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.

Contents

Integers and half-integers

For positive 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 non-positive integers, the Gamma function is not defined.

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,

\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\,.

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 eπ 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.

In particular, Γ(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.

See also

References

External links

Wikipedia content modification information:

  • This page was last modified on 7 November 2008, at 17:51.

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