# Reinhardt domain

###### Definition.

We call an open set $G\subset{\mathbb{C}}^{n}$ a 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 $\theta_{1},\ldots,\theta_{n}$.

The reason for studying these kinds of domains is that logarithmically convex (http://planetmath.org/LogarithmicallyConvexSet) Reinhardt domain are the domains of convergence of power series in several complex variables. Note that in one complex variable, a Reinhardt domain is just a disc.

Note that the intersection of Reinhardt domains is still a Reinhardt domain, so for every Reinhardt domain, there is a smallest Reinhardt domain which contains it.

###### Theorem.

Suppose that $G$ is a Reinhardt domain which contains 0 and that $\tilde{G}$ is the smallest Reinhardt domain such that $G\subset\tilde{G}$. Then any function holomorphic on $G$ has a holomorphic to $\tilde{G}$.

It actually turns out that a Reinhardt domain is a domain of convergence.

examples of 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.

## References

• 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
• 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title Reinhardt domain ReinhardtDomain 2013-03-22 14:29:37 2013-03-22 14:29:37 jirka (4157) jirka (4157) 7 jirka (4157) Definition msc 32A07