OFFSET
1,1
COMMENTS
Close to 3 + (1/e) = 3.367879...
Sum_{n>=0} 1/H(n) = 1/log(2) + e - 1 = 3.1609768... This integral may have a similar representation to the Fransen-Robinson constant.
REFERENCES
J. Hadamard, (1894), Oeuvre de Jacques Hadamard, Centre National de la Recherche Scientifiques, Paris, 1968.
LINKS
Philip J. Davis, Leonhard Euler's Integral: A Historical Profile Of The Gamma Function, Am. Math. Monthly 66, 849-869 (1959).
J. Hadamard, Sur l’Expression du Produit 1*2*3...*(n-1) par une Fonction Entière (in French, obtainable from Peter Luschny's Website).
Hywel Normington, Reciprocal Gamma Integrals, 2023.
Hywel Normington, Julia code, 2023.
Hywel Normington, Python code, 2023.
Wikipedia, Hadamard's gamma function.
FORMULA
Hadamard function definitions:
H(x) = (1/Gamma(1-x)) * (d/dx) log(Gamma(1/2 - x/2)/Gamma(1-x/2)).
H(x) = Gamma(x)*(1 + (sin(Pi*x)/(2*Pi)) * (Psi(x/2) - Psi((x+1)/2))).
Equals Integral_{0..oo} 1/H(x) dx.
EXAMPLE
3.368202929607022792162205962209362548476...
MAPLE
H := x -> 1/((sin(x*Pi)*(Psi(x/2) - Psi(1/2 + x/2)) + 2*Pi) * GAMMA(x)):
evalf[80](2*Pi*Int(H, 0..60, method = _Gquad)); # Peter Luschny, Feb 20 2023
MATHEMATICA
RealDigits[NIntegrate[2*Gamma[1-x]/(PolyGamma[0, 1 - x/2] - PolyGamma[0, 1/2 - x/2]), {x, 0, Infinity}, WorkingPrecision -> 105, MaxRecursion -> Infinity]][[1]] (* Vaclav Kotesovec, Feb 19 2023 *)
PROG
(PARI) default(realprecision, 200); intnum(x=0, [[1], 1], 2*gamma(1-x) / (psi(1-x/2) - psi(1/2-x/2))) \\ (default(realprecision, 200) is enough for 40 valid digits, \p 500 for 71 valid digits, \p 1000 for 110 valid digits, \p 2000 for 171 valid digits). - Vaclav Kotesovec, Feb 19 2023
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Hywel Normington, Feb 04 2023
EXTENSIONS
More digits from Vaclav Kotesovec, Feb 19 2023
STATUS
approved