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.



Computability of finding roots in holomorphic functions.
For example, let $S$ be the open unit disk and $f(z) = -1/2 + \sum\_{j \in A} z^j$ where $A$ is some subset of $\mathbb N$. Then $f$ has a zero if and only if $A$ is nonempty. $A$ could be, say, the set of $n$ such that a given Turing machine with a given input halts in time $\le n$, and there is no algorithm to determine whether $A$ is nonempty. 


