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(1) = (4/3)*Cl2(Pi/3)^2 + (7/6)*zeta(4), where Cl_2 is the Clausen integral.
EXAMPLE
2.636185725224872226546402047919868685533952437408546504962614340...
MATHEMATICA
Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]);
Q[1] = 4/3 Cl2[Pi/3]^2 + 7/6 Zeta[4];
RealDigits[N[Q[1], 103] // Chop][[1]]
PROG
(PARI)
Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(1) \\ Gheorghe Coserea, Sep 30 2018
(PARI)
clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
4/3*clausen(2, Pi/3)^2 + 7/6*zeta(4) \\ Gheorghe Coserea, Sep 30 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jun 23 2016
STATUS
approved