Domain Of Holomorphy

In Mathematics , in the theory of functions of Several Complex Variables , a domain of holomorphy is a set which is maximal in the sense that there exist a Holomorphic Function on this set which cannot be Extended to a bigger set.

Formally, an Open Set

\Omega

in the ''n''-dimensional complex space

{\mathbb{C}}^n

is called a ''domain of holomorphy'' if there do not exist non-empty open sets

U \subset \Omega

and

V \subset {\mathbb{C}}^n

where

V

is Connected ,

V 
ot\subset \Omega

and

U \subset \Omega \cap V

such that for every Holomorphic function

f

\Omega

there exists a holomorphic function

g

V

with

f = g

U

When

n=1

, then every open set is a domain of holomorphy: we can define a holomorphic function which has zeros which Accumulate everywhere on the Boundary of the domain, which must then be a Natural Boundary . For

n \geq 2

this is no longer true, as it follows from Hartogs' Lemma .

REFERENCES

Steven G. Krantz. ''Function Theory of Several Complex Variables'', AMS Chelsea Publishing, Providence, Rhode Island, 1992.

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	CATEGORIES ABOUT DOMAIN OF HOLOMORPHY

	several complex variables

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Domain Of Holomorphy