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 general topology, an isolated singularity of a holomorphic function

Every singularity of a meromorphic function on an open subset

Many important tools of complex analysis such as Laurent series and the residue theorem require that all relevant singularities of the function be isolated.

Other than isolated singularities, complex functions of one variable may exhibit other singular behavior.

Namely, two kinds of nonisolated singularities exist:

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