Essential singularity: Difference between revisions

Latest revision as of 14:32, 14 December 2025

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Plot of the function

exp(1/ z)

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z = 0

Script error: No such module "Check for unknown parameters".. The hue represents the 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).

In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior.

The category essential singularity is a "left-over" or default group of 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 singularities and poles. In practice someScript error: No such module "Unsubst". include non-isolated singularities too; those do not have a residue.

Formal description

Consider an open subset $U$ of the complex plane Template:Tmath. Let $a$ be an element of Template:Tmath, and $f : U ∖ {a} \to ℂ$ a holomorphic function. The point $a$ is called an essential singularity of the function $f$ if the singularity is neither a pole nor a removable singularity.

For example, the function $f (z) = e^{1 / z}$ has an essential singularity at Template:Tmath.

Alternative descriptions

Let $a$ be a complex number, and assume that $f (z)$ is not defined at $a$ but is analytic in some region $U$ of the complex plane, and that every open neighbourhood of $a$ has non-empty intersection with Template:Tmath.

If both $\lim_{z \to a} f (z)$ and $\lim_{z \to a} 1 / f (z)$ exist, then $a$ is a removable singularity of both $f$ and Template:Tmath.
If $\lim_{z \to a} f (z)$ exists but $\lim_{z \to a} 1 / f (z)$ does not exist (Template:Tmath), then $a$ is a zero of $f$ and a pole of Template:Tmath.
If $\lim_{z \to a} f (z)$ does not exist (in fact $\lim_{z \to a} | f (z) | = \infty$ ) but $\lim_{z \to a} 1 / f (z)$ exists, then $a$ is a pole of $f$ and a zero of Template:Tmath.
If neither $\lim_{z \to a} f (z)$ nor $\lim_{z \to a} 1 / f (z)$ exists, then $a$ is an essential singularity of both $f$ and Template:Tmath.

Another way to characterize an essential singularity is that the Laurent series of $f$ at the point $a$ 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 $a$ for which $f (z) (z - a)^{n}$ is not differentiable for any integer Template:Tmath, then $a$ is an essential singularity of Template:Tmath.^[1]

On a Riemann sphere with a point at infinity, Template:Tmath, the function $f (z)$ has an essential singularity at that point if and only if the $f (1 / z)$ has an essential singularity at Template:Tmath: i.e. neither $\lim_{z \to 0} f (1 / z)$ nor $\lim_{z \to 0} 1 / f (1 / z)$ exists.^[2] The Riemann zeta function on the Riemann sphere has only one essential singularity, which is at $\infty_{ℂ}$ .^[3] Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at Template:Tmath.

The behavior of holomorphic functions 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 Template:Tmath, the function $f$ takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function $\exp (1 / z)$ never takes on the value Template:Tmath.)

References

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External links

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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: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.
@@ Line 9: / Line 8: @@
 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==
+== 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}}
 {{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}}

Essential singularity: Difference between revisions

Latest revision as of 14:32, 14 December 2025

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Formal description

Alternative descriptions

References

External links

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Essential singularity: Difference between revisions

Latest revision as of 14:32, 14 December 2025

Formal description

Alternative descriptions

References

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