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. 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 f {displaystyle f} 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.There are three types of isolated singularities: removable singularities, poles and essential singularities.