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

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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.

- 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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Last updated on Saturday September 20, 2008 at 03:04:50 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday September 20, 2008 at 03:04:50 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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