Residue theorem

Let $\mathrm{\Omega }\subset \mathbb{C}$ be a region, and let $z_1,z_2,\dots ,z_n$ be distinct points in $\mathrm{\Omega }$. Let $f$ be a holomorphic function on $\mathrm{\Omega }$ except at the isolated singularities $z_1,z_2,\dots ,z_n$. If $\gamma$ is a curve that does not pass through any of the singularities of $f$ and is homotopic to a point in $\mathrm{\Omega }$, then

where $n(z_j,\gamma )$ is the winding number of $\gamma$ around $z_j$, and $\mathrm{Res}(f,z_j)$ is the residue of $f$ at $z_j$.

Proof
Consider the Laurent series of $f$ around $z_j$.
$◼$

Related: Cauchy integral formula.

Calculate residues

See residue#Residue Calculation for Poles.