A274442 Decimal expansion of Q(4), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).

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

Offset

1,2

Links

Table of n, a(n) for n=1..101.

David J. Broadhurst, Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, arXiv:hep-th/9803091, 1998, p. 12.

Eric Weisstein's MathWorld, Clausen's Integral

Formula

Q(n) = Integral_{0..inf} arccosh((x+2)/2)^2 log((x+1)/x)/(x+n) dx.

Computation is done using the analytical form given by David Broadhurst:

Q(4) = 125/54 zeta(4) + 8 U - 8 V, where U is A255685 and V A274400.

Example

1.87447166949008260118095099948968029705739765892037953480769845119...

Mathematica

digits = 101;
U = A255685 = Pi^4/180 + (Pi^2/12)*Log[2]^2 - (1/12)*Log[2]^4 - 2*PolyLog[4, 1/2];
v[k_] := ((-1)^k*((24*(k - 1)*(3*k - 4))/(3*k - 2)^3 + (8*(3*k*(3*k - 5) + 4))/(27*(k - 1)^3) + PolyGamma[2, (3*k)/2 - 1] - PolyGamma[2, (3*(k - 1))/2]))/(48*(k - 1)*(3*k - 4)*(3*k - 2));
V = A274400 = 3 Zeta[3]/8 - 1/2 + NSum[v[k], {k, 2, Infinity}, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"];
Q[4] = 125/54 Zeta[4] + 8 U - 8 V;
RealDigits[Q[4], 10, digits][[1]]

Prog

(PARI)
Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(4) \\ Gheorghe Coserea, Sep 30 2018
(PARI)
polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
u31=Pi^4/180 + (Pi^2/12)*log(2)^2  - (1/12)*log(2)^4 - 2*polylog(4, 1/2);
v31=3*zeta(3)/8 - 1/2 + sumalt(k=2, (-1)^k*((24*(k-1)*(3*k-4))/(3*k-2)^3 + (8*(3*k*(3*k-5)+4))/(27*(k-1)^3) + polygamma(2, (3*k)/2-1) - polygamma(2, (3*(k-1))/2))/(48*(k-1)*(3*k-4)*(3*k-2)));
125/54*zeta(4) + 8*u31 - 8*v31 \\ Gheorghe Coserea, Sep 30 2018

Crossrefs

Cf. A274438 (Q(0)), A274439 (Q(1)), A274440 (Q(2)), A274441 (Q(3)).

Sequence in context: A260800 A196914 A072102 * A249136 A154815 A085848

Adjacent sequences: A274439 A274440 A274441 * A274443 A274444 A274445

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 23 2016

Status

approved