Rogers L-Function

Author

Eric W. Weisstein
January 18, 2022

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/RogersL-Function.nb.

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

©2022 Wolfram Research, Inc. except for portions noted otherwise

Initialization

RogersL[x_]:=PolyLog[2,x]+Log[x]Log[1-x]/2
RogersL[1]:=Pi^2/6

Plot

Plot[RogersL[x],{x,0,1},AxesLabel->{x,HoldForm[RogersL[x]]}]

Sums

(∑_n^∞)((x^n)/(n^2))+(1/2) log(1-x) log(x)==(PolyLog[2, x])+(1/2) log(1-x) log(x)

Sum[((x^n)/(n^2)),{n,∞}]+(1/2)Log[x]Log[1-x]

Derivative and Integral

Integral

Integrate[RogersL[x],x]//FullSimplify

Integrate[RogersL[x],{x,0,1}]

Derivative

FullSimplify[D[RogersL[x],x],0<x<1]

Identities

L(1-x)+L(x)==1

FullSimplify[RogersL[x]+RogersL[1-x]]

L(x)+L(y)==L(x y)+L(((x (1-y))/(1-x y)))+L((((1-x) y)/(1-x y)))

FullSimplify[Subtract@@{RogersL[x]+RogersL[y],RogersL[x y]+RogersL[((x(1-y))/(1-x y))]+RogersL[((y(1-x))/(1-x y))]}]

FullSimplify[Subtract@@{RogersL[x]+RogersL[y],RogersL[x y]+RogersL[((x(1-y))/(1-x y))]+RogersL[((y(1-x))/(1-x y))]},0<x<1&&0<y<1]

RogersL[x]

Subtract@@@Table[{RogersL[x]+RogersL[y],RogersL[x y]+RogersL[((x(1-y))/(1-x y))]+RogersL[((y(1-x))/(1-x y))]}/.{x->RandomReal[],y->RandomReal[]},{100}]//Chop//Union

2-term identities

L((1/(ϕ ((Sqrt[ϕ])+ϕ))))-L((ϕ/((Sqrt[ϕ])+ϕ)))==-((π^2)/20)

Khoi 2014

Campbell 2021

FullSimplify[RogersL[(1/(GoldenRatio(GoldenRatio+(Sqrt[GoldenRatio]))))]-RogersL[(GoldenRatio/(GoldenRatio+(Sqrt[GoldenRatio])))]==-((π^2)/20)]

N[{RogersL[(1/(GoldenRatio(GoldenRatio+(Sqrt[GoldenRatio]))))]-RogersL[(GoldenRatio/(GoldenRatio+(Sqrt[GoldenRatio])))],-((π^2)/20)},20]

Algebraic

1-Algebraic

ρ=((Sqrt[5])-1)/2;

L(0)==0

Limit[RogersL[x],x->0]

L(1-ρ)==(2/5)

RogersL[1-ρ]//FullSimplify

L((1/2))==(1/2)

RogersL[(1/2)]//FullSimplify

L(ρ)==(3/5)

RogersL[ρ]//FullSimplify

L(1)==1

Limit[RogersL[x],x->1]

2-Algebraic

2 L(1)-6 L((1/4))-2 L((1/8))+L((1/64))==0

RogersL[(1/64)]-2RogersL[(1/8)]-6RogersL[(1/4)]+2RogersL[1]//FullSimplify

N[%]

PossibleZeroQ[%%]

2 L(1)-6 L((1/3))+L((1/9))==0

RogersL[(1/9)]-6RogersL[(1/3)]+2RogersL[1]//FullSimplify

N[%]

PossibleZeroQ[%%]

α, β, γ

α=(1/2)Sec[(2/7)π];
β=(1/2)Sec[(1/7)π];
γ=2Cos[(3/7)π];

N[{α,β,γ}]

FullSimplify[RootReduce[ExpandAll[ToRadicals[(x-α)(x+β)(x+1/γ)]]]]//Expand

Reduce[-1-x+2 (x^2)+(x^3)==0,x,Reals]

L(α)-L((α^2))==(1/7)

{(RogersL[α]-RogersL[(α^2)]),1/7}//N

RogersL[α]-RogersL[(α^2)]-1/7//FullSimplify

PossibleZeroQ[%]

L((β^2))+2 L(β)==(10/7)

2RogersL[β]+RogersL[(β^2)]-10/7//FullSimplify

PossibleZeroQ[%]

L((γ^2))+2 L(γ)==(8/7)

2RogersL[γ]+RogersL[(γ^2)]-8/7//FullSimplify

PossibleZeroQ[%]

δ,κ,λ,μ,a,b,c

δ=(1/2)((Sqrt[3+2(Sqrt[5])])-1);
κ=(1/2)Sec[(π/9)];
λ=(1/2)Sec[(2/9)π];
μ=2Cos[(4/9)π];
a=2(Sqrt[3])Cos[((5π)/18)]-2;
b=2(Sqrt[3])Cos[((11π)/18)]+2;
c=2(Sqrt[3])Cos[((7π)/18)]-1;

5 L((δ^3))-5 L(δ)+L(1)==0

5RogersL[(δ^3)]-5RogersL[δ]+RogersL[1]//FullSimplify

PossibleZeroQ[%]

L((δ^12))-2 L((δ^6))-6 L((δ^4))+4 L((δ^3))+3 L((δ^2))+4 L(δ)-4 L(1)==0

RogersL[(δ^12)]-2RogersL[(δ^6)]-6RogersL[(δ^4)]+4RogersL[(δ^3)]+3RogersL[(δ^2)]+4RogersL[δ]-4RogersL[1]//N

PossibleZeroQ[%]

3 L((κ^3))-9 L((κ^2))-9 L(κ)+7 L(1)==0

3RogersL[(κ^3)]-9RogersL[(κ^2)]-9RogersL[κ]+7RogersL[1]//N

3 L((λ^6))-6 L((λ^3))-27 L((λ^2))+18 L(λ)+2 L(1)==0

3RogersL[(λ^6)]-6RogersL[(λ^3)]-27RogersL[(λ^2)]+18RogersL[λ]+2RogersL[1]//N

3 L((μ^6))-6 L((μ^3))-27 L((μ^2))+18 L(μ)-2 L(1)==0

3RogersL[(μ^6)]-6RogersL[(μ^3)]-27RogersL[(μ^2)]+18RogersL[μ]-2RogersL[1]//N

2 L((a^3))-2 L((a^2))-11 L(a)+3 L(1)==0

2RogersL[(a^3)]-2RogersL[(a^2)]-11RogersL[a]+3RogersL[1]//N

2 L((b^6))-4 L((b^3))-15 L((b^2))+22 L(b)-6 L(1)==0

2RogersL[(b^6)]-4RogersL[(b^3)]-15RogersL[(b^2)]+22RogersL[b]-6RogersL[1]//N

2 L((c^6))-4 L((c^3))-15 L((c^2))+22 L(c)-4 L(1)==0

2RogersL[(c^6)]-4RogersL[(c^3)]-15RogersL[(c^2)]+22RogersL[c]-4RogersL[1]//N