# Reinhardt domain

###### Definition.

We call an open set $G\subset {\u2102}^{n}$ a Reinhardt domain if $({z}_{1},\mathrm{\dots},{z}_{n})\in G$ implies that $({e}^{i{\theta}_{1}}{z}_{1},\mathrm{\dots},{e}^{i{\theta}_{n}}{z}_{n})\in G$ for all real ${\theta}_{1},\mathrm{\dots},{\theta}_{n}$.

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 $\stackrel{\mathrm{~}}{G}$ is the smallest
Reinhardt domain such that $G\mathrm{\subset}\stackrel{\mathrm{~}}{G}$. Then
any function^{} holomorphic on $G$ has a holomorphic
to $\stackrel{\mathrm{~}}{G}$.

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

examples of Reinhardt domains in ${\u2102}^{n}$ are polydiscs such as $\underset{n}{\underset{\u23df}{\mathbb{D}\times \mathrm{\cdots}\times \mathbb{D}}}$ where $\mathbb{D}\subset \u2102$ 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 |
---|---|

Canonical name | ReinhardtDomain |

Date of creation | 2013-03-22 14:29:37 |

Last modified on | 2013-03-22 14:29:37 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 7 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32A07 |