Legendre Relation

Author

Eric W. Weisstein
October 31, 2004

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

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

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

Check

id=EllipticE[m]EllipticK[1-m]-EllipticK[m]EllipticK[1-m]+EllipticE[1-m]EllipticK[m]

Analytic

integrand[x_,y_,m_]:=(1-m Sin[x]^2-(1-m) Sin[y]^2)/Sqrt[1-m Sin[x]^2]/Sqrt[1-(1-m) Sin[y]^2]

v4.2

Integrate[integrand[x,y,m],{y,0,Pi/2},{x,0,Pi/2},GenerateConditions->False]//FunctionExpand//Together

D[%,m]//Together

v5.0

Integrate[integrand[x,y,m],{y,0,Pi/2},{x,0,Pi/2},GenerateConditions->False]//FunctionExpand//Together

D[%,m]//Together

Simplify[%]

v5.1

Integrate[integrand[x,y,m],{y,0,Pi/2},{x,0,Pi/2},GenerateConditions->False]//FunctionExpand//Together

D[%,m]//Together

Numerical

Table[id/.m->Random[Complex,(I+1)5{-1,1},20],{100}]//Chop//Union[#,SameTest->Equal]&