Isolated singularity

Isolated singularity

In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it.

Formally, a complex number z is an isolated singularity of a function f if there exists an open disk D centered at z such that f is holomorphic on D − {z}, that is, on the set obtained from D by taking z out.

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 frac {1} {z} has 0 as an isolated singularity.
  • The cosecant function csc(πz) has every integer as an isolated singularity.
  • The function csc left(frac {1} {pi z}right) 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).

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