Incomplete gamma function - Revision history

imported>Hellacioussatyr at 20:53, 17 November 2025

2025-11-17T20:53:53Z

← Previous revision		Revision as of 20:53, 17 November 2025
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	<math display="block">\gamma(s, z) \asymp z^s \, \Gamma(s) \, \gamma^*(s, 0) = z^s \, \Gamma(s)/\Gamma(s+1) = z^s/s.</math>		<math display="block">\gamma(s, z) \asymp z^s \, \Gamma(s) \, \gamma^*(s, 0) = z^s \, \Gamma(s)/\Gamma(s+1) = z^s/s.</math>

	For positive real {{mvar\|x}}, {{mvar\|y}} and {{mvar\|s}}, {{math\|''x''<sup>''y''</sup>/y → 0}}, when {{math\|(''x'', ''y'') → (0, ''s'')}}. This seems to justify setting {{math\|1=''γ''(''s'', 0) = 0}} for real {{math\|''s'' > 0}}. However, matters are somewhat different in the complex realm. Only if (a) the real part of {{mvar\|s}} is positive, and (b) values {{math\|''u''<sup>''v''</sup>}} are taken from just a finite set of branches, they are guaranteed to converge to zero as {{math\|(''u'', ''v'') → (0, ''s'')}}, and so does {{math\|''γ''(''u'', ''v'')}}. On a single [[branch point\|branch]] of {{math\|''γ''(''b'')}} is naturally fulfilled, so '''there''' {{math\|1=''γ''(''s'', 0) = 0}} for {{mvar\|s}} with positive real part is a [[Continuous function\|continuous limit]]. Also note that such a continuation is by no means an [[analytic continuation\|analytic one]].		For positive real {{mvar\|x}}, {{mvar\|y}} and {{mvar\|s}}, {{math\|''x''<sup>''y''</sup>/y → 0}}, when {{math\|(''x'', ''y'') → (0, ''s'')}}. This seems to justify setting {{math\|1=''γ''(''s'', 0) = 0}} for real {{math\|''s'' > 0}}. However, matters are somewhat different in the complex realm. Only if (a) the real part of {{mvar\|s}} is positive, and (b) values {{math\|''u''<sup>''v''</sup>}} are taken from just a [[finite set]] of branches, they are guaranteed to converge to zero as {{math\|(''u'', ''v'') → (0, ''s'')}}, and so does {{math\|''γ''(''u'', ''v'')}}. On a single [[branch point\|branch]] of {{math\|''γ''(''b'')}} is naturally fulfilled, so '''there''' {{math\|1=''γ''(''s'', 0) = 0}} for {{mvar\|s}} with positive real part is a [[Continuous function\|continuous limit]]. Also note that such a continuation is by no means an [[analytic continuation\|analytic one]].

	=====Algebraic relations=====		=====Algebraic relations=====
	All algebraic relations and differential equations observed by the real {{math\|''γ''(''s'', ''z'')}} hold for its holomorphic counterpart as well. This is a consequence of the identity theorem, stating that equations between holomorphic functions valid on a real interval, hold everywhere. In particular, the recurrence relation <ref name="auto2"/> and {{math\|1=''∂γ''(''s'', ''z'')/''∂z'' = ''z''<sup>''s''−1</sup> ''e''<sup>−''z''</sup>}} <ref name="auto2"/> are preserved on corresponding branches.		All algebraic relations and [[Differential equation\|differential equations]] observed by the real {{math\|''γ''(''s'', ''z'')}} hold for its holomorphic counterpart as well. This is a consequence of the [[identity theorem]], stating that equations between holomorphic functions valid on a real interval, hold everywhere. In particular, the recurrence relation <ref name="auto2"/> and {{math\|1=''∂γ''(''s'', ''z'')/''∂z'' = ''z''<sup>''s''−1</sup> ''e''<sup>−''z''</sup>}} <ref name="auto2"/> are preserved on corresponding branches.

	=====Integral representation=====		=====Integral representation=====
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	To determine the limit, the power series of <math>\gamma^*</math> at {{math\|1=''z'' = 0}} is useful. When replacing <math>e^{-x}</math> by its power series in the integral definition of <math>\gamma</math>, one obtains (assume {{mvar\|x}},{{mvar\|s}} positive reals for now):		To determine the limit, the power series of <math>\gamma^*</math> at {{math\|1=''z'' = 0}} is useful. When replacing <math>e^{-x}</math> by its power series in the integral definition of <math>\gamma</math>, one obtains (assume {{mvar\|x}},{{mvar\|s}} positive reals for now):
	<math display="block">\gamma(s, x) = \int_0^x t^{s-1} e^{-t} \, dt = \int_0^x \sum_{k=0}^\infty \left(-1\right)^k \, \frac{t^{s+k-1}}{k!} \, dt = \sum_{k=0}^\infty \left(-1\right)^k \, \frac{x^{s+k}}{k!(s+k)} = x^s\,\sum_{k=0}^\infty \frac{(-x)^k}{k!(s+k)}</math>		<math display="block">\begin{align}
			\gamma(s, x) &= \int_0^x t^{s-1} e^{-t} \, dt
			= \int_0^x \sum_{k=0}^\infty \left(-1\right)^k \, \frac{t^{s+k-1}}{k!} \, dt \\[1ex]
			&= \sum_{k=0}^\infty \left(-1\right)^k \, \frac{x^{s+k}}{k!(s+k)}
			= x^s\,\sum_{k=0}^\infty \frac{(-x)^k}{k!(s+k)}
			\end{align}</math>
	or<ref name="auto1"/>		or<ref name="auto1"/>
	<math display="block">\gamma^*(s,x) = \sum_{k=0}^\infty \frac{(-x)^k}{k!\,\Gamma(s)(s+k)},</math>		<math display="block">\gamma^*(s,x) = \sum_{k=0}^\infty \frac{(-x)^k}{k!\,\Gamma(s)(s+k)},</math>
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	With its restriction to real values lifted, the series allows the expansion:		With its restriction to real values lifted, the series allows the expansion:
	<math display="block">\gamma(s, z) - \frac{1}{s} = - \frac{1}{s} + z^s\,\sum_{k=0}^\infty \frac{(-z)^k}{k!(s+k)} = \frac{z^s-1}{s} + z^s\, \sum_{k=1}^\infty \frac{\left(-z\right)^k}{k!(s+k)},~~\quad~~ \Re(s) > -1, \,s \ne 0.</math>		<math display="block">\begin{align}
			\gamma(s, z) - \frac{1}{s} &= - \frac{1}{s} + z^s\,\sum_{k=0}^\infty \frac{(-z)^k}{k!(s+k)} \\[1ex]
			&= \frac{z^s-1}{s} + z^s\, \sum_{k=1}^\infty \frac{\left(-z\right)^k}{k!(s+k)}, & \Re(s) > -1, \,s \ne 0.
			\end{align}</math>

	When {{math\|''s'' → 0}}:<ref>[[Gamma function#General\|see last eq.]]</ref>		When {{math\|''s'' → 0}}:<ref>[[Gamma function#General\|see last eq.]]</ref>
	<math display="block">\frac{z^s-1}{s} \to \ln(z),\quad \Gamma(s) - \frac{1}{s} = \frac{1}{s} - \gamma + O(s) - \frac{1}{s} \to -\gamma,</math>		<math display="block">\frac{z^s-1}{s} \to \ln(z),\quad \Gamma(s) - \frac{1}{s} = \frac{1}{s} - \gamma + O(s) - \frac{1}{s} \to -\gamma,</math>
	(<math>\gamma</math> is the [[Euler–Mascheroni constant]] here), hence,		(<math>\gamma</math> is the [[Euler–Mascheroni constant]] here), hence,
	<math display="block">\Gamma(0,z) = \lim_{s \to 0}\left(\Gamma(s) - \tfrac{1}{s} - (\gamma(s, z) - \tfrac{1}{s})\right) = -\gamma - \ln(z) - \sum_{k=1}^\infty \frac{(-z)^k}{k\,(k!)}</math>		<math display="block">\begin{align}
			\Gamma(0,z) &= \lim_{s \to 0}\left(\Gamma(s) - \tfrac{1}{s} - \left(\gamma(s, z) - \tfrac{1}{s}\right)\right) \\
			&= -\gamma - \ln(z) - \sum_{k=1}^\infty \frac{\left(-z\right)^k}{k\,(k!)}
			\end{align}</math>
	is the limiting function to the upper incomplete gamma function as {{math\|''s'' → 0}}, also known as the [[exponential integral]] {{nowrap\|<math>E_1(z)</math>.}}<ref>{{Cite web\|url=https://dlmf.nist.gov/8.4\|title=DLMF: §8.4 Special Values ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions\|website=dlmf.nist.gov}}</ref>		is the limiting function to the upper incomplete gamma function as {{math\|''s'' → 0}}, also known as the [[exponential integral]] {{nowrap\|<math>E_1(z)</math>.}}<ref>{{Cite web\|url=https://dlmf.nist.gov/8.4\|title=DLMF: §8.4 Special Values ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions\|website=dlmf.nist.gov}}</ref>

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	Again with [[confluent hypergeometric functions]] and employing Kummer's identity,		Again with [[confluent hypergeometric functions]] and employing Kummer's identity,
	<math display="block">\begin{align}		<math display="block">\begin{align}
	\Gamma(s,z) &= e^{-z} U(1-s,1-s,z) = \frac{z^s e^{-z}}{\Gamma(1-s)} \int_0^\infty \frac{e^{-u}}{u^s (z+u)} du \\		\Gamma(s,z) &= e^{-z} U(1{-}s, 1{-}s, z)
	&= e^{-z} z^s U(1,1+s,z) = e^{-z} \int_0^\infty e^{-u} (z+u)^{s-1} du = e^{-z} z^s \int_0^\infty e^{-z u} (1+u)^{s-1} du.		= \frac{z^s e^{-z}}{\Gamma(1-s)} \int_0^\infty \frac{e^{-u}}{u^s (z+u)} du \\
			&= e^{-z} z^s U(1,1{+}s,z)
			= e^{-z} \int_0^\infty e^{-u} (z+u)^{s-1} du \\
			&= e^{-z} z^s \int_0^\infty e^{-z u} (1+u)^{s-1} du.
	\end{align}</math>		\end{align}</math>

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	===Multiplication theorem===		===Multiplication theorem===
	The following [[multiplication theorem]] holds true{{Citation needed\|reason=Origin of statement unclear; also clarification of L-function would be helpful, possibly https://en.wikipedia.org/wiki/Laguerre_polynomials#Generalized_Laguerre_polynomials?\|date=November 2024}}:		The following [[multiplication theorem]] holds true{{Citation needed\|reason=Origin of statement unclear; also clarification of L-function would be helpful, possibly https://en.wikipedia.org/wiki/Laguerre_polynomials#Generalized_Laguerre_polynomials?\|date=November 2024}}:
	<math display="block">\Gamma(s,z) = \frac 1 {t^s} \sum_{i=0}^{\infty} \frac{\left(1-\frac 1 t \right)^i}{i!} \Gamma(s+i,t z)		<math display="block">\begin{align}
	= \Gamma(s,t z) -(t z)^s e^{-t z} \sum_{i=1}^{\infty} \frac{\left(\frac 1 t-1 \right)^i}{i} L_{i-1}^{(s-i)}(t z).</math>		\Gamma(s,z) &= \frac 1 {t^s} \sum_{i=0}^{\infty} \frac{\left(1-\frac 1 t \right)^i}{i!} \Gamma(s+i,t z) \\
			&= \Gamma(s,t z) -(t z)^s e^{-t z} \sum_{i=1}^{\infty} \frac{\left(\frac 1 t-1 \right)^i}{i} L_{i-1}^{(s-i)}(t z).
			\end{align}</math>

	===Software implementation===		===Software implementation===
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	This formula can be derived by repeated integration by parts.		This formula can be derived by repeated integration by parts.

	~~In the context of the [[stable count distribution]], the <math> s </math> parameter can be regarded as inverse of Lévy's stability parameter <math> \alpha</math>:~~
	~~<math display="block"> Q(s,x) = \int_0^\infty e^{\left( -{x^s}/{\nu} \right)} \, \mathfrak{N}_{{1}/{s}}\left(\nu\right) \, d\nu , \quad (s > 1)</math>~~
	~~where <math>\mathfrak{N}_{\alpha}(\nu)</math> is a standard stable count distribution of shape <math> \alpha = 1/s < 1</math>.~~

	<math>P(s,x)</math> and <math>Q(s, x)</math> are implemented as <code>gammainc</code><ref>{{Cite web\|url=https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.gammainc.html#scipy.special.gammainc\|title=scipy.special.gammainc — SciPy v1.11.4 Manual\|website=docs.scipy.org}}</ref> and <code>gammaincc</code><ref>{{Cite web\|url=https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.gammaincc.html\|title=scipy.special.gammaincc — SciPy v1.11.4 Manual\|website=docs.scipy.org}}</ref> in [[scipy]].		<math>P(s,x)</math> and <math>Q(s, x)</math> are implemented as <code>gammainc</code><ref>{{Cite web\|url=https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.gammainc.html#scipy.special.gammainc\|title=scipy.special.gammainc — SciPy v1.11.4 Manual\|website=docs.scipy.org}}</ref> and <code>gammaincc</code><ref>{{Cite web\|url=https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.gammaincc.html\|title=scipy.special.gammaincc — SciPy v1.11.4 Manual\|website=docs.scipy.org}}</ref> in [[scipy]].
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	</ref>		</ref>
	<math display="block">\frac{\partial \Gamma (s,x) }{\partial s} = \ln x \Gamma (s,x) + x\,T(3,s,x)</math>		<math display="block">\frac{\partial \Gamma (s,x) }{\partial s} = \ln x \Gamma (s,x) + x\,T(3,s,x)</math>
	and the second derivative by		and the [[second derivative]] by
	<math display="block">\frac{\partial^2 \Gamma (s,x) }{\partial s^2} = \ln^2 x \Gamma (s,x) + 2 x \left[\ln x\,T(3,s,x) + T(4,s,x) \right]</math>		<math display="block">\frac{\partial^2 \Gamma (s,x) }{\partial s^2} = \ln^2 x \Gamma (s,x) + 2 x \left[\ln x\,T(3,s,x) + T(4,s,x) \right]</math>
	where the function <math>T(m,s,x)</math> is a special case of the [[Meijer G-function]]		where the function <math>T(m,s,x)</math> is a special case of the [[Meijer G-function]]

imported>RhubarbJayde: Fix typo (Hartog -> Hartogs)

2025-06-13T09:53:00Z

Fix typo (Hartog -> Hartogs)

← Previous revision		Revision as of 09:53, 13 June 2025
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	Given the rapid growth in [[Absolute value#Complex numbers\|absolute value]] of {{math\|Γ(''z'' + ''k'')}} when {{math\|''k'' → ∞}}, and the fact that the [[Reciprocal Gamma function\|reciprocal of {{math\|Γ(''z'')}}]] is an [[entire function]], the coefficients in the rightmost sum are well-defined, and locally the sum [[Uniform convergence\|converges uniformly]] for all complex {{mvar\|s}} and {{mvar\|x}}. By a theorem of [[Weierstrass]],<ref name="class notes">{{cite web \|url=http://www.math.washington.edu/~marshall/math_534/Notes.pdf \|title=Complex Analysis \| work=Math 534 \|date= Autumn 2009 \| author = Donald E. Marshall \|publisher=University of Washington \|type=student handout \|access-date=2011-04-23 \|url-status=dead \|archive-url=https://web.archive.org/web/20110516005152/http://www.math.washington.edu/~marshall/math_534/Notes.pdf \|archive-date=2011-05-16 \|at= Theorem 3.9 on p.56}}</ref> the limiting function, sometimes denoted as {{nowrap\|<math>\gamma^*</math>,}}<ref name="auto1">{{Cite web\|url=https://dlmf.nist.gov/8.7\|title=DLMF: §8.7 Series Expansions ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions\|website=dlmf.nist.gov}}</ref>		Given the rapid growth in [[Absolute value#Complex numbers\|absolute value]] of {{math\|Γ(''z'' + ''k'')}} when {{math\|''k'' → ∞}}, and the fact that the [[Reciprocal Gamma function\|reciprocal of {{math\|Γ(''z'')}}]] is an [[entire function]], the coefficients in the rightmost sum are well-defined, and locally the sum [[Uniform convergence\|converges uniformly]] for all complex {{mvar\|s}} and {{mvar\|x}}. By a theorem of [[Weierstrass]],<ref name="class notes">{{cite web \|url=http://www.math.washington.edu/~marshall/math_534/Notes.pdf \|title=Complex Analysis \| work=Math 534 \|date= Autumn 2009 \| author = Donald E. Marshall \|publisher=University of Washington \|type=student handout \|access-date=2011-04-23 \|url-status=dead \|archive-url=https://web.archive.org/web/20110516005152/http://www.math.washington.edu/~marshall/math_534/Notes.pdf \|archive-date=2011-05-16 \|at= Theorem 3.9 on p.56}}</ref> the limiting function, sometimes denoted as {{nowrap\|<math>\gamma^*</math>,}}<ref name="auto1">{{Cite web\|url=https://dlmf.nist.gov/8.7\|title=DLMF: §8.7 Series Expansions ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions\|website=dlmf.nist.gov}}</ref>
	<math display="block">\gamma^*(s, z) := e^{-z}\sum_{k=0}^\infty\frac{z^k}{\Gamma(s+k+1)}</math>		<math display="block">\gamma^*(s, z) := e^{-z}\sum_{k=0}^\infty\frac{z^k}{\Gamma(s+k+1)}</math>
	is [[Entire function\|entire]] with respect to both {{mvar\|z}} (for fixed {{mvar\|s}}) and {{mvar\|s}} (for fixed {{mvar\|z}}),<ref name="auto3"/> and, thus, holomorphic on {{math\|'''C''' × '''C'''}} by [[~~Hartog~~'s theorem]].<ref>{{cite web\|author=Paul Garrett\|url=https://www-users.cse.umn.edu/~garrett/m/complex/hartogs.pdf\|title=Hartogs' Theorem: separate analyticity implies joint\|website=cse.umn.edu\|access-date=21 December 2023}}</ref> Hence, the following ''decomposition''<ref name="auto3"/>		is [[Entire function\|entire]] with respect to both {{mvar\|z}} (for fixed {{mvar\|s}}) and {{mvar\|s}} (for fixed {{mvar\|z}}),<ref name="auto3"/> and, thus, holomorphic on {{math\|'''C''' × '''C'''}} by [[Hartogs's theorem on separate holomorphicity\|Hartogs' theorem]].<ref>{{cite web\|author=Paul Garrett\|url=https://www-users.cse.umn.edu/~garrett/m/complex/hartogs.pdf\|title=Hartogs' Theorem: separate analyticity implies joint\|website=cse.umn.edu\|access-date=21 December 2023}}</ref> Hence, the following ''decomposition''<ref name="auto3"/>
	<math display="block">\gamma(s,z) = z^s \, \Gamma(s) \, \gamma^*(s,z),</math>		<math display="block">\gamma(s,z) = z^s \, \Gamma(s) \, \gamma^*(s,z),</math>
	extends the real lower incomplete gamma function as a [[holomorphic function]], both jointly and separately in {{mvar\|z}} and {{mvar\|s}}. It follows from the properties of <math>z^s</math> and the [[Gamma function\|Γ-function]], that the first two factors capture the [[Mathematical singularity\|singularities]] of <math>\gamma(s,z)</math> (at {{math\|1=''z'' = 0}} or {{mvar\|s}} a non-positive integer), whereas the last factor contributes to its zeros.		extends the real lower incomplete gamma function as a [[holomorphic function]], both jointly and separately in {{mvar\|z}} and {{mvar\|s}}. It follows from the properties of <math>z^s</math> and the [[Gamma function\|Γ-function]], that the first two factors capture the [[Mathematical singularity\|singularities]] of <math>\gamma(s,z)</math> (at {{math\|1=''z'' = 0}} or {{mvar\|s}} a non-positive integer), whereas the last factor contributes to its zeros.

imported>Citation bot: Added bibcode. | Use this bot. Report bugs. | Suggested by Dominic3203 | Category:Continued fractions | #UCB_Category 8/29

2025-04-27T06:54:16Z

Added bibcode. | Use this bot. Report bugs. | Suggested by Dominic3203 | Category:Continued fractions | #UCB_Category 8/29

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