Reinhardt area

In the function theory of several variables a designated Reinhardt domain (also Reinhardt'sches area or Reinhardt'scher body called, named after Karl Reinhardt ) an area in which more complex than Association - Tori can be considered. ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle n}$

definition

Be open and cohesive . called Reinhardt domain if for any and all also located. ${\displaystyle \Omega \subseteq \mathbb {C} ^{n}}$ ${\displaystyle \Omega }$ ${\displaystyle z=(z_{1},z_{2},\dots ,z_{n})\in \Omega }$ ${\displaystyle \vartheta _{1},\vartheta _{2},\dots ,\vartheta _{n}\in \mathbb {R} }$ ${\displaystyle \left(e^{i\vartheta _{1}}\cdot z_{1},\;e^{i\vartheta _{2}}\cdot z_{2},\;\dots ,e^{i\vartheta _{n}}\cdot z_{n}\right)\in \Omega }$

A Reinhardt domain is completely when using also the polydisc in is included. ${\displaystyle \Omega }$ ${\displaystyle z=(z_{1},z_{2},\dots ,z_{n})\in \Omega \cap {\mathbb {C} ^{*}}^{n}}$ ${\displaystyle \left\{w=(w_{1},w_{2},\dots ,w_{n})\in \mathbb {C} ^{n}\,:\,\left|w_{j}\right|<\left|z_{j}\right|\right\}}$ ${\displaystyle \Omega }$

Graphical representation

A Reinhardt area has a unique equivalent in , with each point in being mapped to the absolute values ​​of its coordinates . Conversely, every point in a complex torus then corresponds . This means that the Reinhardt area can also be displayed in the higher-dimensional spaces or graphically in or . ${\displaystyle \Omega \subseteq \mathbb {C} ^{n}}$ ${\displaystyle {\mathbb {R} _{0}^{+}}^{n}}$ ${\displaystyle z=(z_{1},z_{2},\dots ,z_{n})\in \Omega }$ ${\displaystyle (\left|z_{1}\right|,\left|z_{2}\right|,\dots ,\left|z_{n}\right|)}$ ${\displaystyle {\mathbb {R} _{0}^{+}}^{n}}$ ${\displaystyle n}$ ${\displaystyle \mathbb {C} ^{2}}$ ${\displaystyle \mathbb {C} ^{3}}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} ^{3}}$

Examples

• complex -dimensional poly cylinder with radii ${\displaystyle n}$ ${\displaystyle \left\{z=(z_{1},z_{2},\dots ,z_{n})\in \mathbb {C} ^{n}\,:\,\left|z_{j}\right|<\rho _{j}\;(j=1,2,\dots n)\right\}}$ ${\displaystyle \rho _{1},\rho _{2},\dots ,\rho _{n}>0}$
• complex -dimensional ball around with radius . ${\displaystyle n}$ ${\displaystyle \left\{z=(z_{1},z_{2},\dots ,z_{n})\in \mathbb {C} ^{n}\,:\,\left|z_{1}-a_{1}\right|^{2}+\left|z_{2}-a_{2}\right|^{2}+\dots +\left|z_{n}-a_{n}\right|^{2}=\rho ^{2}\right\}}$ ${\displaystyle a=(a_{1},a_{2},\dots ,a_{n})\in \mathbb {C} ^{n}}$ ${\displaystyle \rho >0}$

Significance in function theory

The importance of the Reinhardt areas lies in the fact that they are the right areas to consider power or Laurent series. The convergence domain of a power series is a perfect Reinhardt domain. However, not every perfect Reinhardt domain is also a convergence domain of a power series.

Reinhardt domains also play a role in the continuation of holomorphic functions . The following sentence is fundamental:

Let be a Reinhardt domain and a holomorphic function . Then there is a uniquely determined Laurent series which converges to the function on compact subsets of absolute and uniform . ${\displaystyle \Omega \subseteq \mathbb {C} ^{n}}$ ${\displaystyle f\colon \Omega \rightarrow \mathbb {C} }$ ${\displaystyle \textstyle \sum _{\alpha \in \mathbb {Z} ^{n}}a_{\alpha }\cdot z^{\alpha }}$ ${\displaystyle \Omega }$ ${\displaystyle f}$

If it is also true that for each there is a point whose -th coordinate is 0, then the Laurent series is even a power series and the holomorphic function can uniquely be continued on the convergence area of ​​this series. ${\displaystyle j\in \left\{1,2,\dots ,n\right\}}$ ${\displaystyle z\in \Omega }$ ${\displaystyle j}$

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