Hartogs' Lemma

This article explains the theorem about the holomorphic continuation in polycylinders, for other meanings see Hartogs theorem .

In function theory , Hartogs 'lemma (sometimes also Hartogs' continuity theorem ) is a statement according to which a holomorphic function defined in a neighborhood of the edge of a polycylinder can be continued holomorphically into the whole polycylinder.

statement

Let the unit poly-cylinder in , be a neighborhood of the edge such that is connected. Then for every holomorphic function there exists a holomorphic function such that it holds, i.e. represents a holomorphic continuation from to whole . ${\displaystyle P:=\left\{(z_{1},\dots ,z_{n})\in \mathbb {C} ^{n}\,:\,\left|z_{j}\right|<1\;j=1,\dots ,n\right\}}$ ${\displaystyle \mathbb {C} ^{n},\quad n>1}$ ${\displaystyle V\subseteq \mathbb {C} ^{n}}$ ${\displaystyle \partial P}$ ${\displaystyle V\cap P}$ ${\displaystyle f\colon V\rightarrow \mathbb {C} }$ ${\displaystyle F\colon V\cup P\rightarrow \mathbb {C} }$ ${\displaystyle F|_{V}=f}$ ${\displaystyle F}$ ${\displaystyle f}$ ${\displaystyle P}$

meaning

The condition is essential. In the complex one-dimensional case, a corresponding statement is false; z. B. the function is holomorphic in a neighborhood of the edge of the unit disk, but obviously has no holomorphic continuation in the zero point. In the higher-dimensional case, however, this phenomenon can no longer occur because the singularities of holomorphic functions are no longer isolated and would not find a place in any compact unit within the polycylinder, i.e. would also be at the edge, which is excluded according to the assumption of the theorem. ${\displaystyle n>1}$ ${\displaystyle f\colon \mathbb {C} ^{*}\rightarrow \mathbb {C} ,\;z\mapsto {\tfrac {1}{z}}}$

literature

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