= Differential of the third kind
A meromorphic one-form on a <compact Riemann surface> with only simple poles is a differential of the third kind. Prescribed residues can be realized exactly when their sum is zero: in the residue exact sequence for $K_C(S)$, the obstruction in $H^1(C,K_C)\cong\mathbb C$ is the sum of residues. Subtracting a <holomorphic differential form> normalizes its $a$-periods. Integer residues and periods in $2\pi i\mathbb Z$ make the exponential of its integral a single-valued <meromorphic function> with those orders of zeros and poles.
Back to article page
