Liouville's theorem

Liouville's theorem in complex analysis states that every entire function (a holomorphic function f(z) defined on the whole complex plane C) which is bounded (i.e. there exists a real number M such that |f(z)| ≤ M for all z in C) must be constant.

The theorem can be proved by using Cauchy's integral formula to show that the complex derivative f '(z) must be identically zero.

Liouville's theorem can be used to give an elegant short proof for the fundamental theorem of algebra.

The theorem is considerably improved by Picard's little theorem[?], which says that every entire function whose image omits at least two complex numbers must be constant.

In the language of Riemann surfaces, the theorem can be generalized as follows: if M is a parabolic Riemann surface (such as the complex plane C) and N is a hyperbolic one (such as an open disk), then every holomorphic function f : M → N must be constant.