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 general topology, an isolated singularity of a holomorphic function a 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

Nonisolated singularities

Other than isolated singularities, complex functions of one variable may exhibit other singular behavior. Namely, two kinds of nonisolated singularities exist:

Examples

The natural boundary of this power series is the unit circle (read examples).
The natural boundary of this power series is the unit circle (read examples).