Essential singularity - Revision history

imported>1234qwer1234qwer4: link author: Lars Ahlfors (via WP:JWB)

2026-04-24T21:16:01Z

link author: Lars Ahlfors (via WP:JWB)

← Previous revision		Revision as of 21:16, 24 April 2026
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	== Formal description ==		== Formal description ==
	Consider an [[open set\|open subset]] <math>U</math> of the [[complex plane]] {{tmath\| \~~mathbb{~~C} }}. Let <math>a</math> be an element of {{tmath\| U }}, and <math>f: U\smallsetminus\{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]].		Consider an [[open set\|open subset]] <math>U</math> of the [[complex plane]] {{tmath\| \C }}. Let <math>a</math> be an element of {{tmath\| U }}, and <math>f: U\smallsetminus\{a\}\to \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 {{tmath\|1= z=0 }}.		For example, the function <math>f(z)=e^{1/z}</math> has an essential singularity at {{tmath\|1= z=0 }}.
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	* If both <math>\lim_{z \to a}f(z)</math> and <math>\lim_{z \to a} {1}/{f(z)}</math> exist, then <math>a</math> is a ''[[removable singularity]]'' of both <math>f</math> and {{tmath\| {1}/{f} }}.		* If both <math>\lim_{z \to a}f(z)</math> and <math>\lim_{z \to a} {1}/{f(z)}</math> exist, then <math>a</math> is a ''[[removable singularity]]'' of both <math>f</math> and {{tmath\| {1}/{f} }}.
	* If <math>\lim_{z \to a}f(z)</math> exists but <math>\lim_{z \to a} {1}/{f(z)}</math> does not exist ({{tmath\|1= \lim_{z\to a} \left\vert {1}/{f(z)} \right\vert = \infty }}), then <math>a</math> is a [[zero (complex analysis)\|''zero'']] of <math>f</math> and a [[pole (complex analysis)\|''pole'']] of {{tmath\| {1}/{f} }}.		* If <math>\lim_{z \to a}f(z)</math> exists but <math>\lim_{z \to a} {1}/{f(z)}</math> does not exist ({{tmath\|1= \lim_{z\to a} \left\vert {1}/{f(z)} \right\vert = \infty }}), then <math>a</math> is a [[zero (complex analysis)\|''zero'']] of <math>f</math> and a [[pole (complex analysis)\|''pole'']] of {{tmath\| {1}/{f} }}.
	* 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} {1}/{f(z)}</math> exists, then <math>a</math> is a ''pole'' of <math>f</math> and a ''zero'' of {{tmath\| {1}/{f} }}.		* If <math>\lim_{z \to a}f(z)</math> does not exist (in fact {{tmath\| \lim_{z\to a} \vert f(z) \vert = \infty }}) but <math>\lim_{z \to a} {1}/{f(z)}</math> exists, then <math>a</math> is a ''pole'' of <math>f</math> and a ''zero'' of {{tmath\| {1}/{f} }}.
	* If neither <math>\lim_{z \to a} f(z)</math> nor <math>\lim_{z \to a} {1}/{f(z)}</math> exists, then <math>a</math> is an '''essential singularity''' of both <math>f</math> and {{tmath\| {1}/{f} }}.		* If neither <math>\lim_{z \to a} f(z)</math> nor <math>\lim_{z \to a} {1}/{f(z)}</math> exists, then <math>a</math> is an '''essential singularity''' of both <math>f</math> and {{tmath\| {1}/{f} }}.

	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 <math>f(z)(z-a)^n</math> is not differentiable for any integer {{tmath\| n > 0 }}, then <math>a</math> is an essential singularity of {{tmath\| f }}.<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>		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 <math>f(z)(z-a)^n</math> is not differentiable for any integer {{tmath\| n > 0 }}, then <math>a</math> is an essential singularity of {{tmath\| f }}.<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]], {{tmath\| \infty_\~~mathbb{~~C} }}, 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 {{tmath\| 0 }}: i.e. neither <math>\lim_{z \to 0}{f(1/z)}</math> nor <math>\lim_{z \to 0} {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, which is 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 \|arxiv=2007.14661 }}</ref> Indeed, every [[meromorphic]] function aside that is not a [[rational function]] has a unique essential singularity at {{tmath\| \infty_\~~mathbb{~~C} }}.		On a [[Riemann sphere]] with a [[point at infinity]], {{tmath\| \infty_\C }}, 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 {{tmath\| 0 }}: i.e. neither <math>\lim_{z \to 0}{f(1/z)}</math> nor <math>\lim_{z \to 0} {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, which is at {{tmath\| \infty_\C }}.<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 \|arxiv=2007.14661 }}</ref> Indeed, every [[meromorphic]] function aside that is not a [[rational function]] has a unique essential singularity at {{tmath\| \infty_\C }}.

	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 {{tmath\| a }}, 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 {{tmath\| 0 }}.)		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 {{tmath\| a }}, 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 {{tmath\| 0 }}.)
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	{{refbegin}}		{{refbegin}}
	* {{citation \|first1=Lars V. \|last1=Ahlfors \|title=Complex Analysis \|publisher=McGraw-Hill \|date=1979 }}		* {{citation \|first1=Lars V. \|last1=Ahlfors \|author-link=Lars Ahlfors \|title=Complex Analysis \|publisher=McGraw-Hill \|date=1979 }}
	* {{citation \|first1=Rajendra Kumar \|last1=Jain \|first2=S. R. K. \|last2=Iyengar \|title=Advanced Engineering Mathematics \|page=920 \|publisher=Alpha Science International, Limited \|date=2004 \|isbn=1-84265-185-4 }}		* {{citation \|first1=Rajendra Kumar \|last1=Jain \|first2=S. R. K. \|last2=Iyengar \|title=Advanced Engineering Mathematics \|page=920 \|publisher=Alpha Science International, Limited \|date=2004 \|isbn=1-84265-185-4 }}
	{{refend}}		{{refend}}

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

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

imported>Quondum: /* Alternative descriptions */ undoing 2017 rewording that corrupted the meaning

2025-12-14T14:32:15Z

Alternative descriptions: undoing 2017 rewording that corrupted the meaning

← Previous revision		Revision as of 14:32, 14 December 2025
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	{{~~Short~~ description\|Location around which a function displays irregular behavior}}		{{short description\|Location around which a function displays irregular behavior}}
	{{~~For~~\|essential singularities of real valued functions\|Classification of discontinuities}}		{{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: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.		In [[complex analysis]], an '''essential singularity''' of a [[Function (mathematics)\|function]] is a "severe" [[singularity (mathematics)\|singularity]] near which the function exhibits striking behavior.
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	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 – [[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]].		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 – [[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==		== 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]].		Consider an [[open set\|open subset]] <math>U</math> of the [[complex plane]] {{tmath\| \mathbb{C} }}. Let <math>a</math> be an element of {{tmath\| U }}, and <math>f: U\smallsetminus\{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>~~.		For example, the function <math>f(z)=e^{1/z}</math> has an essential singularity at {{tmath\|1= z=0 }}.

	==Alternative descriptions==		== 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>.		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 {{tmath\| U }}.
			* If both <math>\lim_{z \to a}f(z)</math> and <math>\lim_{z \to a} {1}/{f(z)}</math> exist, then <math>a</math> is a ''[[removable singularity]]'' of both <math>f</math> and {{tmath\| {1}/{f} }}.
			* If <math>\lim_{z \to a}f(z)</math> exists but <math>\lim_{z \to a} {1}/{f(z)}</math> does not exist ({{tmath\|1= \lim_{z\to a} \left\vert {1}/{f(z)} \right\vert = \infty }}), then <math>a</math> is a [[zero (complex analysis)\|''zero'']] of <math>f</math> and a [[pole (complex analysis)\|''pole'']] of {{tmath\| {1}/{f} }}.
			* 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} {1}/{f(z)}</math> exists, then <math>a</math> is a ''pole'' of <math>f</math> and a ''zero'' of {{tmath\| {1}/{f} }}.
			* If neither <math>\lim_{z \to a} f(z)</math> nor <math>\lim_{z \to a} {1}/{f(z)}</math> exists, then <math>a</math> is an '''essential singularity''' of both <math>f</math> and {{tmath\| {1}/{f} }}.

	~~: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~~>.		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 <math>f(z)(z-a)^n</math> is not differentiable for any integer {{tmath\| n > 0 }}, then <math>a</math> is an essential singularity of {{tmath\| f }}.<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>

	~~: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>~~.		On a [[Riemann sphere]] with a [[point at infinity]], {{tmath\| \infty_\mathbb{C} }}, 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 {{tmath\| 0 }}: i.e. neither <math>\lim_{z \to 0}{f(1/z)}</math> nor <math>\lim_{z \to 0} {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, which is 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 \|arxiv=2007.14661 }}</ref> Indeed, every [[meromorphic]] function aside that is not a [[rational function]] has a unique essential singularity at {{tmath\| \infty_\mathbb{C} }}.

	~~: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>~~.		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 {{tmath\| a }}, 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 {{tmath\| 0 }}.)

	: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>.		== References ==

	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 \|arxiv=2007.14661 }}</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}}		{{reflist}}

	{{refbegin}}		{{refbegin}}
	*Lars V. Ahlfors~~; ''~~Complex Analysis~~'',~~ McGraw-Hill, 1979		* {{citation \|first1=Lars V. \|last1=Ahlfors \|title=Complex Analysis \|publisher=McGraw-Hill \|date=1979 }}
	*Rajendra Kumar Jain, S. R. K. Iyengar~~; ''~~Advanced Engineering Mathematics~~''. Page~~ 920. Alpha Science International, Limited, 2004~~. {{ISBN~~\|1-84265-185-4}}		* {{citation \|first1=Rajendra Kumar \|last1=Jain \|first2=S. R. K. \|last2=Iyengar \|title=Advanced Engineering Mathematics \|page=920 \|publisher=Alpha Science International, Limited \|date=2004 \|isbn=1-84265-185-4 }}
	{{refend}}		{{refend}}

imported>OAbot: Open access bot: arxiv updated in citation with #oabot.

2025-06-18T16:16:06Z

Open access bot: arxiv updated in citation with #oabot.

← Previous revision		Revision as of 16:16, 18 June 2025
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	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>		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>.		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 \|arxiv=2007.14661 }}</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.)		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.)

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{{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 – [[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]]