Casorati–Weierstrass theorem

The set of Weierstrass Casorati (after Karl Weierstrass and Felice Casorati ) is a set of function theory and is concerned with the behavior of holomorphic functions in environments essential singularities. But he has a weaker statement than the rates of Picard.

The set

Be a point of an area. is an essential singularity of the holomorphic function on exactly when dense for each in -lying area of the image is.

In other words, has a holomorphic function if and only in an essential singularity if in any ( however small ) around any complex number with arbitrary precision as a picture of can be approximated.

Evidence

We show the contrapositive of the statement is accurate then no essential singularity if there is a neighborhood of and a non-empty open set, so that is disjoint from.

First assume that no essential singularity, ie either a removable singularity or a pole. In liftable case ( the continuous extension of ) in an environment of limited beneficial for all. Then to disjoint. If, however, in a pole, so is a natural number and a holomorphic with. In a sufficiently small environment of true and consequently, i.e., is to be disjoint.

Now let conversely, a surrounding area of ​​and open, non-empty and disjoint. Then contains an open disc, so there is one number and one with all. It follows that is limited to through. After the Riemannian Hebbarkeitssatz can be continued to a holomorphic function on all. Since it can not be the zero function, there is a holomorphic and with and. In a possibly smaller neighborhood is also holomorphic. This means

The right side is holomorphic, so has at most a pole of degree.