login
A360375
Decimal expansion of the area under the curve of the reciprocal of the Hadamard gamma function from zero to infinity.
1
3, 3, 6, 8, 2, 0, 2, 9, 2, 9, 6, 0, 7, 0, 2, 2, 7, 9, 2, 1, 6, 2, 2, 0, 5, 9, 6, 2, 2, 0, 9, 3, 6, 2, 5, 4, 8, 4, 7, 6, 1, 0, 6, 4, 8, 8, 7, 6, 1, 0, 3, 1, 2, 1, 9, 4, 7, 0, 2, 8, 7, 5, 2, 0, 2, 6, 1, 6, 1, 6, 0, 5, 1, 3, 3, 6, 1, 3, 1, 4, 4, 2, 0, 3, 0, 2, 5, 3, 9, 3, 9, 8, 4, 1, 2, 4, 4, 3, 8, 1, 3, 8, 1, 7, 2
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).
Hywel Normington, Reciprocal Gamma Integrals, 2023.
Hywel Normington, Julia code, 2023.
Hywel Normington, Python code, 2023.
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
Cf. A058655.
Sequence in context: A298732 A078477 A336077 * A098832 A368151 A107985
KEYWORD
nonn,cons
AUTHOR
Hywel Normington, Feb 04 2023
EXTENSIONS
More digits from Vaclav Kotesovec, Feb 19 2023
STATUS
approved