Knowlpedia

Definition

Residue

The coefficient of the inverse-linear term in a Laurent expansion.

Definition

Let ff be holomorphic on a punctured neighborhood of aa, with

f(z)=∑n=−∞∞cn(z−a)n.f(z)=\sum_{n=-\infty}^{\infty}c_n(z-a)^n.

The residue of ff at aa is

Res⁡(f,a)=c−1=12πi∫∣z−a∣=ρf(z) dz\operatorname{Res}(f,a)=c_{-1} =\frac{1}{2\pi i}\int_{|z-a|=\rho}f(z)\,dz

for any sufficiently small positively oriented circle.

Computation at poles

At a simple pole,

Res⁡(f,a)=lim⁡z→a(z−a)f(z).\operatorname{Res}(f,a)=\lim_{z\to a}(z-a)f(z).

At a pole of order mm,

Res⁡(f,a)=1(m−1)!dm−1dzm−1((z−a)mf(z))∣z=a.\operatorname{Res}(f,a) =\frac{1}{(m-1)!} \left.\frac{d^{m-1}}{dz^{m-1}}\bigl((z-a)^m f(z)\bigr)\right|_{z=a}.
Role

Only the coefficient of (z−a)−1(z-a)^{-1} contributes to a around aa. The turns this local coefficient into a global integration tool.

References
  1. Lars V. Ahlfors, Complex Analysis, 3rd ed., McGraw–Hill, 1979. Relevant: Chapter 5, §§1–2.