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_{0..inf} arccosh((x+2)/2)^2 log((x+1)/x)/(x+n) dx.
Computation is done using the analytical form given by David Broadhurst: Q(0) = 4 Cl_2(Pi/3)^2, where Cl_2 is the Clausen integral.
15 Q(0) + 144 Q(1) - 448 Q(2) + 126 Q(3) + 168 Q(4) = 0.
EXAMPLE
4.1204258576856330093331932058655183968902289805100953379974262667755...
MATHEMATICA
Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]);
Q[0] = 4 Cl2[Pi/3]^2 ;
RealDigits[N[Q[0], 104] // Chop][[1]]
PROG
(PARI)
Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(0) \\ Gheorghe Coserea, Oct 01 2018
(PARI)
clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
4*clausen(2, Pi/3)^2 \\ Gheorghe Coserea, Oct 01 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jun 23 2016
STATUS
approved