Jonquière's Relation

Author

Eric W. Weisstein
February 17, 2008

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

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

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

Identity

General Form

(Subscript[Li, ν])(z)==(((I π) (1-(Sqrt[((z-1)/z)]) (Sqrt[(z/(z-1))])) (log^(ν-1))(z))/(Γ(ν)))+(((E^((π I ν)/2)) (((2 π))^ν) HurwitzZeta(1-ν,((log(-z))/(2 π I))+(1/2)))/(Γ(ν)))-(E^(π I ν)) (Subscript[Li, ν])((1/z))

id=(PolyLog[ν,z]==(((E^((I π ν)/2)) (((2 π))^ν) HurwitzZeta[1-ν,(1/2)-((I Log[-z])/(2 π))])/(Gamma[ν]))+((I π (1-(Sqrt[((-1+z)/z)]) (Sqrt[(z/(-1+z))])) ((Log[z])^(-1+ν)))/(Gamma[ν]))-(E^(I π ν)) PolyLog[ν,(1/z)]);

Except on [0,1]

(Subscript[Li, ν])(z)==(((E^((π I ν)/2)) (((2 π))^ν) HurwitzZeta(1-ν,((log(-z))/(2 π I))+(1/2)))/(Γ(ν)))-(E^(π I ν)) (Subscript[Li, ν])((1/z))/;¬IntervalMemberQ[Interval[{0,1}],z]

FullSimplify[id,z<0||z>1]