Clausen's Integral

Author

Eric W. Weisstein
April 15, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Plots

<<Utilities`Plot`
<<Utilities`Typesetting`

Plot[ClausenCl[2,x],{x,0,2Pi},PlotStyle->Red,AxesLabel->TraditionalForm/@{x,(Subscript[Cl, 2])[x]}]

ComplexPlot3D[ClausenCl[2,z],{z,-5,5},ComplexPlotLabel->(Subscript[Cl, 2])[z]]//Timing

ComplexContourPlot[ClausenCl[2,z],{z,-5,5},ComplexPlotLabel->(Subscript[Cl, 2])[z]]//Timing

Definition

expr1=FullSimplify[-Integrate[Log[2Sin[t/2]],{t,0,x}]]

With[{x=Random[]},{-NIntegrate[Log[2 Sin[1/2 t]],{t,0,x}],((I (π^2))/6)-((I (x^2))/4)-x Log[2]+x Log[1-(E^(I x))]-x Log[Sin[(x/2)]]-I PolyLog[2,(E^(I x))]}]

diff=FullSimplify[ClausenCl[2,x]-expr1,x>0]

diff//TraditionalForm

InputForm[diff]

Plot[Chop[diff],{x,0,2},PlotRange->All,PlotStyle->Red]

FullSimplify[diff,0<x<2π]

Table[diff/.x->Random[Real,{0,2Pi},50],{10}]//Chop//Union

diff/.x->-1.5

Identities

FullSimplify[{expr1,expr2}/.x->Pi/2]

4 (Subscript[Cl, 2])( (π/3))==(Subscript[Cl, 2])( 2 α+π)-3 (Subscript[Cl, 2])( 2 α+((5 π)/3))+2 (Subscript[Cl, 2])( 2 α)

With[{α=ArcTan[Sqrt[3]/9]},
Subtract@@{4ClausenCl[2,Pi/3],2ClausenCl[2,2α]+ClausenCl[2,π+2α]-3ClausenCl[2,5π/3+2α]}
]//FullSimplify

N[%,20]

6 C==-3 (Subscript[Cl, 2])( 2 α-(π/2))+(Subscript[Cl, 2])( 2 α+(π/2))+2 (Subscript[Cl, 2])( 2 α)

With[{α=ArcTan[1/3]},
Subtract@@{6Catalan,2ClausenCl[2,2α]-3ClausenCl[2,2α-Pi/2]+ClausenCl[2,2α+Pi/2]}
]//FullSimplify

N[%,20]

(7/4)(Sqrt[7])(Subscript[L, -7])(2)==3 (Subscript[Cl, 2])( α)-3 (Subscript[Cl, 2])( 2 α)+(Subscript[Cl, 2])( 3 α)

<<NumberTheory`NumberTheoryFunctions`

N[((7Sqrt[7])/4)Sum[KroneckerSymbol[-7,n](n^(-2)),{n,10^5}],20]//Timing

N[With[{α=2ArcTan[Sqrt[7]]},
3ClausenCl[2,α]-3ClausenCl[2,2α]+ClausenCl[2,3α]
],20]