Poincaré residue

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 C"n" (or "n"-dimensional complex manifold, but the definition is local). Along a hypersurface "H" defined by

:"f" = 0

there is the meromorphic 1-form

:"df"/"f".

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 C"n", 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.

ee also

* Grothendieck residue
* Leray residue


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