In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the remarkable behavior of meromorphic functions near essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati.
Formal statement of the theorem.
Start with some open subset 
in the complex plane containing the number 
, and a function 
that is holomorphic on 
, but has an essential singularity at . The Casorati–Weierstrass theorem then states that
if 
is any neighborhood of contained in , then 
is dense in 
.
This can also be stated as follows
for any 
and any complex number 
, there exists a complex number 
in with 
and 
.
Or in still more descriptive terms
comes arbitrarily close to any complex value in every neighbourhood of .
This form of the theorem also applies if is only meromorphic. The theorem is considerably strengthened by Picard’s great theorem, which states, in the notation above, that assumes every complex value, with one possible exception, infinitely often on .
Examples.
The function 
has an essential singularity at 
, but the function 
does not (it has a pole at 
). Consider the function
.
This function has the following Laurent series about the essential singular point at
.
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