Residue

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,

$\oint _{\gamma }f(z)\,dz=2\pi i\sum _{i=1}^{n}\operatorname {I} \ \operatorname {Res} (f,a_{i}).$