A meromorphic one-form on a Riemann surface 
is specified in each local holomorphic
coordinate chart 
by 
, where 
is a meromorphic function.
On an overlap with a coordinate chart 
, the coefficients satisfy
 |
Consequently, the order of each pole does not depend on the coordinate chart. A form with no poles is a holomorphic differential form of degree 1. On a compact Riemann surface, a meromorphic one-form is also called an Abelian differential (Bertola et al. 2024, p. 115).
For example, 
on the Riemann sphere has a simple
pole at 0 and another at infinity. Writing 
gives
 |
so the corresponding complex residues are 1 and 
,
respectively.
