I need to show that if $f$ is an entire function it has a pole at infinity if and only if it is a polynomial. If I start with a polynomial, it is easy to show that it has a pole at infinity, but the other implication is harder.
That is, I want to prove that if $f$ has a pole at infinity it has to be a polynomial:
$f$ entire, pole at infinity $\rightarrow$ $f$ is polynomial
I do know that if $f$ is a polynomial than around zero we have that:
$f(1/z)=z^{-n}\cdot h(z)$, where $h$ is holomorphic and non-vanishing around zero. But how does this help?
I could try contrapositive. That is, if $f$ is not a polynomial, then showing that it can't have a pole at infinity. That is:
$f$ is not polynomial $\rightarrow$ $f(1/z)$ does not have a pole at zero
But this also seems hard, any tips?