$z_0$ is a pole of $f$

Let $f$ be an analytic function on a punctured neighborhood of $x_0\in 𝐂$,that is, $f$ analytic on

for some $\epsilon >0$and such that

$$\underset{z\to z_0}{lim}|f(z)|=\mathrm{\infty }.$$

We say then that $x_0$ is a pole for $f$.