Jacobian Elliptic Functions - 22.18 Mathematical Applications

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
22.18.E1 ( x 2 / a 2 ) + ( y 2 / b 2 ) = 1 superscript π‘₯ 2 superscript π‘Ž 2 superscript 𝑦 2 superscript 𝑏 2 1 {\displaystyle{\displaystyle\left(x^{2}/a^{2}\right)+\left(y^{2}/b^{2}\right)=% 1}}
\left(x^{2}/a^{2}\right)+\left(y^{2}/b^{2}\right) = 1		 

((x)^(2)/(a)^(2))+((y)^(2)/(b)^(2)) = 1
 
((x)^(2)/(a)^(2))+((y)^(2)/(b)^(2)) == 1
 Skipped - no semantic math Skipped - no semantic math - -
22.18#Ex1 x = a ⁒ sn ⁑ ( u , k ) π‘₯ π‘Ž Jacobi-elliptic-sn 𝑒 π‘˜ {\displaystyle{\displaystyle x=a\operatorname{sn}\left(u,k\right)}}
x = a\Jacobiellsnk@{u}{k}		 

x = a*JacobiSN(u, k)

x == a*JacobiSN[u, (k)^2]
Failure Failure
Failed [300 / 300]
Result: 2.688604135+.3653402056*I
Test Values: {a = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1}


Result: 2.519684790-.2348240643e-1*I
Test Values: {a = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[2.688604134627054, 0.3653402057357006]
Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}


Result: Complex[2.5196847900911203, -0.02348240620400443]
Test Values: {Rule[a, -1.5], Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}

... skip entries to safe data
22.18#Ex2 y = b ⁒ cn ⁑ ( u , k ) 𝑦 𝑏 Jacobi-elliptic-cn 𝑒 π‘˜ {\displaystyle{\displaystyle y=b\operatorname{cn}\left(u,k\right)}}
y = b\Jacobiellcnk@{u}{k}		 

y = b*JacobiCN(u, k)

y == b*JacobiCN[u, (k)^2]
Failure Failure
Failed [300 / 300]
Result: -.433885744-.4073155167*I
Test Values: {b = -3/2, u = 1/2*3^(1/2)+1/2*I, y = -3/2, k = 1}


Result: -.399423925+.2175647210e-1*I
Test Values: {b = -3/2, u = 1/2*3^(1/2)+1/2*I, y = -3/2, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.43388574383527945, -0.40731551667372035]
Test Values: {Rule[b, -1.5], Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[y, -1.5]}


Result: Complex[-0.39942392524307424, 0.021756471897004394]
Test Values: {Rule[b, -1.5], Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[y, -1.5]}

... skip entries to safe data
22.18.E4 l ⁒ ( r ) = ( 1 / 2 ) ⁒ arccn ⁑ ( r , 1 / 2 ) 𝑙 π‘Ÿ 1 2 inverse-Jacobi-elliptic-cn π‘Ÿ 1 2 {\displaystyle{\displaystyle l(r)=(1/\sqrt{2})\operatorname{arccn}\left(r,1/% \sqrt{2}\right)}}
l(r) = (1/\sqrt{2})\aJacobiellcnk@{r}{1/\sqrt{2}}		 

l(r) = (1/(sqrt(2)))*InverseJacobiCN(r, 1/(sqrt(2)))

l[r] == (1/(Sqrt[2]))*InverseJacobiCN[r, (1/(Sqrt[2]))^2]
Failure Failure
Failed [18 / 18]
Result: -4.122057553+.6299669258*I
Test Values: {r = -3/2, l = 1}


Result: -5.622057553+.6299669258*I
Test Values: {r = -3/2, l = 2}

... skip entries to safe data
Failed [18 / 18]
Result: Complex[-4.12205755429212, 0.629966925905157]
Test Values: {Rule[l, 1], Rule[r, -1.5]}


Result: Complex[-5.62205755429212, 0.629966925905157]
Test Values: {Rule[l, 2], Rule[r, -1.5]}

... skip entries to safe data
22.18.E5 r = cn ⁑ ( 2 ⁒ l , 1 / 2 ) π‘Ÿ Jacobi-elliptic-cn 2 𝑙 1 2 {\displaystyle{\displaystyle r=\operatorname{cn}\left(\sqrt{2}l,1/\sqrt{2}% \right)}}
r = \Jacobiellcnk@{\sqrt{2}l}{1/\sqrt{2}}		 

r = JacobiCN(sqrt(2)*l, 1/(sqrt(2)))

r == JacobiCN[Sqrt[2]*l, (1/(Sqrt[2]))^2]
Failure Failure
Failed [18 / 18]
Result: -1.810737930
Test Values: {r = -3/2, l = 1}


Result: -.8262668012
Test Values: {r = -3/2, l = 2}

... skip entries to safe data
Failed [18 / 18]
Result: -1.810737930333856
Test Values: {Rule[l, 1], Rule[r, -1.5]}


Result: -0.8262668010254658
Test Values: {Rule[l, 2], Rule[r, -1.5]}

... skip entries to safe data
22.18#Ex3 x = cn ⁑ ( 2 ⁒ l , 1 / 2 ) ⁒ dn ⁑ ( 2 ⁒ l , 1 / 2 ) π‘₯ Jacobi-elliptic-cn 2 𝑙 1 2 Jacobi-elliptic-dn 2 𝑙 1 2 {\displaystyle{\displaystyle x=\operatorname{cn}\left(\sqrt{2}l,1/\sqrt{2}% \right)\operatorname{dn}\left(\sqrt{2}l,1/\sqrt{2}\right)}}
x = \Jacobiellcnk@{\sqrt{2}l}{1/\sqrt{2}}\Jacobielldnk@{\sqrt{2}l}{1/\sqrt{2}}		 

x = JacobiCN(sqrt(2)*l, 1/(sqrt(2)))*JacobiDN(sqrt(2)*l, 1/(sqrt(2)))

x == JacobiCN[Sqrt[2]*l, (1/(Sqrt[2]))^2]*JacobiDN[Sqrt[2]*l, (1/(Sqrt[2]))^2]
Failure Failure
Failed [9 / 9]
Result: 1.269911408
Test Values: {x = 3/2, l = 1}


Result: 2.074437352
Test Values: {x = 3/2, l = 2}

... skip entries to safe data
Failed [9 / 9]
Result: 1.2699114077583538
Test Values: {Rule[l, 1], Rule[x, 1.5]}


Result: 2.0744373520381156
Test Values: {Rule[l, 2], Rule[x, 1.5]}

... skip entries to safe data
22.18.E7 a ⁒ x 2 ⁒ y 2 + b ⁒ ( x 2 ⁒ y + x ⁒ y 2 ) + c ⁒ ( x 2 + y 2 ) + 2 ⁒ d ⁒ x ⁒ y + e ⁒ ( x + y ) + f = 0 π‘Ž superscript π‘₯ 2 superscript 𝑦 2 𝑏 superscript π‘₯ 2 𝑦 π‘₯ superscript 𝑦 2 𝑐 superscript π‘₯ 2 superscript 𝑦 2 2 𝑑 π‘₯ 𝑦 𝑒 π‘₯ 𝑦 𝑓 0 {\displaystyle{\displaystyle ax^{2}y^{2}+b(x^{2}y+xy^{2})+c(x^{2}+y^{2})+2dxy+% e(x+y)+f=0}}
ax^{2}y^{2}+b(x^{2}y+xy^{2})+c(x^{2}+y^{2})+2dxy+e(x+y)+f = 0		 

a*(x)^(2)* (y)^(2)+ b*((x)^(2)* y + x*(y)^(2))+ c*((x)^(2)+ (y)^(2))+ 2*d*x*y + exp(1)*(x + y)+ f = 0
 
a*(x)^(2)* (y)^(2)+ b*((x)^(2)* y + x*(y)^(2))+ c*((x)^(2)+ (y)^(2))+ 2*d*x*y + E*(x + y)+ f == 0
 Skipped - no semantic math Skipped - no semantic math - -
22.18#Ex5 x 3 = x 1 ⁒ y 2 + x 2 ⁒ y 1 1 - k 2 ⁒ x 1 2 ⁒ x 2 2 subscript π‘₯ 3 subscript π‘₯ 1 subscript 𝑦 2 subscript π‘₯ 2 subscript 𝑦 1 1 superscript π‘˜ 2 superscript subscript π‘₯ 1 2 superscript subscript π‘₯ 2 2 {\displaystyle{\displaystyle x_{3}=\frac{x_{1}y_{2}+x_{2}y_{1}}{1-k^{2}x_{1}^{% 2}x_{2}^{2}}}}
x_{3} = \frac{x_{1}y_{2}+x_{2}y_{1}}{1-k^{2}x_{1}^{2}x_{2}^{2}}		 

x[3] = (x[1]*y[2]+ x[2]*y[1])/(1 - (k)^(2)* (x[1])^(2)*(x[2])^(2))
 
Subscript[x, 3] == Divide[Subscript[x, 1]*Subscript[y, 2]+ Subscript[x, 2]*Subscript[y, 1],1 - (k)^(2)* (Subscript[x, 1])^(2)*(Subscript[x, 2])^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
22.18#Ex6 y 3 = y 1 ⁒ y 2 + x 2 ⁒ ( - ( 1 + k 2 ) ⁒ x 1 + 2 ⁒ k 2 ⁒ x 1 3 ) 1 - k 2 ⁒ x 1 2 ⁒ x 2 2 + x 3 ⁒ 2 ⁒ k 2 ⁒ x 1 ⁒ y 1 ⁒ x 2 2 1 - k 2 ⁒ x 1 2 ⁒ x 2 2 subscript 𝑦 3 subscript 𝑦 1 subscript 𝑦 2 subscript π‘₯ 2 1 superscript π‘˜ 2 subscript π‘₯ 1 2 superscript π‘˜ 2 superscript subscript π‘₯ 1 3 1 superscript π‘˜ 2 superscript subscript π‘₯ 1 2 superscript subscript π‘₯ 2 2 subscript π‘₯ 3 2 superscript π‘˜ 2 subscript π‘₯ 1 subscript 𝑦 1 superscript subscript π‘₯ 2 2 1 superscript π‘˜ 2 superscript subscript π‘₯ 1 2 superscript subscript π‘₯ 2 2 {\displaystyle{\displaystyle y_{3}=\frac{y_{1}y_{2}+x_{2}(-(1+k^{2})x_{1}+2k^{% 2}x_{1}^{3})}{1-k^{2}x_{1}^{2}x_{2}^{2}}+x_{3}\frac{2k^{2}x_{1}y_{1}x_{2}^{2}}% {1-k^{2}x_{1}^{2}x_{2}^{2}}}}
y_{3} = \frac{y_{1}y_{2}+x_{2}(-(1+k^{2})x_{1}+2k^{2}x_{1}^{3})}{1-k^{2}x_{1}^{2}x_{2}^{2}}+x_{3}\frac{2k^{2}x_{1}y_{1}x_{2}^{2}}{1-k^{2}x_{1}^{2}x_{2}^{2}}		 

y[3] = (y[1]*y[2]+ x[2]*(-(1 + (k)^(2))*x[1]+ 2*(k)^(2)* (x[1])^(3)))/(1 - (k)^(2)* (x[1])^(2)*(x[2])^(2))+ x[3]*(2*(k)^(2)* x[1]*y[1]*(x[2])^(2))/(1 - (k)^(2)* (x[1])^(2)*(x[2])^(2))
 
Subscript[y, 3] == Divide[Subscript[y, 1]*Subscript[y, 2]+ Subscript[x, 2]*(-(1 + (k)^(2))*Subscript[x, 1]+ 2*(k)^(2)* (Subscript[x, 1])^(3)),1 - (k)^(2)* (Subscript[x, 1])^(2)*(Subscript[x, 2])^(2)]+ Subscript[x, 3]*Divide[2*(k)^(2)* Subscript[x, 1]*Subscript[y, 1]*(Subscript[x, 2])^(2),1 - (k)^(2)* (Subscript[x, 1])^(2)*(Subscript[x, 2])^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
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