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 e1/z, which has an essential singularity at the origin.
Essential singularities are occasionally referred to as "irregular singularities".