Trilogarithm

Author

Eric W. Weisstein
January 3, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Trilogarithm.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Trilogarithm.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Plot

<<Utilities`Plot`

Real Axis

Plot[PolyLog[3,x],{x,-5,1},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,PolyLog[3,x]}]

Complex Plane

ComplexPlot3D[PolyLog[3,z],{z,-5,5}]//Timing

ComplexContourPlot[PolyLog[3,z],{z,-5,5}]//Timing

Special Values

(PolyLog[3, -1])==-((3 (Zeta[3]))/4)

PolyLog[3,-1]

(PolyLog[3, 0])==0

PolyLog[3,0]

(PolyLog[3, (1/2)])==(1/24) (21 (Zeta[3])+4 (log^3)(2)-2 (π^2) log(2))

PolyLog[3,1/2]

(PolyLog[3, 1])==(Zeta[3])

PolyLog[3,1]

(PolyLog[3, (ϕ^(-2))])==((4 (Zeta[3]))/5)+((2 (log^3)(ϕ))/3)-(1/15) 2 (π^2) log(ϕ)

<<Utilities`Simplify`

TranscendentalRecognize[N[PolyLog[3,(GoldenRatio^(-1))],500],{
Zeta[3],
Log[GoldenRatio],((Log[GoldenRatio])^2),((Log[GoldenRatio])^3),π Log[GoldenRatio],(π^2)Log[GoldenRatio],(π^3)Log[GoldenRatio],
π ((Log[GoldenRatio])^2),(π^2)((Log[GoldenRatio])^2),(π^3)((Log[GoldenRatio])^2)}]

TranscendentalRecognize[N[PolyLog[3,(GoldenRatio^(-2))],500],{
Zeta[3],
Log[GoldenRatio],((Log[GoldenRatio])^2),((Log[GoldenRatio])^3),π Log[GoldenRatio],(π^2)Log[GoldenRatio],(π^3)Log[GoldenRatio],
π ((Log[GoldenRatio])^2),(π^2)((Log[GoldenRatio])^2),(π^3)((Log[GoldenRatio])^2)}]

FunctionExpand[PolyLog[3,(GoldenRatio^(-2))]]

FullSimplify[%]

FullSimplify[PolyLog[3,(GoldenRatio^(-2))]-((4/5)Zeta[3]+(2/3)((Log[GoldenRatio])^3)-(2/15)(π^2)Log[GoldenRatio])]

Identities

(PolyLog[3, -z])+(PolyLog[3, z])==((PolyLog[3, (z^2)])/4)

FullSimplify[PolyLog[3,z]+PolyLog[3,-z]]

(PolyLog[3, -z])-(PolyLog[3, -(1/z)])==-(1/6) (log^3)(z)-(1/6) (π^2) log(z)

FullSimplify[PolyLog[3,-z]-PolyLog[3,-(z^(-1))]]

FullSimplify[FunctionExpand[PolyLog[3,-z]-PolyLog[3,-(z^(-1))]],z≠0]

ByteCount/@{PolyLog[3,-z]-PolyLog[3,-(z^(-1))],-(1/6)((Log[z])^3)-(1/6)(π^2)Log[z]}

FullSimplify[Subtract@@{PolyLog[3,-z]-PolyLog[3,-(z^(-1))],-(1/6)((Log[z])^3)-(1/6)(π^2)Log[z]}]

PossibleZeroQ[%]

Subtract@@@Table[{PolyLog[3,-z]-PolyLog[3,-(z^(-1))],-(1/6)((Log[z])^3)-(1/6)(π^2)Log[z]}/.z->RandomComplexDisk[2],{100}]//Chop//Union

Subtract@@@Table[{PolyLog[3,-z]-PolyLog[3,-(z^(-1))],(1/6) Log[-(1/z)] (-2 (π^2)-((3 π z (Sqrt[-((((1+z))^2)/(z^2))]) Log[-(1/z)])/(1+z))+((Log[-(1/z)])^2))}/.z->RandomComplexDisk[2],{100}]//Chop//Union

(1/6) Log[z] ((π^2)+((Log[z])^2))-PolyLog[3,-(1/z)]+PolyLog[3,-z]/.z->2.

Plot[Through[{Re,Im}[(1/6) Log[z] ((π^2)+((Log[z])^2))-PolyLog[3,-(1/z)]+PolyLog[3,-z]]],{z,0,-2,2}]

(PolyLog[3, 1-(1/z)])+(PolyLog[3, 1-z])+(PolyLog[3, z])==(((log^3)(z))/6)-(1/2) log(1-z) (log^2)(z)+(1/6) (π^2) log(z)+(Zeta[3])

FunctionExpand[PolyLog[3,z]+PolyLog[3,1-z]+PolyLog[3,1-(z^(-1))]]

FullSimplify[Subtract@@{PolyLog[3,1-(1/z)]+PolyLog[3,1-z]+PolyLog[3,z],Zeta[3]+(1/6)((Log[z])^3)+(1/6)(π^2)Log[z]-(1/2)((Log[z])^2)Log[1-z]}]

Subtract@@@Table[{PolyLog[3,1-(1/z)]+PolyLog[3,1-z]+PolyLog[3,z],Zeta[3]+(1/6)((Log[z])^3)+(1/6)(π^2)Log[z]-(1/2)((Log[z])^2)Log[1-z]}/.z->RandomComplexDisk[],{100}]//Chop//Union

36 (PolyLog[3, (1/2)])-18 (PolyLog[3, (1/4)])-4 (PolyLog[3, (1/8)])+(PolyLog[3, (1/64)])==((35 (Zeta[3]))/2)-(π^2) log(2)

FullSimplify[FunctionExpand[36PolyLog[3,(1/2)]-18PolyLog[3,(1/4)]-4PolyLog[3,(1/8)]+PolyLog[3,(1/64)]]]

N[%]

(35/2)Zeta[3]-(π^2)Log[2]//N

-24 (PolyLog[3, (1/2)])+18 (PolyLog[3, (1/4)])+4 (PolyLog[3, (1/8)])-(PolyLog[3, (1/64)])==2 (log^3)(2)-7 (Zeta[3])

FullSimplify[FunctionExpand[-24PolyLog[3,(1/2)]+18PolyLog[3,(1/4)]+4PolyLog[3,(1/8)]-PolyLog[3,(1/64)]]]

N[%]

2((Log[2])^3)-7Zeta[3]//N

-48 (PolyLog[3, (1/2)])+54 (PolyLog[3, (1/4)])+12 (PolyLog[3, (1/8)])-3 (PolyLog[3, (1/64)])==10 (log^3)(2)-2 (π^2) log(2)

FullSimplify[FunctionExpand[-48PolyLog[3,(1/2)]+54PolyLog[3,(1/4)]+12PolyLog[3,(1/8)]-3PolyLog[3,(1/64)]]]

N[%]

10((Log[2])^3)-2(π^2)Log[2]//N