|
|
|
|
Reinhardt domain
|
(Definition)
|
|
|
The reason for studying these kinds of domains is that logarithmically convex Reinhardt domain are the domains of convergence of power series in several complex variables. Note that in one complex variable, a logarithmically convex Reinhardt domain is just a disc.
Note that the intersection of logarithmically convex Reinhardt domains is still a logarithmically convex Reinhardt domain, so for every Rienhardt domain, there is a smallest logarithmically convex Reinhardt domain which contains it.
Theorem 1 Suppose that  is a Reinhardt domain which contains 0 and that  is the smallest logarithmically convex Reinhardt domain such that
 . Then any function holomorphic on  has a holomorphic extension to  .
It actually turns out that a logarithmically convex Reinhardt domain is a domain of convergence.
Simple examples of logarithmically convex Reinhardt domains in
 are polydiscs such as
 where
 is the unit disc.
- 1
- Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.
- 2
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
|
"Reinhardt domain" is owned by jirka.
|
|
(view preamble)
Cross-references: unit disc, polydiscs, holomorphic, function, contains, intersection, disc, variable, complex, several complex variables, power series, domains, real, implies, open set
There are 2 references to this entry.
This is version 3 of Reinhardt domain, born on 2004-07-26, modified 2006-09-17.
Object id is 6029, canonical name is ReinhardtDomain.
Accessed 1591 times total.
Classification:
| AMS MSC: | 32A07 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Special domains ) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
