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 .