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(π"z") has every integer as an isolated singularity.

*The function $csc\; left(frac\; \{1\}\; \{pi\; z\}\; ight)$ 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).

External links

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* [http://math.fullerton.edu/mathews/c2003/SingularityZeroPoleMod.html Singularities Zeros, Poles by John H. Mathews]