﻿ Reinhardt domain

# Reinhardt domain

In the theory of functions of several variables, a Reinhardt area designated (also called Reinhardt'sches area or Reinhardt'scher body, named after Karl Reinhardt ) is an area in which can be regarded as a union of complex tori.

## Definition

Be open and connected. is called Reinhardt area, possibly also lies for each and for all.

A Reinhardt area is called perfect if also with the polycylinder is included.

## Graphical representation of

A Reinhardt area has a unique counterpart in, each point is mapped to the absolute values ​​of its coordinates. Conversely, then each point corresponds to a complex torus. This can also Reinhardt area in the higher-dimensional spaces and are shown graphically in still or.

## Examples

• Complex -dimensional polycylinder with radii
• Complex -dimensional ball around with radius.

## Importance in the theory of functions

The importance of Reinhardt areas is that they are the right areas to consider potency and Laurent series. The region of convergence of a power series is a perfect Reinhardt'sches area. However, not every perfect Reinhardt'sche area and region of convergence of a power series.

Reinhardt'sche areas also play a role in the continuation of holomorphic functions. Basically, is the following sentence:

Be a Reinhardt area, and a holomorphic function. Then there exists a uniquely determined Laurent series which converges on compact subsets of absolutely and uniformly to the function.

Applies also that for every one point exists, the - th coordinate is 0, then the Laurent series is even a power series and holomorphic function can be continued uniquely to the region of convergence of this series.

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