A

complex function *f* is said to have an

isolated singularity at a

point **z** if it fails to be

analytic at

**z**, but there exists
a

neighbourhood of

**z** on which

*f* is

analytic everywhere but

**z**. (In other words,

*f* is analytic on some punctured disk centred at

**z**, but not at

**z**.) For example, the function 1 / (z

^{2}+4) has isolated singularities at 2

*i* and -2

*i*.

Isolated singularities are further classified as removable singularities, poles or essential singularities.