Namespaces
Variants
Actions

Difference between revisions of "Carathéodory domain"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (Minot TeX tweak)
 
(4 intermediate revisions by 2 users not shown)
Line 1: Line 1:
A bounded simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c0203101.png" /> in the complex plane such that its boundary is the same as the boundary of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c0203102.png" /> which is the component of the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c0203103.png" /> containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c0203104.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c0203105.png" />:
+
{{TEX|done}}
 +
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 $\{ G_n\}$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c0203106.png" /></td> </tr></table>
+
$$\overline{G}\subset G_{n+1}\subset\overline{G}_{n+1}\subset G_n,\quad n=1,2,\ldots,$$
  
and every domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c0203107.png" /> for which there exists such a sequence is a Carathéodory domain (Carathéodory's theorem, see [[#References|[1]]]).
+
and every domain $G$ for which there exists such a sequence is a Carathéodory domain (Carathéodory's theorem, see {{Cite|Ca}}).
  
====References====
+
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
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Carathéodory,  "Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten"  ''Math. Ann.'' , '''72'''  (1912)  pp. 107–144</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Markushevich,  "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  pp. Chapt. 2  (Translated from Russian)</TD></TR></table>
 
  
 +
$$ E=\{ z:\text{there is a neighbourhood $N$ such that $N\subset G_n$ for all large enough $n$}\}.$$
  
 +
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}}.
  
====Comments====
+
====References====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c0203108.png" /> be a sequence of simply-connected domains in the complex plane. Suppose that each contains a fixed disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c0203109.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c02031010.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c02031011.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c02031012.png" /> is open. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c02031013.png" /> be the component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c02031014.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c02031015.png" />. This domain is called the kernel of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c02031016.png" /> (relative to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c02031017.png" />). The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c02031018.png" /> is said to converge to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c02031019.png" /> if every subsequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c02031020.png" /> has the same kernel relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c02031021.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020310/c02031022.png" /> itself. Cf. [[#References|[2]]].
+
{|
 +
|-
 +
|valign="top"|{{Ref|Ca}}||valign="top"|  C. Carathéodory,  "Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten" ''Math. Ann.'' , '''72'''  (1912)  pp. 107–144
 +
|-
 +
|valign="top"|{{Ref|Ma}}||valign="top"|  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  pp. Chapt. (Translated from Russian)
 +
|-
 +
|}

Latest revision as of 10:07, 22 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

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

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=17212
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article