Suppose f is Complex differentiable everywhere and that Lim|z|→∞}f(z)=1

Prove f(z) is constant

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- October 27th 2009, 09:56 PMthespianLimit question
Suppose f is Complex differentiable everywhere and that Lim|z|→∞}f(z)=1

Prove f(z) is constant - October 28th 2009, 01:49 AMEnrique2
You have to use Liouville theorem. Find a closed ball B such that outside the modulus of f is smaller than 2, and apply that the closed ball is compact to conclude that f is bounded-

- October 28th 2009, 02:35 AMtonio

First proof: Liouville's theorem states that a holomorphic (analytic) function which is bounded must be constant, and that's exactly what happens with our function: in some circle

Second proof: By Cauchy's integral representation and by the estimation lemma: , so constant.

Tonio