OFFSET
1,1
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
FORMULA
Q(n) = Integral_{x>0} arccosh((x+2)/2)^2 log((x+1)/x)/(x+n) dx.
Computation is done using the analytical form given by David Broadhurst:
Q(2) = -Cl2(Pi/3)^2 + 53/16 zeta(4) + 5/2 U, where Cl_2 is the Clausen integral and U is A255685.
EXAMPLE
2.260399248120463689960929066240895031930761500163321388894889042329...
MATHEMATICA
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];
Q[2] = -Cl2[Pi/3]^2 + 53/16 Zeta[4] + 5/2 U;
RealDigits[N[Q[2], 104] // Chop][[1]]
PROG
(PARI)
Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(2) \\ Gheorghe Coserea, Sep 30 2018
(PARI)
clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
u31=Pi^4/180 + (Pi^2/12)*log(2)^2 - (1/12)*log(2)^4 - 2*polylog(4, 1/2);
-clausen(2, Pi/3)^2 + 53/16*zeta(4) + 5/2*u31 \\ Gheorghe Coserea, Sep 30 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jun 23 2016
STATUS
approved