Weierstrass-Casorati theorem
The
Weierstrass-Casorati theorem in
complex analysis describes the remarkable behavior of holomorphic functions near
essential singularities.
Start with an open subset U of the complex plane containing the number z0, and a holomorphic function f defined on U - {z0}. The complex number z0 is called an essential singularity if there is no natural number n such that the limit
exists. For example, the function
f(
z) =
exp(1/
z) has an essential singularity at
z0 = 0, but the function
g(
z) = 1/
z3 does not (it has a
pole at 0).
The Weierstrass-Casorati theorem states that
- if f has an essential singularity at z0, and V is any neighborhood of z0 contained in U, then f(V) is dense in C. Or spelled out: if ε > 0 and w is any complex number, then there exists a complex number z in U with |z - z0| < ε and |f(z) - w| < ε.
The theorem is considerably strengthened by Picard's great theorem, which states, in the notation above, that
f assumes
every complex value, with one possible exception, infinitely often on
V.