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: firstname.lastname@example.org 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
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.