In
complex analysis, a branch of
mathematics, an isolated singularity is one that has no other
singularities close to it. In other words, a
complex numberz0 is an isolated singularity of a function f if there exists an
opendiskD centered at z0 such that f is
holomorphic on D \ {z0}, that is, on the
set obtained from D by taking z0 out.
Formally, and within the general scope of
general topology, an isolated singularity of a
holomorphic function is any
isolated point of the boundary of the domain . In other words, if is an open subset of , and is a holomorphic function, then is an isolated singularity of .
Every singularity of a
meromorphic function on an open subset is isolated, but isolation of singularities alone is not sufficient to guarantee a function is meromorphic. Many important tools of complex analysis such as
Laurent series and the
residue theorem require that all relevant singularities of the function be isolated.
There are three types of isolated singularities:
removable singularities,
poles and
essential singularities.
Examples
The function has 0 as an isolated singularity.
The
cosecant function has every
integer as an isolated singularity.
Nonisolated singularities
Other than isolated singularities, complex functions of one variable may exhibit other singular behavior. Namely, two kinds of nonisolated singularities exist:
Cluster points, i.e.
limit points of isolated singularities: if they are all poles, despite admitting
Laurent series expansions on each of them, no such expansion is possible at its limit.
Natural boundaries, i.e. any non-isolated set (e.g. a curve) around which functions cannot be
analytically continued (or outside them if they are closed curves in the
Riemann sphere).
Examples
The function is
meromorphic on , with simple poles at , for every . Since , every punctured disk centered at has an infinite number of singularities within, so no Laurent expansion is available for around , which is in fact a cluster point of its poles.
The function has a singularity at 0 which is not isolated, since there are additional singularities at the
reciprocal of every
integer, which are located arbitrarily close to 0 (though the singularities at these reciprocals are themselves isolated).
The function defined via the
Maclaurin series converges inside the open unit disk centred at and has the unit circle as its natural boundary.