Lamé Functions - 29.18 Mathematical Applications

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
29.18#Ex1 x = k ⁢ r ⁢ sn ⁡ ( β , k ) ⁢ sn ⁡ ( γ , k ) 𝑥 𝑘 𝑟 Jacobi-elliptic-sn 𝛽 𝑘 Jacobi-elliptic-sn 𝛾 𝑘 {\displaystyle{\displaystyle x=kr\operatorname{sn}\left(\beta,k\right)% \operatorname{sn}\left(\gamma,k\right)}}
x = kr\Jacobiellsnk@{\beta}{k}\Jacobiellsnk@{\gamma}{k}		 

x = k*r*JacobiSN(beta, k)*JacobiSN(gamma, k)

x == k*r*JacobiSN[\[Beta], (k)^2]*JacobiSN[\[Gamma], (k)^2]
Failure Failure
Failed [300 / 300]
Result: 2.206882914
Test Values: {beta = 3/2, gamma = 1/2*3^(1/2)+1/2*I, r = -3/2, x = 3/2, k = 1}


Result: 1.742014676
Test Values: {beta = 3/2, gamma = 1/2*3^(1/2)+1/2*I, r = -3/2, x = 3/2, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[2.5758629567327462, 0.3306870492079255]
Test Values: {Rule[k, 1], Rule[r, -1.5], Rule[x, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.8681877710203056, -0.008479026933090933]
Test Values: {Rule[k, 2], Rule[r, -1.5], Rule[x, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
29.18#Ex2 y = i ⁢ k k ′ ⁢ r ⁢ cn ⁡ ( β , k ) ⁢ cn ⁡ ( γ , k ) 𝑦 imaginary-unit 𝑘 superscript 𝑘 ′ 𝑟 Jacobi-elliptic-cn 𝛽 𝑘 Jacobi-elliptic-cn 𝛾 𝑘 {\displaystyle{\displaystyle y=\mathrm{i}\frac{k}{k^{\prime}}r\operatorname{cn% }\left(\beta,k\right)\operatorname{cn}\left(\gamma,k\right)}}
y = \iunit\frac{k}{k^{\prime}}r\Jacobiellcnk@{\beta}{k}\Jacobiellcnk@{\gamma}{k}		 

y = I*(k)/(sqrt(1 - (k)^(2)))*r*JacobiCN(beta, k)*JacobiCN(gamma, k)

y == I*Divide[k,Sqrt[1 - (k)^(2)]]*r*JacobiCN[\[Beta], (k)^2]*JacobiCN[\[Gamma], (k)^2]
Failure Failure
Failed [300 / 300]
Result: -1.500000000+Float(infinity)*I
Test Values: {beta = 3/2, gamma = 1/2*3^(1/2)+1/2*I, r = -3/2, y = -3/2, k = 1}


Result: .24058897e-1
Test Values: {beta = 3/2, gamma = 1/2*3^(1/2)+1/2*I, r = -3/2, y = -3/2, k = 2}

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


Result: Complex[-0.25004703976217724, 0.0247093927223503]
Test Values: {Rule[k, 2], Rule[r, -1.5], Rule[y, -1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
29.18#Ex3 z = 1 k ′ ⁢ r ⁢ dn ⁡ ( β , k ) ⁢ dn ⁡ ( γ , k ) 𝑧 1 superscript 𝑘 ′ 𝑟 Jacobi-elliptic-dn 𝛽 𝑘 Jacobi-elliptic-dn 𝛾 𝑘 {\displaystyle{\displaystyle z=\frac{1}{k^{\prime}}r\operatorname{dn}\left(% \beta,k\right)\operatorname{dn}\left(\gamma,k\right)}}
z = \frac{1}{k^{\prime}}r\Jacobielldnk@{\beta}{k}\Jacobielldnk@{\gamma}{k}		 

z = (1)/(sqrt(1 - (k)^(2)))*r*JacobiDN(beta, k)*JacobiDN(gamma, k)

z == Divide[1,Sqrt[1 - (k)^(2)]]*r*JacobiDN[\[Beta], (k)^2]*JacobiDN[\[Gamma], (k)^2]
Failure Failure
Failed [300 / 300]
Result: Float(infinity)+.5000000000*I
Test Values: {beta = 3/2, gamma = 1/2*3^(1/2)+1/2*I, r = -3/2, z = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .8660254040+.8623901524*I
Test Values: {beta = 3/2, gamma = 1/2*3^(1/2)+1/2*I, r = -3/2, z = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1], Rule[r, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.6104237277084903, 0.46270316084846885]
Test Values: {Rule[k, 2], Rule[r, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
29.18#Ex4 r ≥ 0 𝑟 0 {\displaystyle{\displaystyle r\geq 0}}
r \geq 0		 

r >= 0
 
r >= 0
 Skipped - no semantic math Skipped - no semantic math - -
29.18#Ex7 0 ≤ γ 0 𝛾 {\displaystyle{\displaystyle 0\leq\gamma}}
0 \leq \gamma		 

0 <= gamma

0 <= \[Gamma]
Failure Failure
Failed [3 / 10]
Result: 0. <= -1.500000000
Test Values: {gamma = -3/2}


Result: 0. <= -.5000000000
Test Values: {gamma = -1/2}

... skip entries to safe data
Failed [7 / 10]
Result: LessEqual[0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: LessEqual[0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
29.18#Ex7 γ ≤ 4 ⁢ K ⁡ 𝛾 4 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle\gamma\leq 4K}}
\gamma \leq 4\compellintKk@@{k}		 

gamma <= 4*EllipticK(k)

\[Gamma] <= 4*EllipticK[(k)^2]
Failure Failure Error
Failed [30 / 30]
Result: LessEqual[Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[]]
Test Values: {Rule[k, 1], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: LessEqual[Complex[0.8660254037844387, 0.49999999999999994], Complex[3.3715007096251925, -4.313031294999287]]
Test Values: {Rule[k, 2], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
29.18.E4 u ⁢ ( r , β , γ ) = u 1 ⁢ ( r ) ⁢ u 2 ⁢ ( β ) ⁢ u 3 ⁢ ( γ ) 𝑢 𝑟 𝛽 𝛾 subscript 𝑢 1 𝑟 subscript 𝑢 2 𝛽 subscript 𝑢 3 𝛾 {\displaystyle{\displaystyle u(r,\beta,\gamma)=u_{1}(r)u_{2}(\beta)u_{3}(% \gamma)}}
u(r,\beta,\gamma) = u_{1}(r)u_{2}(\beta)u_{3}(\gamma)		 

u(r , beta , gamma) = u[1](r)* u[2](beta)* u[3](gamma)
 
u[r , \[Beta], \[Gamma]] == Subscript[u, 1][r]* Subscript[u, 2][\[Beta]]* Subscript[u, 3][\[Gamma]]
 Skipped - no semantic math Skipped - no semantic math - -
29.18.E5 d d r ⁡ ( r 2 ⁢ d u 1 d r ) + ( ω 2 ⁢ r 2 - ν ⁢ ( ν + 1 ) ) ⁢ u 1 = 0 derivative 𝑟 superscript 𝑟 2 derivative subscript 𝑢 1 𝑟 superscript 𝜔 2 superscript 𝑟 2 𝜈 𝜈 1 subscript 𝑢 1 0 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}r}\left(r^{2}\frac{% \mathrm{d}u_{1}}{\mathrm{d}r}\right)+(\omega^{2}r^{2}-\nu(\nu+1))u_{1}=0}}
\deriv{}{r}\left(r^{2}\deriv{u_{1}}{r}\right)+(\omega^{2}r^{2}-\nu(\nu+1))u_{1} = 0		 

diff(((r)^(2)* diff(u[1], r))+((omega)^(2)* (r)^(2)- nu*(nu + 1))*u[1], r) = 0

D[((r)^(2)* D[Subscript[u, 1], r])+(\[Omega]^(2)* (r)^(2)- \[Nu]*(\[Nu]+ 1))*Subscript[u, 1], r] == 0
Failure Failure
Failed [300 / 300]
Result: -.9701216577e-9-3.000000003*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, omega = 1/2*3^(1/2)+1/2*I, r = -3/2, u[1] = 1/2*3^(1/2)+1/2*I}


Result: 3.000000003-.1039230485e-8*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, omega = 1/2*3^(1/2)+1/2*I, r = -3/2, u[1] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-4.440892098500626*^-16, -3.0]
Test Values: {Rule[r, -1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[u, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[3.0, -1.1102230246251565*^-15]
Test Values: {Rule[r, -1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[u, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
29.18.E6 d 2 u 2 d β 2 + ( h - ν ⁢ ( ν + 1 ) ⁢ k 2 ⁢ sn 2 ⁡ ( β , k ) ) ⁢ u 2 = 0 derivative subscript 𝑢 2 𝛽 2 ℎ 𝜈 𝜈 1 superscript 𝑘 2 Jacobi-elliptic-sn 2 𝛽 𝑘 subscript 𝑢 2 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}u_{2}}{{\mathrm{d}\beta}^{2}% }+(h-\nu(\nu+1)k^{2}{\operatorname{sn}^{2}}\left(\beta,k\right))u_{2}=0}}
\deriv[2]{u_{2}}{\beta}+(h-\nu(\nu+1)k^{2}\Jacobiellsnk^{2}@{\beta}{k})u_{2} = 0		 

diff(u[2], [beta$(2)])+(h - nu*(nu + 1)*(k)^(2)* (JacobiSN(beta, k))^(2))*u[2] = 0

D[Subscript[u, 2], {\[Beta], 2}]+(h - \[Nu]*(\[Nu]+ 1)*(k)^(2)* (JacobiSN[\[Beta], (k)^2])^(2))*Subscript[u, 2] == 0
Failure Failure
Failed [300 / 300]
Result: .9035331887e-1-.6627968211*I
Test Values: {beta = 3/2, h = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, u[2] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .4348106217+.6227353309*I
Test Values: {beta = 3/2, h = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, u[2] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.09035331946182387, -0.6627968211359702]
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1], Rule[β, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[u, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.4348106213983929, 0.6227353307293972]
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 2], Rule[β, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[u, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
29.18.E7 d 2 u 3 d γ 2 + ( h - ν ⁢ ( ν + 1 ) ⁢ k 2 ⁢ sn 2 ⁡ ( γ , k ) ) ⁢ u 3 = 0 derivative subscript 𝑢 3 𝛾 2 ℎ 𝜈 𝜈 1 superscript 𝑘 2 Jacobi-elliptic-sn 2 𝛾 𝑘 subscript 𝑢 3 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}u_{3}}{{\mathrm{d}\gamma}^{2% }}+(h-\nu(\nu+1)k^{2}{\operatorname{sn}^{2}}\left(\gamma,k\right))u_{3}=0}}
\deriv[2]{u_{3}}{\gamma}+(h-\nu(\nu+1)k^{2}\Jacobiellsnk^{2}@{\gamma}{k})u_{3} = 0		 

diff(u[3], [gamma$(2)])+(h - nu*(nu + 1)*(k)^(2)* (JacobiSN(gamma, k))^(2))*u[3] = 0

D[Subscript[u, 3], {\[Gamma], 2}]+(h - \[Nu]*(\[Nu]+ 1)*(k)^(2)* (JacobiSN[\[Gamma], (k)^2])^(2))*Subscript[u, 3] == 0
Error Failure -
Failed [300 / 300]
Result: Complex[0.9359870178672973, -0.3879581414973573]
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[u, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.5826053037338313, -2.538844793552361]
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 2], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[u, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
29.18#Ex8 x = k ⁢ sn ⁡ ( α , k ) ⁢ sn ⁡ ( β , k ) ⁢ sn ⁡ ( γ , k ) 𝑥 𝑘 Jacobi-elliptic-sn 𝛼 𝑘 Jacobi-elliptic-sn 𝛽 𝑘 Jacobi-elliptic-sn 𝛾 𝑘 {\displaystyle{\displaystyle x=k\operatorname{sn}\left(\alpha,k\right)% \operatorname{sn}\left(\beta,k\right)\operatorname{sn}\left(\gamma,k\right)}}
x = k\Jacobiellsnk@{\alpha}{k}\Jacobiellsnk@{\beta}{k}\Jacobiellsnk@{\gamma}{k}		 

x = k*JacobiSN(alpha, k)*JacobiSN(beta, k)*JacobiSN(gamma, k)

x == k*JacobiSN[\[Alpha], (k)^2]*JacobiSN[\[Beta], (k)^2]*JacobiSN[\[Gamma], (k)^2]
Failure Failure
Failed [300 / 300]
Result: 1.073444110
Test Values: {alpha = 3/2, beta = 3/2, gamma = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1}


Result: 1.470871115
Test Values: {alpha = 3/2, beta = 3/2, gamma = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.8507896823681017, -0.1995472033956852]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[α, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.4556849214429664, 0.0010205356456730495]
Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[α, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
29.18#Ex9 y = - k k ′ ⁢ cn ⁡ ( α , k ) ⁢ cn ⁡ ( β , k ) ⁢ cn ⁡ ( γ , k ) 𝑦 𝑘 superscript 𝑘 ′ Jacobi-elliptic-cn 𝛼 𝑘 Jacobi-elliptic-cn 𝛽 𝑘 Jacobi-elliptic-cn 𝛾 𝑘 {\displaystyle{\displaystyle y=-\frac{k}{k^{\prime}}\operatorname{cn}\left(% \alpha,k\right)\operatorname{cn}\left(\beta,k\right)\operatorname{cn}\left(% \gamma,k\right)}}
y = -\frac{k}{k^{\prime}}\Jacobiellcnk@{\alpha}{k}\Jacobiellcnk@{\beta}{k}\Jacobiellcnk@{\gamma}{k}		 

y = -(k)/(sqrt(1 - (k)^(2)))*JacobiCN(alpha, k)*JacobiCN(beta, k)*JacobiCN(gamma, k)

y == -Divide[k,Sqrt[1 - (k)^(2)]]*JacobiCN[\[Alpha], (k)^2]*JacobiCN[\[Beta], (k)^2]*JacobiCN[\[Gamma], (k)^2]
Failure Failure
Failed [300 / 300]
Result: Float(infinity)
Test Values: {alpha = 3/2, beta = 3/2, gamma = 1/2*3^(1/2)+1/2*I, y = -3/2, k = 1}


Result: -1.500000000-.9993433457*I
Test Values: {alpha = 3/2, beta = 3/2, gamma = 1/2*3^(1/2)+1/2*I, y = -3/2, k = 2}

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


Result: Complex[-1.4837977605404604, -0.8196088589670207]
Test Values: {Rule[k, 2], Rule[y, -1.5], Rule[α, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
29.18#Ex10 z = i k ⁢ k ′ ⁢ dn ⁡ ( α , k ) ⁢ dn ⁡ ( β , k ) ⁢ dn ⁡ ( γ , k ) 𝑧 imaginary-unit 𝑘 superscript 𝑘 ′ Jacobi-elliptic-dn 𝛼 𝑘 Jacobi-elliptic-dn 𝛽 𝑘 Jacobi-elliptic-dn 𝛾 𝑘 {\displaystyle{\displaystyle z=\frac{\mathrm{i}}{kk^{\prime}}\operatorname{dn}% \left(\alpha,k\right)\operatorname{dn}\left(\beta,k\right)\operatorname{dn}% \left(\gamma,k\right)}}
z = \frac{\iunit}{kk^{\prime}}\Jacobielldnk@{\alpha}{k}\Jacobielldnk@{\beta}{k}\Jacobielldnk@{\gamma}{k}		 

z = (I)/(k*sqrt(1 - (k)^(2)))*JacobiDN(alpha, k)*JacobiDN(beta, k)*JacobiDN(gamma, k)

z == Divide[I,k*Sqrt[1 - (k)^(2)]]*JacobiDN[\[Alpha], (k)^2]*JacobiDN[\[Beta], (k)^2]*JacobiDN[\[Gamma], (k)^2]
Failure Failure
Failed [300 / 300]
Result: .8660254040-Float(infinity)*I
Test Values: {alpha = 3/2, beta = 3/2, gamma = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .7533782555+.5000000000*I
Test Values: {alpha = 3/2, beta = 3/2, gamma = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.8776189378612058, 0.7313924592922922]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
29.18.E10 u ⁢ ( α , β , γ ) = u 1 ⁢ ( α ) ⁢ u 2 ⁢ ( β ) ⁢ u 3 ⁢ ( γ ) 𝑢 𝛼 𝛽 𝛾 subscript 𝑢 1 𝛼 subscript 𝑢 2 𝛽 subscript 𝑢 3 𝛾 {\displaystyle{\displaystyle u(\alpha,\beta,\gamma)=u_{1}(\alpha)u_{2}(\beta)u% _{3}(\gamma)}}
u(\alpha,\beta,\gamma) = u_{1}(\alpha)u_{2}(\beta)u_{3}(\gamma)		 

u(alpha , beta , gamma) = u[1](alpha)* u[2](beta)* u[3](gamma)
 
u[\[Alpha], \[Beta], \[Gamma]] == Subscript[u, 1][\[Alpha]]* Subscript[u, 2][\[Beta]]* Subscript[u, 3][\[Gamma]]
 Skipped - no semantic math Skipped - no semantic math - -
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