Principal part of a meromorphic function

At a point $x$ of a Riemann surface, the principal part of a meromorphic function is the negative-degree part of its Laurent series in a local coordinate. It is equivalently a germ in the quotient ${\mathcal{K}}_x/{\mathcal{O}}_x$.

The Mittag-Leffler problem asks for a global meromorphic function with prescribed principal parts at a discrete collection of points. For a finite collection, those principal parts define a Čech cocycle for the sheaf $\mathcal{O}$; vanishing of its class in ${\check{H}}^1(X,\mathcal{O})$ is exactly the condition that the problem has a solution.