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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