Lemniscate Constant

Author

Eric W. Weisstein
October 6, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/LemniscateConstant.nb.

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Leminscate constant L

Numeric

A062539

N[2 EllipticK[-1],20]

Integrals

(∫_0^π)(1/(Sqrt[(sin^2)(θ)+1]))dθ

Integrate[(1/(Sqrt[1+((Sin[θ])^2)])),{θ,0,π}]

2 (∫_0^1)(1/(Sqrt[1-(x^4)]))dx

2Integrate[(1/(Sqrt[1-(x^4)])),{x,0,1}]

Special functions

π G

N[π G/.G->((2 EllipticK[-1])/π)]

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

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

2 K(-1)

2 EllipticK[-1]//N

(Sqrt[2])(EllipticK[(1/2)])

(Sqrt[2]) EllipticK[(1/2)]//N

(((Gamma[(1/4)])^2)/(2 (Sqrt[2 π])))

(((Gamma[(1/4)])^2)/(2 (Sqrt[2 π])))//N

π ((EllipticTheta[4, 0, (e^(-π))])^2)

N[π ((EllipticTheta[4,0,(E^(-π))])^2)]

((3 π)/(2 (CarlsonRD[0, 2, 1])))

((3π)/(2CarlsonRD[0,2,1]))//N

2 (CarlsonRF[0, 1, 2])

2CarlsonRF[0,1,2]

FunctionExpand[%]

N[%]

π (CarlsonRK[1, 2])

π CarlsonRK[1,2]

FunctionExpand[%]

N[%]

Leminscate constant (Subscript[L, 1]) = (1/2)L

Numeric

A085565

N[EllipticK[-1],20]

Integrals

(1/2)(∫_0^π)(1/(Sqrt[(sin^2)(θ)+1]))dθ

(1/2)Integrate[(1/(Sqrt[1+((Sin[θ])^2)])),{θ,0,π}]

(∫_0^1)(1/(Sqrt[1-(x^4)]))dx

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

Special functions

K(-1)

EllipticK[-1]//N

(2^(-1/2)) (EllipticK[(1/2)])

(2^(-1/2))EllipticK[(1/2)]//N

(((Gamma[(1/4)])^2)/(4 (Sqrt[2 π])))

(((Gamma[(1/4)])^2)/(4 (Sqrt[2 π])))//N

(π/2) ((EllipticTheta[4, 0, (e^(-π))])^2)

N[(π/2)((EllipticTheta[4,0,(E^(-π))])^2)]

((3 π)/(4(CarlsonRD[0, 2, 1])))

((3π)/(4CarlsonRD[0,2,1]))//N

(CarlsonRF[0, 1, 2])

CarlsonRF[0,1,2]

FunctionExpand[%]

N[%]

(π/2) (CarlsonRK[1, 2])

(π/2) CarlsonRK[1,2]

FunctionExpand[%]

N[%]

Leminscate constant (Subscript[L, 2]) = π/(2L)

Numeric

A076390

N[(Pi/(4 EllipticK[-1])),20]

Integrals

(∫_0^1)((x^2)/(Sqrt[1-(x^4)]))dx

Integrate[((x^2)/(Sqrt[1-(x^4)])),{x,0,1}]//FunctionExpand//FullSimplify

Special functions

(π/(2 L))

(π/(2L))/.L->2EllipticK[-1]//N

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

N[((ArithmeticGeometricMean[1,(Sqrt[2])])/2)]

((2 (EllipticK[-1]))/π)

N[(π/(4EllipticK[-1]))]

(π/(2 (Sqrt[2]) (EllipticK[(1/2)])))

N[(π/(2(Sqrt[2]) EllipticK[(1/2)]))]

((π^(3/2))/(8 (Sqrt[2]) ((Gamma[(5/4)])^2)))

N[((π^(3/2))/(8 (Sqrt[2]) ((Gamma[(5/4)])^2)))]

(1/(2 (((2 π))^(-3/2)) ((Gamma[(1/4)])^2)))

N[(1/(2(((2 π))^(-3/2)) ((Gamma[(1/4)])^2)))]

(1/(2 ((EllipticTheta[4, 0, (e^(-π))])^2)))

N[(1/(2((EllipticTheta[4,0,(E^(-π))])^2)))]

(π/(4 (CarlsonRF[0, 1, 2])))

(π/(4CarlsonRF[0,1,2]))//N

((CarlsonRD[0, 2, 1])/3)

(1/3)CarlsonRD[0,2,1]//N

(1/(2 (CarlsonRK[1, 2])))

N[(1/(2CarlsonRK[1,2]))]