Gieseking's Constant

Author

Eric W. Weisstein
August 5, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/GiesekingsConstant.nb.

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

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

Value

Integral

Integrate[Log[2Cos[(x/2)]],{x,0,θ},Assumptions->0<θ<π]//FullSimplify

i[θ_]:=(1/12) I ((π^2)+3 (θ^2)+12 I θ Log[(E^((I θ)/2))]+12 PolyLog[2,-(E^(I θ))])

i[(2/3)π]//FullSimplify

N[%]

Clausen

ClausenCl[2,π/3]

FullSimplify[%]

N[%]

Sum

((3(Sqrt[3]))/(4π))(1-Sum[(1/(((3k+2))^2)),{k,0,∞}]+Sum[(1/(((3k+1))^2)),{k,∞}])

FullSimplify[%]

N[%]

N[π((9-PolyGamma[1,(2/3)]+PolyGamma[1,(4/3)])/(4 (Sqrt[3]) π)),20]

PolyGamma[1,(2/3)]//TraditionalForm

Identities

l={-(1/36) I ((π^2)-36 PolyLog[2,-(((-1))^(2/3))]),(1/2) I (-PolyLog[2,(((-1))^(1/3))]+PolyLog[2,-(((-1))^(2/3))]),((9-PolyGamma[1,(2/3)]+PolyGamma[1,(4/3)])/(4 (Sqrt[3]) ))};

N[l]

FullSimplify[Subtract@@@Subsets[l,{2}]]

N[%,20]

PossibleZeroQ[Last[%45]]