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

  lim  f(z)·(z-z0)n
  z→z0

exists. For example, the function f(z) = exp(1/z) (see exponential function) has an essential singularity at z₀ = 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.