Consider a holomorphic function $f: S \to \mathbb{C}$ where $S$ is a path connected open subset of $\mathbb{C}$ (not necessarily simply connected). Is it then possible to determine if $f$ contains a zero?.

I would suspect that the answer is that it is not possible in general, but has this ever been proven? From Matiyasevich's theorem I have been able to prove this (weaker?) result:

Consider all holomorphic functions $f: S \to \mathbb{C}$ where $S$ is a path connected open subset of $\mathbb{C}$, with an associated analytic function $\tau_f: \mathbb{R} \to S$. Then no algorithm exists that for every $f$ can determine if $f\circ \tau_f$ contains any zeroes.

EDIT: We are only considering functions that are computable at every point inside their domain.