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 points in general.

Integers and half-integers

For non-negative integer arguments, the Gamma function coincides with the factorial, that is,

:$Gamma(n+1)\; =\; n!;,\; quad\; n\; in\; mathbb\{N\}\_0,$

and hence

:$Gamma(1)\; =\; 1,,$:$Gamma(2)\; =\; 1,,$:$Gamma(3)\; =\; 2,,$:$Gamma(4)\; =\; 6,,$:$Gamma(5)\; =\; 24.,$

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

:$Gamma(n/2)\; =\; sqrt\; pi\; frac\{(n-2)!!\}\{2^\{(n-1)/2,,$

or equivalently,

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

where "n"!! denotes the double factorial. In particular,

:

and by means of the reflection formula,

:

General rational arguments

In analogy with the half-integer formula,

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

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

:$Gamma(1/3)\; approx\; 2.6789385347077476337$:$Gamma(1/4)\; approx\; 3.6256099082219083119$:$Gamma(1/5)\; approx\; 4.5908437119988030532$:$Gamma(1/6)\; approx\; 5.5663160017802352043$:$Gamma(1/7)\; approx\; 6.5480629402478244377$

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

The number $Gamma(1/4)$ is related to the lemniscate constant "S" by

:$Gamma(1/4)\; =\; sqrt\{sqrt\{2\; pi\}\; S\},$

and it has been conjectured that

:$Gamma(1/4)\; =\; left(4\; pi^3\; e^\{2\; gamma\; -mathrm\{\; ho\}+1\}\; ight)^\{1/4\}$

where ρ is the Masser-Gramain constant.

Borwein and Zucker have found that $Gamma(n/24)$ can be expressed algebraically in terms of π, $K(k(1))$, $K(k(2))$, $K(k(3))$ and $K(k(6))$ where $K(k(N))$ is a complete elliptic integral of the first kind. This permits efficiently approximating the Gamma function of rational arguments to high precision using quadratically convergent arithmetic-geometric mean iterations. No similar relations are known for $Gamma(1/5)$ or other denominators.

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

Other formulas include the infinite products

:$Gamma(1/4)\; =\; (2\; pi)^\{3/4\}\; prod\_\{k=1\}^infty\; anh\; left(\; frac\{pi\; k\}\{2\}\; ight)$

and

:$Gamma(1/4)\; =\; A^3\; e^\{-G\; /\; pi\}\; sqrt\{pi\}\; 2^\{1/6\}\; prod\_\{k=1\}^infty\; left(1-frac\{1\}\{2k\}\; ight)^\{k(-1)^k\}$

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

:$x\_mathrm\{min\}\; =\; 1.461632144968362341262...,$

with the value

:$Gamma(x\_mathrm\{min\})\; =\; 0.885603194410888...,$

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

ee 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]
*
* 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 Γ(1/4) to 1,000,000 decimal places]