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domain of holomorphy
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(Definition)
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Definition 1 An open set
 is called a domain of holomorphy if there do not exist non-empty open sets
 and
 where  is connected,
 and
 such that for every holomorphic function  on  there exists a holomorphic function  on  such that  on 
When , then every open set is a domain of holomorphy. For an example, assume that the boundary of
is a Jordan curve for simplicity. We can define a holomorphic function which has zeros which accumulate on the boundary of the domain and thus the function cannot be continued past any point in the boundary. If you could extend the function, it would be identically zero.
Alternatively given any open set
and any point
the function
is holomorphic in , but cannot be continued past .
For many domains are not domains of holomorphy. For example if you take
this is no longer a domain of holomorphy by Hartogs's theorem. It turns out that a domain is a domain of holomorphy if and only if the boundary is pseudoconvex. In particular, every convex (in the classical sense) domain is a domain of holomorphy. Simple examples of domains of holomorphy are
an open ball, or a polydisc.
- 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.
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"domain of holomorphy" is owned by jirka.
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Cross-references: polydisc, open ball, convex, pseudoconvex, point, function, domain, Jordan curve, boundary, holomorphic function, connected, open set
There are 10 references to this entry.
This is version 4 of domain of holomorphy, born on 2004-07-25, modified 2008-02-29.
Object id is 6026, canonical name is DomainOfHolomorphy.
Accessed 2344 times total.
Classification:
| AMS MSC: | 32A10 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Holomorphic functions) | | | 32T05 (Several complex variables and analytic spaces :: Pseudoconvex domains :: Domains of holomorphy) |
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Pending Errata and Addenda
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