In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1-forms.
They include for example the hyperelliptic integrals of type where Q is a square-free polynomial of any given degree > 4.
The allowable power k has to be determined by analysis of the possible pole at the point at infinity on the corresponding hyperelliptic curve.
The same type of decomposition exists in general, mutatis mutandis, though the terminology is not completely consistent.
On the other hand, a meromorphic abelian differential of the second kind has traditionally been one with residues at all poles being zero.