OFFSET
0,1
COMMENTS
The probability density function for the standard normal distribution is e^(-x^2/2 + zeta'(0)). - Rick L. Shepherd, Mar 08 2014
For every x > 0, PolyGamma(-2, x+1) - (PolyGamma(-2, x) + x*log(x) - x) equals this constant -zeta'(0), where polygamma functions of negative indices are defined for x > 0 as: PolyGamma(-1, x) = log(Gamma(x)), PolyGamma(-(n+1), x) = Integral_{t=0..x} PolyGamma(-n, x) dx, n >= 1. - Jianing Song, Apr 20 2021
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
J. Sondow and E. W. Weisstein, MathWorld: Wallis Formula.
Eric Weisstein's World of Mathematics, Log Gamma Function.
Eric Weisstein's World of Mathematics, Stirling's Approximation.
Wikipedia, Gamma function.
Wikipedia, Normal curve
FORMULA
Equals Integral_{x=0..1} log(Gamma(x)) dx. - Jean-François Alcover, Apr 29 2013
More generally, equals t-t*log(t)+Integral_{x=t..(t+1)} log(Gamma(x)) dx for any t>=0 (the Raabe formula). - Stanislav Sykora, May 14 2015
Equals lim_{k->oo} log(k!) + k - (k + 1/2)*log(k) (by Stirling's formula). - Amiram Eldar, Aug 21 2020
EXAMPLE
0.91893853320467274178032...
MAPLE
evalf(log(2*Pi)/2, 120); # Muniru A Asiru, Oct 08 2018
MATHEMATICA
Log[Sqrt[2*Pi]] // RealDigits[#, 10, 104] & // First (* Jean-François Alcover, Apr 29 2013 *)
PROG
(PARI) -zeta'(0) \\ Charles R Greathouse IV, Mar 28 2012
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Log(2*Pi(R))/2; // G. C. Greubel, Oct 07 2018
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Benoit Cloitre, Oct 02 2002
EXTENSIONS
Normalized representation (leading zero and offset) R. J. Mathar, Jan 25 2009
STATUS
approved