FeynCalc manual (development version)

Nielsen

Nielsen[i, j, x] denotes Nielsen’s polylogarithm.

See also

Overview, SimplifyPolyLog.

Examples

Nielsen[1, 2, x]

S12(x)S_{12}(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}]

N[Nielsen[1, 2, .45]]

0.07287160.0728716

Some special values are built in.

{Nielsen[1, 2, 0], Nielsen[1, 2, -1], Nielsen[1, 2, 1/2], Nielsen[1, 2, 1]}

{0,ζ(3)8,ζ(3)8,ζ(3)}\left\{0,\frac{\zeta (3)}{8},\frac{\zeta (3)}{8},\zeta (3)\right\}

Nielsen[1, 2, x, PolyLog -> True]

Li3(1x)+Li2(1x)log(1x)+12log(x)log2(1x)+ζ(3)-\text{Li}_3(1-x)+\text{Li}_2(1-x) \log (1-x)+\frac{1}{2} \log (x) \log ^2(1-x)+\zeta (3)

Nielsen[1, 3, x, PolyLog -> True]

Li4(1x)12  Li2(1x)log2(1x)+Li3(1x)log(1x)16log(x)log3(1x)+π490-\text{Li}_4(1-x)-\frac{1}{2} \;\text{Li}_2(1-x) \log ^2(1-x)+\text{Li}_3(1-x) \log (1-x)-\frac{1}{6} \log (x) \log ^3(1-x)+\frac{\pi ^4}{90}

Nielsen[3, 1, x, PolyLog -> True]

Li4(x)\text{Li}_4(x)