Pole (with order)

An isolated singularity where $f(z)\to\infty$ ; its order is the highest negative power in the Laurent series' principal part.

An isolated singularity $z_0$ of $f$ is a pole if $|f(z)|\to\infty$ as $z\to z_0$ -- equivalently, if the principal part of $f$ 's Laurent series at $z_0$ has finitely many nonzero terms, the lowest of which is $a_{-m}(z-z_0)^{-m}$ with $a_{-m}\ne 0$ . That integer $m\ge 1$ is the pole's order: a pole of order $1$ is a simple pole, order $2$ a double pole, and so on, and the order can also be read off as the order of the zero that $1/f$ has at $z_0$ .

Examples

$f(z)=1/z^3$ has a pole of order $3$ at $z=0$ , since its Laurent series there is just $1/z^3$ itself.
$f(z)=\dfrac{1}{(z-2)^2(z+1)}$ has a pole of order $2$ at $z=2$ and a pole of order $1$ (a simple pole) at $z=-1$ .