Suppose $f(z)$ is holomorphic in the whole plane, and that $f(z)$ does not have an essential singularity at $\infty$. Prove that $f(z)$ is a polynomial.
Since $f(z)$ does not have an essential singularity at $\infty$, it has either a removable singularity or a pole. If it has a removable singularity, we can define $f(\infty)$ in such a way that $f$ is holomorphic in the extended plane. Having a pole at point $a$ usually means that we can write $f(z)=\dfrac{g(z)}{(z-a)^k}$ where $g(z)$ is holomorphic. I'm not entirely sure what a pole at infinity means in this case.
