Jacobian Elliptic Functions - 22.16 Related Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
22.16.E1 am ⁡ ( x , k ) = Arcsin ⁡ ( sn ⁡ ( x , k ) ) Jacobi-elliptic-amplitude 𝑥 𝑘 multivalued-inverse-sine Jacobi-elliptic-sn 𝑥 𝑘 {\displaystyle{\displaystyle\operatorname{am}\left(x,k\right)=\operatorname{% Arcsin}\left(\operatorname{sn}\left(x,k\right)\right)}}
\Jacobiamk@{x}{k} = \Asin@{\Jacobiellsnk@{x}{k}}		 

Error

JacobiAmplitude[x, Power[k, 2]] == ArcSin[JacobiSN[x, (k)^2]]
Missing Macro Error Failure -
Failed [1 / 3]
Result: 6.283185307179586
Test Values: {Rule[k, 3], Rule[x, Rational[3, 2]]}

22.16.E2 am ⁡ ( x + 2 ⁢ K , k ) = am ⁡ ( x , k ) + π Jacobi-elliptic-amplitude 𝑥 2 𝐾 𝑘 Jacobi-elliptic-amplitude 𝑥 𝑘 𝜋 {\displaystyle{\displaystyle\operatorname{am}\left(x+2K,k\right)=\operatorname% {am}\left(x,k\right)+\pi}}
\Jacobiamk@{x+2K}{k} = \Jacobiamk@{x}{k}+\pi		 

JacobiAM(x + 2*EllipticK(k), k) = JacobiAM(x, k)+ Pi

JacobiAmplitude[x + 2*EllipticK[(k)^2], Power[k, 2]] == JacobiAmplitude[x, Power[k, 2]]+ Pi
Failure Failure Error
Failed [1 / 3]
Result: Plus[-4.273320998840302, Gudermannian[DirectedInfinity[]]]
Test Values: {Rule[k, 1], Rule[x, Rational[3, 2]]}

22.16.E3 am ⁡ ( x , k ) = ∫ 0 x dn ⁡ ( t , k ) ⁢ d t Jacobi-elliptic-amplitude 𝑥 𝑘 superscript subscript 0 𝑥 Jacobi-elliptic-dn 𝑡 𝑘 𝑡 {\displaystyle{\displaystyle\operatorname{am}\left(x,k\right)=\int_{0}^{x}% \operatorname{dn}\left(t,k\right)\mathrm{d}t}}
\Jacobiamk@{x}{k} = \int_{0}^{x}\Jacobielldnk@{t}{k}\diff{t}		 

JacobiAM(x, k) = int(JacobiDN(t, k), t = 0..x)

JacobiAmplitude[x, Power[k, 2]] == Integrate[JacobiDN[t, (k)^2], {t, 0, x}, GenerateConditions->None]
Failure Successful Successful [Tested: 9] Successful [Tested: 9]
22.16.E4 am ⁡ ( x , 0 ) = x Jacobi-elliptic-amplitude 𝑥 0 𝑥 {\displaystyle{\displaystyle\operatorname{am}\left(x,0\right)=x}}
\Jacobiamk@{x}{0} = x		 

JacobiAM(x, 0) = x

JacobiAmplitude[x, Power[0, 2]] == x
Successful Successful - Successful [Tested: 3]
22.16.E5 am ⁡ ( x , 1 ) = gd ⁡ ( x ) Jacobi-elliptic-amplitude 𝑥 1 Gudermannian 𝑥 {\displaystyle{\displaystyle\operatorname{am}\left(x,1\right)=\operatorname{gd% }\left(x\right)}}
\Jacobiamk@{x}{1} = \Gudermannian@{x}		 - ∞ < x , x < ∞ formulae-sequence 𝑥 𝑥 {\displaystyle{\displaystyle-\infty<x,x<\infty}} 
JacobiAM(x, 1) = arctan(sinh(x))

JacobiAmplitude[x, Power[1, 2]] == Gudermannian[x]
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
22.16.E9 am ⁡ ( x , k ) = π 2 ⁢ K ⁢ x + 2 ⁢ ∑ n = 1 ∞ q n ⁢ sin ⁡ ( 2 ⁢ n ⁢ ζ ) n ⁢ ( 1 + q 2 ⁢ n ) Jacobi-elliptic-amplitude 𝑥 𝑘 𝜋 2 𝐾 𝑥 2 superscript subscript 𝑛 1 superscript 𝑞 𝑛 2 𝑛 𝜁 𝑛 1 superscript 𝑞 2 𝑛 {\displaystyle{\displaystyle\operatorname{am}\left(x,k\right)=\frac{\pi}{2K}x+% 2\sum_{n=1}^{\infty}\frac{q^{n}\sin\left(2n\zeta\right)}{n(1+q^{2n})}}}
\Jacobiamk@{x}{k} = \frac{\pi}{2K}x+2\sum_{n=1}^{\infty}\frac{q^{n}\sin@{2n\zeta}}{n(1+q^{2n})}		 

JacobiAM(x, k) = (Pi)/(2*EllipticK(k))*x + 2*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n)* sin(2*n*zeta))/(n*(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n))), n = 1..infinity)

JacobiAmplitude[x, Power[k, 2]] == Divide[Pi,2*EllipticK[(k)^2]]*x + 2*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n)* Sin[2*n*\[Zeta]],n*(1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n))], {n, 1, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out
Failed [30 / 30]
Result: Plus[Complex[1.9977537490349477, 0.49999999999999994], Times[-1.0, Gudermannian[DirectedInfinity[]]]]
Test Values: {Rule[k, 1], Rule[x, Rational[3, 2]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.6288351638274511, -0.8359897636003678]
Test Values: {Rule[k, 2], Rule[x, Rational[3, 2]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.16.E10 x = F ⁡ ( ϕ , k ) 𝑥 elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle x=F\left(\phi,k\right)}}
x = \incellintFk@{\phi}{k}		 

x = EllipticF(sin(phi), k)

x == EllipticF[\[Phi], (k)^2]
Failure Failure
Failed [90 / 90]
Result: .6791299710-.6773780507*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1}


Result: 1.016811658-.7182528229*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 2}

... skip entries to safe data
Failed [90 / 90]
Result: Complex[0.6791299712710547, -0.6773780505641274]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.0168116579433883, -0.7182528227883367]
Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.16.E11 am ⁡ ( x , k ) = ϕ Jacobi-elliptic-amplitude 𝑥 𝑘 italic-ϕ {\displaystyle{\displaystyle\operatorname{am}\left(x,k\right)=\phi}}
\Jacobiamk@{x}{k} = \phi		 

JacobiAM(x, k) = phi

JacobiAmplitude[x, Power[k, 2]] == \[Phi]
Failure Failure
Failed [90 / 90]
Result: .2657029410-.5000000000*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1}


Result: -.6844899651-.5000000000*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[0.26570294146607043, -0.49999999999999994]
Test Values: {Rule[k, 1], Rule[x, Rational[3, 2]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.6844899649247672, -0.49999999999999994]
Test Values: {Rule[k, 2], Rule[x, Rational[3, 2]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.16.E12 sn ⁡ ( x , k ) = sin ⁡ ϕ Jacobi-elliptic-sn 𝑥 𝑘 italic-ϕ {\displaystyle{\displaystyle\operatorname{sn}\left(x,k\right)=\sin\phi}}
\Jacobiellsnk@{x}{k} = \sin@@{\phi}		 

JacobiSN(x, k) = sin(phi)

JacobiSN[x, (k)^2] == Sin[\[Phi]]
Failure Failure
Failed [90 / 90]
Result: .461679191e-1-.3375964631*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1}


Result: -.6784403409-.3375964631*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 2}

... skip entries to safe data
Failed [90 / 90]
Result: Complex[0.046167919344728525, -0.33759646322287]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.678440340667692, -0.33759646322287]
Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.16.E12 sin ⁡ ϕ = sin ⁡ ( am ⁡ ( x , k ) ) italic-ϕ Jacobi-elliptic-amplitude 𝑥 𝑘 {\displaystyle{\displaystyle\sin\phi=\sin\left(\operatorname{am}\left(x,k% \right)\right)}}
\sin@@{\phi} = \sin@{\Jacobiamk@{x}{k}}		 

sin(phi) = sin(JacobiAM(x, k))

Sin[\[Phi]] == Sin[JacobiAmplitude[x, Power[k, 2]]]
Failure Failure
Failed [90 / 90]
Result: -.461679191e-1+.3375964631*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1}


Result: .6784403409+.3375964631*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-0.046167919344728525, 0.33759646322287]
Test Values: {Rule[k, 1], Rule[x, Rational[3, 2]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.678440340667692, 0.33759646322287]
Test Values: {Rule[k, 2], Rule[x, Rational[3, 2]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.16.E13 cn ⁡ ( x , k ) = cos ⁡ ϕ Jacobi-elliptic-cn 𝑥 𝑘 italic-ϕ {\displaystyle{\displaystyle\operatorname{cn}\left(x,k\right)=\cos\phi}}
\Jacobiellcnk@{x}{k} = \cos@@{\phi}		 

JacobiCN(x, k) = cos(phi)

JacobiCN[x, (k)^2] == Cos[\[Phi]]
Failure Failure
Failed [90 / 90]
Result: -.3054469840+.3969495503*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1}


Result: .2530246253+.3969495503*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 2}

... skip entries to safe data
Failed [90 / 90]
Result: Complex[-0.3054469841149447, 0.3969495502290325]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.2530246251336542, 0.3969495502290325]
Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.16.E13 cos ⁡ ϕ = cos ⁡ ( am ⁡ ( x , k ) ) italic-ϕ Jacobi-elliptic-amplitude 𝑥 𝑘 {\displaystyle{\displaystyle\cos\phi=\cos\left(\operatorname{am}\left(x,k% \right)\right)}}
\cos@@{\phi} = \cos@{\Jacobiamk@{x}{k}}		 

cos(phi) = cos(JacobiAM(x, k))

Cos[\[Phi]] == Cos[JacobiAmplitude[x, Power[k, 2]]]
Failure Failure
Failed [90 / 90]
Result: .3054469840-.3969495503*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1}


Result: -.2530246253-.3969495503*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[0.3054469841149447, -0.3969495502290325]
Test Values: {Rule[k, 1], Rule[x, Rational[3, 2]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.2530246251336542, -0.3969495502290325]
Test Values: {Rule[k, 2], Rule[x, Rational[3, 2]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.16.E33 Z ⁡ ( x + K | k ) = Z ⁡ ( x | k ) - k 2 ⁢ sn ⁡ ( x , k ) ⁢ cd ⁡ ( x , k ) Jacobi-Zeta 𝑥 𝐾 𝑘 Jacobi-Zeta 𝑥 𝑘 superscript 𝑘 2 Jacobi-elliptic-sn 𝑥 𝑘 Jacobi-elliptic-cd 𝑥 𝑘 {\displaystyle{\displaystyle\mathrm{Z}\left(x+K|k\right)=\mathrm{Z}\left(x|k% \right)-k^{2}\operatorname{sn}\left(x,k\right)\operatorname{cd}\left(x,k\right% )}}
\JacobiZetak@{x+K}{k} = \JacobiZetak@{x}{k}-k^{2}\Jacobiellsnk@{x}{k}\Jacobiellcdk@{x}{k}		 

JacobiZeta(x + EllipticK(k), k) = JacobiZeta(x, k)- (k)^(2)* JacobiSN(x, k)*JacobiCD(x, k)

JacobiZeta[x + EllipticK[(k)^2], k] == JacobiZeta[x, k]- (k)^(2)* JacobiSN[x, (k)^2]*JacobiCD[x, (k)^2]
Failure Failure Error
Failed [9 / 9]
Result: Plus[-0.09234673295918805, JacobiZeta[DirectedInfinity[], 1.0]]
Test Values: {Rule[k, 1], Rule[x, 1.5]}


Result: Complex[-2.7319699839312124, -0.6260098794347219]
Test Values: {Rule[k, 2], Rule[x, 1.5]}

... skip entries to safe data
22.16.E34 Z ⁡ ( x + 2 ⁢ K | k ) = Z ⁡ ( x | k ) Jacobi-Zeta 𝑥 2 𝐾 𝑘 Jacobi-Zeta 𝑥 𝑘 {\displaystyle{\displaystyle\mathrm{Z}\left(x+2K|k\right)=\mathrm{Z}\left(x|k% \right)}}
\JacobiZetak@{x+2K}{k} = \JacobiZetak@{x}{k}		 

JacobiZeta(x + 2*EllipticK(k), k) = JacobiZeta(x, k)

JacobiZeta[x + 2*EllipticK[(k)^2], k] == JacobiZeta[x, k]
Successful Failure -
Failed [9 / 9]
Result: Plus[-0.9974949866040544, JacobiZeta[DirectedInfinity[], 1.0]]
Test Values: {Rule[k, 1], Rule[x, 1.5]}


Result: Complex[-0.2870190432201134, -5.437509139287473]
Test Values: {Rule[k, 2], Rule[x, 1.5]}

... skip entries to safe data
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