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(1−x)−21log2(x)+log(1−x)log(x)+iπlog(x)
SimplifyPolyLog[PolyLog[2, x]]
ζ(2)−Li2(1−x)−log(1−x)log(x)
SimplifyPolyLog[PolyLog[2, 1 - x^2]]
−ζ(2)+2Li2(1−x)−2Li2(−x)−2log(x)log(x+1)
SimplifyPolyLog[PolyLog[2, x^2]]
2ζ(2)−2Li2(1−x)+2Li2(−x)−2log(1−x)log(x)
SimplifyPolyLog[PolyLog[2, -x/(1 - x)]]
−ζ(2)+Li2(1−x)−21log2(1−x)+log(x)log(1−x)
SimplifyPolyLog[PolyLog[2, x/(x - 1)]]
−ζ(2)+Li2(1−x)−21log2(1−x)+log(x)log(1−x)
SimplifyPolyLog[Nielsen[1, 2, -x/(1 - x)]]
S12(x)−61log3(1−x)
SimplifyPolyLog[PolyLog[3, -1/x]]
Li3(−x)+ζ(2)log(x)+6log3(x)
SimplifyPolyLog[PolyLog[3, 1 - x]]
Li3(1−x)
SimplifyPolyLog[PolyLog[3, x^2]]
4Li3(−x)−4Li2(1−x)log(x)−4S12(1−x)+4ζ(2)log(x)−2log(1−x)log2(x)+4ζ(3)
SimplifyPolyLog[PolyLog[3, -x/(1 - x)]]
−Li3(1−x)+Li2(1−x)log(x)+S12(1−x)+ζ(2)log(1−x)−ζ(2)log(x)+61log3(1−x)−21log(x)log2(1−x)+21log2(x)log(1−x)
SimplifyPolyLog[PolyLog[3, 1 - 1/x]]
Li2(1−x)log(x)−Li2(1−x)log(1−x)+S12(1−x)+S12(x)+6log3(x)−21log2(1−x)log(x)−ζ(3)
SimplifyPolyLog[PolyLog[4, -x/(1 - x)]]
−Li4(x)+21Li2(1−x)log2(1−x)−Li2(1−x)log(x)log(1−x)−S13(x)+S22(x)−S12(1−x)log(1−x)−S12(x)log(1−x)−21ζ(2)log2(1−x)+ζ(2)log(x)log(1−x)+ζ(3)log(1−x)−241log4(1−x)+21log(x)log3(1−x)−21log2(x)log2(1−x)
SimplifyPolyLog[Log[a + b/c]]
log(cac+b)
SimplifyPolyLog[Log[1/x]]
−log(x)
SimplifyPolyLog[ArcTanh[x]]
21log(−1−xx+1)
SimplifyPolyLog[ArcSinh[x]]
log(x2+1+x)
SimplifyPolyLog[ArcCosh[x]]
log(x2−1+x)