Namespaces
Variants
Actions

Difference between revisions of "Thin set"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092620/t0926201.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092620/t0926202.png" /> such that, for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092620/t0926203.png" />, there exists an open polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092620/t0926204.png" /> and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092620/t0926205.png" /> which is holomorphic, not identically equal to zero, but which vanishes on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092620/t0926206.png" />.
+
<!--
 +
t0926201.png
 +
$#A+1 = 6 n = 0
 +
$#C+1 = 6 : ~/encyclopedia/old_files/data/T092/T.0902620 Thin set
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
 +
{{TEX|auto}}
 +
{{TEX|done}}
  
 +
A subset  $  A $
 +
of a domain  $  D \subset  \mathbf C  ^ {k} $
 +
such that, for each point  $  z \in D $,
 +
there exists an open polydisc  $  \Delta ( z, r) \subset  D $
 +
and a function  $  f $
 +
which is holomorphic, not identically equal to zero, but which vanishes on  $  A \cap \Delta ( z, r) $.
  
 
====Comments====
 
====Comments====

Latest revision as of 08:25, 6 June 2020


A subset $ A $ of a domain $ D \subset \mathbf C ^ {k} $ such that, for each point $ z \in D $, there exists an open polydisc $ \Delta ( z, r) \subset D $ and a function $ f $ which is holomorphic, not identically equal to zero, but which vanishes on $ A \cap \Delta ( z, r) $.

Comments

Usually, being thin means being a subset of an analytic set. Cf. also Thinness of a set.

References

[a1] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) pp. Chapt. 1, Sect. C
[a2] R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. 1, Sect. 3
How to Cite This Entry:
Thin set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thin_set&oldid=11511
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article