View source for Particular values of the Gamma function



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.

==Integers and half-integers==
For non-negative integer arguments, the Gamma function coincides with the [[factorial]], that is,

:<math>\Gamma(n+1) = n! \quad ; \quad n \in \mathbb{N}_0</math>

and hence

:<math>\Gamma(1) = 1\,</math>
:<math>\Gamma(2) = 1\,</math>
:<math>\Gamma(3) = 2\,</math>
:<math>\Gamma(4) = 6\,</math>
:<math>\Gamma(5) = 24.\,</math>

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

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

or equivalently,

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

where ''n''!! denotes the [[double factorial]]. In particular,

:{|
|-
|<math>\Gamma(1/2)\,</math>
|<math>= \sqrt{\pi}\,</math>
|<math>\approx 1.7724538509055160273\,</math>
|-
|<math>\Gamma(3/2)\,</math>
|<math>= \frac {\sqrt{\pi}} {2} \,</math>
|<math>\approx 0.8862269254527580137\,</math>
|-
|<math>\Gamma(5/2)\,</math>
|<math>= \frac {3 \sqrt{\pi}} {4} \,</math>
|<math>\approx 1.3293403881791370205\,</math>
|-
|<math>\Gamma(7/2)\,</math>
|<math>= \frac {15\sqrt{\pi}} {8} \,</math>
|<math>\approx 3.3233509704478425512\,</math>
|}

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

:{|
|-
|<math>\Gamma(-1/2)\,</math>
|<math>= -2\sqrt{\pi}\,</math>
|<math>\approx -3.5449077018110320546\,</math>
|-
|<math>\Gamma(-3/2)\,</math>
|<math>= \frac {4\sqrt{\pi}} {3} \,</math>
|<math>\approx 2.3632718012073547031.\,</math>
|}

==General rational arguments==

In analogy with the half-integer formula,

:<math>\Gamma(n+1/3) =  \Gamma(1/3) \frac{(3n-2)!^{(3)}}{3^n}</math>
:<math>\Gamma(n+1/4) =  \Gamma(1/4) \frac{(4n-3)!^{(4)}}{4^n}</math>
:<math>\Gamma(n+1/p) =  \Gamma(1/p) \frac{(pn-(p-1))!^{(p)}}{p^n}</math>

where <math>n!^{(k)}</math> denotes the ''k'':th [[factorial#Multifactorials|multifactorial]] of ''n''. By exploiting such functional relations, the Gamma function of any rational argument <math>p/q</math> can be expressed in closed algebraic form in terms of <math>\Gamma(1/q)</math>. However, no closed expressions are known for the numbers <math>\Gamma(1/q)</math> where ''q'' > 2. Numerically,

:<math>\Gamma(1/3) \approx 2.6789385347077476337</math>
:<math>\Gamma(1/4) \approx 3.6256099082219083119</math>
:<math>\Gamma(1/5) \approx 4.5908437119988030532</math>
:<math>\Gamma(1/6) \approx 5.5663160017802352043</math>
:<math>\Gamma(1/7) \approx 6.5480629402478244377</math>

It is unknown whether these constants are [[transcendental number|transcendental]] in general, but <math>\Gamma(1/3)</math> was shown to be transcendental by Le Lionnais in 1983 and Chudnovsky showed the transcendence of <math>\Gamma(1/4)</math> in 1984. <math>\Gamma(1/4) / \pi^{-1/4}</math> has also long been known to be transcendental, and [[Yuri V Nesterenko|Yuri Nesterenko]] proved in 1996 that <math>\Gamma(1/4)</math>, <math>\pi</math> and <math>e^{\pi}</math> are [[algebraically independent]]. 

The number <math>\Gamma(1/4)</math> is related to the [[lemniscate constant]] ''S'' by

:<math>\Gamma(1/4) = \sqrt{\sqrt{2 \pi} S},</math>

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

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

where &rho; is the [[Masser-Gramain constant]].

Borwein and Zucker have found that <math>\Gamma(n/24)</math> can be expressed algebraically in terms of &pi;, <math>K(k(1))</math>, <math>K(k(2))</math>, <math>K(k(3))</math> and <math>K(k(6))</math> where <math>K(k(N))</math> 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. No similar relations are known for <math>\Gamma(1/5)</math> or other denominators.

In particular, <math>\Gamma(1/4)</math> is given by
:<math>\Gamma(1/4) = \sqrt \frac{(2 \pi)^{3/2}}{AGM(\sqrt 2, 1)}.</math>

Other formulas include the [[infinite product]]s

:<math>\Gamma(1/4) = (2 \pi)^{3/4} \prod_{k=1}^\infty \tanh \left( \frac{\pi k}{2} \right)</math>

and

:<math>\Gamma(1/4) = A^3 e^{-G / \pi} \sqrt{\pi} 2^{1/6} \prod_{k=1}^\infty \left(1-\frac{1}{2k}\right)^{k(-1)^k}</math>

where ''A'' is the [[Glaisher-Kinkelin constant]] and ''G'' is [[Catalan's constant]].

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

:<math>x_\mathrm{min} = 1.461632144968362341262...\,</math>

with the value

:<math>\Gamma(x_\mathrm{min}) = 0.885603194410888...\,</math>

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

==See also==
*[[Chowla-Selberg relations]]

==References==
* J. M. Borwein & I. J. Zucker ''Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind;'' IMA J. Numerical Analysis 12, 519-526, 1992.
* 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]
* {{MathWorld|title=Gamma Function|urlname=GammaFunction}}
* W. Duke & Ö. Imamoglu. [http://www.math.ucla.edu/~duke/preprints/special-jntb.pdf Special values of multiple gamma functions]
* V. S. Adamchik. [http://www.cs.cmu.edu/~adamchik/articles/rama.pdf Multiple Gamma Function and Its Application to Computation of Series]

==External links==
* [http://www.dd.chalmers.se/~frejohl/math/gamma14_1_000_000.txt Text file of &Gamma;(1/4) to 1,000,000 decimal places]

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