Principal part of a meromorphic section
= Principal part of a meromorphic section
{title2=$\sum_{j=-r}^{-1}a_jz^j\zeta$}
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.