Particular values of the gamma function: Difference between revisions

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{{Short description|Mathematical constants}}

The [[gamma function]] is an important [[special function]] in [[mathematics]]. Its particular values can be expressed in closed form for [[integer]] and, [[half-integer]], and some other rational arguments, but no simple expressions are known for the values at [[rational number|rational]] points in general. Other fractional arguments can be approximated through efficient infinite products, infinite series, and recurrence relations.

==Integers and half-integers==

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and so on. For non-positive integers, the gamma function is not defined.

For positive half-integers <math> \frac{k}{2} </math> where <math> k\in 2\mathbb{N}^*+1 </math> is an odd integer greater or equal <math>3</math>, the function values are given exactly by

:<math>\Gamma \left (\tfrac{nk}{2} \right) = \sqrt \pi \frac{(nk-2)!!}{2^\frac{nk-1}{2}}\,,</math>

or equivalently, for non-negative integer values of&nbsp;{{mvar|n}}:

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\end{align}</math>

where {{math|''n''!^(''q'')}} denotes the {{mvar|q}}th [[Double factorialDouble_factorial#Generalizations|multifactorial]] of {{mvar|n}}. Numerically,

:<math>\Gamma\left(\tfrac13\right) \approx 2.678\,938\,534\,707\,747\,6337</math> {{OEIS2C|A073005}}

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As <math>n</math> tends to infinity,

:<math>\Gamma\left(\tfrac1n\right) \sim n-\gamma </math>

where <math>\gamma</math> is the [[Euler–Mascheroni constant]] and <math>\sim</math> denotes [[Asymptotic analysis|asymptotic equivalence]].

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It is unknown whether these constants are [[transcendental number|transcendental]] in general, but {{math|Γ({{sfrac|1|3}})}} and {{math|Γ({{sfrac|1|4}})}} were shown to be transcendental by [[Chudnovsky brothers|G. V. Chudnovsky]]. {{math|Γ({{sfrac|1|4}}) <big><big>/</big></big> {{radic|π|4}}}} has also long been known to be transcendental, and [[Yuri Valentinovich Nesterenko|Yuri Nesterenko]] proved in 1996 that {{math|Γ({{sfrac|1|4}})}}, {{math|π}}, and {{math|''e''^π}} are [[algebraically independent]].

For <math>n\geq 2</math> at least one of the two numbers <math>\Gamma\left(\tfrac1n\right)</math> and <math>\Gamma\left(\tfrac2n\right)</math> is transcendental.<ref>{{Cite journal |last=Waldschmidt |first=Michel |date=2006 |title=Transcendence of periods: the state of the art |url=https://hal.science/hal-00411301 |journal=Pure and Applied Mathematics Quarterly |volume=2 |issue=2 |pages=435–463|doi=10.4310/PAMQ.2006.v2.n2.a3 }}</ref>

The number {{math|Γ({{sfrac|1|4}})}} is related to the [[lemniscate constant]] {{mvar|ϖ}} by

:The number <math>\Gamma\left(\tfrac14\right)</math> =is related to the [[lemniscate constant]] \sqrt{2\varpi\sqrt{2mvar|<math>\pi}},varpi</math>}} by

:<math>\Gamma \left (\tfrac14 \right ) = \sqrt[4]{4 \pi^3 e^{2 \gamma -varpi\mathrmsqrt{2\delta}+1pi}}</math>

and it has been conjectured by Gramain that

<!-- :<math>\log \Gamma(1/4) = \frac{1}{4}(1 + 3 \log \pi + 2 \log 2 + 2 \gamma - \mathrm{\rho})</math> -->

:<math>\Gamma \left (\tfrac14 \right ) = \sqrt[4]{4 \pi^3 e^{2 \gamma -\mathrm{\delta}+1}}</math>

where {{mvar|δ}} is the [[Masser–Gramain constant]] {{OEIS2C|A086058}}, although numerical work by Melquiond et al. indicates that this conjecture is false.<ref>{{cite journal|doi=10.1090/S0025-5718-2012-02635-4 |first1=Guillaume|last1= Melquiond|first2=W. Georg |last2=Nowak|first3=Paul |last3=Zimmermann|journal=Math. Comp.|title=Numerical approximation of the Masser–Gramain constant to four decimal places|year=2013|volume=82|issue=282|pages=1235–1246|doi-access=free}}</ref>

Borwein and Zucker have found that {{math|Γ({{sfrac|''n''|24}})}} can be expressed algebraically in terms of {{mvar|π}}, {{math|''K''(''k''(1))}}, {{math|''K''(''k''(2))}}, {{math|''K''(''k''(3))}}, and {{math|''K''(''k''(6))}} where {{math|''K''(''k''(''N''))}} is a [[complete elliptic integral of the first kind]]. This permits efficiently approximating the gamma function of rational arguments to high precision using [[quadratic convergence|quadratically convergent]] [[arithmetic–geometric mean]] iterations. For example:

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\Gamma \left(\tfrac14 \right) &= 2\sqrt{K\left( \tfrac 12 \right)\sqrt{\pi}} \\

\Gamma \left(\tfrac13 \right) &= \frac{2^{7/9} \sqrt[3]{\pi K\left(\frac{1}{4}\left(2-\sqrt{3}\right)\right)}}{\sqrt[12]{3}} \\

\Gamma \left(\tfrac 18{1}{8}\right) \Gamma \left(\tfrac 38{3}{8}\right) &= 8 \sqrt[4]{2} \sqrt{\left(\sqrt{2}-1\right) \pi } K\left(3-2 \sqrt{2}\right) \\

\frac{\Gamma \left(\tfrac 18frac{1}{8}\right)}{\Gamma \left(\tfrac 38frac{3}{8}\right)} &= \frac{2 \sqrt{\left(1+\sqrt{2}\right) K\left(\frac{1}{2}\right)}}{\sqrt[4]{\pi }}

\end{align}</math>

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}}</ref>

:<math>\sqrt{\frac{\pi\sqrt{e^\pi}}{2}}\frac{1}{\Gamma^2\left(\frac34\right)^2}=i\sum_{k=-\infty}^\infty e^{\pi(k-2k^2)}\theta_1\left(\frac{i\pi}{2}(2k-1),e^{-\pi}\right),</math>

and

:<math>\sqrt{\frac{\pi}{2}}\frac{1}{\Gamma^2\left(\frac34\right)^2}=\sum_{k=-\infty}^\infty\frac{\theta_4(ik\pi,e^{-\pi})}{e^{2\pi k^2}},</math>

where {{math|''θ''₁}} and {{math|''θ''₄}} are two of the [[Theta function|Jacobi theta functions]].

There also exist a number of [[Carl Johan Malmsten|Malmsten integrals]] for certain values of the gamma function:<ref name=":1">{{Cite journal |last=Blagouchine |first=Iaroslav V. |date=2014-10-01 |title=Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results |url=https://link.springer.com/article/10.1007/s11139-013-9528-5 |journal=The Ramanujan Journal |language=en |volume=35 |issue=1 |pages=21–110 |doi=10.1007/s11139-013-9528-5 |issn=1572-9303|url-access=subscription }}</ref>

Certain values of the gamma function can also be written in terms of the [[hypergeometric function]]. For instance, <math>\Gamma\left(\frac{1}{4}\right)^{4}=\frac{32\pi^{3}}{\sqrt{33}} {}_{3}F_{2}\left(\frac{1}{2},\ \frac{1}{6},\ \frac{5}{6};\ 1,\ 1;\ \frac{8}{1331}\right)</math>

:<math>\int_1^\infty \frac{\ln \ln t}{1+t^2} = \frac\pi4\left(2\ln2 + 3\ln\pi-4\Gamma\left(\tfrac14\right)\right)</math>

and

:<math>\int_1^\infty \frac{\ln \ln t}{1+t+t^2} = \frac\pi{6\sqrt3}\left(8\ln2 -3\ln3 + 8\ln\pi -12\Gamma\left(\tfrac13\right)\right)</math>

<math>\Gamma\left(\frac{1}{3}\right)^{6}=\frac{12\pi^{4}}{\sqrt{10}} {}_{3}F_{2}\left(\frac{1}{2},\ \frac{1}{6},\ \frac{5}{6};\ 1,\ 1;\ -\frac{9}{64000}\right)</math>

however it is an open question whether this is possible for all rational inputs to the gamma function.<ref>Johansson, F. (2023). Arbitrary-precision computation of the gamma function. ''Maple Transactions'', ''3''(1). {{doi|10.5206/mt.v3i1.14591}}</ref>

== Products ==

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:<math> \prod_{r=1}^n \Gamma\left(\tfrac{r}{n+1}\right) = \sqrt{\frac{(2\pi)^n}{n+1}}</math>

From those products can be deduced other values, for example, from the former equations for <math> \prod_{r=1}^3 \Gamma\left(\tfrac{r}{4}\right) </math>, <math>\Gamma\left(\tfrac{1}{4}\right) </math> and <math>\Gamma\left(\tfrac{2}{4}\right) </math>, can be deduced:<ref>{{cite journal |last1=Pascal Sebah |first1=Xavier Gourdon |title=Introduction to the Gamma Function |url=https://www.csie.ntu.edu.tw/~b89089/link/gammaFunction.pdf}}</ref>

<math>\Gamma\left(\tfrac{3}{4}\right) =\left(\tfrac{\pi} {2}\right) ^{\tfrac{1}{4}} {\operatorname{AGM}\left(\sqrt 2, 1\right)}^{\tfrac{1}{2}}</math>

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Here <math>\{\cdot\}</math> denotes the [[fractional part]].

Because of the [[Reflection formula|Euler Reflection Formula]], and the fact that <math>\Gamma(\bar{z})=\bar{\Gamma}(z)</math>, we have an expression for the [[modulus squared]] of the gammaGamma function evaluated on the imaginary axis:

:<math>\left|\Gamma(i\kappa)\right|^2=\frac{\pi}{\kappa\sinh(\pi\kappa)}</math>

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The gamma function has a [[local minimum]] on the positive real axis

:<math>x_{\min} = 1.461\,632\,144\,968\,362\,341\,262\,659\,5423\ldots\,</math> {{OEIS2C|A030169}}

with the value

:<math>\Gamma\left(x_{\min}\right) = 0.885\,603\,194\,410\,888\,700\,278\,815\,9005\ldots\,</math> {{OEIS2C|A030171}}.

Integrating the [[reciprocal gamma function]] along the positive real axis also gives the [[Fransén–Robinson constant]].

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| {{val|−9.7026725400018637360844267649}} || {{0|−}}{{val|0.0000021574161045228505405031}} || {{OEIS2C|A256687}}

|}

The only values of {{math|''x'' > 0}} for which {{math|1=Γ(''x'') = ''x''}} are {{math|1=''x'' = 1}} and {{math|''x'' ≈ {{val|3.5623822853908976914156443427}}}}... {{OEIS2C|A218802}}.

==See also==

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==References==

<references />

==Further reading==

* {{Cite journal

|first1=F.

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}}

* X. Gourdon & P. Sebah. [http://numbers.computation.free.fr/Constants/Miscellaneous/gammaFunction.html Introduction to the Gamma Function]

* S. Finch. [http://www.mathsoft.com/mathsoft_resources/mathsoft_constants/Number_Theory_Constants/2002.asp Euler Gamma Function Constants]{{dead link|date=May 2010}}

* {{MathWorld|title=Gamma Function|urlname=GammaFunction}}

* {{cite journal