Jacobian Elliptic Functions - 22.19 Physical Applications

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
22.19.E1 d 2 θ ⁢ ( t ) d t 2 = - sin ⁡ θ ⁢ ( t ) derivative 𝜃 𝑡 𝑡 2 𝜃 𝑡 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}\theta(t)}{{\mathrm{d}t}^{2}% }=-\sin\theta(t)}}
\deriv[2]{\theta(t)}{t} = -\sin@@{\theta(t)}		 

diff(theta(t), [t$(2)]) = - sin(theta(t))

D[\[Theta][t], {t, 2}] == - Sin[\[Theta][t]]
Failure Failure
Failed [60 / 60]
Result: -1.247168970-.2207308174*I
Test Values: {t = -3/2, theta = 1/2*3^(1/2)+1/2*I}


Result: 1.342338585-1.241300956*I
Test Values: {t = -3/2, theta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [60 / 60]
Result: Complex[-1.247168970138959, -0.22073081765616068]
Test Values: {Rule[t, -1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.3423385844153726, -1.2413009551766627]
Test Values: {Rule[t, -1.5], Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
22.19.E2 sin ⁡ ( 1 2 ⁢ θ ⁢ ( t ) ) = sin ⁡ ( 1 2 ⁢ α ) ⁢ sn ⁡ ( t + K , sin ⁡ ( 1 2 ⁢ α ) ) 1 2 𝜃 𝑡 1 2 𝛼 Jacobi-elliptic-sn 𝑡 𝐾 1 2 𝛼 {\displaystyle{\displaystyle\sin\left(\tfrac{1}{2}\theta(t)\right)=\sin\left(% \frac{1}{2}\alpha\right)\operatorname{sn}\left(t+K,\sin\left(\tfrac{1}{2}% \alpha\right)\right)}}
\sin@{\tfrac{1}{2}\theta(t)} = \sin@{\frac{1}{2}\alpha}\Jacobiellsnk@{t+K}{\sin@{\tfrac{1}{2}\alpha}}		 

sin((1)/(2)*theta(t)) = sin((1)/(2)*alpha)*JacobiSN(t + EllipticK(k), sin((1)/(2)*alpha))

Sin[Divide[1,2]*\[Theta][t]] == Sin[Divide[1,2]*\[Alpha]]*JacobiSN[t + EllipticK[(k)^2], (Sin[Divide[1,2]*\[Alpha]])^2]
Failure Failure Error
Failed [300 / 300]
Result: Plus[Complex[-0.6478293894548304, -0.30568930559799934], Times[-0.6816387600233341, JacobiSN[DirectedInfinity[], 0.4646313991661485]]]
Test Values: {Rule[k, 1], Rule[t, -1.5], Rule[α, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.12355997139036334, 0.2467451262932382]
Test Values: {Rule[k, 2], Rule[t, -1.5], Rule[α, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.19.E3 θ ⁢ ( t ) = 2 ⁢ am ⁡ ( t ⁢ E / 2 , 2 / E ) 𝜃 𝑡 2 Jacobi-elliptic-amplitude 𝑡 𝐸 2 2 𝐸 {\displaystyle{\displaystyle\theta(t)=2\operatorname{am}\left(t\sqrt{E/2},% \sqrt{2/E}\right)}}
\theta(t) = 2\Jacobiamk@{t\sqrt{E/2}}{\sqrt{2/E}}		 

theta(t) = 2*JacobiAM(t*sqrt(E/2), sqrt(2/E))

\[Theta][t] == 2*JacobiAmplitude[t*Sqrt[E/2], Power[Sqrt[2/E], 2]]
Failure Failure
Failed [300 / 300]
Result: .133442481-.3164102922*I
Test Values: {E = 1/2*3^(1/2)+1/2*I, t = -3/2, theta = 1/2*3^(1/2)+1/2*I}


Result: 2.182480587-.8654483982*I
Test Values: {E = 1/2*3^(1/2)+1/2*I, t = -3/2, theta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.13344248094652933, -0.31641029231150586]
Test Values: {Rule[E, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[t, -1.5], Rule[x, Rational[3, 2]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.182480586623187, -0.865448397988164]
Test Values: {Rule[E, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[t, -1.5], Rule[x, Rational[3, 2]], Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
22.19.E5 V ⁢ ( x ) = + 1 2 ⁢ x 2 + 1 4 ⁢ β ⁢ x 4 𝑉 𝑥 1 2 superscript 𝑥 2 1 4 𝛽 superscript 𝑥 4 {\displaystyle{\displaystyle V(x)=+\tfrac{1}{2}x^{2}+\tfrac{1}{4}\beta x^{4}}}
V(x) = +\tfrac{1}{2}x^{2}+\tfrac{1}{4}\beta x^{4}		 

V(x) = +(1)/(2)*(x)^(2)+(1)/(4)*beta*(x)^(4)
 
V[x] == +Divide[1,2]*(x)^(2)+Divide[1,4]*\[Beta]*(x)^(4)
 Skipped - no semantic math Skipped - no semantic math - -
22.19.E6 x ⁢ ( t ) = a ⁢ cn ⁡ ( t ⁢ 1 + 2 ⁢ η , k ) 𝑥 𝑡 𝑎 Jacobi-elliptic-cn 𝑡 1 2 𝜂 𝑘 {\displaystyle{\displaystyle x(t)=a\operatorname{cn}\left(t\sqrt{1+2\eta},k% \right)}}
x(t) = a\Jacobiellcnk@{t\sqrt{1+2\eta}}{k}		 

x(t) = a*JacobiCN(t*sqrt(1 + 2*eta), k)

x[t] == a*JacobiCN[t*Sqrt[1 + 2*\[Eta]], (k)^2]
Failure Failure
Failed [300 / 300]
Result: -2.032489573-.1028075729*I
Test Values: {a = -3/2, eta = 1/2*3^(1/2)+1/2*I, t = -3/2, x = 3/2, k = 1}


Result: -1.103953626-.7415756720e-2*I
Test Values: {a = -3/2, eta = 1/2*3^(1/2)+1/2*I, t = -3/2, x = 3/2, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-2.032489572589819, -0.10280757291863922]
Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[t, -1.5], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.103953626215099, -0.007415756590236153]
Test Values: {Rule[a, -1.5], Rule[k, 2], Rule[t, -1.5], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.19.E7 x ⁢ ( t ) = a ⁢ sn ⁡ ( t ⁢ 1 - η , k ) 𝑥 𝑡 𝑎 Jacobi-elliptic-sn 𝑡 1 𝜂 𝑘 {\displaystyle{\displaystyle x(t)=a\operatorname{sn}\left(t\sqrt{1-\eta},k% \right)}}
x(t) = a\Jacobiellsnk@{t\sqrt{1-\eta}}{k}		 

x(t) = a*JacobiSN(t*sqrt(1 - eta), k)

x[t] == a*JacobiSN[t*Sqrt[1 - \[Eta]], (k)^2]
Failure Failure
Failed [300 / 300]
Result: -3.540811611+.4656977091*I
Test Values: {a = -3/2, eta = 1/2*3^(1/2)+1/2*I, t = -3/2, x = 3/2, k = 1}


Result: -3.440732980-.1498418752e-1*I
Test Values: {a = -3/2, eta = 1/2*3^(1/2)+1/2*I, t = -3/2, x = 3/2, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-3.5408116110434387, 0.46569770889881135]
Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[t, -1.5], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-3.4407329797279083, -0.014984187659583321]
Test Values: {Rule[a, -1.5], Rule[k, 2], Rule[t, -1.5], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.19.E8 x ⁢ ( t ) = a ⁢ dn ⁡ ( t ⁢ η , k ) 𝑥 𝑡 𝑎 Jacobi-elliptic-dn 𝑡 𝜂 𝑘 {\displaystyle{\displaystyle x(t)=a\operatorname{dn}\left(t\sqrt{\eta},k\right% )}}
x(t) = a\Jacobielldnk@{t\sqrt{\eta}}{k}		 

x(t) = a*JacobiDN(t*sqrt(eta), k)

x[t] == a*JacobiDN[t*Sqrt[\[Eta]], (k)^2]
Failure Failure
Failed [300 / 300]
Result: -1.613955183-.2329422536*I
Test Values: {a = -3/2, eta = 1/2*3^(1/2)+1/2*I, t = -3/2, x = 3/2, k = 1}


Result: -3.968457087-.6161541466*I
Test Values: {a = -3/2, eta = 1/2*3^(1/2)+1/2*I, t = -3/2, x = 3/2, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.6139551823182394, -0.23294225362869586]
Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[t, -1.5], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-3.968457085129692, -0.6161541479869231]
Test Values: {Rule[a, -1.5], Rule[k, 2], Rule[t, -1.5], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.19.E9 x ⁢ ( t ) = a ⁢ cn ⁡ ( t ⁢ 2 ⁢ η - 1 , k ) 𝑥 𝑡 𝑎 Jacobi-elliptic-cn 𝑡 2 𝜂 1 𝑘 {\displaystyle{\displaystyle x(t)=a\operatorname{cn}\left(t\sqrt{2\eta-1},k% \right)}}
x(t) = a\Jacobiellcnk@{t\sqrt{2\eta-1}}{k}		 

x(t) = a*JacobiCN(t*sqrt(2*eta - 1), k)

x[t] == a*JacobiCN[t*Sqrt[2*\[Eta]- 1], (k)^2]
Failure Failure
Failed [300 / 300]
Result: -1.736815452-.4365167404*I
Test Values: {a = -3/2, eta = 1/2*3^(1/2)+1/2*I, t = -3/2, x = 3/2, k = 1}


Result: -.272323884+.8696505748*I
Test Values: {a = -3/2, eta = 1/2*3^(1/2)+1/2*I, t = -3/2, x = 3/2, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.7368154521565795, -0.4365167405198458]
Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[t, -1.5], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.27232388329398516, 0.8696505752545954]
Test Values: {Rule[a, -1.5], Rule[k, 2], Rule[t, -1.5], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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