Recall that we proved that  a meromorphic function on the sphere was a rational function, and hence, the sum of the orders of the zeros (including those at oo) equals the sum of the orders of the poles (including those at oo).  There is also a formula for the sum of the residues.
However, it is valid for a meromorphic 1 form rather than a meromorphic function.

A meromorphic 1 form  w  on a Riemann surface is a 1 form which has in a local holomorphic coordinate  z  the form  f(z)dz  where  f  is a meromorphic function.  Thus the poles of  w   (which are in the  z  coordinate neighborhood the poles of  f ) form an isolated subset of the Riemann surface.  The residue of  w at a point  a  of the  z  coordinate neighborhood is the residue of  f  at the corresponding point of the complex plane.

Exercise 1.  Prove the following:

Theorem.  For any meromorphic 1 form  w  on a compact Riemann surface, the sum of the residues of  w  is zero.

Hint.  Apply Stokes' Theorem to w on the complement of a set of   e  disks centered on the simple poles of  w .  Then let  e  approach 0.

Exercise 2.  Prove the following:

Corollary.  For any nonconstant meromorphic function  f  on a compact Riemann surface , the sum of the orders of the zeros of  f  equals the sum of the orders of the poles of  f .

Hint.  Apply the theorem to the 1 form  df / f .