A [[Pole]] of a [[Complex Numbers|Complex]] [[Function]] $f: \mathbb{C} \to \mathbb{\hat C}$ is a value $z_{0}\in \mathbb{C}$ such that the limit holds true: $\huge \begin{align} \lim_{ z \to z_{0} } |f(z_{0})| = \infty \end{align} $ >[!example] > > $\huge \begin{align} > \mathcal{H}(z) &= \left(1+a_{1}z^{-1}\right) \frac{z}{z} \\ > &= \frac{{z+a_{1}}}{z} > \end{align} $ > This function has a zero at $z=-a_{1}$ and a pole at $z=0$.