Editing Isolated singularity (section)

{{Short description|Has no other singularities close to it}}
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In [[complex analysis]], a branch of [[mathematics]], an '''isolated singularity''' is one that has no other [[mathematical singularity|singularities]] close to it. In other words, a [[complex number]] ''z<sub>0</sub>'' is an isolated singularity of a function ''f'' if there exists an [[open set|open]] [[disk (mathematics)|disk]] ''D'' centered at ''z<sub>0</sub>'' such that ''f'' is [[holomorphic function|holomorphic]] on ''D''&nbsp;\&nbsp;{z<sub>0</sub>}, that is, on the [[Set (mathematics)|set]] obtained from ''D'' by taking ''z<sub>0</sub>'' out.

Formally, and within the general scope of [[general topology]], an isolated singularity of a [[holomorphic function]] <math>f: \Omega\to \mathbb {C}</math> is any [[isolated point]] of the boundary <math>\partial \Omega</math> of the domain <math>\Omega</math>. In other words, if <math>U</math> is an open subset of <math>\mathbb {C}</math>, <math>a\in U</math> and <math>f: U\setminus \{a\}\to \mathbb {C}</math> is a holomorphic function, then <math>a</math> is an isolated singularity of <math>f</math>.

Every singularity of a [[meromorphic function]] on an open subset <math>U\subset \mathbb{C}</math> is isolated, but isolation of singularities alone is not 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.
There are three types of isolated singularities: [[Removable singularity|removable singularities]], [[Pole (complex analysis)|poles]] and [[Essential singularity|essential singularities]].