a pole

A pole is the middle child of singularities: worse than a removable hole, but far tamer than an essential one. At a pole the function genuinely blows up — its absolute value runs off to infinity as you approach the point — but it does so in a controlled, predictable way, exactly like 1/(z - z_0) or one of its higher powers. It is the kind of singularity a rational function like 1/(z - 1) has, and the kind that powers the residue theorem.

In Laurent terms, an isolated singularity z_0 is a pole when the principal part is non-empty but FINITE: there are some negative powers, but only down to a smallest one. So f(z) = a_{-m}/(z - z_0)^m + ... + a_{-1}/(z - z_0) + a_0 + a_1 (z - z_0) + ..., with a_{-m} not zero and nothing below the -m term. The number m is the order of the pole. The clean behavioural signature is that |f(z)| goes to infinity as z approaches z_0 — and that limit existing in the sense of going to infinity is exactly what distinguishes a pole from an essential singularity, where no such limit exists.

Poles are the bread and butter of complex analysis precisely because they are so well-behaved: near a pole of order m the function looks like a known multiple of 1/(z - z_0)^m, you can multiply by (z - z_0)^m to remove it and recover a holomorphic function, and the residue (the coefficient a_{-1}) is computable by a simple limit. Functions whose only singularities are poles get their own name — meromorphic — and almost the entire machinery of contour integration is built to harvest the residues at poles.