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Definition 1 Suppose $S \subset {\mathbb{R}}^n$ is an open set (usually connected), then we define \begin{equation*} T_S := \{ z \in {\mathbb{C}}^n \mid \operatorname{Re} z \in S \} . \end{equation*}We call $T_S$ the tube domain associated to $S$
Basically the idea is that these domains give you complete freedom in the imaginary coordinates. An interesting fact about these domains is that $T_S$ is pseudoconvex if and only if $T_S$ is convex in the geometric sense, and this is true if and only if $S$ itself is convex.
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- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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"tube domain" is owned by jirka.
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Cross-references: convex, pseudoconvex, coordinates, imaginary, complete, domains, connected, open set
This is version 3 of tube domain, born on 2004-08-09, modified 2005-03-07.
Object id is 6093, canonical name is TubeDomain.
Accessed 1773 times total.
Classification:
| AMS MSC: | 32A07 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Special domains ) |
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Pending Errata and Addenda
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