Hartogs continuity theorem

This article explains the theorem about the holomorphic continuation in a Hartogs figure, for other meanings see Hartogs theorem

In function theory , Hartogs' continuity theorem is a statement about the continuation of holomorphic functions in so-called Hartogs figures . The continuity theorem represents a generalization of Hartogs ' lemma , which makes an analogous statement about the continuation in poly cylinders.

Hartogsfigur

To formulate the continuity theorem , the concept of the Hartogs figure must first be introduced.

${\displaystyle P:=\left\{(z_{1},\dots ,z_{n})\in \mathbb {C} ^{n}\,:\,\left|z_{j}\right|<1,\;j=1,\dots ,n\right\}}$denote the unit poly-cylinder . be positive real numbers between and . For be as well . The couple is called the Euclidean Hartog figure . ${\displaystyle q_{1},\dots ,q_{n}\in (0,1)}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2\leq k\leq n}$ ${\displaystyle D_{k}:=\left\{(z_{1},\dots ,z_{n})\in P\,:\,\left|z_{1}\right|\leq q_{1}{\mbox{ und }}q_{k}\leq \left|z_{k}\right|<1\right\}}$ ${\displaystyle H:=\bigcap _{k=2}^{n}(P\setminus D_{k})}$ ${\displaystyle (P,H)}$

A common Hartogs figure is the biholomorphic image of a Euclidean Hartogs figure.

Continuity theorem

Let be an open subset and a general Hartogs figure in with as well as a holomorphic function. If is contiguous, it can be continued in a clear way . ${\displaystyle \Omega \subseteq \mathbb {C} ^{n},\quad n>1}$ ${\displaystyle ({\tilde {P}},{\tilde {H}})}$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle {\tilde {H}}\subseteq \Omega }$ ${\displaystyle f\colon \Omega \rightarrow \mathbb {C} }$ ${\displaystyle {\tilde {P}}\cap \Omega }$ ${\displaystyle f}$ ${\displaystyle \Omega \cup {\tilde {P}}}$

literature

• Hans Grauert, Klaus Fritzsche: Introduction to the function theory of several variables . Springer-Verlag, Berlin 1974, ISBN 3-540-06672-1 u. ISBN 0-387-06672-1