proof of Liouville’s theorem

Let f: be a bounded, entire functionMathworldPlanetmath. Then by Taylor’s theorem,

f(z)=n=0cnxn where cn=12πiΓrf(w)wn+1𝑑w

where Γr is the circle of radius r about 0, for r>0. Then cn can be estimated as

|cn|12πlength(Γr)sup{|f(w)wn+1|:wΓr}=12π 2πrMrrn+1=Mrrn

where Mr=sup{|f(w)|:wΓr}.

But f is bounded, so there is M such that MrM for all r. Then |cn|Mrn for all n and all r>0. But since r is arbitrary, this gives cn=0 whenever n>0. So f(z)=c0 for all z, so f is constant.

Title proof of Liouville’s theorem
Canonical name ProofOfLiouvillesTheorem
Date of creation 2013-03-22 12:54:15
Last modified on 2013-03-22 12:54:15
Owner Evandar (27)
Last modified by Evandar (27)
Numerical id 5
Author Evandar (27)
Entry type Proof
Classification msc 30D20