How to compute the "factorial root"?

[David W. Cantrell]

My thread "Inverse gamma function" (sci.math, 2001 Oct. 25)
http://groups.google.com/group/sci.math/msg/075e8828265ed00d
might be helpful. It gives an approximate inverse.

David W. Cantrell

----- Original Message ----- 
From: "Alonso Del Arte" <alonso.delarte at gmail.com>
To: <seqfan at ext.jussieu.fr>
Sent: Wednesday, June 04, 2008 00:27
Subject: How to compute the "factorial root"?


> Perhaps this is off-topic, but:
>
> How do you compute the "factorial root" of an integer that is not a
> factorial? That is, given an integer n, how do you compute the value
> x of Euler's Gamma function such that Gamma(x) = n ?
>
> For example, the "factorial root" of 24 is 5, since 4! = 24, and the
> factorial root of 120 is 6, since 5! = 120. But, say, for 47, I've
> found, by a long series of trials and errors, that Gamma(5.4328989)
> =
> 47.000518252. Surely the line for the Gamma function crosses the
> exact integer 47 at some point.
>
> So is there a formula, or will I just have to settle for a program
> that merely automates my trial and error?
>
> Alonso del Arte