A holomorphic function f on the punctured disc 0 < |z–a| < r is said to have an isolated singularity at a. Such singularities occur in three categories: removable singularities, poles, and essential singularities, classified in terms of the Laurent expansion at a.

More generally, for a holomorphic function f on an open set U with a being a boundary point of U, then a is singular if there is no analytic continuation of f to a.