Liouville’s theorem

A bounded entire functionMathworldPlanetmath is constant. That is, a bounded complex function f: which is holomorphic on the entire complex planeMathworldPlanetmath is always a constant function.

More generally, any holomorphic function f: which satisfies a polynomialMathworldPlanetmathPlanetmathPlanetmath bound condition of the form


for some c, n, and all z with |z| sufficiently large is necessarily equal to a polynomial function.

Liouville’s theorem is a vivid example of how stringent the holomorphicity condition on a complex function really is. One has only to compare the theorem to the corresponding statement for real functions (namely, that a bounded differentiableMathworldPlanetmathPlanetmath real function is constant, a patently false statement) to see how much stronger the complex differentiability condition is compared to real differentiability.

Applications of Liouville’s theorem include proofs of the fundamental theorem of algebraMathworldPlanetmath and of the partial fraction decomposition theorem for rational functionsMathworldPlanetmath.

Title Liouville’s theorem
Canonical name LiouvillesTheorem
Date of creation 2013-03-22 12:04:31
Last modified on 2013-03-22 12:04:31
Owner djao (24)
Last modified by djao (24)
Numerical id 8
Author djao (24)
Entry type Theorem
Classification msc 30D20