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.