cp's OEIS Frontend

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

%I A274441 #14 Feb 16 2025 08:33:36
%S A274441 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,
%T A274441 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,
%U A274441 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
%N A274441 Decimal expansion of Q(3), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).
%H A274441 David J. Broadhurst, <a href="http://arxiv.org/abs/hep-th/9803091">Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity</a>, arXiv:hep-th/9803091, 1998, p. 12.
%H A274441 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/ClausensIntegral.html">Clausen's Integral</a>
%F A274441 Q(n) = Integral_{0..inf} arccosh((x+2)/2)^2 log((x+1)/x)/(x+n) dx.
%F A274441 Computation is done using the analytical form given by David Broadhurst:
%F A274441 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.
%e A274441 2.03436897131720444715410048232706998197695047365128645708443723938...
%t A274441 digits = 101;
%t A274441 Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]);
%t A274441 U = A255685 = Pi^4/180 + (Pi^2/12)*Log[2]^2 - (1/12)*Log[2]^4 - 2*PolyLog[4, 1/2];
%t A274441 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));
%t A274441 V = A274400 = 3 Zeta[3]/8 - 1/2 + NSum[v[k], {k, 2, Infinity}, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"];
%t A274441 Q[3] = -50/9  Cl2[Pi/3]^2 + 596/81 Zeta[4] - 16/9 U + 32/3 V;
%t A274441 RealDigits[N[Q[3], digits] // Chop][[1]]
%o A274441 (PARI)
%o A274441 Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
%o A274441 Q(3) \\ _Gheorghe Coserea_, Sep 30 2018
%o A274441 (PARI)
%o A274441 clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
%o A274441 polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
%o A274441 u31=Pi^4/180 + (Pi^2/12)*log(2)^2  - (1/12)*log(2)^4 - 2*polylog(4, 1/2);
%o A274441 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)));
%o A274441 -50/9*clausen(2, Pi/3)^2 + 596/81*zeta(4) - 16/9*u31 + 32/3*v31 \\ _Gheorghe Coserea_, Sep 30 2018
%Y A274441 Cf. A274438 (Q(0)), A274439 (Q(1)), A274440 (Q(2)), A274442 (Q(4)).
%K A274441 nonn,cons
%O A274441 1,1
%A A274441 _Jean-François Alcover_, Jun 23 2016