View source for Particular values of the gamma function

{{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]] 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==
For positive integer arguments, the gamma function coincides with the [[factorial]]. That is,

:<math>\Gamma(n) = (n-1)!,</math>

and hence

:<math>\begin{align}
\Gamma(1) &= 1, \\
\Gamma(2) &= 1, \\
\Gamma(3) &= 2, \\
\Gamma(4) &= 6, \\
\Gamma(5) &= 24,
\end{align}</math>

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

For positive half-integers, the function values are given exactly by

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

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

:<math>\begin{align}
\Gamma\left(\tfrac12+n\right) &= \frac{(2n-1)!!}{2^n}\, \sqrt{\pi} = \frac{(2n)!}{4^n n!} \sqrt{\pi} \\
\Gamma\left(\tfrac12-n\right) &= \frac{(-2)^n}{(2n-1)!!}\, \sqrt{\pi} = \frac{(-4)^n n!}{(2n)!} \sqrt{\pi} 
\end{align}</math>

where {{math|''n''!!}} denotes the [[double factorial]]. In particular,

:{|
|-
|<math>\Gamma\left(\tfrac12\right)\,</math>
|<math>= \sqrt{\pi}\,</math>
|<math>\approx 1.772\,453\,850\,905\,516\,0273\,,</math>
|{{OEIS2C|A002161}}
|-
|<math>\Gamma\left(\tfrac32\right)\,</math>
|<math>= \tfrac12 \sqrt{\pi}\,</math>
|<math>\approx 0.886\,226\,925\,452\,758\,0137\,,</math>
|{{OEIS2C|A019704}}
|-
|<math>\Gamma\left(\tfrac52\right)\,</math>
|<math>= \tfrac34 \sqrt{\pi}\,</math>
|<math>\approx 1.329\,340\,388\,179\,137\,0205\,,</math>
|{{OEIS2C|A245884}}
|-
|<math>\Gamma\left(\tfrac72\right)\,</math>
|<math>= \tfrac{15}8 \sqrt{\pi}\,</math>
|<math>\approx 3.323\,350\,970\,447\,842\,5512\,,</math>
|{{OEIS2C|A245885}}
|}

and by means of the [[reflection formula]],

:{|
|-
|<math>\Gamma\left(-\tfrac12\right)\,</math>
|<math>= -2\sqrt{\pi}\,</math>
|<math>\approx -3.544\,907\,701\,811\,032\,0546\,,</math>
|{{OEIS2C|A019707}}
|-
|<math>\Gamma\left(-\tfrac32\right)\,</math>
|<math>= \tfrac43 \sqrt{\pi}\,</math>
|<math>\approx 2.363\,271\,801\,207\,354\,7031\,,</math>
|{{OEIS2C|A245886}}
|-
|<math>\Gamma\left(-\tfrac52\right)\,</math>
|<math>= -\tfrac 8{15} \sqrt{\pi}\,</math>
|<math>\approx -0.945\,308\,720\,482\,941\,8812\,,</math>
|{{OEIS2C|A245887}}
|}

==General rational argument==
In analogy with the half-integer formula,

:<math>\begin{align}
\Gamma \left(n+\tfrac13 \right) &= \Gamma \left(\tfrac13 \right) \frac{(3n-2)!!!}{3^n} \\
\Gamma \left(n+\tfrac14 \right) &= \Gamma \left(\tfrac14 \right ) \frac{(4n-3)!!!!}{4^n} \\
\Gamma \left(n+\tfrac{1}{q} \right ) &= \Gamma \left(\tfrac{1}{q} \right ) \frac{\big(qn-(q-1)\big)!^{(q)}}{q^n} \\
\Gamma \left(n+\tfrac{p}{q} \right) &= \Gamma \left(\tfrac{p}{q}\right) \frac{1}{q^n} \prod _{k=1}^n (k q+p-q)
\end{align}</math>

where {{math|''n''!<sup>(''q'')</sup>}} denotes the {{mvar|q}}th [[Double_factorial#Generalizations|multifactorial]] of {{mvar|n}}. Numerically,

:<math>\Gamma\left(\tfrac13\right) \approx 2.678\,938\,534\,707\,747\,6337</math> {{OEIS2C|A073005}}
:<math>\Gamma\left(\tfrac14\right) \approx 3.625\,609\,908\,221\,908\,3119</math> {{OEIS2C|A068466}}
:<math>\Gamma\left(\tfrac15\right) \approx 4.590\,843\,711\,998\,803\,0532</math> {{OEIS2C|A175380}}
:<math>\Gamma\left(\tfrac16\right) \approx 5.566\,316\,001\,780\,235\,2043</math> {{OEIS2C|A175379}}
:<math>\Gamma\left(\tfrac17\right) \approx 6.548\,062\,940\,247\,824\,4377</math> {{OEIS2C|A220086}}
:<math>\Gamma\left(\tfrac18\right) \approx 7.533\,941\,598\,797\,611\,9047</math> {{OEIS2C|A203142}}.

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]].

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''<sup>π</sup>}} 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>\Gamma\left(\tfrac14\right)</math> is related to the [[lemniscate constant]] {{mvar|<math>\varpi</math>}} by

:<math>\Gamma\left(\tfrac14\right) = \sqrt{2\varpi\sqrt{2\pi}}</math>

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:
:<math>\begin{align}
\Gamma \left(\tfrac16 \right) &= \frac{\sqrt{\frac{3}{\pi }} \Gamma\left(\frac{1}{3}\right)^2}{\sqrt[3]{2}} \\
\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{1}{8}\right) \Gamma \left(\tfrac{3}{8}\right) &= 8 \sqrt[4]{2} \sqrt{\left(\sqrt{2}-1\right) \pi } K\left(3-2 \sqrt{2}\right) \\
\frac{\Gamma \left(\frac{1}{8}\right)}{\Gamma \left(\frac{3}{8}\right)} &= \frac{2 \sqrt{\left(1+\sqrt{2}\right) K\left(\frac{1}{2}\right)}}{\sqrt[4]{\pi }}
\end{align}</math>

No similar relations are known for {{math|Γ({{sfrac|1|5}})}} or other denominators.

In particular, where AGM() is the [[arithmetic–geometric mean]], we have<ref>{{cite web|url=https://math.stackexchange.com/q/1631760  |title=Archived copy |accessdate=2015-03-09 }}</ref>
:<math>\Gamma\left(\tfrac13\right) = \frac{2^\frac{7}{9}\cdot \pi^\frac23}{3^\frac{1}{12}\cdot \operatorname{AGM}\left(2,\sqrt{2+\sqrt{3}}\right)^\frac13}</math>
:<math>\Gamma\left(\tfrac14\right) = \sqrt \frac{(2 \pi)^\frac32}{\operatorname{AGM}\left(\sqrt 2, 1\right)}</math>
:<math>\Gamma\left(\tfrac16\right) = \frac{2^\frac{14}{9}\cdot 3^\frac13\cdot \pi^\frac56}{\operatorname{AGM}\left(1+\sqrt{3},\sqrt{8}\right)^\frac23}.</math>

Other formulas include the [[infinite product]]s

:<math>\Gamma\left(\tfrac14\right) = (2 \pi)^\frac34 \prod_{k=1}^\infty \tanh \left( \frac{\pi k}{2} \right)</math>

and

:<math>\Gamma\left(\tfrac14\right) = A^3 e^{-\frac{G}{\pi}} \sqrt{\pi} 2^\frac16 \prod_{k=1}^\infty \left(1-\frac{1}{2k}\right)^{k(-1)^k}</math>

where {{mvar|A}} is the [[Glaisher–Kinkelin constant]] and {{mvar|G}} is [[Catalan's constant]].

The following two representations for  {{math|Γ({{sfrac|3|4}})}} were given by I. Mező<ref name="Mezo2">{{Citation
  | last = Mező
  | first = István
  | title = Duplication formulae involving Jacobi theta functions and Gosper's ''q''-trigonometric functions
  | journal = Proceedings of the American Mathematical Society
  | pages = 2401–2410
  | volume = 141
  | issue = 7
  | year = 2013
  | doi=10.1090/s0002-9939-2013-11576-5
| doi-access = free
  }}</ref>

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

where {{math|''θ''<sub>1</sub>}} and {{math|''θ''<sub>4</sub>}} 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}}</ref>

:<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>
:<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>

== Products ==
Some product identities include:

:<math> \prod_{r=1}^2 \Gamma\left(\tfrac{r}{3}\right) = \frac{2\pi}{\sqrt 3} \approx 3.627\,598\,728\,468\,435\,7012</math> {{OEIS2C|A186706}}
:<math> \prod_{r=1}^3 \Gamma\left(\tfrac{r}{4}\right) = \sqrt{2\pi^3} \approx 7.874\,804\,972\,861\,209\,8721</math> {{OEIS2C|A220610}}
:<math> \prod_{r=1}^4 \Gamma\left(\tfrac{r}{5}\right) = \frac{4\pi^2}{\sqrt 5} \approx 17.655\,285\,081\,493\,524\,2483</math>
:<math> \prod_{r=1}^5 \Gamma\left(\tfrac{r}{6}\right) = 4\sqrt{\frac{\pi^5}3} \approx 40.399\,319\,122\,003\,790\,0785</math>
:<math> \prod_{r=1}^6 \Gamma\left(\tfrac{r}{7}\right) = \frac{8\pi^3}{\sqrt 7} \approx 93.754\,168\,203\,582\,503\,7970</math>
:<math> \prod_{r=1}^7 \Gamma\left(\tfrac{r}{8}\right) = 4\sqrt{\pi^7} \approx 219.828\,778\,016\,957\,263\,6207</math>

In general:
:<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:

<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>

Other rational relations include

:<math>\frac{\Gamma\left(\tfrac15\right)\Gamma\left(\tfrac{4}{15}\right)}{\Gamma\left(\tfrac13\right)\Gamma\left(\tfrac{2}{15}\right)} = \frac{\sqrt{2}\,\sqrt[20]{3}}{\sqrt[6]{5}\,\sqrt[4]{5-\frac{7}{\sqrt 5}+\sqrt{6-\frac{6}{\sqrt 5}}}}</math>

:<math>\frac{\Gamma\left(\tfrac{1}{20}\right)\Gamma\left(\tfrac{9}{20}\right)}{\Gamma\left(\tfrac{3}{20}\right)\Gamma\left(\tfrac{7}{20}\right)} = \frac{\sqrt[4]{5}\left(1+\sqrt{5}\right)}{2}</math><ref>{{MathWorld|GammaFunction|Gamma Function}}</ref>

:<math>\frac{\Gamma\left(\frac{1}{5}\right)^2}{\Gamma\left(\frac{1}{10}\right)\Gamma\left(\frac{3}{10}\right)} = \frac{\sqrt{1+\sqrt{5}}}{2^{\tfrac{7}{10}}\sqrt[4]{5}}</math>

and many more relations for {{math|Γ({{sfrac|''n''|''d''}})}} where the denominator d divides 24 or 60.<ref>[https://arxiv.org/abs/math/0403510 Raimundas Vidūnas, Expressions for Values of the Gamma Function]</ref>

Gamma quotients with algebraic values must be "poised" in the sense that the sum of arguments is the same (modulo 1) for the denominator and the numerator.

A more sophisticated example:
:<math> \frac{ \Gamma\left(\frac{11}{42}\right)\Gamma\left(\frac27\right)}{\Gamma\left(\frac1{21}\right)\Gamma\left(\frac1{2}\right)} = \frac{8 \sin\left(\frac\pi7\right) \sqrt{\sin\left(\frac\pi{21}\right) \sin\left(\frac{4\pi}{21}\right) \sin\left(\frac{5\pi}{21}\right)}}{2^{\frac1{42}}3^{\frac9{28}}7^{\frac13}} </math><ref>[https://math.stackexchange.com/q/2804457 math.stackexchange.com]</ref>

== Imaginary and complex arguments==
The gamma function at the [[imaginary unit]] {{math|''i'' {{=}} {{sqrt|−1}}}} gives {{OEIS2C|A212877}}, {{OEIS2C|A212878}}:
:<math>\Gamma(i) = (-1+i)! \approx -0.1549 - 0.4980i.</math>
It may also be given in terms of the [[Barnes G-function|Barnes {{mvar|G}}-function]]:

:<math>\Gamma(i) = \frac{G(1+i)}{G(i)} = e^{-\log G(i)+ \log G(1+i)}.</math>

Curiously enough, <math>\Gamma(i)</math> appears in the below integral evaluation:<ref>[https://sites.google.com/site/istvanmezo81/monthly-problems The webpage of István Mező]</ref>
:<math>\int_0^{\pi/2}\{\cot(x)\}\,dx=1-\frac{\pi}{2}+\frac{i}{2}\log\left(\frac{\pi}{\sinh(\pi)\Gamma(i)^2}\right).</math>
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 Gamma function evaluated on the imaginary axis:

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

The above integral therefore relates to the phase of <math>\Gamma(i)</math>.

The gamma function with other complex arguments returns
:<math>\Gamma(1 + i) = i\Gamma(i) \approx 0.498 - 0.155i</math>
:<math>\Gamma(1 - i) = -i\Gamma(-i) \approx 0.498 + 0.155i</math>
:<math>\Gamma(\tfrac12 + \tfrac12 i) \approx 0.818\,163\,9995 - 0.763\,313\,8287\, i</math>
:<math>\Gamma(\tfrac12 - \tfrac12 i) \approx 0.818\,163\,9995 + 0.763\,313\,8287\, i</math>
:<math>\Gamma(5 + 3i) \approx 0.016\,041\,8827 - 9.433\,293\,2898\, i</math>
:<math>\Gamma(5 - 3i) \approx 0.016\,041\,8827 + 9.433\,293\,2898\, i.</math>

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

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

with the value

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

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

On the negative real axis, the first local maxima and minima (zeros of the [[digamma function]]) are:

{| class="wikitable"
|+ Approximate local extrema of {{math|Γ(''x'')}}
! {{mvar|x}}!!{{math|Γ(''x'')}}!![[OEIS]]
|-
| {{val|−0.5040830082644554092582693045}} || {{val|−3.5446436111550050891219639933}} || {{OEIS2C|A175472}}
|-
| {{val|−1.5734984731623904587782860437}} || {{0|−}}{{val|2.3024072583396801358235820396}} || {{OEIS2C|A175473}}
|-
| {{val|−2.6107208684441446500015377157}} || {{val|−0.8881363584012419200955280294}} || {{OEIS2C|A175474}}
|-
| {{val|−3.6352933664369010978391815669}} || {{0|−}}{{val|0.2451275398343662504382300889}} || {{OEIS2C|A256681}}
|-
| {{val|−4.6532377617431424417145981511}} || {{val|−0.0527796395873194007604835708}} || {{OEIS2C|A256682}}
|-
| {{val|−5.6671624415568855358494741745}} || {{0|−}}{{val|0.0093245944826148505217119238}} || {{OEIS2C|A256683}}
|-
| {{val|−6.6784182130734267428298558886}} || {{val|−0.0013973966089497673013074887}} || {{OEIS2C|A256684}}
|-
| {{val|−7.6877883250316260374400988918}} || {{0|−}}{{val|0.0001818784449094041881014174}} || {{OEIS2C|A256685}}
|-
| {{val|−8.6957641638164012664887761608}} || {{val|−0.0000209252904465266687536973}} || {{OEIS2C|A256686}}
|-
| {{val|−9.7026725400018637360844267649}} || {{0|−}}{{val|0.0000021574161045228505405031}} || {{OEIS2C|A256687}}
|}

==See also==
*[[Chowla–Selberg formula]]

==References==
<references />

==Further reading==
* {{Cite journal
|first1=F.
|last1=Gramain
|title=Sur le théorème de Fukagawa-Gel'fond
|journal=Invent. Math.
|volume=63
|number=3
|doi=10.1007/BF01389066
|year=1981
|pages=495&ndash;506
|bibcode=1981InMat..63..495G
|s2cid=123079859
}}
* {{Cite journal
|first1=J. M.
|last1=Borwein
|first2=I. J.
|last2=Zucker
|title=Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind
|journal=IMA Journal of Numerical Analysis
|volume=12
|issue=4
|pages=519&ndash;526
|year=1992
|mr=1186733
|doi=10.1093/imanum/12.4.519
}}
* X. Gourdon & P. Sebah. [http://numbers.computation.free.fr/Constants/Miscellaneous/gammaFunction.html Introduction to the Gamma Function]
* {{MathWorld|title=Gamma Function|urlname=GammaFunction}}
* {{cite journal
|first1=Raimundas |last1=Vidunas
|title=Expressions for values of the gamma function
|arxiv=math.CA/0403510
 |doi=10.2206/kyushujm.59.267
 |volume=59
 |issue=2
|journal=Kyushu Journal of Mathematics
 |pages=267–283|year=2005
|s2cid=119623635
}}
* {{cite journal
|first1=Raimundas | last1=Vidunas
|title=Expressions for values of the gamma function
|journal=Kyushu J. Math. |year=2005
|volume=59 | number=2 | pages=267–283 | mr=2188592 | doi=10.2206/kyushujm.59.267|arxiv=math/0403510| s2cid=119623635
}}
* {{Cite journal
|first1=V. S.
|last1=Adamchik
|url=https://www.cs.cmu.edu/~adamchik/articles/rama.pdf
|title=Multiple Gamma Function and Its Application to Computation of Series
|journal=The Ramanujan Journal
|volume=9
|number=3
|doi=10.1007/s11139-005-1868-3
|year=2005
|pages=271&ndash;288
|mr=2173489
|arxiv=math/0308074
|s2cid=15670340
}}
* {{Cite journal
|first1=W. 
|last1=Duke
|first2=Ö. 
|last2=Imamoglu
|url=https://www.math.ucla.edu/~wdduke/preprints/special-jntb.pdf
|title=Special values of multiple gamma functions
|journal=Journal de Théorie des Nombres de Bordeaux
|volume=18
|issue=1
|year=2006
|mr=2245878
|pages=113&ndash;123
|doi=10.5802/jtnb.536
}}

[[Category:Gamma and related functions]]
[[Category:Mathematical constants]]