(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [itex]f[/itex] be entire. Then if [itex]lim_{z\rightarrow \infty}|f(z)|=\infty[/itex] then [itex]f[/itex] must be a non-constant polynomial.

2. Relevant equations

3. The attempt at a solution

So we know f is entire. Thus I suppose it makes sense to go ahead and expand it as a power series centered at zero. Thus what it seems to come down to is showing that if this power series is infinite, then there exists some path to infinity which z can travel such that |f(z)| remains bounded. And then I would take the contrapositive of this statement to prove the claim.

I looked at e^z for a bit of intuition and it's along the complex axis and negative real axis that this function stays bounded. But I just can't see how to generalize this observation.

I was thinking maybe it had something to do with having non-zero derivatives of all orders, but I can't see how to use that either. All in all I'm really stumped.

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# Homework Help: Unbounded Entire Function must be Polynomial

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