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. Given a hypersurface formula_1 defined by a degree formula_2 polynomial formula_3 and a rational formula_4-form formula_5 on formula_6 with a pole of order formula_7 on formula_8, then we can construct a cohomology class formula_9. If formula_10 we recover the classical residue construction. Historical construction. When Poincaré first introduced residues he was studying period integrals of the formformula_11 for formula_12where formula_5  was a rational differential form with poles along a divisor formula_14. He was able to make the reduction of this integral to an integral of the formformula_15 for formula_16where formula_17, sending formula_18 to the boundary of a solid formula_19-tube around formula_18 on the smooth locus formula_21of the divisor. Ifformula_22on an affine chart where formula_23 is irreducible of degree formula_24 and formula_25 (so there is no poles on the line at infinity ). Then, he gave a formula for computing this residue asformula_26which are both cohomologous forms. Construction. Preliminary definition. Given the setup in the introduction, let formula_27 be the space  of meromorphic formula_28-forms on formula_6 which have poles of order up to formula_30. Notice that the standard differential formula_2 sends Define as the rational de-Rham cohomology groups. They form a filtrationformula_34corresponding to the Hodge filtration. Definition of residue. Consider an formula_35-cycle formula_36. We take a tube formula_37 around formula_18 (which is locally isomorphic to formula_39) that lies within the complement of formula_8. Since this is an formula_4-cycle, we can integrate a rational formula_4-form formula_5 and get a number. 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 which we call the residue. Notice if we restrict to the case formula_10, this is just the standard residue from complex analysis (although we extend our meromorphic formula_47-form to all of formula_48. This definition can be summarized as the mapformula_49 Algorithm for computing this class. There is a simple recursive method for computing the residues which reduces to the classical case of formula_10. Recall that the residue of a formula_47-form If we consider a chart containing formula_8 where it is the vanishing locus of formula_54, we can write a meromorphic  formula_4-form with pole on formula_8 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 formula_47 and define the residue of formula_5 as Example. For example, consider the curve formula_63 defined by the polynomial Then, we can apply the previous algorithm to compute the residue of Since and we have that This implies that