. This domain D has 1 2 1 2 the property that every function holomorphic on D continues to a function holo­ morphic on the entire bicylinder. Such a phenomenon never occurs in the theory of one complex variable. In fact, given a domain G c e, there exist functions holomorphic on G which are singular at every boundary point of G.

Keywords

Factor Meromorphic function Riemann surface Steinscher Raum boundary element method eXist function functions set sheaves theorem types variable

Authors and affiliations

  • Hans Grauert
    • 1
  • Reinhold Remmert
    • 2
  1. 1.Mathematisches InstitutUniversität GöttingenGöttingenFederal Republic of Germany
  2. 2.Mathematisches InstitutWestfälischen Wilhelms-UniversitätMünsterFederal Republic of Germany

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4757-4357-9
  • Copyright Information Springer-Verlag New York 1979
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4757-4359-3
  • Online ISBN 978-1-4757-4357-9
  • Series Print ISSN 0072-7830
  • About this book