Pole

Definition

Let $c$ be an isolated singularity of $f$, so that $f$ is holomorphic on $0 < |z - c| < r$ for some $r > 0$. The singularity $c$ is a pole of $f$ if $|f(z)| \to \infty$ as $z \to c$. This means that for each $K > 0$ there is $\delta > 0$ such that $|f(z)| > K$ for each $z$ with $0 < |z - c| < \delta$.

Remarks

The point $c$ is a pole of $f$ if and only if $f(z) = \frac{g(z)}{(z - c)^m}$ near $c$, for an integer $m \ge 1$ and a function $g$ that is holomorphic near $c$ with $g(c) \neq 0$. The integer $m$ is the order of the pole. Equivalently, $1/f$ extends to a holomorphic function near $c$ with a zero at $c$. For example, $\frac{1}{z^2 (z - 1)}$ has a pole of order $2$ at $0$ and a pole of order $1$, a simple pole, at $1$.

Explanations

L1 human

A pole is a point where the function goes to infinity, in the way $1/z$ or $1/z^3$ does at $0$. Near a pole, $1/f$ is a holomorphic function with a zero there.