Particular values of the Gamma function

Jump to navigation Jump to search


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,

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

where ρ is the Masser-Gramain constant.

Borwein and Zucker have found that <math>\Gamma(n/24)</math> can be expressed algebraically in terms of π, <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 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 products

<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

References

External links

Template:WS

Retrieved from "https://www.wikidoc.org/index.php?title=Particular_values_of_the_Gamma_function&oldid=687080"