Legendre's χ-Function

Author

Eric W. Weisstein
October 18, 2008

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

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

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

Sum

(Subscript[χ, n])(x)==(∑_(k=0)^∞)((z^(2 k+1))/(((2 k+1))^ν))

Sum[((z^(2k+1))/(((2k+1))^ν)),{k,0,∞}]

Sum[((z^(2k+1))/(((2k+1))^ν)),{k,0,∞},GenerateConditions->True]

Sum[(1/(((2k+1))^ν)),{k,0,∞},GenerateConditions->True]

Plots

<<Utilities`Plot`

V5

Plot[Evaluate[Table[LegendreChi[n,z],{n,0,5}]],{z,-1,1},PlotStyle->{Black,Red,Yellow,Green,Blue,Violet},
PlotRange->{-1,1}1.5,
AxesLabel->TraditionalForm/@{x,(Subscript[χ, n])[x]}]

V7

Plot[Table[LegendreChi[n,z],{n,0,4}],{z,-1,1},PlotStyle->{Black,Red,Yellow,Green,Blue},
AxesLabel->{x,HoldForm[LegendreChi[n,x]]},Evaluated->True]

Identities

(Subscript[χ, n])(z)==(PolyLog[n, z])-(2^(-n)) (PolyLog[n, (z^2)])

FullSimplify[(2^(-n)) z LerchPhi[(z^2),n,(1/2)]-(PolyLog[n,z]-2^(-n)PolyLog[n,z^2])]

Subtract@@@Table[{(2^(-n)) z LerchPhi[(z^2),n,(1/2)],(PolyLog[n,z]-2^(-n)PolyLog[n,z^2])}/.{z->RandomComplexDisk[],n->RandomComplexDisk[]},{100}]//Chop//Union

(Subscript[χ, n])(z)==(((PolyLog[n, z])-(PolyLog[n, -z]))/2)

FullSimplify[(2^(-n)) z LerchPhi[(z^2),n,(1/2)]-(1/2) (-PolyLog[n,-z]+PolyLog[n,z])]

Subtract@@@Table[{(2^(-n)) z LerchPhi[(z^2),n,(1/2)],(1/2) (-PolyLog[n,-z]+PolyLog[n,z])}/.{z->RandomComplexDisk[],n->RandomComplexDisk[]},{100}]//Chop//Union

Special Values

(Subscript[χ, 2])(i)==i C

LegendreChi[2,I]

(Subscript[χ, 2])((Sqrt[2])-1)==((π^2)/16)-(1/4) (ln^2)((Sqrt[2])+1)

LegendreChi[2,Sqrt[2]-1]//FullSimplify

FunctionExpand[%]

FullSimplify[LegendreChi[2,Sqrt[2]-1]]-(1/16*Pi^2-1/4*Log[Sqrt[2]+1]^2)

N[%]

(Subscript[χ, 2])((Sqrt[5])-2)==((π^2)/24)-(3/4) (ln^2)((1/2) ((Sqrt[5])+1))

FullSimplify[LegendreChi[2,Sqrt[5]-2]]-(1/24*Pi^2-3/4*Log[(Sqrt[5]+1)/2]^2)

FunctionExpand[%]

N[%]

(Subscript[χ, 2])(1)==((π^2)/8)

LegendreChi[2,1]

(Subscript[χ, 2])(-1)==-((π^2)/8)

LegendreChi[2,-1]

(Subscript[χ, n])(1)==λ(n)

V7

LegendreChi[n,1]//FullSimplify

LegendreChi[n,1]//FunctionExpand

FullSimplify[-Zeta[n]+(2^(-n)) (LerchPhi[1,n,(1/2)]+Zeta[n]),Re[n]>1]

Subtract@@@Table[{(2^(-n)) LerchPhi[1,n,(1/2)],(1-(2^(-n))) Zeta[n]}/.n->RandomComplex[{-5I+1,2I+5}],{100}]//Chop//Union

Subtract@@@Table[{(2^(-n)) HurwitzLerchPhi[1,n,(1/2)],(1-(2^(-n))) Zeta[n]}/.n->RandomComplex[{-5I+1,2I+5}],{100}]//Chop//Union

(Subscript[χ, n])(i)==β(n)

LegendreChi[n,I]//FullSimplify

DirichletBeta[n]