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,


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

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,




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

and by means of the reflection formula,




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

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


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

where &rho; is the Masser-Gramain constant.

Borwein and Zucker have found that $$\Gamma(n/24)$$ can be expressed algebraically in terms of &pi;, $$K(k(1))$$, $$K(k(2))$$, $$K(k(3))$$ 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


 * $$\Gamma(1/4) = (2 \pi)^{3/4} \prod_{k=1}^\infty \tanh \left( \frac{\pi k}{2} \right)$$

and


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

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.