Let $\Sigma$ be a compact Riemann surface of genus $g$, and $f: \Sigma \to \mathbb{C}$ a holomorphic function. Take $U \subset \Sigma$ an open disk in $\Sigma$ biholomorphic to a disk in $\mathbb{C}$. Then $f|_{U}$ is a holomorphic function. By the principal of analytic continuation, $f|_{U}$ (or its germ) should determine $f$ and $\Sigma$. Is there any formula or construction for this determination?

More generally, suppose $f$ is a holomorphic global section of some holomorphic vector bundle $E$ over $\Sigma$. Construct $f_{1}|_{{U}}$ similarly in a local trivialization (projected to the first element in the chosen basis). Can we reconstruct the relevant sub-bundle of $E$?

Let $\Sigma$ be a compact Riemann surface of genus $g$, and $f: \Sigma \to \mathbb{C}$ a meromorphic function. Take $U \subset \Sigma$ an open disk in $\Sigma$ biholomorphic to a disk in $\mathbb{C}$, such that $f|_{U}$ is a holomorphic function. By the principal of analytic continuation, $f|_{U}$ (or its germ) should determine $f$ and $\Sigma$. Is there any formula or construction for this determination?

More generally, suppose $f$ is a holomorphic global section of some holomorphic vector bundle $E$ over $\Sigma$. Construct $f_{1}|_{{U}}$ similarly in a local trivialization (projected to the first element in the chosen basis). Can we reconstruct the relevant sub-bundle of $E$?