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Difference between revisions of "Carathéodory domain"

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Let $G_n$ be a sequence of  simply-connected domains in the complex plane. Suppose that each  contains a fixed disc $D$ with centre $z_0$. Let  
 
Let $G_n$ be a sequence of  simply-connected domains in the complex plane. Suppose that each  contains a fixed disc $D$ with centre $z_0$. Let  
  
$$ E=\{ z:\mathrm{there is a neighbourhood }N\mathrm{ such that }N\subset G_n\mathrm{ for all large enough }n\}.$$
+
$$ E=\{ z:\textrm{there is a neighbourhood }N\textrm{ such that }N\subset G_n\textrm{ for all large enough }n\}.$$
  
 
Then $E$ is open. Let $G_{z_0}$ be the component  of $E$ containing $z_0$. This domain is  called the kernel of the sequence $\{ G_n\}$ (relative to the  point $z_0$). The sequence $\{ G_n\}$  is said to  converge to $G_{z_0}$ if every  subsequence of $\{ G_n\}$ has the same  kernel relative to $z_0$ as $\{ G_n\}$ itself. See {{Cite|Ma}}.
 
Then $E$ is open. Let $G_{z_0}$ be the component  of $E$ containing $z_0$. This domain is  called the kernel of the sequence $\{ G_n\}$ (relative to the  point $z_0$). The sequence $\{ G_n\}$  is said to  converge to $G_{z_0}$ if every  subsequence of $\{ G_n\}$ has the same  kernel relative to $z_0$ as $\{ G_n\}$ itself. See {{Cite|Ma}}.

Revision as of 16:56, 21 April 2012

A bounded simply-connected domain $G$ in the complex plane such that its boundary is the same as the boundary of the domain $G_\infty$ which is the component of the complement of $G$ containing the point $\infty$. A domain bounded by a Jordan curve is an example of a Carathéodory domain. Every Carathéodory domain is representable as the kernel of a decreasing convergent sequence of simply-connected domains $\{ G_n\}$:

$$\overline{G}\subset G_{n+1}\subset\overline{G}_{n+1}\subset G_n,\quad n=1,2,\ldots,$$

and every domain $G$ for which there exists such a sequence is a Carathéodory domain (Carathéodory's theorem, see [Ca]).

Let $G_n$ be a sequence of simply-connected domains in the complex plane. Suppose that each contains a fixed disc $D$ with centre $z_0$. Let

$$ E=\{ z:\textrm{there is a neighbourhood }N\textrm{ such that }N\subset G_n\textrm{ for all large enough }n\}.$$

Then $E$ is open. Let $G_{z_0}$ be the component of $E$ containing $z_0$. This domain is called the kernel of the sequence $\{ G_n\}$ (relative to the point $z_0$). The sequence $\{ G_n\}$ is said to converge to $G_{z_0}$ if every subsequence of $\{ G_n\}$ has the same kernel relative to $z_0$ as $\{ G_n\}$ itself. See [Ma].

References

[Ca] C. Carathéodory, "Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten" Math. Ann. , 72 (1912) pp. 107–144
[Ma] A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapt. 2 (Translated from Russian)
How to Cite This Entry:
Carathéodory domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carath%C3%A9odory_domain&oldid=24991
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article