**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 $ \text{Res}(f,a) $. The value of the residue of a pole is given by

- $ \mathrm{Res}(f,c) = \frac{1}{(n-1)!} \lim_{z \to c} \frac{d^{n-1}}{dz^{n-1}}\left( (z-c)^{n}f(z) \right) $

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

- $ \operatorname{Res}(f,c)=\lim_{z\to c}(z-c)f(z) $

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

- $ \operatorname{Res}(f(z),c)=\operatorname{Res}(\tfrac{g(z)}{h(z)},c)=\frac{g(z)}{h'(z)} $

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

- $ 2 \pi i \ \text{Res}(f,a) = \oint_\gamma f(z)\,dz $

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}*,

- $ \oint_\gamma f(z)\, dz = 2\pi i \sum_{i=1}^n \operatorname{I} \ \operatorname{Res}( f, a_i ). $