Legendre Duplication Formula

Author

Eric W. Weisstein
August 11, 2003

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Derivation

Integrate[(u^(m-1))(((1-u))^(n-1)),{u,0,1}]

{2FullSimplify[Integrate[(x^(2m-1))(((1-(x^2)))^(n-1)),{x,0,1},Assumptions->Re[m]>0&&Re[n]>0]],
FunctionExpand[Beta[m,n]]
}

Integrate[(((((1+x)/2)))^(z-1))(((1-((1+x)/2)))^(z-1))/2,{x,-1,1},Assumptions->Re[z]>0]

2 (2^(1-2z))Integrate[(((1-x^2))^(z-1)),{x,0,1},Assumptions->Re[z]>0]

Solve[((Gamma[z]Gamma[z])/(Gamma[2z]))==(2^(1-2z))((Gamma[1/2]Gamma[z])/(Gamma[z+1/2])),Gamma[2z]]

(((2π))^(-1/2))(2^(2z-1/2))Gamma[z]Gamma[z+1/2]

Table[(((2π))^(-1/2))(2^(2z-1/2))Gamma[z]Gamma[z+1/2]-Gamma[2z]/.z->Random[Complex,{-2-2I,2+2I}],{10}]//Chop

FunctionExpand[Gamma[2z]]

FullSimplify[(((2π))^(-1/2))(2^(2z-1/2))Gamma[z]Gamma[z+1/2]]