Reinhardt domain

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multiple-circled domain

A domain in the complex space , , with centre at a point , with the following property: Together with any point , the domain also contains the set A Reinhardt domain with is invariant under the transformations , , . The Reinhardt domains constitute a subclass of the Hartogs domains (cf. Hartogs domain) and a subclass of the circular domains, which are defined by the following condition: Together with any , the domain contains the set i.e. all points of the circle with centre and radius that lie on the complex line through and .

A Reinhardt domain is called a complete Reinhardt domain if together with any point it also contains the polydisc A complete Reinhardt domain is star-like with respect to its centre (cf. Star-like domain).

Examples of complete Reinhardt domains are balls and polydiscs in . A circular domain is called a complete circular domain if together with any pont it also contains the entire disc .

A Reinhardt domain is called logarithmically convex if the image of the set under the mapping is a convex set in the real space . An important property of logarithmically-convex Reinhardt domains is the following: Every such domain in is the interior of the set of points of absolute convergence (i.e. the domain of convergence) of some power series in , and conversely: The domain of convergence of any power series in is a logarithmically-convex Reinhardt domain with centre .

How to Cite This Entry:
Reinhardt domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reinhardt_domain&oldid=16774
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article