FeynCalc manual (development version)

SimplifyPolyLog

SimplifyPolyLog[y] performs several simplifications assuming that the variables occuring in the Log and PolyLog functions are between 0 and 1.

The simplifications will in general not be valid if the arguments are complex or outside the range between 0 and 1.

See also

Overview, Nielsen.

Examples

SimplifyPolyLog[PolyLog[2, 1/x]]

ζ(2)+Li2(1x)12log2(x)+log(1x)log(x)+iπlog(x)\zeta (2)+\text{Li}_2(1-x)-\frac{1}{2} \log ^2(x)+\log (1-x) \log (x)+i \pi \log (x)

SimplifyPolyLog[PolyLog[2, x]]

ζ(2)Li2(1x)log(1x)log(x)\zeta (2)-\text{Li}_2(1-x)-\log (1-x) \log (x)

SimplifyPolyLog[PolyLog[2, 1 - x^2]]

ζ(2)+2  Li2(1x)2  Li2(x)2log(x)log(x+1)-\zeta (2)+2 \;\text{Li}_2(1-x)-2 \;\text{Li}_2(-x)-2 \log (x) \log (x+1)

SimplifyPolyLog[PolyLog[2, x^2]]

2ζ(2)2  Li2(1x)+2  Li2(x)2log(1x)log(x)2 \zeta (2)-2 \;\text{Li}_2(1-x)+2 \;\text{Li}_2(-x)-2 \log (1-x) \log (x)

SimplifyPolyLog[PolyLog[2, -x/(1 - x)]]

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

SimplifyPolyLog[PolyLog[2, x/(x - 1)]]

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

SimplifyPolyLog[Nielsen[1, 2, -x/(1 - x)]]

S12(x)16log3(1x)S_{12}(x)-\frac{1}{6} \log ^3(1-x)

SimplifyPolyLog[PolyLog[3, -1/x]]

Li3(x)+ζ(2)log(x)+log3(x)6\text{Li}_3(-x)+\zeta (2) \log (x)+\frac{\log ^3(x)}{6}

SimplifyPolyLog[PolyLog[3, 1 - x]]

Li3(1x)\text{Li}_3(1-x)

SimplifyPolyLog[PolyLog[3, x^2]]

4  Li3(x)4  Li2(1x)log(x)4S12(1x)+4ζ(2)log(x)2log(1x)log2(x)+4ζ(3)4 \;\text{Li}_3(-x)-4 \;\text{Li}_2(1-x) \log (x)-4 S_{12}(1-x)+4 \zeta (2) \log (x)-2 \log (1-x) \log ^2(x)+4 \zeta (3)

SimplifyPolyLog[PolyLog[3, -x/(1 - x)]]

Li3(1x)+Li2(1x)log(x)+S12(1x)+ζ(2)log(1x)ζ(2)log(x)+16log3(1x)12log(x)log2(1x)+12log2(x)log(1x)-\text{Li}_3(1-x)+\text{Li}_2(1-x) \log (x)+S_{12}(1-x)+\zeta (2) \log (1-x)-\zeta (2) \log (x)+\frac{1}{6} \log ^3(1-x)-\frac{1}{2} \log (x) \log ^2(1-x)+\frac{1}{2} \log ^2(x) \log (1-x)

SimplifyPolyLog[PolyLog[3, 1 - 1/x]]

Li2(1x)log(x)Li2(1x)log(1x)+S12(1x)+S12(x)+log3(x)612log2(1x)log(x)ζ(3)\text{Li}_2(1-x) \log (x)-\text{Li}_2(1-x) \log (1-x)+S_{12}(1-x)+S_{12}(x)+\frac{\log ^3(x)}{6}-\frac{1}{2} \log ^2(1-x) \log (x)-\zeta (3)

SimplifyPolyLog[PolyLog[4, -x/(1 - x)]]

Li4(x)+12  Li2(1x)log2(1x)Li2(1x)log(x)log(1x)S13(x)+S22(x)S12(1x)log(1x)S12(x)log(1x)12ζ(2)log2(1x)+ζ(2)log(x)log(1x)+ζ(3)log(1x)124log4(1x)+12log(x)log3(1x)12log2(x)log2(1x)-\text{Li}_4(x)+\frac{1}{2} \;\text{Li}_2(1-x) \log ^2(1-x)-\text{Li}_2(1-x) \log (x) \log (1-x)-S_{13}(x)+S_{22}(x)-S_{12}(1-x) \log (1-x)-S_{12}(x) \log (1-x)-\frac{1}{2} \zeta (2) \log ^2(1-x)+\zeta (2) \log (x) \log (1-x)+\zeta (3) \log (1-x)-\frac{1}{24} \log ^4(1-x)+\frac{1}{2} \log (x) \log ^3(1-x)-\frac{1}{2} \log ^2(x) \log ^2(1-x)

SimplifyPolyLog[Log[a + b/c]]

log(ac+bc)\log \left(\frac{a c+b}{c}\right)

SimplifyPolyLog[Log[1/x]]

log(x)-\log (x)

SimplifyPolyLog[ArcTanh[x]]

12log(x+11x)\frac{1}{2} \log \left(-\frac{x+1}{1-x}\right)

SimplifyPolyLog[ArcSinh[x]]

log(x2+1+x)\log \left(\sqrt{x^2+1}+x\right)

SimplifyPolyLog[ArcCosh[x]]

log(x21+x)\log \left(\sqrt{x^2-1}+x\right)