Periodic holomorphic descent through the exponential map

If a holomorphic function of $w$ is periodic with period one, then $F((\log z)/(2\pi i))$ is independent of the choice of a local logarithm: two choices differ by $2\pi ik$, hence their arguments differ by an integer. Consequently it defines a holomorphic function on the image of the original domain under $w\mapsto e^{2\pi iw}$, provided that domain is invariant under integer shifts. Local logarithms prove holomorphy across an arbitrary global branch cut. Singularities on an integer lattice all map to $z=1$.