Isolated singularity

In complex analysis, a branch of mathematics, an isolated singularity is a singularity of a function f at a point z such that there exists an open disk centered at z within which f is analytic at every point except z.

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.

Examples

• The function 1/z contains an isolated singularity at 0

• The cosecant function csc (π z) contains an isolated singularity at every integer

• The function csc (1/(π z)) 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).