A360375 Decimal expansion of the area under the curve of the reciprocal of the Hadamard gamma function from zero to infinity.

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

Table of n, a(n) for n=1..105.

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

Cf. A058655.

Sequence in context: A298732 A078477 A336077 * A098832 A368151 A107985

Adjacent sequences: A360372 A360373 A360374 * A360376 A360377 A360378

Keyword

nonn,cons

Author

Hywel Normington, Feb 04 2023

Extensions

More digits from Vaclav Kotesovec, Feb 19 2023

Status

approved