Holomorphic area

The holomorphic area is considered in the multi-dimensional function theory. On each holomorphy there is a holomorphic function which does not have the area continues to be.

definition

The quantities in the definition

An open set is called a holomorphic domain if there are no open subsets and in with the following properties: ${\displaystyle \Omega \subset \mathbb {C} ^{n}}$ ${\displaystyle \Omega _{1}}$ ${\displaystyle \Omega _{2}}$ ${\displaystyle \mathbb {C} ^{n}}$

1. ${\displaystyle \emptyset \neq \Omega _{1}\subset \Omega _{2}\cap \Omega }$.
2. ${\displaystyle \Omega _{2}}$is contiguous and not included in. ${\displaystyle \Omega }$
3. For every holomorphic function there is a (necessarily unique) holomorphic function , so that in holds. ${\displaystyle u\colon \Omega \to \mathbb {C} }$ ${\displaystyle u_{2}\colon \Omega _{2}\to \mathbb {C} }$ ${\displaystyle u=u_{2}}$ ${\displaystyle \Omega _{1}}$

Examples

• Simple examples are the , the open sphere or the poly cylinder . ${\displaystyle \mathbb {C} ^{n}}$
• Every convex set is a holomorphic domain.
• A region is a holomorphic region if and only if it is pseudoconvex .
• In the case , every open subset is a holomorphic domain. If you choose a holomorphic function with only zeros on all boundary points of , you can not continue beyond. The lemma of Hartogs shows that a similar statement for is wrong. In particular, there is no holomorphism, where poly-cylinder denotes. ${\displaystyle n=1}$ ${\displaystyle \Omega \subset \mathbb {C} }$ ${\displaystyle f\in {\mathcal {O}}(\mathbb {C} )}$ ${\displaystyle \Omega }$ ${\displaystyle {\tfrac {1}{f}}}$ ${\displaystyle \Omega }$ ${\displaystyle n>1}$ ${\displaystyle \Delta (0,0;2,2)-{\overline {\Delta (0,0;1,1)}}\subset \mathbb {C} ^{2}}$ ${\displaystyle \Delta (\ldots ;\ldots )}$

literature

• Lars Hörmander: An Introduction to Complex Analysis in Several Variables. North Holland Pub. Co., Amsterdam; American Elsevier Pub. Co., New York 1973, ISBN 9780444105233 .