Cyclic complex functions and uniqueness

[bo198214]

In the case of complex domain, the uniqueness of tetration \( F \) seems to be provided by the axiom about the asymptotic behavior of \( F(z) \) at \( \Im(z) \rightarrow +\infty \):

So that would imply that any holomorphic function periodic on the real axis goes to infinity on the imaginary axis. Is that true? (My knowledge about complex analysis is not that exhaustive.)