**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 *a _{i}*,