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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 ai,

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