Reflection Relation

Author

Eric W. Weisstein
January 1, 2009

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

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

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

Cases

(Gamma[z]) (Gamma[1-z])==π csc(π z)

FullSimplify[Gamma[z]Gamma[1-z]]

(Zeta[1-z])==(Zeta[z]) χ(z)

χ[z_]:=2(((2π))^(-z))Cos[((π z)/2)]Gamma[z]

FullSimplify[χ[z]Zeta[z]]

Subtract@@@Table[{(((2 π))^(-z)) Csc[((π z)/2)] Gamma[z] Sin[π z] Zeta[z],Zeta[1-z]}/.z->RandomComplexDisk[],{100}]//Chop//Union

ξ(1-z)==ξ(z)

XiFunction[z]

FullSimplify[XiFunction[1-z]-XiFunction[z]]

Subtract@@@Table[{XiFunction[1-z],XiFunction[z]}/.z->RandomComplexDisk[],{100}]//Chop//Union

(BarnesG[z]) (Gamma[z])==(BarnesG[z+1])

FullSimplify[BarnesG[z]Gamma[z]]

FullSimplify[BarnesG[z]Gamma[z]-BarnesG[z+1]]

Subtract@@@Table[{BarnesG[z]Gamma[z],BarnesG[z+1]}/.z->RandomComplexDisk[],{100}]//Chop//Union

((RamanujanTauL(s) (Gamma[s]))/(((2 π))^s))==((RamanujanTauL(12-s) (Gamma[12-s]))/(((2 π))^(12-s)))

FullSimplify[((RamanujanTauL[s]Gamma[s])/(((2π))^s))-((RamanujanTauL[12-s]Gamma[12-s])/(((2π))^(12-s)))]

Table[-(((2 π))^(-12+s)) Gamma[12-s] RamanujanTauL[12-s]+(((2 π))^(-s)) Gamma[s] RamanujanTauL[s]/.s->RandomComplexDisk[],{100}]//Chop//Union