Hyperbolic Lemniscate Function

Author

Eric W. Weisstein
September 19, 2008

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

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

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

Definitions

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

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

Sums

FunctionExpand[(2/π)EllipticK[(1/2)]]

Integrate[(1/(Sqrt[1+(t^4)])),{t,0,v}]

2 (tan^(-1))(v)==(∑_n^∞)((sin(2 n θ))/(n cosh(π n)))+θ

With[{θ=π ((v Hypergeometric2F1[(1/4),(1/2),(5/4),-(v^4)])/(EllipticK[(1/2)]))},θ+Sum[((Sin[2n θ])/(n Cosh[n π])),{n,∞}]]

With[{θ=π ((v Hypergeometric2F1[(1/4),(1/2),(5/4),-(v^4)])/(EllipticK[(1/2)]))},{2ArcTan[v],θ+Sum[((Sin[2n θ])/(n Cosh[n π])),{n,200}]}]/.v->2.1

(π/8)-(1/2) (tan^(-1))((v^2))==(∑_(n=0)^∞)(((((-1))^n) cos((2 n+1) θ))/((2 n+1) cosh((1/2) π (2 n+1))))

With[{θ=π ((v Hypergeometric2F1[(1/4),(1/2),(5/4),-(v^4)])/(EllipticK[(1/2)]))},{(π/8)-(1/2)ArcTan[(v^2)],Sum[(((((-1))^n)Cos[(2n+1) θ])/((2n+1)Cosh[(2n+1) π/2])),{n,0,∞}]}]

With[{θ=π ((v Hypergeometric2F1[(1/4),(1/2),(5/4),-(v^4)])/(EllipticK[(1/2)]))},{(π/8)-(1/2)ArcTan[(v^2)],Sum[(((((-1))^n)Cos[(2n+1) θ])/((2n+1)Cosh[(2n+1) π/2])),{n,0,200}]}/.v->2.1]

ln(((v+1)/(1-v)))==4 (∑_(n=0)^∞)(((((-1))^n) sin((2 n+1) θ))/((2 n+1) (exp(π (2 n+1))-1)))+ln(tan((θ/2)+(π/4)))

With[{θ=π ((v Hypergeometric2F1[(1/4),(1/2),(5/4),-(v^4)])/(EllipticK[(1/2)]))},{Log[((1+v)/(1-v))],Log[Tan[(π/4)+(1/2)θ]]+4Sum[(((((-1))^n)Sin[(2n+1) θ])/((2n+1)(Exp[(2n+1) π]-1))),{n,0,∞}]}]

With[{θ=π ((v Hypergeometric2F1[(1/4),(1/2),(5/4),-(v^4)])/(EllipticK[(1/2)]))},{Log[((1+v)/(1-v))],Log[Tan[(π/4)+(1/2)θ]]+4Sum[(((((-1))^n)Sin[(2n+1) θ])/((2n+1)(Exp[(2n+1) π]-1))),{n,0,200}]}/.v->2.1]