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A268895
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Decimal expansion of the upper bound of 1/Gamma(x) - x on the unit interval x = [0,1].
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4
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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, 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, 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
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OFFSET
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-1,1
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COMMENTS
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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).
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LINKS
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FORMULA
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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).
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EXAMPLE
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0.0721862796822481164934370114884600281187017754898161...
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MAPLE
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Digits:= 500; x0:=fsolve(Psi(x)+GAMMA(x)=0, x): evalf(1/GAMMA(x0)-x0, 120);
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MATHEMATICA
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y = FindRoot[Gamma[x]+PolyGamma[x]==0, {x, 0.6}, WorkingPrecision->120][[1, 2]]; N[1/Gamma[y] - y, 120] // RealDigits[#, 10, 104] & // First
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PROG
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(PARI) default(realprecision, 500); x0=solve(x = 0.60, 0.68, gamma(x)+psi(x)); 1/gamma(x0)-x0
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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