What are the stalks of holomorphic functions wrt the analytic Zariski topology?

Question

Let $U \subset \mathbb C^n$ be an open subset, and take $x \in U$. Let $\mathcal O_U$ be the sheaf of holomorphic functions in $U$. Then I know that the stalk at $x$ is isomorphic to the ring of power series in $n$ variables: $$\mathcal O_{U, x} = \lim_{x \in V \subset U\\\,\,\,\text{open}} \mathcal O_U(V) \cong \mathbb C\{x_1, \dotsc, x_n\}.$$ I wonder what the stalk with respect to the analytic Zariski topology is. A set $V \subset U$ is analytically open if it is the complement of an analytic set $Z$, and an analytic set is locally the vanishing locus of holomorphic functions, $$Z \cap U' = V(f_1, \dotsc, f_k) = \{z : f_i(z) = 0 \forall i\} \subset U' \subset U \quad \text{for} \quad f_1, \dotsc, f_k \in \mathcal O_U(U').$$ So what is $$\lim_{\,\,\,\,\,\,\,x \in V \subset U\\\text{analytically open}} \mathcal O_U(V) \quad ?$$