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| <div style="width: 100%; height: 75vh; overflow: auto;"> | | <div style="-moz-column-count:2; column-count:2;"> |
| {| class="wikitable sortable" style="margin: 0;"
| | ; Notation : [[29.1|29.1 Special Notation]]<br> |
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| | ; Lamé Functions : [[29.2|29.2 Differential Equations]]<br>[[29.3|29.3 Definitions and Basic Properties]]<br>[[29.4|29.4 Graphics]]<br>[[29.5|29.5 Special Cases and Limiting Forms]]<br>[[29.6|29.6 Fourier Series]]<br>[[29.7|29.7 Asymptotic Expansions]]<br>[[29.8|29.8 Integral Equations]]<br>[[29.9|29.9 Stability]]<br>[[29.10|29.10 Lamé Functions with Imaginary Periods]]<br>[[29.11|29.11 Lamé Wave Equation]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | DLMF
| | ; Lamé Polynomials : [[29.12|29.12 Definitions]]<br>[[29.13|29.13 Graphics]]<br>[[29.14|29.14 Orthogonality]]<br>[[29.15|29.15 Fourier Series and Chebyshev Series]]<br>[[29.16|29.16 Asymptotic Expansions]]<br>[[29.17|29.17 Other Solutions]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | Formula
| | ; Applications : [[29.18|29.18 Mathematical Applications]]<br>[[29.19|29.19 Physical Applications]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | Constraints
| | ; Computation : [[29.20|29.20 Methods of Computation]]<br>[[29.21|29.21 Tables]]<br>[[29.22|29.22 Software]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | Maple
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| ! scope="col" style="position: sticky; top: 0;" | Mathematica
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| ! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
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| ! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
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| ! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
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| ! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| | [https://dlmf.nist.gov/29.2.E1 29.2.E1] || [[Item:Q8574|<math>\deriv[2]{w}{z}+(h-\nu(\nu+1)k^{2}\Jacobiellsnk^{2}@{z}{k})w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{z}+(h-\nu(\nu+1)k^{2}\Jacobiellsnk^{2}@{z}{k})w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [z$(2)])+(h - nu*(nu + 1)*(k)^(2)* (JacobiSN(z, k))^(2))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 2}]+(h - \[Nu]*(\[Nu]+ 1)*(k)^(2)* (JacobiSN[z, (k)^2])^(2))*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9359870183-.3879581426*I
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| 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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5826053060-2.538844794*I
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| 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}</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[0.9359870178672973, -0.3879581414973573]
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| 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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.5826053037338313, -2.538844793552361]
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| 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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/29.2.E2 29.2.E2] || [[Item:Q8575|<math>\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</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || <syntaxhighlight lang=mathematica>(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</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .3643094905+3.048781532*I
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| 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}</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[0.9804044230224559, 0.49853856488927895]
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| 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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.36430949083593944, 3.048781532678858]
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| 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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/29.2.E4 29.2.E4] || [[Item:Q8577|<math>(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</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(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</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(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</syntaxhighlight> || <syntaxhighlight lang=mathematica>(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</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</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[0.09035331946182407, -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]]]}</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[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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.2#Ex1 29.2#Ex1] || [[Item:Q8579|<math>e_{1}+e_{2}+e_{3} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>e_{1}+e_{2}+e_{3} = 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">e[1]+ e[2]+ e[3] = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[e, 1]+ Subscript[e, 2]+ Subscript[e, 3] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.2#Ex2 29.2#Ex2] || [[Item:Q8580|<math>\ifrac{(e_{2}-e_{3})}{(e_{1}-e_{3})} = k^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\ifrac{(e_{2}-e_{3})}{(e_{1}-e_{3})} = k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(e[2]- e[3])/(e[1]- e[3]) = (k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Divide[Subscript[e, 2]- Subscript[e, 3],Subscript[e, 1]- Subscript[e, 3]] == (k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/29.2.E10 29.2.E10] || [[Item:Q8584|<math>{\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</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\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</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Failure || Missing Macro Error || Error || -
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| | [https://dlmf.nist.gov/29.3#Ex3 29.3#Ex3] || [[Item:Q8602|<math>\alpha_{p} = \tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\alpha_{p} = \tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">alpha[p] = (1)/(2)*(nu - 2*p - 2)*(nu + 2*p + 3)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Alpha], p] == Divide[1,2]*(\[Nu]- 2*p - 2)*(\[Nu]+ 2*p + 3)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/29.3#Ex4 29.3#Ex4] || [[Item:Q8603|<math>\gamma_{p} = \tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{p} = \tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p + 1)*(nu + 2*p)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p + 1)*(\[Nu]+ 2*p)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/29.3#Ex5 29.3#Ex5] || [[Item:Q8605|<math>\alpha_{p} = \tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\alpha_{p} = \tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">alpha[p] = (1)/(2)*(nu - 2*p - 2)*(nu + 2*p + 3)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Alpha], p] == Divide[1,2]*(\[Nu]- 2*p - 2)*(\[Nu]+ 2*p + 3)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.3#Ex6 29.3#Ex6] || [[Item:Q8606|<math>\gamma_{p} = \tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{p} = \tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p + 1)*(nu + 2*p)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p + 1)*(\[Nu]+ 2*p)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/29.3#Ex7 29.3#Ex7] || [[Item:Q8607|<math>\alpha_{p} = \tfrac{1}{2}(\nu-2p-3)(\nu+2p+4)k^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\alpha_{p} = \tfrac{1}{2}(\nu-2p-3)(\nu+2p+4)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">alpha[p] = (1)/(2)*(nu - 2*p - 3)*(nu + 2*p + 4)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Alpha], p] == Divide[1,2]*(\[Nu]- 2*p - 3)*(\[Nu]+ 2*p + 4)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/29.3#Ex8 29.3#Ex8] || [[Item:Q8608|<math>\beta_{p} = (2p+2)^{2}(2-k^{2})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\beta_{p} = (2p+2)^{2}(2-k^{2})</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">beta[p] = (2*p + 2)^(2)*(2 - (k)^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Beta], p] == (2*p + 2)^(2)*(2 - (k)^(2))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/29.3#Ex9 29.3#Ex9] || [[Item:Q8609|<math>\gamma_{p} = \tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{p} = \tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p)*(nu + 2*p + 1)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p)*(\[Nu]+ 2*p + 1)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.6.E5 29.6.E5] || [[Item:Q8631|<math>\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{\infty}A_{2p}^{2} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{\infty}A_{2p}^{2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1)/(2)*(A[0])^(2)+ sum((A[2*p])^(2), p = 1..infinity) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Divide[1,2]*(Subscript[A, 0])^(2)+ Sum[(Subscript[A, 2*p])^(2), {p, 1, Infinity}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E6 29.6.E6] || [[Item:Q8632|<math>\tfrac{1}{2}A_{0}+\sum_{p=1}^{\infty}A_{2p} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\tfrac{1}{2}A_{0}+\sum_{p=1}^{\infty}A_{2p} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1)/(2)*A[0]+ sum(A[2*p], p = 1..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Divide[1,2]*Subscript[A, 0]+ Sum[Subscript[A, 2*p], {p, 1, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E7 29.6.E7] || [[Item:Q8633|<math>\lim_{p\to\infty}\frac{A_{2p+2}}{A_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{A_{2p+2}}{A_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n, \nu = 2n, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((A[2*p + 2])/(A[2*p]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[A, 2*p + 2],Subscript[A, 2*p]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6#Ex2 29.6#Ex2] || [[Item:Q8638|<math>\beta_{p} = 4p^{2}(2-k^{2})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\beta_{p} = 4p^{2}(2-k^{2})</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">beta[p] = 4*(p)^(2)*(2 - (k)^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Beta], p] == 4*(p)^(2)*(2 - (k)^(2))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6#Ex3 29.6#Ex3] || [[Item:Q8639|<math>\gamma_{p} = \tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{p} = \tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p + 1)*(nu + 2*p)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p + 1)*(\[Nu]+ 2*p)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E12 29.6.E12] || [[Item:Q8640|<math>\left(1-\tfrac{1}{2}k^{2}\right)\left(\tfrac{1}{2}C_{0}^{2}+\sum_{p=1}^{\infty}C_{2p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{\infty}C_{2p}C_{2p+2} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\left(1-\tfrac{1}{2}k^{2}\right)\left(\tfrac{1}{2}C_{0}^{2}+\sum_{p=1}^{\infty}C_{2p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{\infty}C_{2p}C_{2p+2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1 -(1)/(2)*(k)^(2))*((1)/(2)*(C[0])^(2)+ sum((C[2*p])^(2), p = 1..infinity))-(1)/(2)*(k)^(2)* sum(C[2*p]*C[2*p + 2], p = 0..infinity) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1 -Divide[1,2]*(k)^(2))*(Divide[1,2]*(Subscript[C, 0])^(2)+ Sum[(Subscript[C, 2*p])^(2), {p, 1, Infinity}, GenerateConditions->None])-Divide[1,2]*(k)^(2)* Sum[Subscript[C, 2*p]*Subscript[C, 2*p + 2], {p, 0, Infinity}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E13 29.6.E13] || [[Item:Q8641|<math>\tfrac{1}{2}C_{0}+\sum_{p=1}^{\infty}C_{2p} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\tfrac{1}{2}C_{0}+\sum_{p=1}^{\infty}C_{2p} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1)/(2)*C[0]+ sum(C[2*p], p = 1..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Divide[1,2]*Subscript[C, 0]+ Sum[Subscript[C, 2*p], {p, 1, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E14 29.6.E14] || [[Item:Q8642|<math>\lim_{p\to\infty}\frac{C_{2p+2}}{C_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{C_{2p+2}}{C_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n+1, \nu = 2n+1, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((C[2*p + 2])/(C[2*p]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[C, 2*p + 2],Subscript[C, 2*p]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E20 29.6.E20] || [[Item:Q8648|<math>\sum_{p=0}^{\infty}A_{2p+1}^{2} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}A_{2p+1}^{2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((A[2*p + 1])^(2), p = 0..infinity) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(Subscript[A, 2*p + 1])^(2), {p, 0, Infinity}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E21 29.6.E21] || [[Item:Q8649|<math>\sum_{p=0}^{\infty}A_{2p+1} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}A_{2p+1} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum(A[2*p + 1], p = 0..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[Subscript[A, 2*p + 1], {p, 0, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E22 29.6.E22] || [[Item:Q8650|<math>\lim_{p\to\infty}\frac{A_{2p+1}}{A_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{A_{2p+1}}{A_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n+1, \nu = 2n+1, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((A[2*p + 1])/(A[2*p - 1]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[A, 2*p + 1],Subscript[A, 2*p - 1]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6#Ex4 29.6#Ex4] || [[Item:Q8654|<math>\alpha_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\alpha_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">alpha[p] = (1)/(2)*(nu - 2*p - 1)*(nu + 2*p + 2)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Alpha], p] == Divide[1,2]*(\[Nu]- 2*p - 1)*(\[Nu]+ 2*p + 2)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6#Ex6 29.6#Ex6] || [[Item:Q8656|<math>\gamma_{p} = \tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{p} = \tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p)*(nu + 2*p + 1)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p)*(\[Nu]+ 2*p + 1)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E28 29.6.E28] || [[Item:Q8658|<math>\sum_{p=0}^{\infty}C_{2p+1} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}C_{2p+1} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum(C[2*p + 1], p = 0..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[Subscript[C, 2*p + 1], {p, 0, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E29 29.6.E29] || [[Item:Q8659|<math>\lim_{p\to\infty}\frac{C_{2p+1}}{C_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{C_{2p+1}}{C_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n+2, \nu = 2n+2, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((C[2*p + 1])/(C[2*p - 1]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[C, 2*p + 1],Subscript[C, 2*p - 1]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E35 29.6.E35] || [[Item:Q8665|<math>\sum_{p=0}^{\infty}B_{2p+1}^{2} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}B_{2p+1}^{2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((B[2*p + 1])^(2), p = 0..infinity) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(Subscript[B, 2*p + 1])^(2), {p, 0, Infinity}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E36 29.6.E36] || [[Item:Q8666|<math>\sum_{p=0}^{\infty}(2p+1)B_{2p+1} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}(2p+1)B_{2p+1} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((2*p + 1)*B[2*p + 1], p = 0..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(2*p + 1)*Subscript[B, 2*p + 1], {p, 0, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E37 29.6.E37] || [[Item:Q8667|<math>\lim_{p\to\infty}\frac{B_{2p+1}}{B_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{B_{2p+1}}{B_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n+1, \nu = 2n+1, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((B[2*p + 1])/(B[2*p - 1]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[B, 2*p + 1],Subscript[B, 2*p - 1]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6#Ex7 29.6#Ex7] || [[Item:Q8671|<math>\alpha_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\alpha_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">alpha[p] = (1)/(2)*(nu - 2*p - 1)*(nu + 2*p + 2)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Alpha], p] == Divide[1,2]*(\[Nu]- 2*p - 1)*(\[Nu]+ 2*p + 2)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6#Ex9 29.6#Ex9] || [[Item:Q8673|<math>\gamma_{p} = \tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{p} = \tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p)*(nu + 2*p + 1)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p)*(\[Nu]+ 2*p + 1)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E43 29.6.E43] || [[Item:Q8675|<math>\sum_{p=0}^{\infty}(2p+1)D_{2p+1} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}(2p+1)D_{2p+1} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((2*p + 1)*D[2*p + 1], p = 0..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(2*p + 1)*Subscript[D, 2*p + 1], {p, 0, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E44 29.6.E44] || [[Item:Q8676|<math>\lim_{p\to\infty}\frac{D_{2p+1}}{D_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{D_{2p+1}}{D_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n+2, \nu = 2n+2, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((D[2*p + 1])/(D[2*p - 1]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[D, 2*p + 1],Subscript[D, 2*p - 1]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E50 29.6.E50] || [[Item:Q8682|<math>\sum_{p=1}^{\infty}B_{2p}^{2} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=1}^{\infty}B_{2p}^{2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((B[2*p])^(2), p = 1..infinity) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(Subscript[B, 2*p])^(2), {p, 1, Infinity}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.6.E51 29.6.E51] || [[Item:Q8683|<math>\sum_{p=0}^{\infty}(2p+2)B_{2p+2} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}(2p+2)B_{2p+2} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((2*p + 2)*B[2*p + 2], p = 0..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(2*p + 2)*Subscript[B, 2*p + 2], {p, 0, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.6.E52 29.6.E52] || [[Item:Q8684|<math>\lim_{p\to\infty}\frac{B_{2p+2}}{B_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{B_{2p+2}}{B_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n+2, \nu = 2n+2, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((B[2*p + 2])/(B[2*p]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[B, 2*p + 2],Subscript[B, 2*p]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.6#Ex10 29.6#Ex10] || [[Item:Q8688|<math>\alpha_{p} = \tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\alpha_{p} = \tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">alpha[p] = (1)/(2)*(nu - 2*p - 2)*(nu + 2*p + 3)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Alpha], p] == Divide[1,2]*(\[Nu]- 2*p - 2)*(\[Nu]+ 2*p + 3)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.6#Ex11 29.6#Ex11] || [[Item:Q8689|<math>\beta_{p} = (2p+2)^{2}(2-k^{2})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\beta_{p} = (2p+2)^{2}(2-k^{2})</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">beta[p] = (2*p + 2)^(2)*(2 - (k)^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Beta], p] == (2*p + 2)^(2)*(2 - (k)^(2))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.6#Ex12 29.6#Ex12] || [[Item:Q8690|<math>\gamma_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p - 1)*(nu + 2*p + 2)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p - 1)*(\[Nu]+ 2*p + 2)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.6.E57 29.6.E57] || [[Item:Q8691|<math>\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=1}^{\infty}D_{2p}^{2}-\tfrac{1}{2}k^{2}\sum_{p=1}^{\infty}D_{2p}D_{2p+2} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=1}^{\infty}D_{2p}^{2}-\tfrac{1}{2}k^{2}\sum_{p=1}^{\infty}D_{2p}D_{2p+2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1 -(1)/(2)*(k)^(2))*sum((D[2*p])^(2), p = 1..infinity)-(1)/(2)*(k)^(2)* sum(D[2*p]*D[2*p + 2], p = 1..infinity) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1 -Divide[1,2]*(k)^(2))*Sum[(Subscript[D, 2*p])^(2), {p, 1, Infinity}, GenerateConditions->None]-Divide[1,2]*(k)^(2)* Sum[Subscript[D, 2*p]*Subscript[D, 2*p + 2], {p, 1, Infinity}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.6.E58 29.6.E58] || [[Item:Q8692|<math>\sum_{p=0}^{\infty}(2p+2)D_{2p+2} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}(2p+2)D_{2p+2} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((2*p + 2)*D[2*p + 2], p = 0..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(2*p + 2)*Subscript[D, 2*p + 2], {p, 0, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.6.E59 29.6.E59] || [[Item:Q8693|<math>\lim_{p\to\infty}\frac{D_{2p+2}}{D_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{D_{2p+2}}{D_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n+3, \nu = 2n+3, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((D[2*p + 2])/(D[2*p]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[D, 2*p + 2],Subscript[D, 2*p]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.7.E3 29.7.E3] || [[Item:Q8698|<math>\tau_{0} = \frac{1}{2^{3}}(1+k^{2})(1+p^{2})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\tau_{0} = \frac{1}{2^{3}}(1+k^{2})(1+p^{2})</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">tau[0] = (1)/((2)^(3))*(1 + (k)^(2))*(1 + (p)^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Tau], 0] == Divide[1,(2)^(3)]*(1 + (k)^(2))*(1 + (p)^(2))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.7.E4 29.7.E4] || [[Item:Q8699|<math>\tau_{1} = \frac{p}{2^{6}}((1+k^{2})^{2}(p^{2}+3)-4k^{2}(p^{2}+5))</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\tau_{1} = \frac{p}{2^{6}}((1+k^{2})^{2}(p^{2}+3)-4k^{2}(p^{2}+5))</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">tau[1] = (p)/((2)^(6))*((1 + (k)^(2))^(2)*((p)^(2)+ 3)- 4*(k)^(2)*((p)^(2)+ 5))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Tau], 1] == Divide[p,(2)^(6)]*((1 + (k)^(2))^(2)*((p)^(2)+ 3)- 4*(k)^(2)*((p)^(2)+ 5))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.7.E6 29.7.E6] || [[Item:Q8701|<math>\tau_{2} = \frac{1}{2^{10}}(1+k^{2})(1-k^{2})^{2}(5p^{4}+34p^{2}+9)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\tau_{2} = \frac{1}{2^{10}}(1+k^{2})(1-k^{2})^{2}(5p^{4}+34p^{2}+9)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">tau[2] = (1)/((2)^(10))*(1 + (k)^(2))*(1 - (k)^(2))^(2)*(5*(p)^(4)+ 34*(p)^(2)+ 9)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Tau], 2] == Divide[1,(2)^(10)]*(1 + (k)^(2))*(1 - (k)^(2))^(2)*(5*(p)^(4)+ 34*(p)^(2)+ 9)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.7.E7 29.7.E7] || [[Item:Q8702|<math>\tau_{3} = \frac{p}{2^{14}}((1+k^{2})^{4}(33p^{4}+410p^{2}+405)-24k^{2}(1+k^{2})^{2}(7p^{4}+90p^{2}+95)+16k^{4}(9p^{4}+130p^{2}+173))</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\tau_{3} = \frac{p}{2^{14}}((1+k^{2})^{4}(33p^{4}+410p^{2}+405)-24k^{2}(1+k^{2})^{2}(7p^{4}+90p^{2}+95)+16k^{4}(9p^{4}+130p^{2}+173))</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">tau[3] = (p)/((2)^(14))*((1 + (k)^(2))^(4)*(33*(p)^(4)+ 410*(p)^(2)+ 405)- 24*(k)^(2)*(1 + (k)^(2))^(2)*(7*(p)^(4)+ 90*(p)^(2)+ 95)+ 16*(k)^(4)*(9*(p)^(4)+ 130*(p)^(2)+ 173))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Tau], 3] == Divide[p,(2)^(14)]*((1 + (k)^(2))^(4)*(33*(p)^(4)+ 410*(p)^(2)+ 405)- 24*(k)^(2)*(1 + (k)^(2))^(2)*(7*(p)^(4)+ 90*(p)^(2)+ 95)+ 16*(k)^(4)*(9*(p)^(4)+ 130*(p)^(2)+ 173))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.7.E8 29.7.E8] || [[Item:Q8703|<math>\tau_{4} = \frac{1}{2^{16}}((1+k^{2})^{5}(63p^{6}+1260p^{4}+2943p^{2}+486)-8k^{2}(1+k^{2})^{3}(49p^{6}+1010p^{4}+2493p^{2}+432)+16k^{4}(1+k^{2})(35p^{6}+760p^{4}+2043p^{2}+378))</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\tau_{4} = \frac{1}{2^{16}}((1+k^{2})^{5}(63p^{6}+1260p^{4}+2943p^{2}+486)-8k^{2}(1+k^{2})^{3}(49p^{6}+1010p^{4}+2493p^{2}+432)+16k^{4}(1+k^{2})(35p^{6}+760p^{4}+2043p^{2}+378))</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">tau[4] = (1)/((2)^(16))*((1 + (k)^(2))^(5)*(63*(p)^(6)+ 1260*(p)^(4)+ 2943*(p)^(2)+ 486)- 8*(k)^(2)*(1 + (k)^(2))^(3)*(49*(p)^(6)+ 1010*(p)^(4)+ 2493*(p)^(2)+ 432)+ 16*(k)^(4)*(1 + (k)^(2))*(35*(p)^(6)+ 760*(p)^(4)+ 2043*(p)^(2)+ 378))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Tau], 4] == Divide[1,(2)^(16)]*((1 + (k)^(2))^(5)*(63*(p)^(6)+ 1260*(p)^(4)+ 2943*(p)^(2)+ 486)- 8*(k)^(2)*(1 + (k)^(2))^(3)*(49*(p)^(6)+ 1010*(p)^(4)+ 2493*(p)^(2)+ 432)+ 16*(k)^(4)*(1 + (k)^(2))*(35*(p)^(6)+ 760*(p)^(4)+ 2043*(p)^(2)+ 378))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/29.8.E1 29.8.E1] || [[Item:Q8704|<math>x = k^{2}\Jacobiellsnk@{z}{k}\Jacobiellsnk@{z_{1}}{k}\Jacobiellsnk@{z_{2}}{k}\Jacobiellsnk@{z_{3}}{k}-\frac{k^{2}}{{k^{\prime}}^{2}}\Jacobiellcnk@{z}{k}\Jacobiellcnk@{z_{1}}{k}\Jacobiellcnk@{z_{2}}{k}\Jacobiellcnk@{z_{3}}{k}+\frac{1}{{k^{\prime}}^{2}}\Jacobielldnk@{z}{k}\Jacobielldnk@{z_{1}}{k}\Jacobielldnk@{z_{2}}{k}\Jacobielldnk@{z_{3}}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x = k^{2}\Jacobiellsnk@{z}{k}\Jacobiellsnk@{z_{1}}{k}\Jacobiellsnk@{z_{2}}{k}\Jacobiellsnk@{z_{3}}{k}-\frac{k^{2}}{{k^{\prime}}^{2}}\Jacobiellcnk@{z}{k}\Jacobiellcnk@{z_{1}}{k}\Jacobiellcnk@{z_{2}}{k}\Jacobiellcnk@{z_{3}}{k}+\frac{1}{{k^{\prime}}^{2}}\Jacobielldnk@{z}{k}\Jacobielldnk@{z_{1}}{k}\Jacobielldnk@{z_{2}}{k}\Jacobielldnk@{z_{3}}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>x = (k)^(2)* JacobiSN(x + y*I, k)*JacobiSN(x + y*I[1], k)*JacobiSN(x + y*I[2], k)*JacobiSN(x + y*I[3], k)-((k)^(2))/(1 - (k)^(2))*JacobiCN(x + y*I, k)*JacobiCN(x + y*I[1], k)*JacobiCN(x + y*I[2], k)*JacobiCN(x + y*I[3], k)+(1)/(1 - (k)^(2))*JacobiDN(x + y*I, k)*JacobiDN(x + y*I[1], k)*JacobiDN(x + y*I[2], k)*JacobiDN(x + y*I[3], k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>x == (k)^(2)* JacobiSN[x + y*I, (k)^2]*JacobiSN[Subscript[x + y*I, 1], (k)^2]*JacobiSN[Subscript[x + y*I, 2], (k)^2]*JacobiSN[Subscript[x + y*I, 3], (k)^2]-Divide[(k)^(2),1 - (k)^(2)]*JacobiCN[x + y*I, (k)^2]*JacobiCN[Subscript[x + y*I, 1], (k)^2]*JacobiCN[Subscript[x + y*I, 2], (k)^2]*JacobiCN[Subscript[x + y*I, 3], (k)^2]+Divide[1,1 - (k)^(2)]*JacobiDN[x + y*I, (k)^2]*JacobiDN[Subscript[x + y*I, 1], (k)^2]*JacobiDN[Subscript[x + y*I, 2], (k)^2]*JacobiDN[Subscript[x + y*I, 3], (k)^2]</syntaxhighlight> || Failure || Aborted || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [54 / 54]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
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| Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[1.5, Times[Complex[1.6461554600232724, 0.45267954815584505], JacobiCN[Subscript[Complex[1.5, -1.5], 1], 4.0], JacobiCN[Subscript[Complex[1.5, -1.5], 2], 4.0], JacobiCN[Subscript[Complex[1.5, -1.5], 3], 4.0]], Times[Complex[0.619203045121549, 0.30086290583863873], JacobiDN[Subscript[Complex[1.5, -1.5], 1], 4.0], JacobiDN[Subscript[Complex[1.5, -1.5], 2], 4.0], JacobiDN[Subscript[Complex[1.5, -1.5], 3], 4.0]], Times[Complex[-2.0469921952210957, 3.2763330530501245], JacobiSN[Subscript[Complex[1.5, -1.5], 1], 4.0], JacobiSN[Subscript[Complex[1.5, -1.5], 2], 4.0], JacobiSN[Subscript[Complex[1.5, -1.5], 3], 4.0]]]
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| Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/29.8.E2 29.8.E2] || [[Item:Q8705|<math>\mu w(z_{1})w(z_{2})w(z_{3}) = \int_{-2\compellintKk@@{k}}^{2\compellintKk@@{k}}\FerrersP[]{\nu}@{x}w(z)\diff{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\mu w(z_{1})w(z_{2})w(z_{3}) = \int_{-2\compellintKk@@{k}}^{2\compellintKk@@{k}}\FerrersP[]{\nu}@{x}w(z)\diff{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>mu*w*(x + y*I[1])*w*(x + y*I[2])*w*(x + y*I[3]) = int(LegendreP(nu, x)*w*((x + y*I)), (x + y*I) = - 2*EllipticK(k)..2*EllipticK(k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Mu]*w*(Subscript[x + y*I, 1])*w*(Subscript[x + y*I, 2])*w*(Subscript[x + y*I, 3]) == Integrate[LegendreP[\[Nu], x]*w*((x + y*I)), {(x + y*I), - 2*EllipticK[(k)^2], 2*EllipticK[(k)^2]}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[-1.0, NIntegrate[Complex[2.8644916021274596, 0.010098545944192239]
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| Test Values: {Complex[1.5, -1.5], DirectedInfinity[], DirectedInfinity[]}]], Times[Complex[-0.49999999999999994, 0.8660254037844387], Subscript[Complex[1.5, -1.5], 1], Subscript[Complex[1.5, -1.5], 2], Subscript[Complex[1.5, -1.5], 3]]], {Rule[k, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, -1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[-1.0, NIntegrate[Complex[2.8644916021274596, 0.010098545944192239]
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| Test Values: {Complex[1.5, -1.5], Times[-2, EllipticK[4]], Times[2, EllipticK[4]]}]], Times[Complex[-0.49999999999999994, 0.8660254037844387], Subscript[Complex[1.5, -1.5], 1], Subscript[Complex[1.5, -1.5], 2], Subscript[Complex[1.5, -1.5], 3]]], {Rule[k, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, -1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/29.8.E3 29.8.E3] || [[Item:Q8706|<math>\mu = \frac{2\sigma\tau}{\Wronskian@{w,w_{2}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\mu = \frac{2\sigma\tau}{\Wronskian@{w,w_{2}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>mu = (2*sigma*tau)/((w)*diff(w[2], w)-diff(w, w)*(w[2]))</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Mu] == Divide[2*\[Sigma]*\[Tau],Wronskian[{w, Subscript[w, 2]}, w]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.598076212+1.500000000*I
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| Test Values: {mu = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, w[2] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.866025404-1.232050808*I
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| Test Values: {mu = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, w[2] = -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: Plus[Complex[0.8660254037844387, 0.49999999999999994], Times[Complex[-1.0000000000000002, -1.7320508075688772], Power[Plus[Complex[-0.8660254037844387, -0.49999999999999994], Times[Complex[0.8660254037844387, 0.49999999999999994], Derivative[1, 0][Subscript][Complex[0.8660254037844387, 0.49999999999999994], 2.0]]], -1]]]
| |
| Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.8660254037844387, 0.49999999999999994], Times[Complex[-1.0000000000000002, -1.7320508075688772], Power[Plus[Complex[0.4999999999999998, -0.8660254037844387], Times[Complex[0.8660254037844387, 0.49999999999999994], Derivative[1, 0][Subscript][Complex[0.8660254037844387, 0.49999999999999994], 2.0]]], -1]]]
| |
| Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/29.8#Ex1 29.8#Ex1] || [[Item:Q8707|<math>w(z+2\compellintKk@@{k}) = \sigma w(z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w(z+2\compellintKk@@{k}) = \sigma w(z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>w(z + 2*EllipticK(k)) = sigma*w(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>w[z + 2*EllipticK[(k)^2]] == \[Sigma]*w[z]</syntaxhighlight> || Failure || Failure || Error || <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[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[3.038160455456161, -1.1586967532026022]
| |
| Test Values: {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[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/29.8#Ex2 29.8#Ex2] || [[Item:Q8708|<math>w_{2}(z+2\compellintKk@@{k}) = \tau w(z)+\sigma w_{2}(z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w_{2}(z+2\compellintKk@@{k}) = \tau w(z)+\sigma w_{2}(z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>w[2](z + 2*EllipticK(k)) = tau*w(z)+ sigma*w[2](z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[w, 2][z + 2*EllipticK[(k)^2]] == \[Tau]*w[z]+ \[Sigma]*Subscript[w, 2][z]</syntaxhighlight> || Failure || Failure || Error || <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[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[3.038160455456161, -2.1586967532026025]
| |
| Test Values: {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[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/29.8.E6 29.8.E6] || [[Item:Q8710|<math>y = \frac{1}{k^{\prime}}\Jacobielldnk@{z}{k}\Jacobielldnk@{z_{1}}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>y = \frac{1}{k^{\prime}}\Jacobielldnk@{z}{k}\Jacobielldnk@{z_{1}}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>y = (1)/(sqrt(1 - (k)^(2)))*JacobiDN(x + y*I, k)*JacobiDN(x + y*I[1], k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>y == Divide[1,Sqrt[1 - (k)^(2)]]*JacobiDN[x + y*I, (k)^2]*JacobiDN[Subscript[x + y*I, 1], (k)^2]</syntaxhighlight> || Failure || Aborted || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [54 / 54]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
| |
| Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-1.5, Times[Complex[-0.5211098390253335, 1.072491134351887], JacobiDN[Subscript[Complex[1.5, -1.5], 1], 4.0]]]
| |
| Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/29.11.E1 29.11.E1] || [[Item:Q8721|<math>\deriv[2]{w}{z}+(h-\nu(\nu+1)k^{2}\Jacobiellsnk^{2}@{z}{k}+k^{2}\omega^{2}\Jacobiellsnk^{4}@{z}{k})w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{z}+(h-\nu(\nu+1)k^{2}\Jacobiellsnk^{2}@{z}{k}+k^{2}\omega^{2}\Jacobiellsnk^{4}@{z}{k})w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [z$(2)])+(h - nu*(nu + 1)*(k)^(2)* (JacobiSN(z, k))^(2)+ (k)^(2)* (omega)^(2)* (JacobiSN(z, k))^(4))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 2}]+(h - \[Nu]*(\[Nu]+ 1)*(k)^(2)* (JacobiSN[z, (k)^2])^(2)+ (k)^(2)* \[Omega]^(2)* (JacobiSN[z, (k)^2])^(4))*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .4970479804-.2136667430*I
| |
| Test Values: {h = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, omega = 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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5039614158-1.687364305*I
| |
| Test Values: {h = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, omega = 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}</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[0.4970479802306743, -0.21366674241821534]
| |
| 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]]], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.5039614145885605, -1.6873643054323533]
| |
| 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]]], Rule[ω, 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/29.12.E10 29.12.E10] || [[Item:Q8731|<math>0 < \xi_{1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>0 < \xi_{1}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">0 < xi[1]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">0 < Subscript[\[Xi], 1]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.12.E12 29.12.E12] || [[Item:Q8733|<math>0 \leq t_{1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>0 \leq t_{1}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">0 <= t[1]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">0 <= Subscript[t, 1]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/29.14.E3 29.14.E3] || [[Item:Q8737|<math>w(s,t) = \Jacobiellsnk^{2}@{\compellintKk@@{k}+\iunit t}{k}-\Jacobiellsnk^{2}@{s}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w(s,t) = \Jacobiellsnk^{2}@{\compellintKk@@{k}+\iunit t}{k}-\Jacobiellsnk^{2}@{s}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>w(s , t) = (JacobiSN(EllipticK(k)+ I*t, k))^(2)- (JacobiSN(s, k))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>w[s , t] == (JacobiSN[EllipticK[(k)^2]+ I*t, (k)^2])^(2)- (JacobiSN[s, (k)^2])^(2)</syntaxhighlight> || Failure || Failure || Error || Error
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.15.E5 29.15.E5] || [[Item:Q8750|<math>\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{n}A_{2p}^{2} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{n}A_{2p}^{2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1)/(2)*(A[0])^(2)+ sum((A[2*p])^(2), p = 1..n) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Divide[1,2]*(Subscript[A, 0])^(2)+ Sum[(Subscript[A, 2*p])^(2), {p, 1, n}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.15.E6 29.15.E6] || [[Item:Q8751|<math>\tfrac{1}{2}A_{0}+\sum_{p=1}^{n}A_{2p} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\tfrac{1}{2}A_{0}+\sum_{p=1}^{n}A_{2p} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1)/(2)*A[0]+ sum(A[2*p], p = 1..n) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Divide[1,2]*Subscript[A, 0]+ Sum[Subscript[A, 2*p], {p, 1, n}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.15.E10 29.15.E10] || [[Item:Q8755|<math>\sum_{p=0}^{n}A_{2p+1}^{2} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{n}A_{2p+1}^{2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((A[2*p + 1])^(2), p = 0..n) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(Subscript[A, 2*p + 1])^(2), {p, 0, n}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.15.E11 29.15.E11] || [[Item:Q8756|<math>\sum_{p=0}^{n}A_{2p+1} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{n}A_{2p+1} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum(A[2*p + 1], p = 0..n) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[Subscript[A, 2*p + 1], {p, 0, n}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.15.E15 29.15.E15] || [[Item:Q8760|<math>\sum_{p=0}^{n}B_{2p+1}^{2} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{n}B_{2p+1}^{2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((B[2*p + 1])^(2), p = 0..n) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(Subscript[B, 2*p + 1])^(2), {p, 0, n}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.15.E16 29.15.E16] || [[Item:Q8761|<math>\sum_{p=0}^{n}(2p+1)B_{2p+1} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{n}(2p+1)B_{2p+1} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((2*p + 1)*B[2*p + 1], p = 0..n) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(2*p + 1)*Subscript[B, 2*p + 1], {p, 0, n}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.15.E20 29.15.E20] || [[Item:Q8765|<math>\left(1-\tfrac{1}{2}k^{2}\right)\left(\tfrac{1}{2}C_{0}^{2}+\sum_{p=1}^{n}C_{2p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{n-1}C_{2p}C_{2p+2} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\left(1-\tfrac{1}{2}k^{2}\right)\left(\tfrac{1}{2}C_{0}^{2}+\sum_{p=1}^{n}C_{2p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{n-1}C_{2p}C_{2p+2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1 -(1)/(2)*(k)^(2))*((1)/(2)*(C[0])^(2)+ sum((C[2*p])^(2), p = 1..n))-(1)/(2)*(k)^(2)* sum(C[2*p]*C[2*p + 2], p = 0..n - 1) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1 -Divide[1,2]*(k)^(2))*(Divide[1,2]*(Subscript[C, 0])^(2)+ Sum[(Subscript[C, 2*p])^(2), {p, 1, n}, GenerateConditions->None])-Divide[1,2]*(k)^(2)* Sum[Subscript[C, 2*p]*Subscript[C, 2*p + 2], {p, 0, n - 1}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.15.E21 29.15.E21] || [[Item:Q8766|<math>\tfrac{1}{2}C_{0}+\sum_{p=1}^{n}C_{2p} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\tfrac{1}{2}C_{0}+\sum_{p=1}^{n}C_{2p} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1)/(2)*C[0]+ sum(C[2*p], p = 1..n) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Divide[1,2]*Subscript[C, 0]+ Sum[Subscript[C, 2*p], {p, 1, n}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.15.E25 29.15.E25] || [[Item:Q8770|<math>\sum_{p=0}^{n}B_{2p+2}^{2} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{n}B_{2p+2}^{2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((B[2*p + 2])^(2), p = 0..n) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(Subscript[B, 2*p + 2])^(2), {p, 0, n}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.15.E26 29.15.E26] || [[Item:Q8771|<math>\sum_{p=0}^{n}(2p+2)B_{2p+2} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{n}(2p+2)B_{2p+2} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((2*p + 2)*B[2*p + 2], p = 0..n) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(2*p + 2)*Subscript[B, 2*p + 2], {p, 0, n}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.15.E31 29.15.E31] || [[Item:Q8776|<math>\sum_{p=0}^{n}C_{2p+1} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{n}C_{2p+1} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum(C[2*p + 1], p = 0..n) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[Subscript[C, 2*p + 1], {p, 0, n}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.15.E36 29.15.E36] || [[Item:Q8781|<math>\sum_{p=0}^{n}(2p+1)D_{2p+1} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{n}(2p+1)D_{2p+1} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((2*p + 1)*D[2*p + 1], p = 0..n) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(2*p + 1)*Subscript[D, 2*p + 1], {p, 0, n}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.15.E40 29.15.E40] || [[Item:Q8785|<math>\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}D_{2p+2}^{2}-\tfrac{1}{2}k^{2}\sum_{p=1}^{n}D_{2p}D_{2p+2} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}D_{2p+2}^{2}-\tfrac{1}{2}k^{2}\sum_{p=1}^{n}D_{2p}D_{2p+2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1 -(1)/(2)*(k)^(2))*sum((D[2*p + 2])^(2), p = 0..n)-(1)/(2)*(k)^(2)* sum(D[2*p]*D[2*p + 2], p = 1..n) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1 -Divide[1,2]*(k)^(2))*Sum[(Subscript[D, 2*p + 2])^(2), {p, 0, n}, GenerateConditions->None]-Divide[1,2]*(k)^(2)* Sum[Subscript[D, 2*p]*Subscript[D, 2*p + 2], {p, 1, n}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/29.15.E41 29.15.E41] || [[Item:Q8786|<math>\sum_{p=0}^{n}(2p+2)D_{2p+2} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{n}(2p+2)D_{2p+2} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((2*p + 2)*D[2*p + 2], p = 0..n) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(2*p + 2)*Subscript[D, 2*p + 2], {p, 0, n}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/29.18#Ex1 29.18#Ex1] || [[Item:Q8798|<math>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] || [[Item:Q8799|<math>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] || [[Item:Q8800|<math>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] || [[Item:Q8801|<math>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] || [[Item:Q8804|<math>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] || [[Item:Q8804|<math>\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] || [[Item:Q8805|<math>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 || - || -
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| |-
| |
| | [https://dlmf.nist.gov/29.18.E5 29.18.E5] || [[Item:Q8806|<math>\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] || [[Item:Q8807|<math>\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] || [[Item:Q8808|<math>\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] || [[Item:Q8809|<math>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] || [[Item:Q8810|<math>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)
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| 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
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| 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[]
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| 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]
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| 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>
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| | [https://dlmf.nist.gov/29.18#Ex10 29.18#Ex10] || [[Item:Q8811|<math>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
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| 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
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| 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[]
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| 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]
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| 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>
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/29.18.E10 29.18.E10] || [[Item:Q8814|<math>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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| |}
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| </div> | | </div> |