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| |- | | ; Notation : [[33.1|33.1 Special Notation]]<br> |
| ! DLMF !! Formula !! Constraints !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
| | ; Variables <math>\rho,\eta</math> : [[33.2|33.2 Definitions and Basic Properties]]<br>[[33.3|33.3 Graphics]]<br>[[33.4|33.4 Recurrence Relations and Derivatives]]<br>[[33.5|33.5 Limiting Forms for Small <math>\rho</math> , Small <math>|\eta|</math> , or Large <math>\ell</math>]]<br>[[33.6|33.6 Power-Series Expansions in <math>\rho</math>]]<br>[[33.7|33.7 Integral Representations]]<br>[[33.8|33.8 Continued Fractions]]<br>[[33.9|33.9 Expansions in Series of Bessel Functions]]<br>[[33.10|33.10 Limiting Forms for Large <math>\rho</math> or Large <math>\abs{\eta}</math>]]<br>[[33.11|33.11 Asymptotic Expansions for Large <math>\rho</math>]]<br>[[33.12|33.12 Asymptotic Expansions for Large <math>\eta</math>]]<br>[[33.13|33.13 Complex Variable and Parameters]]<br> |
| |-
| | ; Variables <math>r,\epsilon</math> : [[33.14|33.14 Definitions and Basic Properties]]<br>[[33.15|33.15 Graphics]]<br>[[33.16|33.16 Connection Formulas]]<br>[[33.17|33.17 Recurrence Relations and Derivatives]]<br>[[33.18|33.18 Limiting Forms for Large <math>\ell</math>]]<br>[[33.19|33.19 Power-Series Expansions in <math>r</math>]]<br>[[33.20|33.20 Expansions for Small <math>|\epsilon|</math>]]<br>[[33.21|33.21 Asymptotic Approximations for Large <math>|r|</math>]]<br> |
| | [https://dlmf.nist.gov/33.2.E1 33.2.E1] || [[Item:Q9496|<math>\deriv[2]{w}{\rho}+\left(1-\frac{2\eta}{\rho}-\frac{\ell(\ell+1)}{\rho^{2}}\right)w = 0</math>]] || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [rho$(2)])+(1 -(2*eta)/(rho)-(ell*(ell + 1))/((rho)^(2)))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {\[Rho], 2}]+(1 -Divide[2*\[Eta],\[Rho]]-Divide[\[ScriptL]*(\[ScriptL]+ 1),\[Rho]^(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: -11.25833025+5.499999998*I
| | ; Physical Applications : [[33.22|33.22 Particle Scattering and Atomic and Molecular Spectra]]<br> |
| Test Values: {eta = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, ell = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -5.499999998-11.25833025*I
| | ; Computation : [[33.23|33.23 Methods of Computation]]<br>[[33.24|33.24 Tables]]<br>[[33.25|33.25 Approximations]]<br>[[33.26|33.26 Software]]<br> |
| Test Values: {eta = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, w = -1/2+1/2*I*3^(1/2), ell = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [294 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-11.258330249197703, 5.5]
| | </div> |
| Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 3], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[10.2583302491977, -3.767949192431125]
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| Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 3], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/33.4.E2 33.4.E2] || [[Item:Q9514|<math>R_{\ell}X_{\ell-1}-T_{\ell}X_{\ell}+R_{\ell+1}X_{\ell+1} = 0</math>]] || <math>\ell \geq 1</math> || <syntaxhighlight lang=mathematica>R[ell]*X[ell - 1]-(S[ell]+ S[ell + 1])*X[ell]+ R[ell + 1]*X[ell + 1] = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[R, \[ScriptL]]*Subscript[X, \[ScriptL]- 1]-(Subscript[S, \[ScriptL]]+ Subscript[S, \[ScriptL]+ 1])*Subscript[X, \[ScriptL]]+ Subscript[R, \[ScriptL]+ 1]*Subscript[X, \[ScriptL]+ 1] == 0</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/33.5#Ex7 33.5#Ex7] || [[Item:Q9523|<math>\regCoulombF{\ell}@{0}{\rho} = (\pi\rho/2)^{1/2}\BesselJ{\ell+\frac{1}{2}}@{\rho}</math>]] || <math>\realpart@@{\ell+\frac{1}{2}+k+1} > 0</math> || <syntaxhighlight lang=mathematica>CoulombF(ell, 0, rho) = (Pi*rho/2)^(1/2)* BesselJ(ell +(1)/(2), rho)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Failure || Missing Macro Error || Error || -
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| | [https://dlmf.nist.gov/33.5#Ex9 33.5#Ex9] || [[Item:Q9525|<math>\regCoulombF{0}@{0}{\rho} = \sin@@{\rho}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>CoulombF(0, 0, rho) = sin(rho)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Successful || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/33.5.E6 33.5.E6] || [[Item:Q9528|<math>\frac{2^{\ell}\ell!}{(2\ell+1)!} = \frac{1}{(2\ell+1)!!}</math>]] || <math>\realpart@@{\ell+1+\iunit\eta} > 0</math> || <syntaxhighlight lang=mathematica>((2)^(ell)* factorial(ell))/(factorial(2*ell + 1)) = (1)/(doublefactorial(2*ell + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[(2)^\[ScriptL]* (\[ScriptL])!,(2*\[ScriptL]+ 1)!] == Divide[1,(2*\[ScriptL]+ 1)!!]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[Power[2.0, ℓ], Factorial[ℓ], Power[Factorial[Plus[1.0, Times[2.0, ℓ]]], -1]], Times[-1.0, Power[Factorial2[Plus[1.0, Times[2.0, ℓ]]], -1]]]
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| Test Values: {}</syntaxhighlight><br></div></div>
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| | [https://dlmf.nist.gov/33.6.E3 33.6.E3] || [[Item:Q9535|<math>(k+\ell)(k-\ell-1)A_{k}^{\ell} = 2\eta A_{k-1}^{\ell}-A_{k-2}^{\ell}</math>]] || <math>k = \ell+3</math> || <syntaxhighlight lang=mathematica>(k + ell)*(k - ell - 1)*(A[k])^(ell) = 2*eta*(A[k - 1])^(ell)- (A[k - 2])^(ell)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(k + \[ScriptL])*(k - \[ScriptL]- 1)*(Subscript[A, k])^\[ScriptL] == 2*\[Eta]*(Subscript[A, k - 1])^\[ScriptL]- (Subscript[A, k - 2])^\[ScriptL]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/33.6.E4 33.6.E4] || [[Item:Q9536|<math>A_{k}^{\ell}(\eta) = \dfrac{(-\iunit)^{k-\ell-1}}{(k-\ell-1)!}\*\genhyperF{2}{1}@{\ell+1-k,\ell+1-\iunit\eta}{2\ell+2}{2}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>(A[k])^(ell)(eta) = ((- I)^(k - ell - 1))/(factorial(k - ell - 1))* hypergeom([ell + 1 - k , ell + 1 - I*eta], [2*ell + 2], 2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Subscript[A, k])^\[ScriptL][\[Eta]] == Divide[(- I)^(k - \[ScriptL]- 1),(k - \[ScriptL]- 1)!]* HypergeometricPFQ[{\[ScriptL]+ 1 - k , \[ScriptL]+ 1 - I*\[Eta]}, {2*\[ScriptL]+ 2}, 2]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [293 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.5000000000000001, 0.8660254037844386]
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| Test Values: {Rule[k, 1], Rule[ℓ, 1], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.0, 1.0]
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| Test Values: {Rule[k, 1], Rule[ℓ, 2], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, k], 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/33.7.E1 33.7.E1] || [[Item:Q9538|<math>\regCoulombF{\ell}@{\eta}{\rho} = \frac{\rho^{\ell+1}2^{\ell}e^{\iunit\rho-(\pi\eta/2)}}{|\EulerGamma@{\ell+1+\iunit\eta}|}\int_{0}^{1}e^{-2\iunit\rho t}t^{\ell+\iunit\eta}(1-t)^{\ell-\iunit\eta}\diff{t}</math>]] || <math>\realpart@@{\ell+1+\iunit\eta} > 0</math> || <syntaxhighlight lang=mathematica>CoulombF(ell, eta, rho) = ((rho)^(ell + 1)* (2)^(ell)* exp(I*rho -(Pi*eta/2)))/(abs(GAMMA(ell + 1 + I*eta)))*int(exp(- 2*I*rho*t)*(t)^(ell + I*eta)*(1 - t)^(ell - I*eta), t = 0..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Failure || Missing Macro Error || Error || -
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| | [https://dlmf.nist.gov/33.8#Ex4 33.8#Ex4] || [[Item:Q9547|<math>\regCoulombF{\ell} = +(q^{-1}(u-p)^{2}+q)^{-1/2}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>CoulombF(ell, =, +)*((q)^(- 1)*(u - p)^(2)+ q)^(- 1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Translation Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/33.8#Ex4 33.8#Ex4] || [[Item:Q9547|<math>\regCoulombF{\ell} = -(q^{-1}(u-p)^{2}+q)^{-1/2}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>CoulombF(ell, =, -)*((q)^(- 1)*(u - p)^(2)+ q)^(- 1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Translation Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/33.9.E2 33.9.E2] || [[Item:Q9552|<math>\frac{k(k+2\ell+1)}{2k+2\ell+1}a_{k}-2\eta a_{k-1}+\frac{(k-2)(k+2\ell-1)}{2k+2\ell-3}a_{k-2} = 0</math>]] || <math></math> || <syntaxhighlight lang=mathematica>(k*(k + 2*ell + 1))/(2*k + 2*ell + 1)*a[k]- 2*eta*a[k - 1]+((k - 2)*(k + 2*ell - 1))/(2*k + 2*ell - 3)*a[k - 2] = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[k*(k + 2*\[ScriptL]+ 1),2*k + 2*\[ScriptL]+ 1]*Subscript[a, k]- 2*\[Eta]*Subscript[a, k - 1]+Divide[(k - 2)*(k + 2*\[ScriptL]- 1),2*k + 2*\[ScriptL]- 3]*Subscript[a, k - 2] == 0</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/33.9.E5 33.9.E5] || [[Item:Q9555|<math>4\eta^{2}(k-2\ell)b_{k+1}+kb_{k-1}+b_{k-2} = 0</math>]] || <math>k = 2\ell+2</math> || <syntaxhighlight lang=mathematica>4*(eta)^(2)*(k - 2*ell)*b[k + 1]+ k*b[k - 1]+ b[k - 2] = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>4*\[Eta]^(2)*(k - 2*\[ScriptL])*Subscript[b, k + 1]+ k*Subscript[b, k - 1]+ Subscript[b, k - 2] == 0</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/33.12#Ex6 33.12#Ex6] || [[Item:Q9601|<math>B_{1} = -\tfrac{1}{5}x</math>]] || <math></math> || <syntaxhighlight lang=mathematica>B[1] = -(1)/(5)*x</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[B, 1] == -Divide[1,5]*x</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/33.12#Ex7 33.12#Ex7] || [[Item:Q9602|<math>B_{2} = \tfrac{1}{350}(7x^{5}-30x^{2})</math>]] || <math></math> || <syntaxhighlight lang=mathematica>B[2] = (1)/(350)*(7*(x)^(5)- 30*(x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[B, 2] == Divide[1,350]*(7*(x)^(5)- 30*(x)^(2))</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/33.12#Ex8 33.12#Ex8] || [[Item:Q9603|<math>B_{3} = \tfrac{1}{15750}(264x^{6}-290x^{3}-560)</math>]] || <math></math> || <syntaxhighlight lang=mathematica>B[3] = (1)/(15750)*(264*(x)^(6)- 290*(x)^(3)- 560)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[B, 3] == Divide[1,15750]*(264*(x)^(6)- 290*(x)^(3)- 560)</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/33.12.E8 33.12.E8] || [[Item:Q9606|<math>\deriv[2]{w}{z} = \left(4\eta^{2}\left(\frac{1-z}{z}\right)+\frac{\ell(\ell+1)}{z^{2}}\right)w</math>]] || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [z$(2)]) = (4*(eta((1 - z)/(z)))^(2)+(ell*(ell + 1))/((z)^(2)))*w</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 2}] == (4*(\[Eta][Divide[1 - z,z]])^(2)+Divide[\[ScriptL]*(\[ScriptL]+ 1),(z)^(2)])*w</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [296 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-3.7320508075688767, 1.5358983848622458]
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| Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 1], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-7.196152422706632, 3.535898384862246]
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| Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 2], Rule[η, 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/33.14.E1 33.14.E1] || [[Item:Q9609|<math>\deriv[2]{w}{r}+\left(\epsilon+\frac{2}{r}-\frac{\ell(\ell+1)}{r^{2}}\right)w = 0</math>]] || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [r$(2)])+(epsilon +(2)/(r)-(ell*(ell + 1))/((r)^(2)))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {r, 2}]+(\[Epsilon]+Divide[2,r]-Divide[\[ScriptL]*(\[ScriptL]+ 1),(r)^(2)])*w == 0</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.0584754935143141, -0.611111111111111]
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| Test Values: {Rule[r, Rational[-3, 2]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 1], Rule[ϵ, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.19245008972987526, -0.11111111111111109]
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| Test Values: {Rule[r, Rational[-3, 2]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 1], Rule[ϵ, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/33.14#Ex1 33.14#Ex1] || [[Item:Q9610|<math>r = -\eta\rho</math>]] || <math></math> || <syntaxhighlight lang=mathematica>r = - eta*rho</syntaxhighlight> || <syntaxhighlight lang=mathematica>r == - \[Eta]*\[Rho]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/33.14#Ex2 33.14#Ex2] || [[Item:Q9611|<math>\epsilon = 1/\eta^{2}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>epsilon = 1/(eta)^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Epsilon] == 1/\[Eta]^(2)</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/33.14.E12 33.14.E12] || [[Item:Q9622|<math>A(\epsilon,\ell) = \frac{\EulerGamma@{1+\ell+\kappa}}{\EulerGamma@{\kappa-\ell}}\kappa^{-2\ell-1}</math>]] || <math>\realpart@@{1+\ell+\kappa} > 0, \realpart@@{\kappa-\ell} > 0</math> || <syntaxhighlight lang=mathematica>(product(1 + epsilon*(k)^(2), k = 0..ell)) = (GAMMA(1 + ell + kappa))/(GAMMA(kappa - ell))*(kappa)^(- 2*ell - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Product[1 + \[Epsilon]*(k)^(2), {k, 0, \[ScriptL]}, GenerateConditions->None]) == Divide[Gamma[1 + \[ScriptL]+ \[Kappa]],Gamma[\[Kappa]- \[ScriptL]]]*\[Kappa]^(- 2*\[ScriptL]- 1)</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.4444444444444444
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| Test Values: {Rule[ℓ, 1], Rule[ϵ, 1], Rule[κ, Rational[3, 2]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[2.4444444444444446, 0.0]
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| Test Values: {Rule[ℓ, 1], Rule[ϵ, 2], Rule[κ, Rational[3, 2]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/33.19.E4 33.19.E4] || [[Item:Q9662|<math>\gamma_{k}-\gamma_{k-1}+\tfrac{1}{4}(k-1)(k-2\ell-2)\epsilon\gamma_{k-2} = 0</math>]] || <math></math> || <syntaxhighlight lang=mathematica>gamma[k]- gamma[k - 1]+(1)/(4)*(k - 1)*(k - 2*ell - 2)*epsilon*gamma[k - 2] = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Gamma], k]- Subscript[\[Gamma], k - 1]+Divide[1,4]*(k - 1)*(k - 2*\[ScriptL]- 2)*\[Epsilon]*Subscript[\[Gamma], k - 2] == 0</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/33.19.E6 33.19.E6] || [[Item:Q9665|<math>k(k+2\ell+1)\delta_{k}+2\delta_{k-1}+\epsilon\delta_{k-2}+2(2k+2\ell+1)A(\epsilon,\ell)\alpha_{k} = 0</math>]] || <math></math> || <syntaxhighlight lang=mathematica>k*(k + 2*ell + 1)*delta[k]+ 2*delta[k - 1]+ epsilon*delta[k - 2]+ 2*(2*k + 2*ell + 1)*(product(1 + epsilon*(k)^(2), k = 0..ell))*alpha[k] = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>k*(k + 2*\[ScriptL]+ 1)*Subscript[\[Delta], k]+ 2*Subscript[\[Delta], k - 1]+ \[Epsilon]*Subscript[\[Delta], k - 2]+ 2*(2*k + 2*\[ScriptL]+ 1)*(Product[1 + \[Epsilon]*(k)^(2), {k, 0, \[ScriptL]}, GenerateConditions->None])*Subscript[\[Alpha], k] == 0</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/33.19.E7 33.19.E7] || [[Item:Q9666|<math>\beta_{k}-\beta_{k-1}+\tfrac{1}{4}(k-1)(k-2\ell-2)\epsilon\beta_{k-2}+\tfrac{1}{2}(k-1)\epsilon\gamma_{k-2} = 0</math>]] || <math></math> || <syntaxhighlight lang=mathematica>beta[k]- beta[k - 1]+(1)/(4)*(k - 1)*(k - 2*ell - 2)*epsilon*beta[k - 2]+(1)/(2)*(k - 1)*epsilon*gamma[k - 2] = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Beta], k]- Subscript[\[Beta], k - 1]+Divide[1,4]*(k - 1)*(k - 2*\[ScriptL]- 2)*\[Epsilon]*Subscript[\[Beta], k - 2]+Divide[1,2]*(k - 1)*\[Epsilon]*Subscript[\[Gamma], k - 2] == 0</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/33.20.E4 33.20.E4] || [[Item:Q9672|<math>{\sf F}_{k}(\ell;r) = \sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}\BesselJ{2\ell+1+p}@{\sqrt{8r}}</math>]] || <math>r > 0, \realpart@@{2\ell+1+p+k+1} > 0</math> || <syntaxhighlight lang=mathematica>F[k](ell ; r) = sum((2*r)^((p + 1)/2)* C[k , p]*BesselJ(2*ell + 1 + p, sqrt(8*r)), p = 2*k..3*k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[F, k][\[ScriptL]; r] == Sum[(2*r)^((p + 1)/2)* Subscript[C, k , p]*BesselJ[2*\[ScriptL]+ 1 + p, Sqrt[8*r]], {p, 2*k, 3*k}, GenerateConditions->None]</syntaxhighlight> || Translation Error || Translation Error || - || -
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| | [https://dlmf.nist.gov/33.20.E5 33.20.E5] || [[Item:Q9673|<math>{\sf F}_{k}(\ell;r) = \sum_{p=2k}^{3k}(-1)^{\ell+1+p}(2|r|)^{(p+1)/2}C_{k,p}\modBesselI{2\ell+1+p}@{\sqrt{8|r|}}</math>]] || <math>r < 0, \realpart@@{2\ell+1+p+k+1} > 0</math> || <syntaxhighlight lang=mathematica>F[k](ell ; r) = sum((- 1)^(ell + 1 + p)*(2*abs(r))^((p + 1)/2)* C[k , p]*BesselI(2*ell + 1 + p, sqrt(8*abs(r))), p = 2*k..3*k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[F, k][\[ScriptL]; r] == Sum[(- 1)^(\[ScriptL]+ 1 + p)*(2*Abs[r])^((p + 1)/2)* Subscript[C, k , p]*BesselI[2*\[ScriptL]+ 1 + p, Sqrt[8*Abs[r]]], {p, 2*k, 3*k}, GenerateConditions->None]</syntaxhighlight> || Translation Error || Translation Error || - || -
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| | [https://dlmf.nist.gov/33.20#Ex5 33.20#Ex5] || [[Item:Q9674|<math>C_{k,p} = 0</math>]] || <math>p < 2k, p > 3k</math> || <syntaxhighlight lang=mathematica>C[k , p] = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[C, k , p] == 0</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
| |
| | [https://dlmf.nist.gov/33.20#Ex6 33.20#Ex6] || [[Item:Q9675|<math>C_{k,p} = \left(-(2\ell+p)C_{k-1,p-2}+C_{k-1,p-3}\right)/(4p)</math>]] || <math>k > 0, 2k \leq p, p \leq 3k</math> || <syntaxhighlight lang=mathematica>C[k , p] = (-(2*ell + p)*C[k - 1 , p - 2]+ C[k - 1 , p - 3])/(4*p)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[C, k , p] == (-(2*\[ScriptL]+ p)*Subscript[C, k - 1 , p - 2]+ Subscript[C, k - 1 , p - 3])/(4*p)</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
| |
| | [https://dlmf.nist.gov/33.20.E8 33.20.E8] || [[Item:Q9677|<math>{\sf H}_{k}(\ell;r) = \sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}\BesselY{2\ell+1+p}@{\sqrt{8r}}</math>]] || <math>r > 0, \realpart@@{2\ell+1+p+k+1} > 0, \realpart@@{-2\ell+1+p+k+1} > 0</math> || <syntaxhighlight lang=mathematica>H[k](ell ; r) = sum((2*r)^((p + 1)/2)* C[k , p]*BesselY(2*ell + 1 + p, sqrt(8*r)), p = 2*k..3*k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[H, k][\[ScriptL]; r] == Sum[(2*r)^((p + 1)/2)* Subscript[C, k , p]*BesselY[2*\[ScriptL]+ 1 + p, Sqrt[8*r]], {p, 2*k, 3*k}, GenerateConditions->None]</syntaxhighlight> || Translation Error || Translation Error || - || -
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| |-
| |
| | [https://dlmf.nist.gov/33.20.E9 33.20.E9] || [[Item:Q9678|<math>{\sf H}_{k}(\ell;r) = (-1)^{\ell+1}\frac{2}{\pi}\sum_{p=2k}^{3k}(2|r|)^{(p+1)/2}C_{k,p}\modBesselK{2\ell+1+p}@{\sqrt{8|r|}}</math>]] || <math>r < 0</math> || <syntaxhighlight lang=mathematica>H[k](ell ; r) = (- 1)^(ell + 1)*(2)/(Pi)*sum((2*abs(r))^((p + 1)/2)* C[k , p]*BesselK(2*ell + 1 + p, sqrt(8*abs(r))), p = 2*k..3*k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[H, k][\[ScriptL]; r] == (- 1)^(\[ScriptL]+ 1)*Divide[2,Pi]*Sum[(2*Abs[r])^((p + 1)/2)* Subscript[C, k , p]*BesselK[2*\[ScriptL]+ 1 + p, Sqrt[8*Abs[r]]], {p, 2*k, 3*k}, GenerateConditions->None]</syntaxhighlight> || Translation Error || Translation Error || - || -
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| |-
| |
| | [https://dlmf.nist.gov/33.22.E3 33.22.E3] || [[Item:Q9687|<math>\deriv[2]{w}{x}+\left({\sf k}^{2}-\frac{2Z}{x}-\frac{\ell(\ell+1)}{x^{2}}\right)w = 0</math>]] || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [x$(2)])+((k)^(2)-(2*Z)/(x)-(ell*(ell + 1))/((x)^(2)))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {x, 2}]+((k)^(2)-Divide[2*Z,x]-Divide[\[ScriptL]*(\[ScriptL]+ 1),(x)^(2)])*w == 0</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [297 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.5704416218017292, -1.0991449828236957]
| |
| Test Values: {Rule[k, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, Rational[3, 2]], Rule[Z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-2.110042339640732, -1.9880338717125847]
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| Test Values: {Rule[k, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, Rational[3, 2]], Rule[Z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| |-
| |
| | [https://dlmf.nist.gov/33.22#Ex10 33.22#Ex10] || [[Item:Q9694|<math>r = -\eta\rho</math>]] || <math></math> || <syntaxhighlight lang=mathematica>r = - eta*rho</syntaxhighlight> || <syntaxhighlight lang=mathematica>r == - \[Eta]*\[Rho]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
| |
| | [https://dlmf.nist.gov/33.22#Ex11 33.22#Ex11] || [[Item:Q9695|<math>\epsilon = 1/\eta^{2}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>epsilon = 1/(eta)^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Epsilon] == 1/\[Eta]^(2)</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
| |
| | [https://dlmf.nist.gov/33.22#Ex12 33.22#Ex12] || [[Item:Q9696|<math>z = 2\iunit\rho</math>]] || <math></math> || <syntaxhighlight lang=mathematica>z = 2*I*rho</syntaxhighlight> || <syntaxhighlight lang=mathematica>z == 2*I*\[Rho]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.866025404-1.232050808*I
| |
| Test Values: {rho = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .5000000000-.8660254040*I
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| Test Values: {rho = 1/2*3^(1/2)+1/2*I, z = -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 [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.8660254037844386, -1.2320508075688774]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[2.598076211353316, 1.4999999999999996]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| |-
| |
| | [https://dlmf.nist.gov/33.22#Ex13 33.22#Ex13] || [[Item:Q9697|<math>\kappa = \iunit\eta</math>]] || <math></math> || <syntaxhighlight lang=mathematica>kappa = I*eta</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Kappa] == I*\[Eta]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [96 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.366025404-.3660254040*I
| |
| Test Values: {eta = 1/2*3^(1/2)+1/2*I, kappa = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.000000000-1.732050808*I
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| Test Values: {eta = 1/2*3^(1/2)+1/2*I, kappa = 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 [96 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.3660254037844386, -0.36602540378443876]
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| Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.0, -1.7320508075688772]
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| Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/33.22#Ex14 33.22#Ex14] || [[Item:Q9698|<math>\rho = z/(2\iunit)</math>]] || <math></math> || <syntaxhighlight lang=mathematica>rho = z/(2*I)</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Rho] == z/(2*I)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .6160254040+.9330127020*I
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| Test Values: {rho = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .4330127020+.2500000000*I
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| Test Values: {rho = 1/2*3^(1/2)+1/2*I, z = -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 [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.6160254037844387, 0.9330127018922193]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.7499999999999998, 1.299038105676658]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| |-
| |
| | [https://dlmf.nist.gov/33.22#Ex15 33.22#Ex15] || [[Item:Q9699|<math>\eta = \kappa/\iunit</math>]] || <math></math> || <syntaxhighlight lang=mathematica>eta = kappa/I</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Eta] == \[Kappa]/I</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [96 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .3660254040+1.366025404*I
| |
| Test Values: {eta = 1/2*3^(1/2)+1/2*I, kappa = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.732050808+1.000000000*I
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| Test Values: {eta = 1/2*3^(1/2)+1/2*I, kappa = 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 [96 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.36602540378443876, 1.3660254037844386]
| |
| Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.7320508075688772, 1.0]
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| Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/33.22#Ex16 33.22#Ex16] || [[Item:Q9700|<math>r = \kappa z/2</math>]] || <math></math> || <syntaxhighlight lang=mathematica>r = kappa*z/2</syntaxhighlight> || <syntaxhighlight lang=mathematica>r == \[Kappa]*z/2</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
| |
| | [https://dlmf.nist.gov/33.22#Ex17 33.22#Ex17] || [[Item:Q9701|<math>\epsilon = -1/\kappa^{2}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>epsilon = - 1/(kappa)^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Epsilon] == - 1/\[Kappa]^(2)</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
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| | [https://dlmf.nist.gov/33.22#Ex18 33.22#Ex18] || [[Item:Q9702|<math>\eta = +\epsilon^{-1/2}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>eta = + (epsilon)^(- 1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Eta] == + \[Epsilon]^(- 1/2)</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
| |
| | [https://dlmf.nist.gov/33.22#Ex19 33.22#Ex19] || [[Item:Q9703|<math>\rho = -r/\eta</math>]] || <math></math> || <syntaxhighlight lang=mathematica>rho = - r/eta</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Rho] == - r/\[Eta]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/33.22#Ex20 33.22#Ex20] || [[Item:Q9704|<math>\kappa = +(-\epsilon)^{-1/2}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>kappa = +(- epsilon)^(- 1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Kappa] == +(- \[Epsilon])^(- 1/2)</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
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| | [https://dlmf.nist.gov/33.22#Ex21 33.22#Ex21] || [[Item:Q9705|<math>z = 2r/\kappa</math>]] || <math></math> || <syntaxhighlight lang=mathematica>z = 2*r/kappa</syntaxhighlight> || <syntaxhighlight lang=mathematica>z == 2*r/\[Kappa]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |}
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