```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 articlein this thread.Many thanks to Bill Gosper for kindly sending mea copy of a pertinent e-mail, reproduced below mysignature. (As I had indicated previously, thenon-principal branches of the inverse of the gammafunction are reasonably easily dealt with.)His e-mail answers a question asked by Leroy Queton 1999/07/31 from the thread "Inverse Factorial":"I wonder if there is an integral or seriesrepresentation for the inverse of Gamma(x) for anyparticular range of x's."Regards,  David Cantrell----------------------------------->Date: Fri, 16 Feb 1996 00:45-0800>Subject: inverse factorial>To: math-fun@cs.arizona.eduIn-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 issufficiently large that the turning points of ! are smallerin magnitude than |f|.  This suggests that the "radius ofdivergence" (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-- -------------------- http://NewsReader.Com/ --------------------                    Usenet Newsgroup Service
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