login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Decimal expansion of Q(3), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).
4

%I #13 Oct 01 2018 04:10:38

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

%H 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 Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/ClausensIntegral.html">Clausen's Integral</a>

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

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

%F 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 2.03436897131720444715410048232706998197695047365128645708443723938...

%t digits = 101;

%t Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]);

%t U = A255685 = Pi^4/180 + (Pi^2/12)*Log[2]^2 - (1/12)*Log[2]^4 - 2*PolyLog[4, 1/2];

%t 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 V = A274400 = 3 Zeta[3]/8 - 1/2 + NSum[v[k], {k, 2, Infinity}, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"];

%t Q[3] = -50/9 Cl2[Pi/3]^2 + 596/81 Zeta[4] - 16/9 U + 32/3 V;

%t RealDigits[N[Q[3], digits] // Chop][[1]]

%o (PARI)

%o Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));

%o Q(3) \\ _Gheorghe Coserea_, Sep 30 2018

%o (PARI)

%o clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));

%o polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));

%o u31=Pi^4/180 + (Pi^2/12)*log(2)^2 - (1/12)*log(2)^4 - 2*polylog(4, 1/2);

%o 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 -50/9*clausen(2, Pi/3)^2 + 596/81*zeta(4) - 16/9*u31 + 32/3*v31 \\ _Gheorghe Coserea_, Sep 30 2018

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

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Jun 23 2016