In general, a singularity is a point at which an equation, surface, etc., blows up or becomes degenerate. Singularities are often also called singular points.
Singularities are extremely important in complex analysis, where they characterize the possible behaviors of analytic functions.
Complex singularities are points 
in the domain of a function 
where 
fails to be analytic.
Isolated singularities may be classified
as poles, essential
singularities, logarithmic singularities,
or removable singularities. Nonisolated
singularities may arise as natural boundaries
or branch cuts.
Consider the second-order ordinary differential equation
 |
If 
and 
remain finite at 
, then 
is called an ordinary point.
If either 
or 
diverges as 
, then 
is called a singular point. Singular points are further
classified as follows:
1. If either 
or 
diverges as 
but 
and 
remain finite as 
, then 
is called a regular
singular point (or nonessential singularity).
2. If 
diverges more quickly than 
, so 
approaches infinity
as 
, or 
diverges more quickly than 
so that 
goes to infinity
as 
, then 
is called an irregular
singularity (or essential singularity).
A pole of order 
is a point 
of 
such that the Laurent series
of 
has 
for 
and 
.
Essential singularities are poles of infinite order. A pole of
order 
is a singularity 
of 
for which the function 
is nonsingular and for which 
is singular for 
, 1, ..., 
.
A logarithmic singularity is a singularity of an analytic function whose main 
-dependent
term is of order 
,
, etc.
Removable singularities are singularities for which it is possible to assign a complex number
in such a way that 
becomes analytic. For example, the function
has a removable
singularity at 0, since 
everywhere but 0, and 
can be set equal to 0 at 
. Removable singularities
are not poles.
For example, the function 
has the following singularities: poles at 
, and a nonisolated singularity at 0.
