Isolated singularity: Difference between revisions

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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''&nbsp;\&nbsp;{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).
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*  [[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]]
[[Category:Complex analysis]]

Latest revision as of 21:17, 14 December 2025

Template:Short description Template:Complex analysis sidebar

In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number Template:Tmath is an isolated singularity of a function Template:Tmath if there exists an open disk Template:Tmath centered at Template:Tmath such that f is holomorphic on Template:Tmath, that is, on the set obtained from Template:Tmath by removing Template:Tmath .

Formally, and within the general scope of general topology, an isolated singularity of a holomorphic function Template:Tmath is any isolated point of the boundary ∂Ω of the domain Template:Tmath. In other words, if U is an open subset of Template:Tmath, Template:Tmath and Template:Tmath is a holomorphic function, then a is an isolated singularity of Template:Tmath.

Every singularity of a meromorphic function on an open subset U⊂ℂ 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. Isolated singularities may be classified into three distinct types: removable singularities, poles and essential singularities.

Examples

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 analytically continued (or outside them if they are closed curves in the Riemann sphere).

Examples

File:Natural boundary example.gif
The natural boundary of this power series is the unit circle (read examples).

External links

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