Hartogs' extension theorem

In the theory of functions commonly referred to as Hartogs lemma (sometimes also continuity theorem of Hartogs ) refers to a statement that a function defined in a neighborhood of the boundary of a holomorphic function in the whole Polyzylinders polycylinder can be continued holomorphically.

Statement

Is the unit polycylinder in a vicinity of the edge such that is connected. Then for every holomorphic function is a holomorphic function such that the following holds, ie a holomorphic continuation to the whole of representing.

Importance

The condition is essential. In the complex one-dimensional case, a corresponding statement is false; e.g. is the function holomorphic in a neighborhood of the boundary of the unit disk, but apparently has no holomorphic continuation at zero. However, in the higher-dimensional case, this phenomenon can not occur because the singularities of holomorphic functions are no longer isolated and could fit within the Polyzylinders in any compact set, would therefore also lie on the edge, but this is excluded by the hypothesis of the theorem.