Let $C$ be a compact Riemann surfaces of genus $g$. Let $p$ be a point, $\Delta$ a disc around $p$, and $\Delta^\*$ the disc minus $p$. Let $\omega$ be a holomorphic one form defined on $\Delta^\*$.


Given a meromorphic function $f$ on $C$, regular outside $p$, we can consider the residue of $f\omega$ at $p$.


If $\omega$ is the restriction of a regular form on $C\setminus p$ to $\Delta^\*$, then this residue is zero by the classical Residue Theorem.


I read that also the converse is true. So, if the residue of $f\omega$ at $p$ is zero for every meromorphic function $f$ on $C$, regular outside $p$, then $\omega$ is the restriction of a regular form on $C\setminus p$ to $\Delta^\*$.


In the book "Vertex algebras and Algebraic Curves" this goes under the name of strong residue theorem (9.2.9). I struggle to find a proof in the literature, this could be something relatively elementary to be found in some book on Riemann surfaces. It should follow from a variant of Serre duality.


I would like to have a proof and/or a reference about this result. I am interested both in the case $C$ smooth and $C$ nodal. The latter case I think makes really sense just when $C$ is irreducible, but I am not sure.


Thanks 



Strong (Inverse of) Residue Theorem
Indeed, this follows from Serre duality: Take your favorite local holomorphic coordinate $z$ centered at $p$, and use this as the transition function $g\_{0,\infty}=z$ of the line bundle $L(p)$ with global holomorphic section $s\_p$. Consider the section $$\omega\otimes s\_{-np}$$ and take a $C^\infty$ cut-off away from $p$ as follows: Let $\varphi$ be a function with support in $\Delta$ which is constantly 1 near $p$. This gives a global section $\tilde\omega\_n=\varphi \omega\otimes s\_{n-p}\in \Gamma(C\setminus\{p\},K\otimes L(-np))$ which is meromorphic near $p$. Apply the $\bar\partial$ operator to obtain a smooth section $$\bar\partial\tilde\omega\_n$$ of $\bar K KL(-np)$ with support in an annulus $A\subset\Delta,$ $p\notin A$. 
Consider a section $s\in H^0(C,L(np)).$ Then $$\int\_C \bar\partial(\tilde\omega\_n) s=\int\_C(\bar\partial \varphi) \omega s\_{-np} s=\int\_{\partial A}\varphi \omega s\_{-np} s=-\int\_\gamma \omega s\_{-np}s=-2\pi i Res\_p(f\omega),$$
where $\gamma$ is a small curve around $p$ along which $\varphi$ is 1, and $f$ is the meromorphic function $f=s\_{-np}s.$ Therefore, by your assumption, the pairing of $\bar\partial\tilde\omega\_n$ with any holomorphic section in $L(np)$ vanishes, and Serre duality 
yields a smooth section $s$ of $KL(-np)$ such that $$\bar\partial\tilde \omega\_n=\bar\partial s\_n,$$ hence $$\tilde{\omega}\_n-s\_n$$ is a global meromorphic section of $KL(-np),$ or equivalently, a meromorphic 1-form $\omega\_n$ with prescribed behavior up to order $n$ around $p.$ For $n$ big enough ($\geq 2g-2$), all $\omega\_n$ are the same (e.g., by the easy part of the Serre duality) and hence coincide with $\omega $ on $\Delta\setminus\{p\}$.


