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	{{~~Short~~ description\|Has no other singularities close to it}}		{{short description\|Has no other singularities close to it}}
	{{~~Complex~~ analysis sidebar}}		{{complex analysis sidebar}}

	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~~'' ~~\~~ ~~{~~z<sub>0</sub>~~}, that is, on the [[Set (mathematics)\|set]] obtained from ''D'' by ~~taking ''z<sub>0</sub>'' out~~.		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]] {{tmath\| z_0 }} is an isolated singularity of a function {{tmath\| f }} if there exists an [[open set\|open]] [[disk (mathematics)\|disk]] {{tmath\| D }} centered at {{tmath\| z_0 }} such that ''f'' is [[holomorphic function\|holomorphic]] on {{tmath\| D \smallsetminus \{z_0\} }}, that is, on the [[Set (mathematics)\|set]] obtained from {{tmath\| D }} by removing {{tmath\| z_0 }} .

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

	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.		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]].		Isolated singularities may be classified into three distinct types: [[Removable singularity\|removable singularities]], [[Pole (complex analysis)\|poles]] and [[Essential singularity\|essential singularities]].

	==Examples==		== Examples ==
			* The function {{tmath\| \textstyle \frac{1}{z} }} has {{tmath\| 0 }} as an isolated singularity.
			* The [[cosecant]] function {{tmath\| \csc \left(\pi z\right) }} has every [[integer]] as an isolated singularity.

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

	== External links ==		== External links ==
			* {{citation \|author-link=Lars Ahlfors \|last1=Ahlfors \|first1=L. \|title=Complex Analysis \|edition=3rd \|publisher=McGraw-Hill \|date=1979 }}
			* {{citation \|author-link=Walter Rudin \|last1=Rudin \|first1=W. \|title=Real and Complex Analysis \|edition=3rd \|publisher=McGraw-Hill \|date=1986 }}
			* {{MathWorld \|urlname= Singularity \|title= Singularity}}

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

	~~{{Authority~~ control}}

	[[Category:Complex analysis]]		[[Category:Complex analysis]]

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2024-01-22T14:43:02Z

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{{Short description|Has no other singularities close to it}}
{{Complex analysis sidebar}}

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]] ''z0'' is an isolated singularity of a function ''f'' if there exists an [[open set|open]] [[disk (mathematics)|disk]] ''D'' centered at ''z0'' such that ''f'' is [[holomorphic function|holomorphic]] on ''D'' \ {z0}, that is, on the [[Set (mathematics)|set]] obtained from ''D'' by taking ''z0'' 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}}

{{Authority control}}

[[Category:Complex analysis]]

Isolated singularity - Revision history

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