Residue

$\operatorname{Res}_{z=z_0} f = a_{-1}$ , the coefficient of $(z-z_0)^{-1}$ in the Laurent series at $z_0$ .

The residue of $f$ at an isolated singularity $z_0$ , written $\operatorname{Res}_{z=z_0} f$ , is the coefficient $a_{-1}$ of $(z-z_0)^{-1}$ in $f$ 's Laurent series there. It is singled out from every other Laurent coefficient because it alone survives integration -- $\oint_C f(z)\,dz = 2\pi i\,\operatorname{Res}_{z=z_0} f$ for a small positively oriented circle $C$ around $z_0$ containing no other singularity, since every other power of $(z-z_0)$ integrates to zero around a closed loop.

Examples

For $f(z)=e^z/z^2$ , the Laurent series at $0$ is $\frac1{z^2}+\frac1z+\frac12+\cdots$ , so $\operatorname{Res}_{z=0}f=1$ .
A function analytic at $z_0$ (no singularity there at all) has residue $0$ at $z_0$ , since its Laurent series is just its Taylor series, with no $(z-z_0)^{-1}$ term.