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Topic:
Inverse gamma function
Replies:
1
Last Post:
Nov 4, 2001 11:52 PM




Re: Inverse gamma function
Posted:
Nov 4, 2001 11:52 PM


This is a small followup to my previous article in this thread.
Many thanks to Bill Gosper for kindly sending me a copy of a pertinent email, reproduced below my signature. (As I had indicated previously, the nonprincipal branches of the inverse of the gamma function are reasonably easily dealt with.) His email 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 00:450800 >Subject: inverse factorial >To: mathfun@cs.arizona.edu InReplyTo: <19960215084321.6.RWG@SRINI.macsyma.com> MessageId: <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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