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.
with a pole of order
, then we can construct a cohomology class
Res ( ω ) ∈
we recover the classical residue construction.
When Poincaré first introduced residues[1] he was studying period integrals of the form
was a rational differential form with poles along a divisor
He was able to make the reduction of this integral to an integral of the form
to the boundary of a solid
on an affine chart where
(so there is no poles on the line at infinity[2] page 150).
Then, he gave a formula for computing this residue as
{\displaystyle {\text{Res}}(\omega )=-{\frac {qdx}{\partial p/\partial y}}={\frac {qdy}{\partial p/\partial x}}}
Notice that the standard differential
sends Define
as the rational de-Rham cohomology groups.
(which is locally isomorphic to
-cycle, we can integrate a rational
If we write this as then we get a linear transformation on the homology classes.
Homology/cohomology duality implies that this is a cohomology class
Notice if we restrict to the case
, this is just the standard residue from complex analysis (although we extend our meromorphic
This definition can be summarized as the map
There is a simple recursive method for computing the residues which reduces to the classical case of
, we can write a meromorphic
as Then we can write it out as This shows that the two cohomology classes are equal.
We have thus reduced the order of the pole hence we can use recursion to get a pole of order
and define the residue of
defined by the polynomial Then, we can apply the previous algorithm to compute the residue of Since and we have that This implies that