I'm currently reviewing for a qualifying exam and I was looking at old exams, and the following question had stumped me for a couple of days now, and I wanted to see if anyone had a hint; it's for a complex analysis qual. The Statement:
Find all entire functions $f$ such that $f(z+i) = f(z)$ for all $z \in \mathbb{C}$ and $f(z+1) = e^{2 \pi} f(z)$ for all $z \in \mathbb{C}$.
A hint would be really appreciated; I don't want a full solution since I'm really trying to figure this out, and it just doesn't seem like it be a good studying habit to have the solution given to me. Thanks!
Hint: $e^{-2\pi z}f(z)$ is doubly periodic.