Editing Essential singularity

{{Short description|Location around which a function displays irregular behavior}}
{{For|essential singularities of real valued functions|Classification of discontinuities}}

[[File:Essential singularity.png|right|220px|thumb|Plot of the function {{math|exp(1/''z'')}}, centered on the essential singularity at {{math|1=''z'' = 0}}. The hue represents the [[Arg (mathematics)|complex argument]], the luminance represents the [[absolute value]]. This plot shows how approaching the essential singularity from different directions yields different behaviors (as opposed to a pole, which, approached from any direction, would be uniformly white).]]
[[File:Modell des Graphen von 6w=eˆ(1-6z) -Schilling XIV, 6 - 312- (2).jpg|thumb|Model illustrating essential singularity of a complex function {{math|1=6''w'' = exp(1/(6''z''))}}]]

In [[complex analysis]], an '''essential singularity''' of a [[Function (mathematics)|function]] is a "severe" [[singularity (mathematics)|singularity]] near which the function exhibits striking behavior. 

The category ''essential singularity'' is a "left-over" or default group of [[Isolated singularity|isolated singularities]] that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner &ndash; [[removable singularity|removable singularities]] and [[pole (complex analysis)|pole]]s. In practice some{{Who?|date=January 2022}} include non-isolated singularities too; those do not have a [[Residue (complex analysis)|residue]].

==Formal description==
Consider an [[open set|open subset]] <math>U</math> of the [[complex plane]] <math>\mathbb{C}</math>. Let <math>a</math> be an element of <math>U</math>, and <math>f\colon U\setminus\{a\}\to \mathbb{C}</math> a [[holomorphic function]]. The point <math>a</math> is called an ''essential singularity'' of the function <math>f</math> if the singularity is neither a [[pole (complex analysis)|pole]] nor a [[removable singularity]].

For example, the function <math>f(z)=e^{1/z}</math> has an essential singularity at <math>z=0</math>.

==Alternative descriptions==
Let <math>a</math> be a [[complex number]], and assume that <math>f(z)</math> is not defined at <math>a</math> but is [[Analytic function|analytic]] in some region <math>U</math> of the complex plane, and that every [[Open set|open]] [[neighbourhood (mathematics)|neighbourhood]] of <math>a</math> has non-empty intersection with <math>U</math>. 

:If both <math>\lim_{z \to a}f(z)</math> and <math>\lim_{z \to a}\frac{1}{f(z)}</math> exist, then <math>a</math> is a ''[[removable singularity]]'' of both <math>f</math> and <math>\frac{1}{f}</math>. 

:If <math>\lim_{z \to a}f(z)</math> exists but <math>\lim_{z \to a}\frac{1}{f(z)}</math> does not exist (in fact <math>\lim_{z\to a}|1/f(z)|=\infty</math>), then <math>a</math> is a [[zero (complex analysis)|''zero'']] of <math>f</math> and a [[pole (complex analysis)|''pole'']] of <math>\frac{1}{f}</math>.

:Similarly, if <math>\lim_{z \to a}f(z)</math> does not exist (in fact <math>\lim_{z\to a}|f(z)|=\infty</math>) but <math>\lim_{z \to a}\frac{1}{f(z)}</math> exists, then <math>a</math> is a ''pole'' of <math>f</math> and a ''zero'' of <math>\frac{1}{f}</math>.

:If neither <math>\lim_{z \to a}f(z)</math> nor <math>\lim_{z \to a}\frac{1}{f(z)}</math> exists, then <math>a</math> is an '''essential singularity''' of both <math>f</math> and <math>\frac{1}{f}</math>.

Another way to characterize an essential singularity is that the [[Laurent series]] of <math>f</math> at the point <math>a</math> has infinitely many negative degree terms (i.e., the [[principal part]] of the Laurent series is an infinite sum). A related definition is that if there is a point <math>a</math> for which no derivative of <math>f(z)(z-a)^n</math> converges to a limit as <math>z</math> tends to <math>a</math>, then <math>a</math> is an essential singularity of <math>f</math>.<ref>{{cite web |last=Weisstein |first=Eric W. |title=Essential Singularity |url=http://mathworld.wolfram.com/EssentialSingularity.html |website=MathWorld |publisher=Wolfram |access-date=11 February 2014}}</ref>

On a [[Riemann sphere]] with a [[point at infinity]], <math>\infty_\mathbb{C}</math>, the function <math>{f(z)}</math> has an essential singularity at that point if and only if the <math>{f(1/z)}</math> has an essential singularity at 0: i.e. neither <math>\lim_{z \to 0}{f(1/z)}</math> nor <math>\lim_{z \to 0}\frac{1}{f(1/z)}</math> exists.<ref>{{Cite web|title=Infinity as an Isolated Singularity|url=https://people.math.gatech.edu/~xchen/teach/comp_analysis/note-sing-infinity.pdf|access-date=2022-01-06}}</ref> The [[Riemann zeta function]] on the Riemann sphere has only one essential singularity, at <math>\infty_\mathbb{C}</math>.<ref>{{Cite journal |last1=Steuding |first1=Jörn |last2=Suriajaya |first2=Ade Irma |date=2020-11-01 |title=Value-Distribution of the Riemann Zeta-Function Along Its Julia Lines |journal=Computational Methods and Function Theory |language=en |volume=20 |issue=3 |pages=389–401 |doi=10.1007/s40315-020-00316-x |issn=2195-3724|doi-access=free |hdl=2324/4483207 |hdl-access=free }}</ref> Indeed, every [[meromorphic]] function aside that is not a [[rational function]] has a unique essential singularity at <math>\infty_\mathbb{C}</math>.

The behavior of [[holomorphic function]]s near their essential singularities is described by the [[Casorati–Weierstrass theorem]] and by the considerably stronger [[Picard's great theorem]]. The latter says that in every neighborhood of an essential singularity <math>a</math>, the function <math>f</math> takes on ''every'' complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function <math>\exp(1/z)</math> never takes on the value 0.)

==References==
{{reflist}}

{{refbegin}}
*Lars V. Ahlfors; ''Complex Analysis'', McGraw-Hill, 1979
*Rajendra Kumar Jain, S. R. K. Iyengar; ''Advanced Engineering Mathematics''. Page 920. Alpha Science International, Limited, 2004. {{ISBN|1-84265-185-4}}
{{refend}}

== External links ==
* '' [http://demonstrations.wolfram.com/AnEssentialSingularity/ An Essential Singularity]'' by [[Stephen Wolfram]], [[Wolfram Demonstrations Project]].
* [http://planetmath.org/essentialsingularity Essential Singularity on Planet Math]

[[Category:Complex analysis]]