Particular values of the Gamma function

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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.

1 Integers and half-integers
2 General rational arguments
3 Other constants
4 See also
5 References
6 External links

[edit] Integers and half-integers

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

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

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,

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

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

$\Gamma(1/2)\,$	$= \sqrt{\pi}\,$	$\approx 1.7724538509055160273\,$
$\Gamma(3/2)\,$	$= \frac {\sqrt{\pi}} {2} \,$	$\approx 0.8862269254527580137\,$
$\Gamma(5/2)\,$	$= \frac {3 \sqrt{\pi}} {4} \,$	$\approx 1.3293403881791370205\,$
$\Gamma(7/2)\,$	$= \frac {15\sqrt{\pi}} {8} \,$	$\approx 3.3233509704478425512\,$

and by means of the reflection formula,

$\Gamma(-1/2)\,$	$= -2\sqrt{\pi}\,$	$\approx -3.5449077018110320546\,$
$\Gamma(-3/2)\,$	$= \frac {4\sqrt{\pi}} {3} \,$	$\approx 2.3632718012073547031.\,$

[edit] 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 $Γ(1 / q)$ . However, no closed expressions are known for the numbers $Γ(1 / q)$ 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 $Γ(1 / 3)$ was shown to be transcendental by Le Lionnais in 1983 and Chudnovsky showed the transcendence of $Γ(1 / 4)$ in 1984. $Γ(1 / 4) / π - 1 / 4$ has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that $Γ(1 / 4)$ , $π$ and $e π$ are algebraically independent.

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

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

and it has been conjectured that

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

where ρ is the Masser-Gramain constant.

Borwein and Zucker have found that $Γ(n / 24)$ 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 $Γ(1 / 5)$ or other denominators.