Weierstrass-Casorati theorem

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The Weierstrass-Casorati theorem (after Karl Weierstrass and Felice Casorati ) is a theorem from function theory and deals with the behavior of holomorphic functions in the vicinity of essential singularities . But it has a weaker statement than Picard's sentences .

The sentence

Be a point of an area . is an essential singularity of on holomorphic function if and only if for every in lying area of the image close in lies.

In other words: A holomorphic function has an essential singularity if and only if in every (however small) neighborhood of every complex number can be approximated as precisely as an image of .

proof

We show the contraposition of the statement: is not an essential singularity if and only if there is a neighborhood of and a nonempty open set such that is disjoint to .

First of all, don't be an essential singularity, i.e. either a liftable singularity or a pole . In the liftable case (the continuous continuation of) is limited in a neighborhood of , for example for all . Then it is disjoint too . If in contrast has a pole, then is for a natural number and a holomorphic with . In a sufficiently small neighborhood of , and consequently , i. H. is disjoint to .

Conversely, now be an environment of and open, not empty and disjoint . Then contains an open circular disc, so there is a number and a with for all . It follows that is limited to by . According to Riemann's theorem of liftability , a fully holomorphic function can be continued. Since the null function can not be, there is an and holomorphic with and . In a possibly smaller neighborhood of is also holomorphic. this means

for everyone .

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

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