Residue Integration

Suppose $f(z)$ has a singularity at $z=z_0$ inside a simple closed curve $C$ but is otherwise analytic on $C$ and inside $C.$

Then, $f(z)$ has a @laurent-series

$$f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n(z-z_0{)}^n+\frac{b_1}{z-z_0}+\frac{b_2}{(z-z_0{)}^2}+\cdots$$

that converges for all points near $z=z_0$ (except at $z=z_0$ itself,) in some domain of the form

The coefficient $b_1$ of the first negative power $1/(z-z_0)$ of this @laurent-series is given by the @Cauchy-integral-formula as

$$b_1=\frac{1}{2\ast pi\ast i}{\oint }_{C}f(z)dz.$$

Now, we can use this to find the value of the integral without using any of the integral formulas:

$${\oint }_{C}f(z)dz=2\pi i{b}_1.$$

This is a CCW integral around a simple closed path $C$ that contains $z=z_0$ in its interior (but no other singularities of $f(z)$ on or inside C.)

Residue Formulas

Instead of finding the @laurent-series, we can use these handy formulas.

Simple pole at $z_0$

Poles of any Order at $z_0$

Several Singularities Inside the Contour