A274440 - OEIS

%I #11 Oct 01 2018 02:39:56

%S 2,2,6,0,3,9,9,2,4,8,1,2,0,4,6,3,6,8,9,9,6,0,9,2,9,0,6,6,2,4,0,8,9,5,

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

%U 8,5,7,4,8,5,6,8,7,2,5,7,0,5,8,8,7,5,0,4,7,0,4,6,7,8,6,2,0,3,7,4,5,0,7,5

%N Decimal expansion of Q(2), 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_{x>0} 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(2) = -Cl2(Pi/3)^2 + 53/16 zeta(4) + 5/2 U, where Cl_2 is the Clausen integral and U is A255685.

%e 2.260399248120463689960929066240895031930761500163321388894889042329...

%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 Q[2] = -Cl2[Pi/3]^2 + 53/16 Zeta[4] + 5/2 U;

%t RealDigits[N[Q[2], 104] // Chop][[1]]

%o (PARI)

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

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

%o -clausen(2, Pi/3)^2 + 53/16*zeta(4) + 5/2*u31 \\ _Gheorghe Coserea_, Sep 30 2018

%Y Cf. A274438 (Q(0)), A274439 (Q(1)), A274441 (Q(3)), A274442 (Q(4)).

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Jun 23 2016