A360809 - OEIS

%I #18 Feb 24 2023 04:49:05

%S 2,5,8,6,7,0,5,0,5,9,7,8,6,8,0,8,2,2,7,7,7,8,1,0,6,8,7,2,9,4,6,9,6,0,

%T 2,1,3,5,7,3,0,9,6,2,7,4,2,4,8,9,3,6,1,2,4,4,6,7,0,8,2,4,2,2,5,8,5,9,

%U 4,0,4,5,5,6,0,6,6,4,3,4,2,6,4,2,8,8,2,7,7,7,5,6,7,5,3,9,0,8,8,7,6,4,4,6,9,9,8,1

%N Decimal expansion of the area under the curve of the reciprocal of the Luschny factorial function from zero to infinity.

%H Peter Luschny, <a href="https://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunctionMJ.html">Is the Gamma-function misdefined? Or: Hadamard vs Euler - Who found the better Gamma function?</a>.

%H Hywel Normington, <a href="https://github.com/Horep/ReciprocalLuschnyIntegrals/blob/main/ReciprocalLuschnyIntegral.py">Python Code</a>, 2023.

%H Hywel Normington, <a href="https://github.com/Horep/ReciprocalHadamardIntegral/blob/main/Reciprocal_Gamma_Integrals.pdf">Reciprocal Gamma Integrals</a>, 2023.

%F L(x) = Gamma(x+1)P(x), where P(x) = 1 - g(x)*sin(Pi*x)/(Pi*x) and g(x) = (x/2)*(Psi((x+1)/2) - Psi(x/2)) - 1/2.

%F Equals Integral_{0..oo} 1/L(x) dx.

%e 2.58670505978680822777810687294696021357309627424893612446708242258594...

%p L := proc(x) local G, S, y; if x = 0 then return 0.5 fi; y := x * 0.5; if is(x < 0) then y := -y fi; G := y * (Psi(y + 0.5) - Psi(y)) - 0.5; if is(x < 0) then return G/(-x)! fi; y := Pi * x; S := sin(y) / y; (1 - S * G) * x! end: RL := x -> 1 / L(x):

%p IntRL := n -> evalf[n](Int(RL, 0..n, method = _Gquad)): IntRL(40); # _Peter Luschny_, Feb 22 2023

%t RealDigits[NIntegrate[1 / (Gamma[x+1] * (1 - (x/2 * (PolyGamma[0, (x+1)/2] - PolyGamma[0, x/2]) - 1/2) * Sin[Pi*x]/(Pi*x))), {x, 0, Infinity}, WorkingPrecision -> 110, MaxRecursion -> Infinity]][[1]] (* _Vaclav Kotesovec_, Feb 22 2023 *)

%o (PARI) default(realprecision, 500); intnum(x=0, [[1], 1], 1 / (gamma(x+1) * (1 - (x/2 * (psi((x+1)/2) - psi(x/2)) - 1/2) * sin(Pi*x)/(Pi*x)))) \\ (default(realprecision, 200) is enough for 59 valid digits, \p 500 for 102 valid digits, \p 1000 for 148 valid digits). - _Vaclav Kotesovec_, Feb 22 2023

%Y Cf. A360375, A058655.

%K nonn,cons

%O 1,1

%A _Hywel Normington_, Feb 21 2023

%E More digits from _Vaclav Kotesovec_, Feb 22 2023