Jacobian Elliptic Functions - 22.20 Methods of Computation

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
22.20#Ex1 a n = 1 2 ⁒ ( a n - 1 + b n - 1 ) subscript π‘Ž 𝑛 1 2 subscript π‘Ž 𝑛 1 subscript 𝑏 𝑛 1 {\displaystyle{\displaystyle a_{n}=\tfrac{1}{2}\left(a_{n-1}+b_{n-1}\right)}}
a_{n} = \tfrac{1}{2}\left(a_{n-1}+b_{n-1}\right)		 

a[n] = (1)/(2)*(a[n - 1]+ b[n - 1])
 
Subscript[a, n] == Divide[1,2]*(Subscript[a, n - 1]+ Subscript[b, n - 1])
 Skipped - no semantic math Skipped - no semantic math - -
22.20#Ex2 b n = ( a n - 1 ⁒ b n - 1 ) 1 / 2 subscript 𝑏 𝑛 superscript subscript π‘Ž 𝑛 1 subscript 𝑏 𝑛 1 1 2 {\displaystyle{\displaystyle b_{n}=\left(a_{n-1}b_{n-1}\right)^{1/2}}}
b_{n} = \left(a_{n-1}b_{n-1}\right)^{1/2}		 

b[n] = (a[n - 1]*b[n - 1])^(1/2)
 
Subscript[b, n] == (Subscript[a, n - 1]*Subscript[b, n - 1])^(1/2)
 Skipped - no semantic math Skipped - no semantic math - -
22.20#Ex3 c n = 1 2 ⁒ ( a n - 1 - b n - 1 ) subscript 𝑐 𝑛 1 2 subscript π‘Ž 𝑛 1 subscript 𝑏 𝑛 1 {\displaystyle{\displaystyle c_{n}=\tfrac{1}{2}\left(a_{n-1}-b_{n-1}\right)}}
c_{n} = \tfrac{1}{2}\left(a_{n-1}-b_{n-1}\right)		 

c[n] = (1)/(2)*(a[n - 1]- b[n - 1])
 
Subscript[c, n] == Divide[1,2]*(Subscript[a, n - 1]- Subscript[b, n - 1])
 Skipped - no semantic math Skipped - no semantic math - -
22.20.E3 Ο• N = 2 N ⁒ a N ⁒ x subscript italic-Ο• 𝑁 superscript 2 𝑁 subscript π‘Ž 𝑁 π‘₯ {\displaystyle{\displaystyle\phi_{N}=2^{N}a_{N}x}}
\phi_{N} = 2^{N}a_{N}x		 

phi[N] = (2)^(N)* a[N]*x
 
Subscript[\[Phi], N] == (2)^(N)* Subscript[a, N]*x
 Skipped - no semantic math Skipped - no semantic math - -
22.20.E4 Ο• n - 1 = 1 2 ⁒ ( Ο• n + arcsin ⁑ ( c n a n ⁒ sin ⁑ Ο• n ) ) subscript italic-Ο• 𝑛 1 1 2 subscript italic-Ο• 𝑛 subscript 𝑐 𝑛 subscript π‘Ž 𝑛 subscript italic-Ο• 𝑛 {\displaystyle{\displaystyle\phi_{n-1}=\frac{1}{2}\left(\phi_{n}+\operatorname% {arcsin}\left(\frac{c_{n}}{a_{n}}\sin\phi_{n}\right)\right)}}
\phi_{n-1} = \frac{1}{2}\left(\phi_{n}+\asin@{\frac{c_{n}}{a_{n}}\sin@@{\phi_{n}}}\right)		 

phi[n - 1] = (1)/(2)*(phi[n]+ arcsin((c[n])/(a[n])*sin(phi[n])))

Subscript[\[Phi], n - 1] == Divide[1,2]*(Subscript[\[Phi], n]+ ArcSin[Divide[Subscript[c, n],Subscript[a, n]]*Sin[Subscript[\[Phi], n]]])
Failure Failure
Failed [276 / 300]
Result: -1.366025404+.3660254040*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, a[n] = 1/2*3^(1/2)+1/2*I, c[n] = 1/2*3^(1/2)+1/2*I, phi[n] = 1/2*3^(1/2)+1/2*I, phi[-1+n] = -1/2+1/2*I*3^(1/2), n = 1}


Result: -1.366025404+.3660254040*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, a[n] = 1/2*3^(1/2)+1/2*I, c[n] = 1/2*3^(1/2)+1/2*I, phi[n] = 1/2*3^(1/2)+1/2*I, phi[-1+n] = -1/2+1/2*I*3^(1/2), n = 2}

... skip entries to safe data
Failed [276 / 300]
Result: Complex[1.3660254037844384, -0.36602540378443876]
Test Values: {Rule[n, 1], Rule[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ο•, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ο•, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[1.3660254037844384, -0.36602540378443876]
Test Values: {Rule[n, 2], Rule[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ο•, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ο•, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
22.20#Ex4 sn ⁑ ( x , k ) = sin ⁑ Ο• 0 Jacobi-elliptic-sn π‘₯ π‘˜ subscript italic-Ο• 0 {\displaystyle{\displaystyle\operatorname{sn}\left(x,k\right)=\sin\phi_{0}}}
\Jacobiellsnk@{x}{k} = \sin@@{\phi_{0}}		 

JacobiSN(x, k) = sin(phi[0])

JacobiSN[x, (k)^2] == Sin[Subscript[\[Phi], 0]]
Failure Failure
Failed [300 / 300]
Result: .461679191e-1-.3375964631*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, phi[0] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.6784403409-.3375964631*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, phi[0] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.046167919344728525, -0.33759646322287]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ο•, 0], 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[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ο•, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.20#Ex5 cn ⁑ ( x , k ) = cos ⁑ Ο• 0 Jacobi-elliptic-cn π‘₯ π‘˜ subscript italic-Ο• 0 {\displaystyle{\displaystyle\operatorname{cn}\left(x,k\right)=\cos\phi_{0}}}
\Jacobiellcnk@{x}{k} = \cos@@{\phi_{0}}		 

JacobiCN(x, k) = cos(phi[0])

JacobiCN[x, (k)^2] == Cos[Subscript[\[Phi], 0]]
Failure Failure
Failed [300 / 300]
Result: -.3054469840+.3969495503*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, phi[0] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .2530246253+.3969495503*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, phi[0] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.3054469841149447, 0.3969495502290325]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ο•, 0], 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[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ο•, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.20#Ex6 dn ⁑ ( x , k ) = cos ⁑ Ο• 0 cos ⁑ ( Ο• 1 - Ο• 0 ) Jacobi-elliptic-dn π‘₯ π‘˜ subscript italic-Ο• 0 subscript italic-Ο• 1 subscript italic-Ο• 0 {\displaystyle{\displaystyle\operatorname{dn}\left(x,k\right)=\frac{\cos\phi_{% 0}}{\cos\left(\phi_{1}-\phi_{0}\right)}}}
\Jacobielldnk@{x}{k} = \frac{\cos@@{\phi_{0}}}{\cos@{\phi_{1}-\phi_{0}}}		 

JacobiDN(x, k) = (cos(phi[0]))/(cos(phi[1]- phi[0]))

JacobiDN[x, (k)^2] == Divide[Cos[Subscript[\[Phi], 0]],Cos[Subscript[\[Phi], 1]- Subscript[\[Phi], 0]]]
Failure Failure
Failed [300 / 300]
Result: -.3054469840+.3969495503*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, phi[0] = 1/2*3^(1/2)+1/2*I, phi[1] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -1.663077867+.3969495503*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, phi[0] = 1/2*3^(1/2)+1/2*I, phi[1] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.3054469841149447, 0.3969495502290325]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ο•, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ο•, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.6630778670906836, 0.3969495502290325]
Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ο•, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ο•, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.20#Ex7 K = Ο€ 2 ⁒ M ⁒ ( 1 , k β€² ) 𝐾 πœ‹ 2 𝑀 1 superscript π‘˜ β€² {\displaystyle{\displaystyle K=\frac{\pi}{2M(1,k^{\prime})}}}
K = \frac{\pi}{2M(1,k^{\prime})}		 

EllipticK(k) = (Pi)/(2*M(1 ,sqrt(1 - (k)^(2))))
 
EllipticK[(k)^2] == Divide[Pi,2*M[1 ,Sqrt[1 - (k)^(2)]]]
 Skipped - no semantic math Skipped - no semantic math - -
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