Value of Gamma(1/3)??

[Gottfried Barthel]

Question: Is there any nice expression for the value of Gamma(1/3) other
than the formula Gamma(1/3)*Gamma(2/3) = 2*Pi*sqrt(3)/3 ?

Maple says GAMMA(1/3) = 2*Pi*sqrt(3)/(3*GAMMA(2/3)) and returns
GAMMA(2/3) = GAMMA(2/3) which isn't too helpful.

Similarly, what about Gamma(1/4)?

Thanks for any useful hints!

Gottfried Barthel

Fachbereich Mathematik und Statistik, Universitaet Konstanz

[Matthias Mahnke]

Gottfried Barthel wrote:
>
> Question: Is there any nice expression for the value of Gamma(1/3) other
> than the formula Gamma(1/3)*Gamma(2/3) = 2*Pi*sqrt(3)/3 ?
>
> Maple says GAMMA(1/3) = 2*Pi*sqrt(3)/(3*GAMMA(2/3)) and returns
> GAMMA(2/3) = GAMMA(2/3) which isn't too helpful.
>
> Similarly, what about Gamma(1/4)?
>

Some time ago in de.sci.math i had a similar question about
the minimal value of the Gamma function.

Back then people assured me that the only really known values
(those which can be expressed by simpler means than gamma values)
are Gamma(z) and Gamma(z+1/2) where z is a whole number.

Up to now I didn't find anything contradictory.
If you do, please let me know.

HTH Matthias
--

[Thomas Kellar]

Maybe you have seen the below. I hope it is okay
to quote from a web page. There is a similar page
for Gamma[1/4]. (gamma14.txt and look at index.html)

Quoting from http://www.lacim.uqam.ca/piDATA/gamma13.txt

Thomas

quote --

Gamma(1/3) to 250000 digits computed by Greg J. Fee and Simon Plouffe
on November 22, 1996 on a SGI R10000 machine at 194 Mhz.

The computation took 105 minutes.

We used the following formula
2/3 4/9 3/4
PI 2 3
Gamma(1/3) =1/3 ---------------
1/3
agm(1, v)
Where v = (sqrt(3)-1)/2*sqrt(2)) ,

References :

Fast evaluation of the gamma function for small rational fractions
using complete elliptic integrals of the first kind, by J.M. Borwein
and I.J. Zucker, IMA Journal of Numerical Analysis (1992) 12, 519-526.

also the classic : Pi and the AGM, by J.M. and P.B. Borwein.

--unquote

[Gottfried Barthel]

Thomas Kellar wrote:
>
> Maybe you have seen the below. I hope it is okay
> to quote from a web page. There is a similar page
> for Gamma[1/4]. (gamma14.txt and look at index.html)
>
> Quoting from http://www.lacim.uqam.ca/piDATA/gamma13.txt
>
> Thomas
>
> quote --

[...]
> --unquote

Thomas:

Thank you for the interesting link; I was not aware of it. In fact, I would
have been more interested in formulae expressing Gamma(1/3) in terms of other
"familiar" constants or special values of "familiar" functions. Raymond
Manzoni <ray...@club-internet.fr> on Thu, 05 Jul 2001 12:57:02 +0200
answered my posting to sci.math, giving another interesting link to
S. Finch's page :

http://pauillac.inria.fr/algo/bsolve/constant/gamma/gamma.html

On that page, one finds a reference for the algebraic independence of
Gamma(1/3) and pi, so there is no hope of having such a nice formula for
Gamma(1/3) like the one for Gamma(1/2) ...

Regards from Konstanz,

Gottfried Barthel