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. Other fractional arguments can be approximated through efficient infinite products, infinite series, and recurrence relations.

1 Integers and half-integers
2 General rational arguments
3 Imaginary unit
4 Complex Arguments
5 Other constants
6 See also
7 References
8 External links

Integers and half-integers

For positive integer arguments, the Gamma function coincides with the factorial, that is,

$\Gamma(n%2B1) = n!\;, \quad n \in \mathbb{N}_0,$

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

$\Gamma(\tfrac12 n) = \sqrt \pi \frac{(n-2)!!}{2^{(n-1)/2}}\,,$

or equivalently, for non-negative integer values of n:

$\Gamma\left(\frac{1}{2}%2Bn\right) = \frac{(2n-1)!!}{2^n}\, \sqrt{\pi} = {(2n)! \over 4^n n!} \sqrt{\pi}$

$\Gamma\left(\frac{1}{2}-n\right) = \frac{(-2)^n}{(2n-1)!!}\, \sqrt{\pi} = {(-4)^n n! \over (2n)!} \sqrt{\pi}$

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

$\Gamma(\tfrac12)\,$	$= \sqrt{\pi}\,$	$\approx 1.7724538509055160273\,,$	A002161
$\Gamma(\tfrac32)\,$	$= \frac {1}{2} \sqrt{\pi}\,$	$\approx 0.8862269254527580137\,,$	A019704
$\Gamma(\tfrac52)\,$	$= \frac {3}{4} \sqrt{\pi}\,$	$\approx 1.3293403881791370205\,,$
$\Gamma(\tfrac72)\,$	$= \frac {15}{8} \sqrt{\pi}\,$	$\approx 3.3233509704478425512\,,$

and by means of the reflection formula,

$\Gamma(-\tfrac12)\,$	$= -2\sqrt{\pi}\,$	$\approx -3.5449077018110320546\,,$	A019707
$\Gamma(-\tfrac32)\,$	$= \frac {4}{3} \sqrt{\pi}\,$	$\approx 2.3632718012073547031\,,$
$\Gamma(-\tfrac52)\,$	$= -\frac {8}{15} \sqrt{\pi}\,$	$\approx -0.9453087204829418812\,.$

General rational arguments

In analogy with the half-integer formula,

$\Gamma(n%2B\tfrac13) = \Gamma(\tfrac13) \frac{(3n-2)!^{(3)}}{3^n}$

$\Gamma(n%2B\tfrac14) = \Gamma(\tfrac14) \frac{(4n-3)!^{(4)}}{4^n}$

$\Gamma(n%2B1/p) = \Gamma(1/p) \frac{(pn-(p-1))!^{(p)}}{p^n}$

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,

$\Gamma(\tfrac13) \approx 2.6789385347077476337$ A073005

$\Gamma(\tfrac14) \approx 3.6256099082219083119$ A068466

$\Gamma(\tfrac15) \approx 4.5908437119988030532$ A175380

$\Gamma(\tfrac16) \approx 5.5663160017802352043$ A175379

$\Gamma(\tfrac17) \approx 6.5480629402478244377$

It is unknown whether these constants are transcendental in general, but $\Gamma(\tfrac13)$ and $\Gamma(\tfrac14)$ were shown to be transcendental by Chudnovsky. $\Gamma(\tfrac14) / \pi^{-1/4}$ has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that $\Gamma(\tfrac14)$ , $\pi$ , and $e^{\pi}$ are algebraically independent.

The number $\Gamma(\tfrac14)$ is related to the lemniscate constant S by

$\Gamma(\tfrac14) = \sqrt{\sqrt{2 \pi} S},$

and it has been conjectured that

$\Gamma(\tfrac14) = \left(4 \pi^3 e^{2 \gamma -\mathrm{\rho}%2B1}\right)^{1/4}$

where ρ is the Masser-Gramain constant A086058.

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

In particular, $\Gamma(\tfrac14)$ is given by

$\Gamma(\tfrac14) = \sqrt \frac{(2 \pi)^{3/2}}{AGM(\sqrt 2, 1)}.$

Other formulas include the infinite products

$\Gamma(\tfrac14) = (2 \pi)^{3/4} \prod_{k=1}^\infty \tanh \left( \frac{\pi k}{2} \right)$

and

$\Gamma(\tfrac14) = A^3 e^{-G / \pi} \sqrt{\pi} 2^{1/6} \prod_{k=1}^\infty \left(1-\frac{1}{2k}\right)^{k(-1)^k}$

where A is the Glaisher-Kinkelin constant and G is Catalan's constant.

C. H. Brown derived rapidly converging infinite series for particular values of the gamma function ^[1].:

$\frac{[\Gamma(\tfrac13)]^6\sqrt{10}}{12\pi^4}=\sum_{k = 0}^{\infty} \frac{(6k)!(-1)^k}{(k!)^{3}(3k)! 3^{k}160^{3k}}$

as well as,

$\frac{[\Gamma(\tfrac14)]^4}{128\pi^3} = \frac{1}{\sqrt{u}} \sum_{k = 0}^{\infty} \frac{(6k)!(2w)^k}{(k!)^{3}(3k)! 6486^{3k}}$

where,

$\begin{align} u &= 273%2B180\sqrt{2}\\ w &= 538359129\sqrt{2}-761354780\\ \end{align}$

or, since 2w/6486^3 in fact is a cube involving u,

$\frac{[\Gamma(\tfrac14)]^4}{128\pi^3} = \frac{1}{\sqrt{u}} \sum_{k = 0}^{\infty} \frac{(6k)!}{(k!)^{3}(3k)!} \frac{1}{(u\sqrt{2}(1%2B\sqrt{2})^2)^{3k}}$

Imaginary unit

The gamma function on the imaginary unit $i = \sqrt{-1}$ returns

$\Gamma(i) = (-1%2Bi)! \approx -0.1549 - 0.4980i.$

It may also be given in terms of the Barnes G-function:

$\Gamma(i) = \frac{G(1%2Bi)}{G(i)} = e^{-\log G(i)%2B \log G(1%2Bi)}.$

Complex Arguments

The gamma function with the complex Arguments $i = \sqrt{-1}$ returns

$\Gamma(1 %2B i) \approx 0.498 - 0.155i$

$\Gamma(1 - i) \approx 0.498 %2B 0.155i$

$\Gamma(0.5 %2B 0.5i) \approx 0.851 - 0.761i$

$\Gamma(0.5 - 0.5i) \approx 0.851 %2B 0.761i$

$\Gamma(5 %2B 3i) \approx 0.016 - 9.433i$

$\Gamma(5 - 3i) \approx 0.016 %2B 9.433i$

Other constants

The Gamma function has a local minimum on the positive real axis

$x_\mathrm{min} = 1.461632144968362341262\ldots\,$ A030169

with the value

$\Gamma(x_\mathrm{min}) = 0.885603194410888\ldots\,$ A030171

Integrating the reciprocal Gamma function along the positive real axis also gives the Fransén-Robinson constant.

On the negative real axis, the first local maxima and minima (zeros of the Digamma function) are:

Approximate local extrema of Γ(x)
x	Γ(x)
-0.5040830082644554092582693045	-3.5446436111550050891219639933
-1.5734984731623904587782860437	2.3024072583396801358235820396
-2.6107208684441446500015377157	-0.8881363584012419200955280294
-3.6352933664369010978391815669	0.2451275398343662504382300889
-4.6532377617431424417145981511	-0.0527796395873194007604835708
-5.6671624415568855358494741745	0.0093245944826148505217119238
-6.6784182130734267428298558886	-0.0013973966089497673013074887
-7.6877883250316260374400988918	0.0001818784449094041881014174
-8.6957641638164012664887761608	-0.0000209252904465266687536973
-9.7026725400018637360844267649	0.0000021574161045228505405031

References

^ Cetin Hakimgolu-Brown : iamned.com math page

Borwein, J. M.; Zucker, I. J. (1992). "Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind". IMA J. Numerical Analysis 12 (4): 519–526. MR 1186733.
X. Gourdon & P. Sebah. Introduction to the Gamma Function
S. Finch. Euler Gamma Function Constants
Weisstein, Eric W., "Gamma Function" from MathWorld.
Duke, W.; Imamoglu, Ö. (2006). "Special values of multiple gamma functions". J. Theor. Nombres Bordeaux 18 (1): 113–123. MR 2245878. http://www.math.ucla.edu/~wdduke/preprints/special-jntb.pdf.
Adamchik, V. S. (2005). "Multiple Gamma Function and Its Application to Computation of Series". Ramanujan Journal 9: 271–288. MR 2173489. http://www.cs.cmu.edu/~adamchik/articles/rama.pdf.