Let $f$ be an entire function. And assume $f(z)$ is real on $Im(z)=0$ and $Im(z)=\pi$ lines. Show $f$ is $2\pi i$ periodic.

Question

Let $f$ be an entire function. And assume $f(z)$ is real on $Im(z)=0$ and $Im(z)=\pi$ lines. Show $f$ is $2\pi i$ periodic.

No clue how to start this problem. I was thinking of maybe equating it to some iteration of $e^z$ but that got me nowhere.

Answer 1

Let $g(z)=\overline {f(\overline {z})}$. The $g$ is entire and $g=f$ on the real axis. Hence $g=f$ everywhere by the Identity Theorem. Now use the second part of the hypothesis to conclude that $f(x-\pi i)=f(x+\pi i)$ for $x$ real.