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,
- <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
- 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. Introduction to the Gamma Function
- S. Finch. Euler Gamma Function Constants
- Template:MathWorld
- W. Duke & Ö. Imamoglu. Special values of multiple gamma functions
- V. S. Adamchik. Multiple Gamma Function and Its Application to Computation of Series