Let the complex function $f(z)$ be holomorphic in a deleted neighbourhood of the point $z = z_0$ of the closed complex plane $\mathbb{C}\cup\{\infty\}$ . This point is

• a regular point (or a removable singularity) iff $f(z)$ is bounded in a neighbourhood of $z_0$ ,
• a pole iff $\displaystyle\lim_{z \to z_0}|f(z)| \,=\, +\infty$ ,
• an essential singularity iff there is neither of the above cases.