Main Page | See live article | Alphabetical index

Liouville's theorem (complex analysis)

Liouville's theorem in complex analysis states that every entire function (a holomorphic function f(z) defined on the whole complex plane C) that 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 : MN must be constant.

See also Joseph Liouville.