OFFSET
-1,1
COMMENTS
Gamma(x) stands for the gamma function (Euler's integral of the second kind).
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.
Numerically, the value of C is quite close to the first Stieltjes constant with the opposite sign (see A082633).
LINKS
Iaroslav V. Blagouchine, 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, arXiv:1408.3902 [math.NT], 2014-2016.
FORMULA
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).
EXAMPLE
0.0721862796822481164934370114884600281187017754898161...
MAPLE
Digits:= 500; x0:=fsolve(Psi(x)+GAMMA(x)=0, x): evalf(1/GAMMA(x0)-x0, 120);
MATHEMATICA
y = FindRoot[Gamma[x]+PolyGamma[x]==0, {x, 0.6}, WorkingPrecision->120][[1, 2]]; N[1/Gamma[y] - y, 120] // RealDigits[#, 10, 104] & // First
PROG
(PARI) default(realprecision, 500); x0=solve(x = 0.60, 0.68, gamma(x)+psi(x)); 1/gamma(x0)-x0
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Iaroslav V. Blagouchine, Feb 15 2016
STATUS
approved