Poincaré residue
Encyclopedia
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Poincaré residue is a generalization, to several complex variables and complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
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
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is 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 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.