In the interest of analytic solutions to functional equations I was considering the case where instead of $F(x+\omega) = F(x)$ we have $F(x+\omega) = G(F(x))$ for some $G,F$ analytical functions and $\omega \in \mathbb{C}$, or to simplify it just have $\omega \in \mathbb{R}$.
I was wondering if anyone has researched this topic and if there are any interesting results about it. If there would, how would you figure out solutions to such problems?
For example a solution to equations like $F(x+\omega) = F(x) + k$ for some constant k.
maybe a function similar to $\sin(x)$ or $\cos(x)$, but bent sideways, which would create some kind of wavy diagonal function.
Thanks!