Simple pole

A pole of order $1$ -- the Laurent series' principal part is a single term, $a_{-1}(z-z_0)^{-1}$ .

A simple pole is a pole of order $1$ : the principal part of the Laurent series at $z_0$ consists of exactly one term, $\dfrac{a_{-1}}{z-z_0}$ . Simple poles are by far the most common singularity encountered in practice, and they admit the fastest residue shortcuts of any pole order, since $a_{-1}$ can often be read off with a single limit rather than a full Laurent expansion.

Examples

$f(z)=\dfrac{1}{z-3}$ has a simple pole at $z=3$ .
A rational function $p(z)/q(z)$ in lowest terms has a simple pole at any root of $q$ that is not also a root of $p$ and is a simple root of $q$ .