Principal part of a meromorphic section (${\sum }_{j=-r}^{-1}{a}_j{z}^j\zeta$)

For a holomorphic line bundle on a complex curve, a meromorphic section modulo sections holomorphic through a point is its principal part. In a local coordinate and holomorphic local frame it is represented by finitely many negative powers. A vanishing $H^1(X,\mathcal{O}(E))$ permits prescribed principal parts, by a cutoff followed by solving a global Dolbeault operator equation.