Isolated singularity
From Wikipedia, the free encyclopedia.

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).

The original article can be found at: http://en.wikipedia.org/wiki/Isolated_singularity.
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