Isolated singularity

Short description: Has no other singularities close to it

Mathematical analysis → Complex analysis

Complex analysis
Complex numbers

* Real number
* Imaginary number
* Complex plane
* Complex conjugate
* Unit complex number

Complex functions

* Complex-valued function
* Analytic function
* Holomorphic function
* Cauchy–Riemann equations
* Formal power series

Basic Theory

* Zeros and poles
* Cauchy's integral theorem
* Local primitive
* Cauchy's integral formula
* Winding number
* Laurent series
* Isolated singularity
* Residue theorem
* Conformal map
* Schwarz lemma
* Harmonic function
* Laplace's equation

Geometric function theory
People

* Augustin-Louis Cauchy
* Leonhard Euler
* Carl Friedrich Gauss
* Jacques Hadamard
* Kiyoshi Oka
* Bernhard Riemann
* Karl Weierstrass

* v
* t
* e

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 z0 is an isolated singularity of a function f if there exists an open disk D centered at z0 such that f is holomorphic on D \ {z0}, that is, on the 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]\displaystyle{ f: \Omega\to \mathbb {C} }[/math] is any isolated point of the boundary [math]\displaystyle{ \partial \Omega }[/math] of the domain [math]\displaystyle{ \Omega }[/math]. In other words, if [math]\displaystyle{ U }[/math] is an open subset of [math]\displaystyle{ \mathbb {C} }[/math], [math]\displaystyle{ a\in U }[/math] and [math]\displaystyle{ f: U\setminus \\{a\\}\to \mathbb {C} }[/math] is a holomorphic function, then [math]\displaystyle{ a }[/math] is an isolated singularity of [math]\displaystyle{ f }[/math].

Every singularity of a meromorphic function on an open subset [math]\displaystyle{ 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 singularities, poles and essential singularities.

## Contents

* 1 Examples
* 2 Nonisolated singularities
* 2.1 Examples
* 3 External links

## Examples

* The function [math]\displaystyle{ \frac {1} {z} }[/math] has 0 as an isolated singularity.
* The cosecant function [math]\displaystyle{ \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 analytically continued (or outside them if they are closed curves in the Riemann sphere).

### Examples

The natural boundary of this power series is the unit circle (read examples).

* The function [math]\displaystyle{ \tan\left(\frac{1}{z}\right) }[/math] is meromorphic on [math]\displaystyle{ \mathbb{C}\setminus\\{0\\} }[/math], with simple poles at [math]\displaystyle{ z_n = \left(\frac{\pi}{2}+n\pi\right)^{-1} }[/math], for every [math]\displaystyle{ n\in\mathbb{N}_0 }[/math]. Since [math]\displaystyle{ z_n\rightarrow 0 }[/math], every punctured disk centered at [math]\displaystyle{ 0 }[/math] has an infinite number of singularities within, so no Laurent expansion is available for [math]\displaystyle{ \tan\left(\frac{1}{z}\right) }[/math] around [math]\displaystyle{ 0 }[/math], which is in fact a cluster point of its poles.
* The function [math]\displaystyle{ \csc \left(\frac {\pi} {z}\right) }[/math] 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).
* The function defined via the Maclaurin series [math]\displaystyle{ \sum_{n=0}^{\infty}z^{2^n} }[/math] converges inside the open unit disk centred at [math]\displaystyle{ 0 }[/math] and has the unit circle as its natural boundary.

## External links

* Ahlfors, L., Complex Analysis, 3 ed. (McGraw-Hill, 1979).
* Rudin, W., Real and Complex Analysis, 3 ed. (McGraw-Hill, 1986).
* Weisstein, Eric W.. "Singularity". http://mathworld.wolfram.com/Singularity.html.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Isolated singularity. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Isolated_singularity&oldid=3374520"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template