View source for 29.18

{{DISPLAYTITLE:Lamé Functions - 29.18 Mathematical Applications}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/29.18#Ex1 29.18#Ex1] || <math qid="Q8798">x = kr\Jacobiellsnk@{\beta}{k}\Jacobiellsnk@{\gamma}{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x = kr\Jacobiellsnk@{\beta}{k}\Jacobiellsnk@{\gamma}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>x = k*r*JacobiSN(beta, k)*JacobiSN(gamma, k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>x == k*r*JacobiSN[\[Beta], (k)^2]*JacobiSN[\[Gamma], (k)^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/29.18#Ex2 29.18#Ex2] || <math qid="Q8799">y = \iunit\frac{k}{k^{\prime}}r\Jacobiellcnk@{\beta}{k}\Jacobiellcnk@{\gamma}{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>y = \iunit\frac{k}{k^{\prime}}r\Jacobiellcnk@{\beta}{k}\Jacobiellcnk@{\gamma}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>y = I*(k)/(sqrt(1 - (k)^(2)))*r*JacobiCN(beta, k)*JacobiCN(gamma, k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>y == I*Divide[k,Sqrt[1 - (k)^(2)]]*r*JacobiCN[\[Beta], (k)^2]*JacobiCN[\[Gamma], (k)^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/29.18#Ex3 29.18#Ex3] || <math qid="Q8800">z = \frac{1}{k^{\prime}}r\Jacobielldnk@{\beta}{k}\Jacobielldnk@{\gamma}{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z = \frac{1}{k^{\prime}}r\Jacobielldnk@{\beta}{k}\Jacobielldnk@{\gamma}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z = (1)/(sqrt(1 - (k)^(2)))*r*JacobiDN(beta, k)*JacobiDN(gamma, k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>z == Divide[1,Sqrt[1 - (k)^(2)]]*r*JacobiDN[\[Beta], (k)^2]*JacobiDN[\[Gamma], (k)^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/29.18#Ex4 29.18#Ex4] || <math qid="Q8801">r \geq 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>r \geq 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">r >= 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">r >= 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- 
| [https://dlmf.nist.gov/29.18#Ex7 29.18#Ex7] || <math qid="Q8804">0 \leq \gamma</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>0 \leq \gamma</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>0 <= gamma</syntaxhighlight> || <syntaxhighlight lang=mathematica>0 <= \[Gamma]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 0. <= -1.500000000
Test Values: {gamma = -3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 0. <= -.5000000000
Test Values: {gamma = -1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/29.18#Ex7 29.18#Ex7] || <math qid="Q8804">\gamma \leq 4\compellintKk@@{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\gamma \leq 4\compellintKk@@{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>gamma <= 4*EllipticK(k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Gamma] <= 4*EllipticK[(k)^2]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[]]
Test Values: {Rule[k, 1], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/29.18.E4 29.18.E4] || <math qid="Q8805">u(r,\beta,\gamma) = u_{1}(r)u_{2}(\beta)u_{3}(\gamma)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>u(r,\beta,\gamma) = u_{1}(r)u_{2}(\beta)u_{3}(\gamma)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">u(r , beta , gamma) = u[1](r)* u[2](beta)* u[3](gamma)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">u[r , \[Beta], \[Gamma]] == Subscript[u, 1][r]* Subscript[u, 2][\[Beta]]* Subscript[u, 3][\[Gamma]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- 
| [https://dlmf.nist.gov/29.18.E5 29.18.E5] || <math qid="Q8806">\deriv{}{r}\left(r^{2}\deriv{u_{1}}{r}\right)+(\omega^{2}r^{2}-\nu(\nu+1))u_{1} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{r}\left(r^{2}\deriv{u_{1}}{r}\right)+(\omega^{2}r^{2}-\nu(\nu+1))u_{1} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(((r)^(2)* diff(u[1], r))+((omega)^(2)* (r)^(2)- nu*(nu + 1))*u[1], r) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[((r)^(2)* D[Subscript[u, 1], r])+(\[Omega]^(2)* (r)^(2)- \[Nu]*(\[Nu]+ 1))*Subscript[u, 1], r] == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/29.18.E6 29.18.E6] || <math qid="Q8807">\deriv[2]{u_{2}}{\beta}+(h-\nu(\nu+1)k^{2}\Jacobiellsnk^{2}@{\beta}{k})u_{2} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{u_{2}}{\beta}+(h-\nu(\nu+1)k^{2}\Jacobiellsnk^{2}@{\beta}{k})u_{2} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(u[2], [beta$(2)])+(h - nu*(nu + 1)*(k)^(2)* (JacobiSN(beta, k))^(2))*u[2] = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Subscript[u, 2], {\[Beta], 2}]+(h - \[Nu]*(\[Nu]+ 1)*(k)^(2)* (JacobiSN[\[Beta], (k)^2])^(2))*Subscript[u, 2] == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/29.18.E7 29.18.E7] || <math qid="Q8808">\deriv[2]{u_{3}}{\gamma}+(h-\nu(\nu+1)k^{2}\Jacobiellsnk^{2}@{\gamma}{k})u_{3} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{u_{3}}{\gamma}+(h-\nu(\nu+1)k^{2}\Jacobiellsnk^{2}@{\gamma}{k})u_{3} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(u[3], [gamma$(2)])+(h - nu*(nu + 1)*(k)^(2)* (JacobiSN(gamma, k))^(2))*u[3] = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Subscript[u, 3], {\[Gamma], 2}]+(h - \[Nu]*(\[Nu]+ 1)*(k)^(2)* (JacobiSN[\[Gamma], (k)^2])^(2))*Subscript[u, 3] == 0</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/29.18#Ex8 29.18#Ex8] || <math qid="Q8809">x = k\Jacobiellsnk@{\alpha}{k}\Jacobiellsnk@{\beta}{k}\Jacobiellsnk@{\gamma}{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x = k\Jacobiellsnk@{\alpha}{k}\Jacobiellsnk@{\beta}{k}\Jacobiellsnk@{\gamma}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>x = k*JacobiSN(alpha, k)*JacobiSN(beta, k)*JacobiSN(gamma, k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>x == k*JacobiSN[\[Alpha], (k)^2]*JacobiSN[\[Beta], (k)^2]*JacobiSN[\[Gamma], (k)^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/29.18#Ex9 29.18#Ex9] || <math qid="Q8810">y = -\frac{k}{k^{\prime}}\Jacobiellcnk@{\alpha}{k}\Jacobiellcnk@{\beta}{k}\Jacobiellcnk@{\gamma}{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>y = -\frac{k}{k^{\prime}}\Jacobiellcnk@{\alpha}{k}\Jacobiellcnk@{\beta}{k}\Jacobiellcnk@{\gamma}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>y = -(k)/(sqrt(1 - (k)^(2)))*JacobiCN(alpha, k)*JacobiCN(beta, k)*JacobiCN(gamma, k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>y == -Divide[k,Sqrt[1 - (k)^(2)]]*JacobiCN[\[Alpha], (k)^2]*JacobiCN[\[Beta], (k)^2]*JacobiCN[\[Gamma], (k)^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/29.18#Ex10 29.18#Ex10] || <math qid="Q8811">z = \frac{\iunit}{kk^{\prime}}\Jacobielldnk@{\alpha}{k}\Jacobielldnk@{\beta}{k}\Jacobielldnk@{\gamma}{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z = \frac{\iunit}{kk^{\prime}}\Jacobielldnk@{\alpha}{k}\Jacobielldnk@{\beta}{k}\Jacobielldnk@{\gamma}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z = (I)/(k*sqrt(1 - (k)^(2)))*JacobiDN(alpha, k)*JacobiDN(beta, k)*JacobiDN(gamma, k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>z == Divide[I,k*Sqrt[1 - (k)^(2)]]*JacobiDN[\[Alpha], (k)^2]*JacobiDN[\[Beta], (k)^2]*JacobiDN[\[Gamma], (k)^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/29.18.E10 29.18.E10] || <math qid="Q8814">u(\alpha,\beta,\gamma) = u_{1}(\alpha)u_{2}(\beta)u_{3}(\gamma)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>u(\alpha,\beta,\gamma) = u_{1}(\alpha)u_{2}(\beta)u_{3}(\gamma)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">u(alpha , beta , gamma) = u[1](alpha)* u[2](beta)* u[3](gamma)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">u[\[Alpha], \[Beta], \[Gamma]] == Subscript[u, 1][\[Alpha]]* Subscript[u, 2][\[Beta]]* Subscript[u, 3][\[Gamma]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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