Lamé Functions - 29.2 Differential Equations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
29.2.E1 d 2 w d z 2 + ( h - ν ⁢ ( ν + 1 ) ⁢ k 2 ⁢ sn 2 ⁡ ( z , k ) ) ⁢ w = 0 derivative 𝑤 𝑧 2 ℎ 𝜈 𝜈 1 superscript 𝑘 2 Jacobi-elliptic-sn 2 𝑧 𝑘 𝑤 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+(h-\nu% (\nu+1)k^{2}{\operatorname{sn}^{2}}\left(z,k\right))w=0}}
\deriv[2]{w}{z}+(h-\nu(\nu+1)k^{2}\Jacobiellsnk^{2}@{z}{k})w = 0		 

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

D[w, {z, 2}]+(h - \[Nu]*(\[Nu]+ 1)*(k)^(2)* (JacobiSN[z, (k)^2])^(2))*w == 0
Failure Failure
Failed [300 / 300]
Result: .9359870183-.3879581426*I
Test Values: {h = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.5826053060-2.538844794*I
Test Values: {h = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 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: Complex[0.9359870178672973, -0.3879581414973573]
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 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[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
29.2.E2 d 2 w d ξ 2 + 1 2 ⁢ ( 1 ξ + 1 ξ - 1 + 1 ξ - k - 2 ) ⁢ d w d ξ + h ⁢ k - 2 - ν ⁢ ( ν + 1 ) ⁢ ξ 4 ⁢ ξ ⁢ ( ξ - 1 ) ⁢ ( ξ - k - 2 ) ⁢ w = 0 derivative 𝑤 𝜉 2 1 2 1 𝜉 1 𝜉 1 1 𝜉 superscript 𝑘 2 derivative 𝑤 𝜉 ℎ superscript 𝑘 2 𝜈 𝜈 1 𝜉 4 𝜉 𝜉 1 𝜉 superscript 𝑘 2 𝑤 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\xi}^{2}}+% \frac{1}{2}\left(\frac{1}{\xi}+\frac{1}{\xi-1}+\frac{1}{\xi-k^{-2}}\right)% \frac{\mathrm{d}w}{\mathrm{d}\xi}+\frac{hk^{-2}-\nu(\nu+1)\xi}{4\xi(\xi-1)(\xi% -k^{-2})}w=0}}
\deriv[2]{w}{\xi}+\frac{1}{2}\left(\frac{1}{\xi}+\frac{1}{\xi-1}+\frac{1}{\xi-k^{-2}}\right)\deriv{w}{\xi}+\frac{hk^{-2}-\nu(\nu+1)\xi}{4\xi(\xi-1)(\xi-k^{-2})}w = 0		 

subs( temp=((JacobiSN(z, k))^(2)), diff( w, temp$(2) ) )+(1)/(2)*((1)/((JacobiSN(z, k))^(2))+(1)/(((JacobiSN(z, k))^(2))- 1)+(1)/(((JacobiSN(z, k))^(2))- (k)^(- 2)))*subs( temp=((JacobiSN(z, k))^(2)), diff( w, temp$(1) ) )+(h*(k)^(- 2)- nu*(nu + 1)*((JacobiSN(z, k))^(2)))/(4*((JacobiSN(z, k))^(2))*(((JacobiSN(z, k))^(2))- 1)*(((JacobiSN(z, k))^(2))- (k)^(- 2)))*w = 0

(D[w, {temp, 2}]/.temp-> ((JacobiSN[z, (k)^2])^(2)))+Divide[1,2]*(Divide[1,(JacobiSN[z, (k)^2])^(2)]+Divide[1,((JacobiSN[z, (k)^2])^(2))- 1]+Divide[1,((JacobiSN[z, (k)^2])^(2))- (k)^(- 2)])*(D[w, {temp, 1}]/.temp-> ((JacobiSN[z, (k)^2])^(2)))+Divide[h*(k)^(- 2)- \[Nu]*(\[Nu]+ 1)*((JacobiSN[z, (k)^2])^(2)),4*((JacobiSN[z, (k)^2])^(2))*(((JacobiSN[z, (k)^2])^(2))- 1)*(((JacobiSN[z, (k)^2])^(2))- (k)^(- 2))]*w == 0
Failure Failure
Failed [300 / 300]
Result: .9804044245+.4985385652*I
Test Values: {h = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .3643094905+3.048781532*I
Test Values: {h = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 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: Complex[0.9804044230224559, 0.49853856488927895]
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.36430949083593944, 3.048781532678858]
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
29.2.E4 ( 1 - k 2 ⁢ cos 2 ⁡ ϕ ) ⁢ d 2 w d ϕ 2 + k 2 ⁢ cos ⁡ ϕ ⁢ sin ⁡ ϕ ⁢ d w d ϕ + ( h - ν ⁢ ( ν + 1 ) ⁢ k 2 ⁢ cos 2 ⁡ ϕ ) ⁢ w = 0 1 superscript 𝑘 2 2 italic-ϕ derivative 𝑤 italic-ϕ 2 superscript 𝑘 2 italic-ϕ italic-ϕ derivative 𝑤 italic-ϕ ℎ 𝜈 𝜈 1 superscript 𝑘 2 2 italic-ϕ 𝑤 0 {\displaystyle{\displaystyle(1-k^{2}{\cos^{2}}\phi)\frac{{\mathrm{d}}^{2}w}{{% \mathrm{d}\phi}^{2}}+k^{2}\cos\phi\sin\phi\frac{\mathrm{d}w}{\mathrm{d}\phi}+(% h-\nu(\nu+1)k^{2}{\cos^{2}}\phi)w=0}}
(1-k^{2}\cos^{2}@@{\phi})\deriv[2]{w}{\phi}+k^{2}\cos@@{\phi}\sin@@{\phi}\deriv{w}{\phi}+(h-\nu(\nu+1)k^{2}\cos^{2}@@{\phi})w = 0		 

(1 - (k)^(2)* (cos((1)/(2)*Pi - JacobiAM(z, k)))^(2))*subs( temp=((1)/(2)*Pi - JacobiAM(z, k)), diff( w, temp$(2) ) )+ (k)^(2)* cos((1)/(2)*Pi - JacobiAM(z, k))*sin((1)/(2)*Pi - JacobiAM(z, k))*subs( temp=((1)/(2)*Pi - JacobiAM(z, k)), diff( w, temp$(1) ) )+(h - nu*(nu + 1)*(k)^(2)* (cos((1)/(2)*Pi - JacobiAM(z, k)))^(2))*w = 0

(1 - (k)^(2)* (Cos[Divide[1,2]*Pi - JacobiAmplitude[z, Power[k, 2]]])^(2))*(D[w, {temp, 2}]/.temp-> (Divide[1,2]*Pi - JacobiAmplitude[z, Power[k, 2]]))+ (k)^(2)* Cos[Divide[1,2]*Pi - JacobiAmplitude[z, Power[k, 2]]]*Sin[Divide[1,2]*Pi - JacobiAmplitude[z, Power[k, 2]]]*(D[w, {temp, 1}]/.temp-> (Divide[1,2]*Pi - JacobiAmplitude[z, Power[k, 2]]))+(h - \[Nu]*(\[Nu]+ 1)*(k)^(2)* (Cos[Divide[1,2]*Pi - JacobiAmplitude[z, Power[k, 2]]])^(2))*w == 0
Failure Failure
Failed [300 / 300]
Result: .9359870183-.3879581426*I
Test Values: {h = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.5826053060-2.538844794*I
Test Values: {h = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 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: Complex[0.09035331946182407, -0.66279682113597]
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, Rational[3, 2]], Rule[z, Rational[3, 2]], Rule[ν, 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[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, Rational[3, 2]], Rule[z, Rational[3, 2]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
29.2#Ex1 e 1 + e 2 + e 3 = 0 subscript 𝑒 1 subscript 𝑒 2 subscript 𝑒 3 0 {\displaystyle{\displaystyle e_{1}+e_{2}+e_{3}=0}}
e_{1}+e_{2}+e_{3} = 0		 

e[1]+ e[2]+ e[3] = 0
 
Subscript[e, 1]+ Subscript[e, 2]+ Subscript[e, 3] == 0
 Skipped - no semantic math Skipped - no semantic math - -
29.2#Ex2 ( e 2 - e 3 ) / ( e 1 - e 3 ) = k 2 subscript 𝑒 2 subscript 𝑒 3 subscript 𝑒 1 subscript 𝑒 3 superscript 𝑘 2 {\displaystyle{\displaystyle\ifrac{(e_{2}-e_{3})}{(e_{1}-e_{3})}=k^{2}}}
\ifrac{(e_{2}-e_{3})}{(e_{1}-e_{3})} = k^{2}		 

(e[2]- e[3])/(e[1]- e[3]) = (k)^(2)
 
Divide[Subscript[e, 2]- Subscript[e, 3],Subscript[e, 1]- Subscript[e, 3]] == (k)^(2)
 Skipped - no semantic math Skipped - no semantic math - -
29.2.E10 d 2 w d ζ 2 + 1 2 ⁢ ( 1 ζ - e 1 + 1 ζ - e 2 + 1 ζ - e 3 ) ⁢ d w d ζ + g - ν ⁢ ( ν + 1 ) ⁢ ζ 4 ⁢ ( ζ - e 1 ) ⁢ ( ζ - e 2 ) ⁢ ( ζ - e 3 ) ⁢ w = 0 derivative 𝑤 𝜁 2 1 2 1 𝜁 subscript 𝑒 1 1 𝜁 subscript 𝑒 2 1 𝜁 subscript 𝑒 3 derivative 𝑤 𝜁 𝑔 𝜈 𝜈 1 𝜁 4 𝜁 subscript 𝑒 1 𝜁 subscript 𝑒 2 𝜁 subscript 𝑒 3 𝑤 0 {\displaystyle{\displaystyle{\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\zeta}^{2}}+% \frac{1}{2}\left(\frac{1}{\zeta-e_{1}}+\frac{1}{\zeta-e_{2}}+\frac{1}{\zeta-e_% {3}}\right)\frac{\mathrm{d}w}{\mathrm{d}\zeta}}+\frac{g-\nu(\nu+1)\zeta}{4(% \zeta-e_{1})(\zeta-e_{2})(\zeta-e_{3})}w=0}}
{\deriv[2]{w}{\zeta}+\frac{1}{2}\left(\frac{1}{\zeta-e_{1}}+\frac{1}{\zeta-e_{2}}+\frac{1}{\zeta-e_{3}}\right)\deriv{w}{\zeta}}+\frac{g-\nu(\nu+1)\zeta}{4(\zeta-e_{1})(\zeta-e_{2})(\zeta-e_{3})}w = 0		 

subs( temp=(WeierstrassP((e[1]- e[3])^(- 1/2)*(z - I*EllipticCK(k)), g[2], 4*e[1]*e[2]*e[3])), diff( w, temp$(2) ) )+(1)/(2)*((1)/((WeierstrassP((e[1]- e[3])^(- 1/2)*(z - I*EllipticCK(k)), g[2], 4*e[1]*e[2]*e[3]))- e[1])+(1)/((WeierstrassP((e[1]- e[3])^(- 1/2)*(z - I*EllipticCK(k)), g[2], 4*e[1]*e[2]*e[3]))- e[2])+(1)/((WeierstrassP((e[1]- e[3])^(- 1/2)*(z - I*EllipticCK(k)), g[2], 4*e[1]*e[2]*e[3]))- e[3]))*subs( temp=(WeierstrassP((e[1]- e[3])^(- 1/2)*(z - I*EllipticCK(k)), g[2], 4*e[1]*e[2]*e[3])), diff( w, temp$(1) ) )+(((e[1]- e[3])*h + nu*(nu + 1)*e[3])- nu*(nu + 1)*(WeierstrassP((e[1]- e[3])^(- 1/2)*(z - I*EllipticCK(k)), g[2], 4*e[1]*e[2]*e[3])))/(4*((WeierstrassP((e[1]- e[3])^(- 1/2)*(z - I*EllipticCK(k)), g[2], 4*e[1]*e[2]*e[3]))- e[1])*((WeierstrassP((e[1]- e[3])^(- 1/2)*(z - I*EllipticCK(k)), g[2], 4*e[1]*e[2]*e[3]))- e[2])*((WeierstrassP((e[1]- e[3])^(- 1/2)*(z - I*EllipticCK(k)), g[2], 4*e[1]*e[2]*e[3]))- e[3]))*w = 0

Error
Failure Missing Macro Error Error -
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