Residue is a complex number proportional to a contour integral over a path around a pole a of a meromorphic function f. It is denoted as . The value of the residue of a pole is given by

In the case of a simple pole (a pole with a multiplicity of 1) this formula reduces to

If f(z) can be expressed as the quotient of two holomorphic functions, and h'(c) ≠0, this formula further reduces to

By Cauchy's integral formula, residue is related to the contour integral by

By the residue theorem, the value of a closed contour integral is equal to the sum of the residues inside of it. For a positively oriented (the integral is taken counter-clockwise) curve γ with winding number I (the number of times the curve loops around the pole) around n poles, located at ai,

Community content is available under CC-BY-SA unless otherwise noted.