Carathéodory domain

A bounded simply-connected domain in the complex plane such that its boundary is the same as the boundary of the domain which is the component of the complement of containing the point . 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 :

and every domain for which there exists such a sequence is a Carathéodory domain (Carathéodory's theorem, see [1]).

References

[1]	C. Carathéodory, "Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten" Math. Ann. , 72 (1912) pp. 107–144
[2]	A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapt. 2 (Translated from Russian)

Comments

Let be a sequence of simply-connected domains in the complex plane. Suppose that each contains a fixed disc with centre . Let . Then is open. Let be the component of containing . This domain is called the kernel of the sequence (relative to the point ). The sequence is said to converge to if every subsequence of has the same kernel relative to as itself. Cf. [2].