Nielsen
Nielsen[i, j, x]
denotes Nielsen’s polylogarithm.
See also
Overview, SimplifyPolyLog.
Examples
S12(x)
Numerical evaluation is done via N[Nielsen[n_,p_,x_]] := (-1)^(n+p-1)/(n-1)!/p! NIntegrate[Log[1-x t]^p Log[t]^(n-1)/t,{t,0,1}]
0.0728716
Some special values are built in.
{Nielsen[1, 2, 0], Nielsen[1, 2, -1], Nielsen[1, 2, 1/2], Nielsen[1, 2, 1]}
{0,8ζ(3),8ζ(3),ζ(3)}
Nielsen[1, 2, x, PolyLog -> True]
−Li3(1−x)+Li2(1−x)log(1−x)+21log(x)log2(1−x)+ζ(3)
Nielsen[1, 3, x, PolyLog -> True]
−Li4(1−x)−21Li2(1−x)log2(1−x)+Li3(1−x)log(1−x)−61log(x)log3(1−x)+90π4
Nielsen[3, 1, x, PolyLog -> True]
Li4(x)