Reinhardt domain
ReinhardtDomain
2004-07-26 13:02:31
2008-12-16 12:43:10
Definition
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\begin{defn}
We call an open set $G \subset {\mathbb{C}}^n$ a {\em Reinhardt domain}
if $(z_1,\ldots,z_n) \in G$ implies that
$(e^{i\theta_1}z_1,\ldots,e^{i\theta_n}z_n) \in G$ for all real
$\theta_1,\ldots,\theta_n$.
\end{defn}
The reason for studying these kinds of domains is that
\PMlinkname{logarithmically convex}{LogarithmicallyConvexSet}
Reinhardt domain are the domains of convergence of power series in
several complex variables. Note that in one complex variable, a
\PMlinkescapetext{logarithmically convex}
Reinhardt domain is just a disc.
Note that the intersection of
\PMlinkescapetext{logarithmically convex}
Reinhardt domains is still a
\PMlinkescapetext{logarithmically convex}
Reinhardt domain, so for every Reinhardt domain, there is a smallest
\PMlinkescapetext{logarithmically convex}
Reinhardt domain which contains it.
\begin{thm}
Suppose that $G$ is a Reinhardt domain which contains 0 and
that $\tilde{G}$ is the smallest
\PMlinkescapetext{logarithmically convex}
Reinhardt domain such that $G \subset \tilde{G}$. Then
any function holomorphic on $G$ has a holomorphic \PMlinkescapetext{extension}
to $\tilde{G}$.
\end{thm}
It actually turns out that a
\PMlinkescapetext{logarithmically convex}
Reinhardt domain is a domain of convergence.
\PMlinkescapetext{Simple} examples of
\PMlinkescapetext{logarithmically convex}
Reinhardt domains in ${\mathbb{C}}^n$ are polydiscs such as
$\underbrace{{\mathbb{D}} \times \cdots \times {\mathbb{D}}}_n$
where ${\mathbb{D}} \subset {\mathbb{C}}$ is the unit disc.
\begin{thebibliography}{9}
\bibitem{Hormander:several}
Lars H\"ormander.
{\em \PMlinkescapetext{An Introduction to Complex Analysis in Several
Variables}},
North-Holland Publishing Company, New York, New York, 1973.
\bibitem{Krantz:several}
Steven~G.\@ Krantz.
{\em \PMlinkescapetext{Function Theory of Several Complex Variables}},
AMS Chelsea Publishing, Providence, Rhode Island, 1992.
\end{thebibliography}