The function
has an **isolated singularity** at
if it is analytic on a deleted open neighborhood of
, but is not analytic at
.

- has an isolated singularity at 0.
- has an isolated singularity at 2.

- If there exists a function
, analytic on an open neighborhood
of
and such that
, then the singularity of
at
is called
**removable**. - Suppose that for every
,
, where
verify the following conditions:
- and are analytic at ;
- and ;

**pole**at . If is a zero of order of , we say that has a pole of order at . - If
is neither a pole nor a removable singularity of
, we say that
has an
**essential singularity**at .

Therefore

and the singularity is removable.

and the singularity is not removable.

Recall that in Calculus, you would have proven that the real valued function of the real variable given by has two infinite one-sided limits at 2.

- If there exists a negative integer such that for every integer and , then has a pole of order at .
- If the Laurent series expansion does contain negative powers of , but there is no such , the point is an essential singularity of .

A Taylor expansion of about 3 is:

Thus

By substitution we get a Laurent expansion of about 0:

The negative powers are not bounded below.

Noah Dana-Picard 2007-12-24