Domain of holomorphy

The holomorphy is considered in the multi-dimensional function theory. On each domain of holomorphy, there is a holomorphic function which can not be continued over the area.

Definition

An open set is called domain of holomorphy if there is no open subsets and are having the following characteristics:

Examples

• Simple examples are of the open ball or polycylinder.
• Every convex set is a domain of holomorphy.
• A field is a domain of holomorphy if and only if it is pseudoconvex.
• In the case of every open subset is a domain of holomorphy. Choose a holomorphic function with zeros at all boundary points of, so can not continue beyond it. The lemma of Hartogs shows that an analogous statement for false.