Editing Isolated singularity

{{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]].

==Examples==

*The function <math>\frac {1} {z}</math> has 0 as an isolated singularity.
*The [[cosecant]] function <math>\csc \left(\pi z\right)</math> has every [[integer]] as an isolated singularity.

==Nonisolated singularities==
Other than isolated singularities, complex functions of one variable may exhibit other singular behavior. Namely, two kinds of nonisolated singularities exist:

* '''Cluster points''', i.e. [[limit points]] of isolated singularities: if they are all poles, despite admitting [[Laurent series]] expansions on each of them, no such expansion is possible at its limit.
* '''Natural boundaries''', i.e. any non-isolated set (e.g. a curve) around which functions cannot be [[analytic continuation|analytically continued]] (or outside them if they are closed curves in the [[Riemann sphere]]).

===Examples===
[[Image:Natural_boundary_example.gif|thumb|right|256px|The natural boundary of this power series is the unit circle (read examples).]]
*The function <math display="inline">\tan\left(\frac{1}{z}\right)</math> is [[meromorphic]] on <math>\mathbb{C}\setminus\{0\}</math>, with simple poles at <math display="inline">z_n = \left(\frac{\pi}{2}+n\pi\right)^{-1}</math>, for every <math> n\in\mathbb{N}_0</math>. Since <math>z_n\rightarrow 0</math>, every punctured disk centered at <math>0</math> has an infinite number of singularities within, so no Laurent expansion is available for <math display="inline">\tan\left(\frac{1}{z}\right)</math> around <math>0</math>, which is in fact a cluster point of its poles.
*The function <math display="inline">\csc \left(\frac {\pi} {z}\right)</math> has a singularity at 0 which is ''not'' isolated, since there are additional singularities at the [[Multiplicative inverse|reciprocal]] of every [[integer]], which are located arbitrarily close to 0 (though the singularities at these reciprocals are themselves isolated).
*The function defined via the [[Maclaurin series]] <math display="inline">\sum_{n=0}^{\infty}z^{2^n}</math> converges inside the open unit disk centred at <math>0</math> and has the unit circle as its natural boundary.

== External links ==

*  [[Lars Ahlfors|Ahlfors, L.]], ''Complex Analysis, 3 ed.'' (McGraw-Hill, 1979).
*  [[Walter Rudin|Rudin, W.]], ''Real and Complex Analysis, 3 ed.'' (McGraw-Hill, 1986).
* {{MathWorld | urlname= Singularity | title= Singularity}}

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[[Category:Complex analysis]]