If $f:\mathbb{C}\rightarrow\mathbb{C}$ is a differentiable function and $f(x+2\pi)=f(x)$ for all $x\in \mathbb{R}$, would $f(z+2\pi)=f(z)$ for all $z\in \mathbb{C}$?
Is there any theorem/lemma concerning this? Are there any examples/counter examples for this?