Elliptic Integral Singular Value

Author

Eric W. Weisstein
January 20, 2004

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

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

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

Analytic Results

sv0=#[[1,1,1]]&/@DownValues[EllipticLambda]

sv=Select[sv0,IntegerQ]

Orders of λ(n)

RootReduce/@EllipticLambda/@Range[18]

Switch[#,
_Root,Exponent[#[[1]][x],x],
_,2]&/@RootReduce/@EllipticLambda/@Range[18]

{#,EllipticLambda[#]}&/@sv//TableForm//TraditionalForm

TableForm[{#,FullSimplify[RootReduce[EllipticLambda[#]]]}&/@Take[sv,20]]//TraditionalForm

{#,EllipticK[SingularValue[#]]}&/@sv//TableForm//TraditionalForm

Check

MapIndexed[EllipticKSingular[#2[[1]]]-EllipticK[#1^2]&,ks]//N//Chop

(Subscript[k, 1])

i1=Integrate[1/Sqrt[(1-t^2)(1-t^2/2)],{t,0,1}]

i2=Integrate[1/Sqrt[1-t^4],{t,0,1}]Sqrt[2]

N[FullSimplify[FunctionExpand[i1-i2]],100]

N[Gamma[1/4]^2/4/Sqrt[Pi]-EllipticK[(1/2)],100]