Grundlehren der mathematischen Wissenschaften

Theory of Stein Spaces

Authors: Grauert, H., Remmert, R.

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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.

Table of contents (9 chapters)

Table of contents (9 chapters)

Buy this book

eBook $74.99
price for USA in USD (gross)
  • ISBN 978-1-4757-4357-9
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
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Bibliographic Information

Bibliographic Information
Book Title
Theory of Stein Spaces
Authors
Translated by
Huckleberry, A.
Series Title
Grundlehren der mathematischen Wissenschaften
Series Volume
236
Copyright
1979
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-1-4757-4357-9
DOI
10.1007/978-1-4757-4357-9
Series ISSN
0072-7830
Edition Number
1
Number of Pages
XXI, 252
Number of Illustrations
2 b/w illustrations
Additional Information
Original German edition published as volume 227 in the same series
Topics