Date: Nov 4, 2001 11:52 PM
Author: David W. Cantrell
Subject: Re: Inverse gamma function



This is a small follow-up to my previous article
in this thread.

Many thanks to Bill Gosper for kindly sending me
a copy of a pertinent e-mail, reproduced below my
signature. (As I had indicated previously, the
non-principal branches of the inverse of the gamma
function are reasonably easily dealt with.)
His e-mail answers a question asked by Leroy Quet
on 1999/07/31 from the thread "Inverse Factorial":
"I wonder if there is an integral or series
representation for the inverse of Gamma(x) for any
particular range of x's."

Regards,
David Cantrell
-----------------------------------
>Date: Fri, 16 Feb 1996 00:45-0800
>Subject: inverse factorial
>To: math-fun@cs.arizona.edu

In-Reply-To: <19960215084321.6.RWG@SRINI.macsyma.com>
Message-Id: <19960216084525.7.RWG@SRINI.macsyma.com>

Someone recently complained that Macsyma couldn't solve x! = 720.
In general, there are infinitely many solutions to x! = f, the
one near -n being
n psi (n)
(- 1) 0
(d1114) x = - n - ---------- + ------------
(n - 1)! f 2 2
(n - 1)! f

2 2 n
(3 psi (n) - 9 psi (n) - %pi ) (- 1)
1 0
+ -------------------------------------
3 3
(6 (n - 1)! ) f

3 2
psi (n) - 12 psi (n) psi (n) + 16 psi (n) + 4 %pi psi (n)
2 0 1 0 0
+ ---------------------------------------------------------- + . . .
4 4
(6 (n - 1)! ) f

n = 1, 2, 3, ...

For |f| sufficiently small, this fails to converge until n is
sufficiently large that the turning points of ! are smaller
in magnitude than |f|. This suggests that the "radius of
divergence" (1/radius of convergence) of the series =
|(turning pt nearest to -n)!|. This would give equal radii
(|f| = 3.5446436111550050891219639932755823752019
= - (-1.5040830082644554092582693045333024989554)!)
for n=1 and n=2, whose series indeed have the same denominators.

--Bill Gosper

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