proof of extended Liouville’s theorem


This is a proof of the second, more general, form of Liouville’s theorem given in the parent (http://planetmath.org/LiouvillesTheorem2) article.

Let f: be a holomorphic functionMathworldPlanetmath such that

|f(z)|<c|z|n

for some c and for z with |z| sufficiently large. Consider

g(z)={f(z)-f(0)zz0f(0)z=0

Since f is holomorphic, g is as well, and by the bound on f, we have

|g(z)|<c1+c2|z|n-1<c|z|n-1

again for |z| sufficiently large.

By inductionMathworldPlanetmath, g is a polynomial of degree at most n-1, and thus f is a polynomial of degree at most n.

Title proof of extended Liouville’s theorem
Canonical name ProofOfExtendedLiouvillesTheorem
Date of creation 2013-03-22 16:18:31
Last modified on 2013-03-22 16:18:31
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 8
Author rm50 (10146)
Entry type Proof
Classification msc 30D20