When a

complex function **f** has an

isolated singularity at a

point **a** which is neither a

pole nor a

removable singularity, it is said to have an

**essential singularity** at

**a**. This is equivalent to saying that the

Laurent series of

**f** at

**a** contains

infinitely many terms involving

negative powers of (

**z**-

**a**), so that

**f** (

**z**-

**a**)

^{n} fails to be

differentiable at

**a** for all

*n*.

An example of a function with an essential singularity is *e*^{1/z}, which has an essential singularity at the origin.

Essential singularities are occasionally referred to as "irregular singularities".