Poincaré residue

In mathematics, the Poincaré residue is a generalization, to several complex variables and complex manifold theory, of the residue at a pole of complex function theory. It is just one of a number of such possible extensions.

The theory assumes given a meromorphic complex form ω of degree n on Cⁿ (or n-dimensional complex manifold, but the definition is local). Along a hypersurface H defined by

there is the meromorphic 1-form

The Poincaré residue ρ along H is by definition a holomorphic (n − 1)-form on the hypersurface, for which there is an extension ρ′, locally to Cⁿ, such that ω is the wedge product of df/f with ρ′. While ρ′ is not necessary unique, as a holomorphic extension of ρ, it is the case that ρ is uniquely defined.