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%I #16 Feb 26 2016 05:06:18
%S 7,2,1,8,6,2,7,9,6,8,2,2,4,8,1,1,6,4,9,3,4,3,7,0,1,1,4,8,8,4,6,0,0,2,
%T 8,1,1,8,7,0,1,7,7,5,4,8,9,8,1,6,1,3,9,3,8,7,4,7,3,5,8,8,3,4,8,3,9,3,
%U 8,1,4,5,8,9,1,9,3,8,6,7,2,1,5,3,3,6,3,8,9,0,2,2,0,0,8,4,8,7,6,0,7,1,0,6
%N Decimal expansion of the upper bound of 1/Gamma(x) - x on the unit interval x = [0,1].
%C Gamma(x) stands for the gamma function (Euler's integral of the second kind).
%C On the unit interval the function 1/Gamma(x) may be bounded from below and from above as follows: x <= 1/Gamma(x) <= x + C, where C = 0.072186279... is the constant which we introduced above. Numerical simulations show that these lower and upper bounds are both quite accurate. Some other bounds for 1/Gamma(x) may be found in the reference given below.
%C Numerically, the value of C is quite close to the first Stieltjes constant with the opposite sign (see A082633).
%H Iaroslav V. Blagouchine, <a href="http://arxiv.org/abs/1408.3902">Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi</a>, arXiv:1408.3902 [math.NT], 2014-2016.
%F Equals 1/Gamma(x_0) - x_0, where x_0 is the unique positive root of the equation Gamma(x) + Psi(x) = 0 (see A268893).
%e 0.0721862796822481164934370114884600281187017754898161...
%p Digits:= 500; x0:=fsolve(Psi(x)+GAMMA(x)=0, x): evalf(1/GAMMA(x0)-x0, 120);
%t y = FindRoot[Gamma[x]+PolyGamma[x]==0, {x, 0.6}, WorkingPrecision->120][[1, 2]]; N[1/Gamma[y] - y, 120] // RealDigits[#, 10, 104] & // First
%o (PARI) default(realprecision, 500); x0=solve(x = 0.60, 0.68, gamma(x)+psi(x)); 1/gamma(x0)-x0
%Y Cf. A268893, A268911.
%K nonn,cons
%O -1,1
%A _Iaroslav V. Blagouchine_, Feb 15 2016
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