Isolated singularity
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 number z0 is an isolated singularity of a function f if there exists an open disk D 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 functional analysis, an isolated singularity for a function 
is any topologically isolated point within an open set where the function is defined.
Every singularity of a meromorphic function is isolated, but isolation of singularities is not alone 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.
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Examples
- The function
has 0 as an isolated singularity.
has every Nonisolated singularities
Other than isolated sigularities, complex functions of one variable may exhibit other singular behaviour. 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) which functions can not be analytically continued around (or outside them if they are closed curves in the Riemann sphere).
Examples
- The function
is meromorphic in 
, with simple poles in 
, for every 
. Since 
, every punctured disk centred in 
has an infinite number of singularities within, so no Laurent espansion is available for around , which is in fact a cluster point of its.
- 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 here defined as the Maclaurin series
converges inside the unit circle centred in and has it boundarying circumference as natural boundary