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Let $X$ be a compact Riemann surface and $D \subset X$ a compact domain with boundary $\partial D$. Let $\omega$ be a meromorphic $1$-form in a neighborhood of $D$ which does not have neither zeros, nor poles on $\partial D$. Is it possible to compute the degree of the divisor of $\omega$ in $D$ (i.e. the sum of the orders of zeros minus the sum of the orders of poles) by using the integration along $\partial D$? In the case of a planar domain $D$ we could just divide $\omega$ by $dz$ and apply the argument principle for meromorphic functions.

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