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

<math>\lim_{z \to z_0} f(z) \cdot (z - z_0)^n</math>

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.