Reinhardt domain


We call an open set Gn a Reinhardt domain if (z1,,zn)G implies that (eiθ1z1,,eiθnzn)G for all real θ1,,θn.

The reason for studying these kinds of domains is that logarithmically convex ( Reinhardt domain are the domains of convergence of power seriesMathworldPlanetmath 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.


Suppose that G is a Reinhardt domain which contains 0 and that G~ is the smallest Reinhardt domain such that GG~. Then any functionMathworldPlanetmath holomorphic on G has a holomorphic to G~.

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

examples of Reinhardt domains in n are polydiscs such as 𝔻××𝔻n where 𝔻 is the unit disc.


  • 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