Periodic holomorphic function on the right half-plane

Question

If $G(z)$ is holomorphic and $2\pi i$-periodic on the right half-plane $\{Re(z)>0\}$ then can we always write $G(z) = g(e^{-z})$ for some holomorphic function g defined on $D(0,1)-\{0\}$?

Answer 1

Yes, for each $w \in D(0,1)-\{0\}$ you can define $g(w)$ as $$ g(w) = G(-\log w) = G(-\log \lvert z \rvert - i \arg w) $$ and the definition does not depend on the choice of $\arg w$ (because $G$ is $2 \pi i$-periodic).

$g$ is holomorphic because locally $g(w) = G(-\log w)$ for some holomorphic branch of the logarithm.