A274441 Decimal expansion of Q(3), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).

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

Offset: 1

Author

Jean-François Alcover, Jun 23 2016

Keywords

• nonn
• cons

Examples

			2.03436897131720444715410048232706998197695047365128645708443723938...
		

Links

• 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

Crossrefs

Cf. A274438 (Q(0)), A274439 (Q(1)), A274440 (Q(2)), A274442 (Q(4)).

Programs

Mathematica

digits = 101;
Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]);
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[3] = -50/9  Cl2[Pi/3]^2 + 596/81 Zeta[4] - 16/9 U + 32/3 V;
RealDigits[N[Q[3], digits] // Chop][[1]]

PARI

Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(3) \\ Gheorghe Coserea, Sep 30 2018

PARI

clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
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)));
-50/9*clausen(2, Pi/3)^2 + 596/81*zeta(4) - 16/9*u31 + 32/3*v31 \\ Gheorghe Coserea, Sep 30 2018

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(3) = -50/9 Cl2(Pi/3)^2+596/81 zeta(4)-16/9 U+32/3 V, where Cl_2 is the Clausen integral, U A255685 and V A274400.