Lemniscate Function

Author

Eric W. Weisstein
April 23, 2008

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

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

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

L

2 ((∫_0^1)(1/(((1-(t^4)))^(1/2)))dt)

L=2Integrate[(((1-(t^4)))^(-1/2)),{t,0,1}]

<<Utilities`Simplify`

%//GammaSimplify

(π/((agm(1,(2^(1/2))))))

FunctionExpand[(π/(ArithmeticGeometricMean[1,(Sqrt[2])]))]

Integrals

ϕ(x)==(∫_0^x)(1/(Sqrt[1-(t^4)]))dt

V5

Integrate[(1/(Sqrt[1-(t^4)])),{t,0,x},Assumptions->0<x<1]

V7

Integrate[(1/(Sqrt[1-(t^4)])),{t,0,x},Assumptions->0<x<1]

ϕ'(x)==(∫_x^1)(1/(Sqrt[1-(t^4)]))dt

V5

Integrate[(1/(Sqrt[1-(t^4)])),{t,x,1}]

FullSimplify[-EllipticF[ArcSin[x],-1]+EllipticK[-1]]

FunctionExpand[-EllipticF[ArcSin[x],-1]+EllipticK[-1]]

FullSimplify[%144,0<x<1]

Integrate[(1/(Sqrt[1-(t^4)])),{t,x,1},Assumptions->0<x<1]

Subtract@@@Table[{(((Sqrt[π]) Gamma[(5/4)])/(Gamma[(3/4)]))-x Hypergeometric2F1[(1/4),(1/2),(5/4),(x^4)],-EllipticF[ArcSin[x],-1]+EllipticK[-1]}/.x->RandomReal[],{100}]//Chop//Union

V7

Integrate[(1/(Sqrt[1-(t^4)])),{t,x,1},Assumptions->0<x<1]

FullSimplify[%]

FunctionExpand[EllipticK[-1]]

<<Utilities`Simplify`

GammaSimplify[(((Sqrt[π]) Gamma[(5/4)])/(Gamma[(3/4)]))]

FunctionExpand[-EllipticF[ArcSin[x],-1]+EllipticK[-1]]//FullSimplify

Sums

ϕ(x)==(∑_(n=0)^∞)(((Pochhammer[(1/2), n]) (x^(4 n+1)))/((4 n+1) n!))

V5

Sum[((Pochhammer[1/2,n](x^(4n+1)))/(n!(4n+1))),{n,0,∞}]

FullSimplify[%]

V7

Sum[((Pochhammer[(1/2),n](x^(4n+1)))/(n!(4n+1))),{n,0,∞}]

FullSimplify[%]