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| | <div style="-moz-column-count:2; column-count:2;"> |
| |- | | ; Notation : [[34.1|34.1 Special Notation]]<br> |
| ! DLMF !! Formula !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
| | ; Properties : [[34.2|34.2 Definition: <math>3j</math> Symbol]]<br>[[34.3|34.3 Basic Properties: <math>3j</math> Symbol]]<br>[[34.4|34.4 Definition: <math>6j</math> Symbol]]<br>[[34.5|34.5 Basic Properties: <math>6j</math> Symbol]]<br>[[34.6|34.6 Definition: <math>9j</math> Symbol]]<br>[[34.7|34.7 Basic Properties: <math>9j</math> Symbol]]<br>[[34.8|34.8 Approximations for Large Parameters]]<br>[[34.9|34.9 Graphical Method]]<br>[[34.10|34.10 Zeros]]<br>[[34.11|34.11 Higher-Order <math>3nj</math> Symbols]]<br> |
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| | ; Applications : [[34.12|34.12 Physical Applications]]<br> |
| | [https://dlmf.nist.gov/34.1.E1 34.1.E1] || [[Item:Q9709|<math>\ClebschGordan{j_{1}}{m_{1}}{j_{2}}{m_{2}}{j_{3}}{m_{3}} = (-1)^{j_{1}-j_{2}+m_{3}}(2j_{3}+1)^{\frac{1}{2}}\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{-m_{3}}</math>]] || <code>Error</code> || <code>ClebschGordan[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}] == (- 1)^(Subscript[j, 1]- Subscript[j, 2]+ Subscript[m, 3])*(2*Subscript[j, 3]+ 1)^(Divide[1,2])* ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], - Subscript[m, 3]}]</code> || Missing Macro Error || Failure || - || Successful [Tested: 300]
| | ; Computation : [[34.13|34.13 Methods of Computation]]<br>[[34.14|34.14 Tables]]<br>[[34.15|34.15 Software]]<br> |
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| | </div> |
| | [https://dlmf.nist.gov/34.2.E1 34.2.E1] || [[Item:Q9710|<math>|j_{r}-j_{s}| \leq j_{t}</math>]] || <code>abs(j[r]- j[s]) <= j[t]</code> || <code>Abs[Subscript[j, r]- Subscript[j, s]] <= Subscript[j, t]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/34.2.E3 34.2.E3] || [[Item:Q9712|<math>m_{1}+m_{2}+m_{3} = 0</math>]] || <code>m[1]+ m[2]+ m[3] = 0</code> || <code>Subscript[m, 1]+ Subscript[m, 2]+ Subscript[m, 3] == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/34.2.E4 34.2.E4] || [[Item:Q9713|<math>\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}} = {(-1)^{j_{1}-j_{2}-m_{3}}}\Delta(j_{1}j_{2}j_{3})\left((j_{1}+m_{1})!(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!(j_{3}+m_{3})!(j_{3}-m_{3})!\right)^{\frac{1}{2}}\*\sum_{s}\frac{(-1)^{s}}{s!(j_{1}+j_{2}-j_{3}-s)!(j_{1}-m_{1}-s)!(j_{2}+m_{2}-s)!(j_{3}-j_{2}+m_{1}+s)!(j_{3}-j_{1}-m_{2}+s)!}</math>]] || <code>Error</code> || <code>ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}] == (- 1)^(Subscript[j, 1]- Subscript[j, 2]- Subscript[m, 3])*\[CapitalDelta]*(Subscript[j, 1]*Subscript[j, 2]*Subscript[j, 3])*((Subscript[j, 1]+ Subscript[m, 1])!*(Subscript[j, 1]- Subscript[m, 1])!*(Subscript[j, 2]+ Subscript[m, 2])!*(Subscript[j, 2]- Subscript[m, 2])!*(Subscript[j, 3]+ Subscript[m, 3])!*(Subscript[j, 3]- Subscript[m, 3])!)^(Divide[1,2])* Sum[Divide[(- 1)^(s),(s)!*(Subscript[j, 1]+ Subscript[j, 2]- Subscript[j, 3]- s)!*(Subscript[j, 1]- Subscript[m, 1]- s)!*(Subscript[j, 2]+ Subscript[m, 2]- s)!*(Subscript[j, 3]- Subscript[j, 2]+ Subscript[m, 1]+ s)!*(Subscript[j, 3]- Subscript[j, 1]- Subscript[m, 2]+ s)!], {s, - Infinity, Infinity}, GenerateConditions->None]</code> || Missing Macro Error || Translation Error || - || -
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| | [https://dlmf.nist.gov/34.2.E6 34.2.E6] || [[Item:Q9715|<math>\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}} = {(-1)^{j_{2}-m_{1}+m_{3}}}\frac{(j_{1}+j_{2}+m_{3})!(j_{2}+j_{3}-m_{1})!}{\Delta(j_{1}j_{2}j_{3})(j_{1}+j_{2}+j_{3}+1)!}\left(\frac{(j_{1}+m_{1})!(j_{3}-m_{3})!}{(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!(j_{3}+m_{3})!}\right)^{\frac{1}{2}}\*{\genhyperF{3}{2}@{-j_{1}-j_{2}-j_{3}-1,-j_{1}+m_{1},-j_{3}-m_{3}}{-j_{1}-j_{2}-m_{3},-j_{2}-j_{3}+m_{1}}{1}}</math>]] || <code>Error</code> || <code>ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}] == (- 1)^(Subscript[j, 2]- Subscript[m, 1]+ Subscript[m, 3])*Divide[(Subscript[j, 1]+ Subscript[j, 2]+ Subscript[m, 3])!*(Subscript[j, 2]+ Subscript[j, 3]- Subscript[m, 1])!,\[CapitalDelta]*(Subscript[j, 1]*Subscript[j, 2]*Subscript[j, 3])*(Subscript[j, 1]+ Subscript[j, 2]+ Subscript[j, 3]+ 1)!]*(Divide[(Subscript[j, 1]+ Subscript[m, 1])!*(Subscript[j, 3]- Subscript[m, 3])!,(Subscript[j, 1]- Subscript[m, 1])!*(Subscript[j, 2]+ Subscript[m, 2])!*(Subscript[j, 2]- Subscript[m, 2])!*(Subscript[j, 3]+ Subscript[m, 3])!])^(Divide[1,2])*HypergeometricPFQ[{- Subscript[j, 1]- Subscript[j, 2]- Subscript[j, 3]- 1 , - Subscript[j, 1]+ Subscript[m, 1], - Subscript[j, 3]- Subscript[m, 3]}, {- Subscript[j, 1]- Subscript[j, 2]- Subscript[m, 3], - Subscript[j, 2]- Subscript[j, 3]+ Subscript[m, 1]}, 1]</code> || Missing Macro Error || Translation Error || - || -
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| | [https://dlmf.nist.gov/34.3.E1 34.3.E1] || [[Item:Q9716|<math>\Wignerthreejsym{j}{j}{0}{m}{-m}{0} = \frac{(-1)^{j-m}}{(2j+1)^{\frac{1}{2}}}</math>]] || <code>Error</code> || <code>ThreeJSymbol[{j, m}, {j, - m}, {m, 0}] == Divide[(- 1)^(j - m),(2*j + 1)^(Divide[1,2])]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><code>{-0.16910197872576277 <- {Rule[j, 1], Rule[m, 1]}</code><br><code>0.5773502691896258 <- {Rule[j, 1], Rule[m, 2]}</code><br></div></div>
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| | [https://dlmf.nist.gov/34.3.E2 34.3.E2] || [[Item:Q9717|<math>\Wignerthreejsym{j}{j}{1}{m}{-m}{0} = (-1)^{j-m}\frac{2m}{\left(2j(2j+1)(2j+2)\right)^{\frac{1}{2}}}</math>]] || <code>Error</code> || <code>ThreeJSymbol[{j, m}, {j, - m}, {m, 0}] == (- 1)^(j - m)*Divide[2*m,(2*j*(2*j + 1)*(2*j + 2))^(Divide[1,2])]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 9]<div class="mw-collapsible-content"><code>{0.816496580927726 <- {Rule[j, 1], Rule[m, 2]}</code><br><code>-1.224744871391589 <- {Rule[j, 1], Rule[m, 3]}</code><br></div></div>
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| | [https://dlmf.nist.gov/34.3.E3 34.3.E3] || [[Item:Q9718|<math>\Wignerthreejsym{j}{j}{1}{m}{-m-1}{1} = (-1)^{j-m}\left(\frac{2(j-m)(j+m+1)}{2j(2j+1)(2j+2)}\right)^{\frac{1}{2}}</math>]] || <code>Error</code> || <code>ThreeJSymbol[{j, m}, {j, - m - 1}, {m, 1}] == (- 1)^(j - m)*(Divide[2*(j - m)*(j + m + 1),2*j*(2*j + 1)*(2*j + 2)])^(Divide[1,2])</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 9]<div class="mw-collapsible-content"><code>{Complex[0.0, 0.5773502691896258] <- {Rule[j, 1], Rule[m, 2]}</code><br><code>Complex[0.0, -0.9128709291752769] <- {Rule[j, 1], Rule[m, 3]}</code><br></div></div>
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| | [https://dlmf.nist.gov/34.3.E6 34.3.E6] || [[Item:Q9721|<math>\Wignerthreejsym{j_{1}}{j_{2}}{j_{1}+j_{2}}{m_{1}}{m_{2}}{-m_{1}-m_{2}} = (-1)^{j_{1}-j_{2}+m_{1}+m_{2}}\left(\frac{(2j_{1})!(2j_{2})!(j_{1}+j_{2}+m_{1}+m_{2})!(j_{1}+j_{2}-m_{1}-m_{2})!}{(2j_{1}+2j_{2}+1)!(j_{1}+m_{1})!(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!}\right)^{\frac{1}{2}}</math>]] || <code>Error</code> || <code>ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], - Subscript[m, 1]- Subscript[m, 2]}] == (- 1)^(Subscript[j, 1]- Subscript[j, 2]+ Subscript[m, 1]+ Subscript[m, 2])*(Divide[(2*Subscript[j, 1])!*(2*Subscript[j, 2])!*(Subscript[j, 1]+ Subscript[j, 2]+ Subscript[m, 1]+ Subscript[m, 2])!*(Subscript[j, 1]+ Subscript[j, 2]- Subscript[m, 1]- Subscript[m, 2])!,(2*Subscript[j, 1]+ 2*Subscript[j, 2]+ 1)!*(Subscript[j, 1]+ Subscript[m, 1])!*(Subscript[j, 1]- Subscript[m, 1])!*(Subscript[j, 2]+ Subscript[m, 2])!*(Subscript[j, 2]- Subscript[m, 2])!])^(Divide[1,2])</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [277 / 300]<div class="mw-collapsible-content"><code>{Complex[-0.009681373425206639, 0.01697152361235145] <- {Rule[Subscript[j, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[m, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[m, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-0.0017082454309239698, -0.0034991197231335393] <- {Rule[Subscript[j, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[m, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[m, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/34.3.E7 34.3.E7] || [[Item:Q9722|<math>\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{j_{1}}{-j_{1}-m_{3}}{m_{3}} = (-1)^{j_{1}-j_{2}-m_{3}}\left(\frac{(2j_{1})!(-j_{1}+j_{2}+j_{3})!(j_{1}+j_{2}+m_{3})!(j_{3}-m_{3})!}{(j_{1}+j_{2}+j_{3}+1)!(j_{1}-j_{2}+j_{3})!(j_{1}+j_{2}-j_{3})!(-j_{1}+j_{2}-m_{3})!(j_{3}+m_{3})!}\right)^{\frac{1}{2}}</math>]] || <code>Error</code> || <code>ThreeJSymbol[{Subscript[j, 1], Subscript[j, 1]}, {Subscript[j, 2], - Subscript[j, 1]- Subscript[m, 3]}, {Subscript[j, 1], Subscript[m, 3]}] == (- 1)^(Subscript[j, 1]- Subscript[j, 2]- Subscript[m, 3])*(Divide[(2*Subscript[j, 1])!*(- Subscript[j, 1]+ Subscript[j, 2]+ Subscript[j, 3])!*(Subscript[j, 1]+ Subscript[j, 2]+ Subscript[m, 3])!*(Subscript[j, 3]- Subscript[m, 3])!,(Subscript[j, 1]+ Subscript[j, 2]+ Subscript[j, 3]+ 1)!*(Subscript[j, 1]- Subscript[j, 2]+ Subscript[j, 3])!*(Subscript[j, 1]+ Subscript[j, 2]- Subscript[j, 3])!*(- Subscript[j, 1]+ Subscript[j, 2]- Subscript[m, 3])!*(Subscript[j, 3]+ Subscript[m, 3])!])^(Divide[1,2])</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [265 / 300]<div class="mw-collapsible-content"><code>{Complex[1.73830318129122, -1.18098937472798] <- {Rule[Subscript[j, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[m, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-2.022477625790687, -8.26930711198898] <- {Rule[Subscript[j, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[m, 3], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/34.3.E8 34.3.E8] || [[Item:Q9723|<math>\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}} = \Wignerthreejsym{j_{2}}{j_{3}}{j_{1}}{m_{2}}{m_{3}}{m_{1}}</math>]] || <code>Error</code> || <code>ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}] == ThreeJSymbol[{Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}, {Subscript[m, 2], Subscript[m, 1]}]</code> || Missing Macro Error || Failure || - || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/34.3.E8 34.3.E8] || [[Item:Q9723|<math>\Wignerthreejsym{j_{2}}{j_{3}}{j_{1}}{m_{2}}{m_{3}}{m_{1}} = \Wignerthreejsym{j_{3}}{j_{1}}{j_{2}}{m_{3}}{m_{1}}{m_{2}}</math>]] || <code>Error</code> || <code>ThreeJSymbol[{Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}, {Subscript[m, 2], Subscript[m, 1]}] == ThreeJSymbol[{Subscript[j, 3], Subscript[m, 3]}, {Subscript[j, 1], Subscript[m, 1]}, {Subscript[m, 3], Subscript[m, 2]}]</code> || Missing Macro Error || Failure || - || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/34.3.E9 34.3.E9] || [[Item:Q9724|<math>\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}} = (-1)^{j_{1}+j_{2}+j_{3}}\Wignerthreejsym{j_{2}}{j_{1}}{j_{3}}{m_{2}}{m_{1}}{m_{3}}</math>]] || <code>Error</code> || <code>ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}] == (- 1)^(Subscript[j, 1]+ Subscript[j, 2]+ Subscript[j, 3])* ThreeJSymbol[{Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 1], Subscript[m, 1]}, {Subscript[m, 2], Subscript[m, 3]}]</code> || Missing Macro Error || Failure || - || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/34.3.E10 34.3.E10] || [[Item:Q9725|<math>\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}} = (-1)^{j_{1}+j_{2}+j_{3}}\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{-m_{1}}{-m_{2}}{-m_{3}}</math>]] || <code>Error</code> || <code>ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}] == (- 1)^(Subscript[j, 1]+ Subscript[j, 2]+ Subscript[j, 3])* ThreeJSymbol[{Subscript[j, 1], - Subscript[m, 1]}, {Subscript[j, 2], - Subscript[m, 2]}, {- Subscript[m, 1], - Subscript[m, 3]}]</code> || Missing Macro Error || Failure || - || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/34.3.E11 34.3.E11] || [[Item:Q9726|<math>\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}} = \Wignerthreejsym{j_{1}}{\frac{1}{2}(j_{2}+j_{3}+m_{1})}{\frac{1}{2}(j_{2}+j_{3}-m_{1})}{j_{2}-j_{3}}{\frac{1}{2}(j_{3}-j_{2}+m_{1})+m_{2}}{\frac{1}{2}(j_{3}-j_{2}+m_{1})+m_{3}}</math>]] || <code>Error</code> || <code>ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}] == ThreeJSymbol[{Subscript[j, 1], Subscript[j, 2]- Subscript[j, 3]}, {Divide[1,2]*(Subscript[j, 2]+ Subscript[j, 3]+ Subscript[m, 1]), Divide[1,2]*(Subscript[j, 3]- Subscript[j, 2]+ Subscript[m, 1])+ Subscript[m, 2]}, {Subscript[j, 2]- Subscript[j, 3], Divide[1,2]*(Subscript[j, 3]- Subscript[j, 2]+ Subscript[m, 1])+ Subscript[m, 3]}]</code> || Missing Macro Error || Failure || - || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/34.3.E12 34.3.E12] || [[Item:Q9727|<math>\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}} = \Wignerthreejsym{\frac{1}{2}(j_{1}+j_{2}-m_{3})}{\frac{1}{2}(j_{2}+j_{3}-m_{1})}{\frac{1}{2}(j_{1}+j_{3}-m_{2})}{j_{3}-\frac{1}{2}(j_{1}+j_{2}+m_{3})}{j_{1}-\frac{1}{2}(j_{2}+j_{3}+m_{1})}{j_{2}-\frac{1}{2}(j_{1}+j_{3}+m_{2})}</math>]] || <code>Error</code> || <code>ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}] == ThreeJSymbol[{Divide[1,2]*(Subscript[j, 1]+ Subscript[j, 2]- Subscript[m, 3]), Subscript[j, 3]-Divide[1,2]*(Subscript[j, 1]+ Subscript[j, 2]+ Subscript[m, 3])}, {Divide[1,2]*(Subscript[j, 2]+ Subscript[j, 3]- Subscript[m, 1]), Subscript[j, 1]-Divide[1,2]*(Subscript[j, 2]+ Subscript[j, 3]+ Subscript[m, 1])}, {Subscript[j, 3]-Divide[1,2]*(Subscript[j, 1]+ Subscript[j, 2]+ Subscript[m, 3]), Subscript[j, 2]-Divide[1,2]*(Subscript[j, 1]+ Subscript[j, 3]+ Subscript[m, 2])}]</code> || Missing Macro Error || Failure || - || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/34.3.E13 34.3.E13] || [[Item:Q9728|<math>\left((j_{1}+j_{2}+j_{3}+1)(-j_{1}+j_{2}+j_{3})\right)^{\frac{1}{2}}\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}} = \left((j_{2}+m_{2})(j_{3}-m_{3})\right)^{\frac{1}{2}}\Wignerthreejsym{j_{1}}{j_{2}-\frac{1}{2}}{j_{3}-\frac{1}{2}}{m_{1}}{m_{2}-\frac{1}{2}}{m_{3}+\frac{1}{2}}-\left((j_{2}-m_{2})(j_{3}+m_{3})\right)^{\frac{1}{2}}\Wignerthreejsym{j_{1}}{j_{2}-\frac{1}{2}}{j_{3}-\frac{1}{2}}{m_{1}}{m_{2}+\frac{1}{2}}{m_{3}-\frac{1}{2}}</math>]] || <code>Error</code> || <code>((Subscript[j, 1]+ Subscript[j, 2]+ Subscript[j, 3]+ 1)*(- Subscript[j, 1]+ Subscript[j, 2]+ Subscript[j, 3]))^(Divide[1,2])* ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}] == ((Subscript[j, 2]+ Subscript[m, 2])*(Subscript[j, 3]- Subscript[m, 3]))^(Divide[1,2])* ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2]-Divide[1,2], Subscript[m, 2]-Divide[1,2]}, {Subscript[m, 1], Subscript[m, 3]+Divide[1,2]}]-((Subscript[j, 2]- Subscript[m, 2])*(Subscript[j, 3]+ Subscript[m, 3]))^(Divide[1,2])* ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2]-Divide[1,2], Subscript[m, 2]+Divide[1,2]}, {Subscript[m, 1], Subscript[m, 3]-Divide[1,2]}]</code> || Missing Macro Error || Failure || - || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/34.3.E14 34.3.E14] || [[Item:Q9729|<math>\left(j_{1}(j_{1}+1)-j_{2}(j_{2}+1)-j_{3}(j_{3}+1)-2m_{2}m_{3}\right)\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}} = \left((j_{2}-m_{2})(j_{2}+m_{2}+1)(j_{3}-m_{3}+1)(j_{3}+m_{3})\right)^{\frac{1}{2}}\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}+1}{m_{3}-1}+\left((j_{2}-m_{2}+1)(j_{2}+m_{2})(j_{3}-m_{3})(j_{3}+m_{3}+1)\right)^{\frac{1}{2}}\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}-1}{m_{3}+1}</math>]] || <code>Error</code> || <code>(Subscript[j, 1]*(Subscript[j, 1]+ 1)- Subscript[j, 2]*(Subscript[j, 2]+ 1)- Subscript[j, 3]*(Subscript[j, 3]+ 1)- 2*Subscript[m, 2]*Subscript[m, 3])* ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}] == ((Subscript[j, 2]- Subscript[m, 2])*(Subscript[j, 2]+ Subscript[m, 2]+ 1)*(Subscript[j, 3]- Subscript[m, 3]+ 1)*(Subscript[j, 3]+ Subscript[m, 3]))^(Divide[1,2])* ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]+ 1}, {Subscript[m, 1], Subscript[m, 3]- 1}]+((Subscript[j, 2]- Subscript[m, 2]+ 1)*(Subscript[j, 2]+ Subscript[m, 2])*(Subscript[j, 3]- Subscript[m, 3])*(Subscript[j, 3]+ Subscript[m, 3]+ 1))^(Divide[1,2])* ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]- 1}, {Subscript[m, 1], Subscript[m, 3]+ 1}]</code> || Missing Macro Error || Failure || - || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/34.3.E15 34.3.E15] || [[Item:Q9730|<math>(2j_{1}+1)\left((j_{2}(j_{2}+1)-j_{3}(j_{3}+1))m_{1}-j_{1}(j_{1}+1)(m_{3}-m_{2})\right)\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}} = (j_{1}+1)\left(j_{1}^{2}-(j_{2}-j_{3})^{2}\right)^{\frac{1}{2}}\left((j_{2}+j_{3}+1)^{2}-j_{1}^{2}\right)^{\frac{1}{2}}\left(j_{1}^{2}-m_{1}^{2}\right)^{\frac{1}{2}}\Wignerthreejsym{j_{1}-1}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}}+j_{1}\left((j_{1}+1)^{2}-(j_{2}-j_{3})^{2}\right)^{\frac{1}{2}}\left((j_{2}+j_{3}+1)^{2}-(j_{1}+1)^{2}\right)^{\frac{1}{2}}\left((j_{1}+1)^{2}-m_{1}^{2}\right)^{\frac{1}{2}}\Wignerthreejsym{j_{1}+1}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}}</math>]] || <code>Error</code> || <code>(2*Subscript[j, 1]+ 1)*((Subscript[j, 2]*(Subscript[j, 2]+ 1)- Subscript[j, 3]*(Subscript[j, 3]+ 1))*Subscript[m, 1]- Subscript[j, 1]*(Subscript[j, 1]+ 1)*(Subscript[m, 3]- Subscript[m, 2]))* ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}] (j(Subscript[j, 1])^(2)- m(Subscript[m, 1])^(2))^(Divide[1,2])* ThreeJSymbol[{Subscript[j, 1]- 1, Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}]+ Subscript[j, 1]*((Subscript[j, 1]+ 1)^(2)-(Subscript[j, 2]- Subscript[j, 3])^(2))^(Divide[1,2])*((Subscript[j, 1]+ 1)^(2)- m(Subscript[m, 1])^(2))^(Divide[1,2])* ThreeJSymbol[{Subscript[j, 1]+ 1, Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}]</code> || Missing Macro Error || Failure || - || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/34.3.E18 34.3.E18] || [[Item:Q9733|<math>\sum_{m_{1}m_{2}m_{3}}\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}}\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}} = 1</math>]] || <code>Error</code> || <code>Sum[ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}]*ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}], {Subscript[m, 1]*Subscript[m, 2]*Subscript[m, 3], - Infinity, Infinity}, GenerateConditions->None] == 1</code> || Translation Error || Translation Error || - || -
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| | [https://dlmf.nist.gov/34.3.E19 34.3.E19] || [[Item:Q9734|<math>\assLegendreP[]{l_{1}}@{\cos@@{\theta}}\assLegendreP[]{l_{2}}@{\cos@@{\theta}} = \sum_{l}(2l+1)\Wignerthreejsym{l_{1}}{l_{2}}{l}{0}{0}{0}^{2}\assLegendreP[]{l}@{\cos@@{\theta}}</math>]] || <code>Error</code> || <code>LegendreP[Subscript[l, 1], 0, 3, Cos[\[Theta]]]*LegendreP[Subscript[l, 2], 0, 3, Cos[\[Theta]]] == Sum[(2*l + 1)* (ThreeJSymbol[{Subscript[l, 1], 0}, {Subscript[l, 2], 0}, {0, 0}])^(2)* LegendreP[l, 0, 3, Cos[\[Theta]]], {l, - Infinity, Infinity}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/34.3.E20 34.3.E20] || [[Item:Q9735|<math>\sphharmonicY{l_{1}}{m_{1}}@{\theta}{\phi}\sphharmonicY{l_{2}}{m_{2}}@{\theta}{\phi} = \sum_{l,m}\left(\frac{(2l_{1}+1)(2l_{2}+1)(2l+1)}{4\pi}\right)^{\frac{1}{2}}\Wignerthreejsym{l_{1}}{l_{2}}{l}{m_{1}}{m_{2}}{m}\conj{\sphharmonicY{l}{m}@{\theta}{\phi}}\Wignerthreejsym{l_{1}}{l_{2}}{l}{0}{0}{0}</math>]] || <code>Error</code> || <code>SphericalHarmonicY[Subscript[l, 1], Subscript[m, 1], \[Theta], \[Phi]]*SphericalHarmonicY[Subscript[l, 2], Subscript[m, 2], \[Theta], \[Phi]] == Sum[Sum[(Divide[(2*Subscript[l, 1]+ 1)*(2*Subscript[l, 2]+ 1)*(2*l + 1),4*Pi])^(Divide[1,2])* ThreeJSymbol[{Subscript[l, 1], Subscript[m, 1]}, {Subscript[l, 2], Subscript[m, 2]}, {Subscript[m, 1], m}]*Conjugate[SphericalHarmonicY[l, m, \[Theta], \[Phi]]]*ThreeJSymbol[{Subscript[l, 1], 0}, {Subscript[l, 2], 0}, {0, 0}], {m, - Infinity, Infinity}, GenerateConditions->None], {l, - Infinity, Infinity}, GenerateConditions->None]</code> || Translation Error || Translation Error || - || -
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| | [https://dlmf.nist.gov/34.3.E21 34.3.E21] || [[Item:Q9736|<math>\int_{0}^{\pi}\assLegendreP[]{l_{1}}@{\cos@@{\theta}}\assLegendreP[]{l_{2}}@{\cos@@{\theta}}\assLegendreP[]{l_{3}}@{\cos@@{\theta}}\sin@@{\theta}\diff{\theta} = 2\Wignerthreejsym{l_{1}}{l_{2}}{l_{3}}{0}{0}{0}^{2}</math>]] || <code>Error</code> || <code>Integrate[LegendreP[Subscript[l, 1], 0, 3, Cos[\[Theta]]]*LegendreP[Subscript[l, 2], 0, 3, Cos[\[Theta]]]*LegendreP[Subscript[l, 3], 0, 3, Cos[\[Theta]]]*Sin[\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None] == 2*(ThreeJSymbol[{Subscript[l, 1], 0}, {Subscript[l, 2], 0}, {0, 0}])^(2)</code> || Missing Macro Error || Failure || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/34.3.E22 34.3.E22] || [[Item:Q9737|<math>\int_{0}^{2\pi}\!\int_{0}^{\pi}\sphharmonicY{l_{1}}{m_{1}}@{\theta}{\phi}\sphharmonicY{l_{2}}{m_{2}}@{\theta}{\phi}\sphharmonicY{l_{3}}{m_{3}}@{\theta}{\phi}\sin@@{\theta}\diff{\theta}\diff{\phi} = \left(\frac{(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi}\right)^{\frac{1}{2}}\Wignerthreejsym{l_{1}}{l_{2}}{l_{3}}{0}{0}{0}\Wignerthreejsym{l_{1}}{l_{2}}{l_{3}}{m_{1}}{m_{2}}{m_{3}}</math>]] || <code>Error</code> || <code>Integrate[Integrate[SphericalHarmonicY[Subscript[l, 1], Subscript[m, 1], \[Theta], \[Phi]]*SphericalHarmonicY[Subscript[l, 2], Subscript[m, 2], \[Theta], \[Phi]]*SphericalHarmonicY[Subscript[l, 3], Subscript[m, 3], \[Theta], \[Phi]]*Sin[\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None], {\[Phi], 0, 2*Pi}, GenerateConditions->None] == (Divide[(2*Subscript[l, 1]+ 1)*(2*Subscript[l, 2]+ 1)*(2*Subscript[l, 3]+ 1),4*Pi])^(Divide[1,2])* ThreeJSymbol[{Subscript[l, 1], 0}, {Subscript[l, 2], 0}, {0, 0}]*ThreeJSymbol[{Subscript[l, 1], Subscript[m, 1]}, {Subscript[l, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}]</code> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/34.4.E2 34.4.E2] || [[Item:Q9739|<math>\Wignersixjsym{j_{1}}{j_{2}}{j_{3}}{l_{1}}{l_{2}}{l_{3}} = \Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})\Delta(l_{1}j_{2}l_{3})\Delta(l_{1}l_{2}j_{3})\*\sum_{s}\frac{(-1)^{s}(s+1)!}{(s-j_{1}-j_{2}-j_{3})!(s-j_{1}-l_{2}-l_{3})!(s-l_{1}-j_{2}-l_{3})!(s-l_{1}-l_{2}-j_{3})!}\*\frac{1}{(j_{1}+j_{2}+l_{1}+l_{2}-s)!(j_{2}+j_{3}+l_{2}+l_{3}-s)!(j_{3}+j_{1}+l_{3}+l_{1}-s)!}</math>]] || <code>Error</code> || <code>SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {Subscript[l, 1], Subscript[l, 2], Subscript[l, 3]}] == \[CapitalDelta]*(Subscript[j, 1]*Subscript[j, 2]*Subscript[j, 3])* \[CapitalDelta]*(Subscript[j, 1]*Subscript[l, 2]*Subscript[l, 3])* \[CapitalDelta]*(Subscript[l, 1]*Subscript[j, 2]*Subscript[l, 3])* \[CapitalDelta]*(Subscript[l, 1]*Subscript[l, 2]*Subscript[j, 3])* Sum[Divide[(- 1)^(s)*(s + 1)!,(s - Subscript[j, 1]- Subscript[j, 2]- Subscript[j, 3])!*(s - Subscript[j, 1]- Subscript[l, 2]- Subscript[l, 3])!*(s - Subscript[l, 1]- Subscript[j, 2]- Subscript[l, 3])!*(s - Subscript[l, 1]- Subscript[l, 2]- Subscript[j, 3])!]*Divide[1,(Subscript[j, 1]+ Subscript[j, 2]+ Subscript[l, 1]+ Subscript[l, 2]- s)!*(Subscript[j, 2]+ Subscript[j, 3]+ Subscript[l, 2]+ Subscript[l, 3]- s)!*(Subscript[j, 3]+ Subscript[j, 1]+ Subscript[l, 3]+ Subscript[l, 1]- s)!], {s, - Infinity, Infinity}, GenerateConditions->None]</code> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/34.4.E3 34.4.E3] || [[Item:Q9740|<math>\Wignersixjsym{j_{1}}{j_{2}}{j_{3}}{l_{1}}{l_{2}}{l_{3}} = {(-1)^{j_{1}+j_{3}+l_{1}+l_{3}}}\frac{\Delta(j_{1}j_{2}j_{3})\Delta(j_{2}l_{1}l_{3})(j_{1}-j_{2}+l_{1}+l_{2})!(-j_{2}+j_{3}+l_{2}+l_{3})!(j_{1}+j_{3}+l_{1}+l_{3}+1)!}{\Delta(j_{1}l_{2}l_{3})\Delta(j_{3}l_{1}l_{2})(j_{1}-j_{2}+j_{3})!(-j_{2}+l_{1}+l_{3})!(j_{1}+l_{2}+l_{3}+1)!(j_{3}+l_{1}+l_{2}+1)!}\*\genhyperF{4}{3}@@{-j_{1}+j_{2}-j_{3},j_{2}-l_{1}-l_{3},-j_{1}-l_{2}-l_{3}-1,-j_{3}-l_{1}-l_{2}-1}{-j_{1}+j_{2}-l_{1}-l_{2},j_{2}-j_{3}-l_{2}-l_{3},-j_{1}-j_{3}-l_{1}-l_{3}-1}{1}</math>]] || <code>Error</code> || <code>SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {Subscript[l, 1], Subscript[l, 2], Subscript[l, 3]}] == (- 1)^(Subscript[j, 1]+ Subscript[j, 3]+ Subscript[l, 1]+ Subscript[l, 3])*Divide[\[CapitalDelta]*(Subscript[j, 1]*Subscript[j, 2]*Subscript[j, 3])* \[CapitalDelta]*(Subscript[j, 2]*Subscript[l, 1]*Subscript[l, 3])*(Subscript[j, 1]- Subscript[j, 2]+ Subscript[l, 1]+ Subscript[l, 2])!*(- Subscript[j, 2]+ Subscript[j, 3]+ Subscript[l, 2]+ Subscript[l, 3])!*(Subscript[j, 1]+ Subscript[j, 3]+ Subscript[l, 1]+ Subscript[l, 3]+ 1)!,\[CapitalDelta]*(Subscript[j, 1]*Subscript[l, 2]*Subscript[l, 3])* \[CapitalDelta]*(Subscript[j, 3]*Subscript[l, 1]*Subscript[l, 2])*(Subscript[j, 1]- Subscript[j, 2]+ Subscript[j, 3])!*(- Subscript[j, 2]+ Subscript[l, 1]+ Subscript[l, 3])!*(Subscript[j, 1]+ Subscript[l, 2]+ Subscript[l, 3]+ 1)!*(Subscript[j, 3]+ Subscript[l, 1]+ Subscript[l, 2]+ 1)!]* HypergeometricPFQ[{- Subscript[j, 1]+ Subscript[j, 2]- Subscript[j, 3], Subscript[j, 2]- Subscript[l, 1]- Subscript[l, 3], - Subscript[j, 1]- Subscript[l, 2]- Subscript[l, 3]- 1 , - Subscript[j, 3]- Subscript[l, 1]- Subscript[l, 2]- 1}, {- Subscript[j, 1]+ Subscript[j, 2]- Subscript[l, 1]- Subscript[l, 2], Subscript[j, 2]- Subscript[j, 3]- Subscript[l, 2]- Subscript[l, 3], - Subscript[j, 1]- Subscript[j, 3]- Subscript[l, 1]- Subscript[l, 3]- 1}, 1]</code> || Missing Macro Error || Failure || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/34.5.E1 34.5.E1] || [[Item:Q9741|<math>\Wignersixjsym{j_{1}}{j_{2}}{j_{3}}{0}{j_{3}}{j_{2}} = \frac{(-1)^{J}}{\left((2j_{2}+1)(2j_{3}+1)\right)^{\frac{1}{2}}}</math>]] || <code>Error</code> || <code>SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {0, Subscript[j, 3], Subscript[j, 2]}] == Divide[(- 1)^(J),((2*Subscript[j, 2]+ 1)*(2*Subscript[j, 3]+ 1))^(Divide[1,2])]</code> || Missing Macro Error || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Plus[Complex[-1.3200489060811386*^-4, -0.0030849476982290776], Piecewise[{{Complex[-23.42903161903947, -30.25069364758892], Or[And[Greater[Complex[0.8660254037844387, 0.49999999999999994], 0.0], Inequality[0.0, Less, Complex[0.8660254037844387, 0.49999999999999994], Less, Complex[0.8660254037844387, 0.49999999999999994]]], And[Greater[Complex[0.8660254037844387, 0.49999999999999994], 0.0], Inequality[Complex[0.43301270189221935, 0.24999999999999997], Less, Complex[0.8660254037844387, 0.49999999999999994], Less, Complex[0.8660254037844387, 0.49999999999999994]], Inequality[0.0, LessEqual, Complex[0.8660254037844387, 0.49999999999999994], Less, Complex[0.8660254037844387, 0.49999999999999994]]], And[Greater[Complex[0.8660254037844387, 0.49999999999999994], 0.0], Inequality[0.0, Less, Complex[0.8660254037844387, 0.49999999999999994], Less, Complex[0.43301270189221935, 0.24999999999999997]], Inequality[0.0, LessEqual, Complex[0.8660254037844387, 0.49999999999999994], LessEqual, Comp</div></div>
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| | [https://dlmf.nist.gov/34.5.E2 34.5.E2] || [[Item:Q9742|<math>\Wignersixjsym{j_{1}}{j_{2}}{j_{3}}{\frac{1}{2}}{j_{3}-\frac{1}{2}}{j_{2}+\frac{1}{2}} = (-1)^{J}\left(\frac{(j_{1}+j_{3}-j_{2})(j_{1}+j_{2}-j_{3}+1)}{(2j_{2}+1)(2j_{2}+2)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}}</math>]] || <code>Error</code> || <code>SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {Divide[1,2], Subscript[j, 3]-Divide[1,2], Subscript[j, 2]+Divide[1,2]}] == (- 1)^(J)*(Divide[(Subscript[j, 1]+ Subscript[j, 3]- Subscript[j, 2])*(Subscript[j, 1]+ Subscript[j, 2]- Subscript[j, 3]+ 1),(2*Subscript[j, 2]+ 1)*(2*Subscript[j, 2]+ 2)* 2*Subscript[j, 3]*(2*Subscript[j, 3]+ 1)])^(Divide[1,2])</code> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/34.5.E3 34.5.E3] || [[Item:Q9743|<math>\Wignersixjsym{j_{1}}{j_{2}}{j_{3}}{\frac{1}{2}}{j_{3}-\frac{1}{2}}{j_{2}-\frac{1}{2}} = (-1)^{J}\left(\frac{(j_{2}+j_{3}-j_{1})(j_{1}+j_{2}+j_{3}+1)}{2j_{2}(2j_{2}+1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}}</math>]] || <code>Error</code> || <code>SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {Divide[1,2], Subscript[j, 3]-Divide[1,2], Subscript[j, 2]-Divide[1,2]}] == (- 1)^(J)*(Divide[(Subscript[j, 2]+ Subscript[j, 3]- Subscript[j, 1])*(Subscript[j, 1]+ Subscript[j, 2]+ Subscript[j, 3]+ 1),2*Subscript[j, 2]*(2*Subscript[j, 2]+ 1)* 2*Subscript[j, 3]*(2*Subscript[j, 3]+ 1)])^(Divide[1,2])</code> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/34.5.E4 34.5.E4] || [[Item:Q9744|<math>\Wignersixjsym{j_{1}}{j_{2}}{j_{3}}{1}{j_{3}-1}{j_{2}-1} = (-1)^{J}\left(\frac{J(J+1)(J-2j_{1})(J-2j_{1}-1)}{(2j_{2}-1)2j_{2}(2j_{2}+1)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}}</math>]] || <code>Error</code> || <code>SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {1, Subscript[j, 3]- 1, Subscript[j, 2]- 1}] == (- 1)^(J)*(Divide[J*(J + 1)*(J - 2*Subscript[j, 1])*(J - 2*Subscript[j, 1]- 1),(2*Subscript[j, 2]- 1)* 2*Subscript[j, 2]*(2*Subscript[j, 2]+ 1)*(2*Subscript[j, 3]- 1)* 2*Subscript[j, 3]*(2*Subscript[j, 3]+ 1)])^(Divide[1,2])</code> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/34.5.E5 34.5.E5] || [[Item:Q9745|<math>\Wignersixjsym{j_{1}}{j_{2}}{j_{3}}{1}{j_{3}-1}{j_{2}} = (-1)^{J}\left(\frac{2(J+1)(J-2j_{1})(J-2j_{2})(J-2j_{3}+1)}{2j_{2}(2j_{2}+1)(2j_{2}+2)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}}</math>]] || <code>Error</code> || <code>SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {1, Subscript[j, 3]- 1, Subscript[j, 2]}] == (- 1)^(J)*(Divide[2*(J + 1)*(J - 2*Subscript[j, 1])*(J - 2*Subscript[j, 2])*(J - 2*Subscript[j, 3]+ 1),2*Subscript[j, 2]*(2*Subscript[j, 2]+ 1)*(2*Subscript[j, 2]+ 2)*(2*Subscript[j, 3]- 1)* 2*Subscript[j, 3]*(2*Subscript[j, 3]+ 1)])^(Divide[1,2])</code> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/34.5.E6 34.5.E6] || [[Item:Q9746|<math>\Wignersixjsym{j_{1}}{j_{2}}{j_{3}}{1}{j_{3}-1}{j_{2}+1} = (-1)^{J}\left(\frac{(J-2j_{2}-1)(J-2j_{2})(J-2j_{3}+1)(J-2j_{3}+2)}{(2j_{2}+1)(2j_{2}+2)(2j_{2}+3)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}}</math>]] || <code>Error</code> || <code>SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {1, Subscript[j, 3]- 1, Subscript[j, 2]+ 1}] == (- 1)^(J)*(Divide[(J - 2*Subscript[j, 2]- 1)*(J - 2*Subscript[j, 2])*(J - 2*Subscript[j, 3]+ 1)*(J - 2*Subscript[j, 3]+ 2),(2*Subscript[j, 2]+ 1)*(2*Subscript[j, 2]+ 2)*(2*Subscript[j, 2]+ 3)*(2*Subscript[j, 3]- 1)* 2*Subscript[j, 3]*(2*Subscript[j, 3]+ 1)])^(Divide[1,2])</code> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/34.5.E7 34.5.E7] || [[Item:Q9747|<math>\Wignersixjsym{j_{1}}{j_{2}}{j_{3}}{1}{j_{3}}{j_{2}} = (-1)^{J+1}\frac{2(j_{2}(j_{2}+1)+j_{3}(j_{3}+1)-j_{1}(j_{1}+1))}{\left(2j_{2}(2j_{2}+1)(2j_{2}+2)2j_{3}(2j_{3}+1)(2j_{3}+2)\right)^{\frac{1}{2}}}</math>]] || <code>Error</code> || <code>SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {1, Subscript[j, 3], Subscript[j, 2]}] == (- 1)^(J + 1)*Divide[2*(Subscript[j, 2]*(Subscript[j, 2]+ 1)+ Subscript[j, 3]*(Subscript[j, 3]+ 1)- Subscript[j, 1]*(Subscript[j, 1]+ 1)),(2*Subscript[j, 2]*(2*Subscript[j, 2]+ 1)*(2*Subscript[j, 2]+ 2)*2*Subscript[j, 3]*(2*Subscript[j, 3]+ 1)*(2*Subscript[j, 3]+ 2))^(Divide[1,2])]</code> || Missing Macro Error || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Plus[Complex[6.600244530405688*^-5, 0.0015424738491145388], Piecewise[{{Complex[11.71451580951971, 15.125346823794441], Or[And[Inequality[0.5, Less, Complex[0.8660254037844387, 0.49999999999999994], LessEqual, 1.0], Inequality[0.5, LessEqual, Complex[0.8660254037844387, 0.49999999999999994], Less, Complex[0.8660254037844387, 0.49999999999999994]], Inequality[Complex[0.8660254037844387, 0.49999999999999994], LessEqual, Complex[0.8660254037844387, 0.49999999999999994], LessEqual, Complex[1.7320508075688774, 0.9999999999999999]]], And[Inequality[0.5, LessEqual, Complex[0.8660254037844387, 0.49999999999999994], LessEqual, 1.0], Inequality[1.0, Less, Complex[0.8660254037844387, 0.49999999999999994], Less, 1.5], Inequality[Complex[0.8660254037844387, 0.49999999999999994], LessEqual, Complex[0.8660254037844387, 0.49999999999999994], LessEqual, 1.5]], And[Inequality[1.0, Less, Complex[0.8660254037844387, 0.49999999999999994], Less, 1.5], Inequality[Complex[1.3660254037844388, 0.49999999</div></div>
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| | [https://dlmf.nist.gov/34.5.E8 34.5.E8] || [[Item:Q9748|<math>\Wignersixjsym{j_{1}}{j_{2}}{j_{3}}{l_{1}}{l_{2}}{l_{3}} = \Wignersixjsym{j_{2}}{j_{1}}{j_{3}}{l_{2}}{l_{1}}{l_{3}}</math>]] || <code>Error</code> || <code>SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {Subscript[l, 1], Subscript[l, 2], Subscript[l, 3]}] == SixJSymbol[{Subscript[j, 2], Subscript[j, 1], Subscript[j, 3]}, {Subscript[l, 2], Subscript[l, 1], Subscript[l, 3]}]</code> || Missing Macro Error || Failure || - || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/34.5.E8 34.5.E8] || [[Item:Q9748|<math>\Wignersixjsym{j_{2}}{j_{1}}{j_{3}}{l_{2}}{l_{1}}{l_{3}} = \Wignersixjsym{j_{1}}{l_{2}}{l_{3}}{l_{1}}{j_{2}}{j_{3}}</math>]] || <code>Error</code> || <code>SixJSymbol[{Subscript[j, 2], Subscript[j, 1], Subscript[j, 3]}, {Subscript[l, 2], Subscript[l, 1], Subscript[l, 3]}] == SixJSymbol[{Subscript[j, 1], Subscript[l, 2], Subscript[l, 3]}, {Subscript[l, 1], Subscript[j, 2], Subscript[j, 3]}]</code> || Missing Macro Error || Failure || - || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/34.5.E9 34.5.E9] || [[Item:Q9749|<math>\Wignersixjsym{j_{1}}{j_{2}}{j_{3}}{l_{1}}{l_{2}}{l_{3}} = \Wignersixjsym{j_{1}}{\frac{1}{2}(j_{2}+l_{2}+j_{3}-l_{3})}{\frac{1}{2}(j_{2}-l_{2}+j_{3}+l_{3})}{l_{1}}{\frac{1}{2}(j_{2}+l_{2}-j_{3}+l_{3})}{\frac{1}{2}(-j_{2}+l_{2}+j_{3}+l_{3})}</math>]] || <code>Error</code> || <code>SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {Subscript[l, 1], Subscript[l, 2], Subscript[l, 3]}] == SixJSymbol[{Subscript[j, 1], Divide[1,2]*(Subscript[j, 2]+ Subscript[l, 2]+ Subscript[j, 3]- Subscript[l, 3]), Divide[1,2]*(Subscript[j, 2]- Subscript[l, 2]+ Subscript[j, 3]+ Subscript[l, 3])}, {Subscript[l, 1], Divide[1,2]*(Subscript[j, 2]+ Subscript[l, 2]- Subscript[j, 3]+ Subscript[l, 3]), Divide[1,2]*(- Subscript[j, 2]+ Subscript[l, 2]+ Subscript[j, 3]+ Subscript[l, 3])}]</code> || Missing Macro Error || Failure || - || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/34.5.E10 34.5.E10] || [[Item:Q9750|<math>\Wignersixjsym{j_{1}}{j_{2}}{j_{3}}{l_{1}}{l_{2}}{l_{3}} = \Wignersixjsym{\frac{1}{2}(j_{2}+l_{2}+j_{3}-l_{3})}{\frac{1}{2}(j_{1}-l_{1}+j_{3}+l_{3})}{\frac{1}{2}(j_{1}+l_{1}+j_{2}-l_{2})}{\frac{1}{2}(j_{2}+l_{2}-j_{3}+l_{3})}{\frac{1}{2}(-j_{1}+l_{1}+j_{3}+l_{3})}{\frac{1}{2}(j_{1}+l_{1}-j_{2}+l_{2})}</math>]] || <code>Error</code> || <code>SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {Subscript[l, 1], Subscript[l, 2], Subscript[l, 3]}] == SixJSymbol[{Divide[1,2]*(Subscript[j, 2]+ Subscript[l, 2]+ Subscript[j, 3]- Subscript[l, 3]), Divide[1,2]*(Subscript[j, 1]- Subscript[l, 1]+ Subscript[j, 3]+ Subscript[l, 3]), Divide[1,2]*(Subscript[j, 1]+ Subscript[l, 1]+ Subscript[j, 2]- Subscript[l, 2])}, {Divide[1,2]*(Subscript[j, 2]+ Subscript[l, 2]- Subscript[j, 3]+ Subscript[l, 3]), Divide[1,2]*(- Subscript[j, 1]+ Subscript[l, 1]+ Subscript[j, 3]+ Subscript[l, 3]), Divide[1,2]*(Subscript[j, 1]+ Subscript[l, 1]- Subscript[j, 2]+ Subscript[l, 2])}]</code> || Missing Macro Error || Failure || - || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/34.5.E11 34.5.E11] || [[Item:Q9751|<math>{(2j_{1}+1)\left((J_{3}+J_{2}-J_{1})(L_{3}+L_{2}-J_{1})-2(J_{3}L_{3}+J_{2}L_{2}-J_{1}L_{1})\right)\Wignersixjsym{j_{1}}{j_{2}}{j_{3}}{l_{1}}{l_{2}}{l_{3}}}\\ = j_{1}E(j_{1}+1)\Wignersixjsym{j_{1}+1}{j_{2}}{j_{3}}{l_{1}}{l_{2}}{l_{3}}+(j_{1}+1)E(j_{1})\Wignersixjsym{j_{1}-1}{j_{2}}{j_{3}}{l_{1}}{l_{2}}{l_{3}}</math>]] || <code>Error</code> || <code>(2*Subscript[j, 1]+ 1)*((Subscript[J, 3]+ Subscript[J, 2]- Subscript[J, 1])*(Subscript[L, 3]+ Subscript[L, 2]- Subscript[J, 1])- 2*(Subscript[J, 3]*Subscript[L, 3]+ Subscript[J, 2]*Subscript[L, 2]- Subscript[J, 1]*Subscript[L, 1]))* SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {Subscript[l, 1], Subscript[l, 2], Subscript[l, 3]}] == Subscript[j, 1]*E*(Subscript[j, 1]+ 1)* SixJSymbol[{Subscript[j, 1]+ 1, Subscript[j, 2], Subscript[j, 3]}, {Subscript[l, 1], Subscript[l, 2], Subscript[l, 3]}]+(Subscript[j, 1]+ 1)* E*(Subscript[j, 1])* SixJSymbol[{Subscript[j, 1]- 1, Subscript[j, 2], Subscript[j, 3]}, {Subscript[l, 1], Subscript[l, 2], Subscript[l, 3]}]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"></div></div>
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| | [https://dlmf.nist.gov/34.5.E19 34.5.E19] || [[Item:Q9760|<math>\sum_{l}\Wignersixjsym{j_{1}}{j_{2}}{l}{j_{2}}{j_{1}}{j} = 0</math>]] || <code>Error</code> || <code>Sum[SixJSymbol[{Subscript[j, 1], Subscript[j, 2], l}, {Subscript[j, 2], Subscript[j, 1], j}], {l, - Infinity, Infinity}, GenerateConditions->None] == 0</code> || Missing Macro Error || Failure || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/34.5.E20 34.5.E20] || [[Item:Q9761|<math>\sum_{l}(-1)^{l+j}\Wignersixjsym{j_{1}}{j_{2}}{l}{j_{1}}{j_{2}}{j} = \frac{(-1)^{2\mu}}{2j+1}</math>]] || <code>Error</code> || <code>Sum[(- 1)^(l + j)* SixJSymbol[{Subscript[j, 1], Subscript[j, 2], l}, {Subscript[j, 1], Subscript[j, 2], j}], {l, - Infinity, Infinity}, GenerateConditions->None] == Divide[(- 1)^(2*\[Mu]),2*j + 1]</code> || Missing Macro Error || Failure || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/34.5.E21 34.5.E21] || [[Item:Q9762|<math>\sum_{l}(-1)^{l+j+j_{1}+j_{2}}\Wignersixjsym{j_{1}}{j_{2}}{l}{j_{2}}{j_{1}}{j} = \frac{1}{2j+1}\left(\frac{(2j_{1}-j)!(2j_{2}+j+1)!}{(2j_{2}-j)!(2j_{1}+j+1)!}\right)^{\frac{1}{2}}</math>]] || <code>Error</code> || <code>Sum[(- 1)^(l + j + Subscript[j, 1]+ Subscript[j, 2])* SixJSymbol[{Subscript[j, 1], Subscript[j, 2], l}, {Subscript[j, 2], Subscript[j, 1], j}], {l, - Infinity, Infinity}, GenerateConditions->None] == Divide[1,2*j + 1]*(Divide[(2*Subscript[j, 1]- j)!*(2*Subscript[j, 2]+ j + 1)!,(2*Subscript[j, 2]- j)!*(2*Subscript[j, 1]+ j + 1)!])^(Divide[1,2])</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [38 / 63]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[j, 1], Rule[Subscript[j, 1], -1.5], Rule[Subscript[j, 2], -1.5]}</code><br><code>Indeterminate <- {Rule[j, 2], Rule[Subscript[j, 1], -1.5], Rule[Subscript[j, 2], -1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/34.5.E22 34.5.E22] || [[Item:Q9763|<math>\sum_{l}(-1)^{l+j+j_{1}+j_{2}}\frac{1}{l(l+1)}\Wignersixjsym{j_{1}}{j_{2}}{l}{j_{2}}{j_{1}}{j} = \frac{1}{j_{1}(j_{1}+1)-j_{2}(j_{2}+1)}\left(\frac{(2j_{1}-j)!(2j_{2}+j+1)!}{(2j_{2}-j)!(2j_{1}+j+1)!}\right)^{\frac{1}{2}}</math>]] || <code>Error</code> || <code>Sum[(- 1)^(l + j + Subscript[j, 1]+ Subscript[j, 2])*Divide[1,l*(l + 1)]*SixJSymbol[{Subscript[j, 1], Subscript[j, 2], l}, {Subscript[j, 2], Subscript[j, 1], j}], {l, - Infinity, Infinity}, GenerateConditions->None] == Divide[1,Subscript[j, 1]*(Subscript[j, 1]+ 1)- Subscript[j, 2]*(Subscript[j, 2]+ 1)]*(Divide[(2*Subscript[j, 1]- j)!*(2*Subscript[j, 2]+ j + 1)!,(2*Subscript[j, 2]- j)!*(2*Subscript[j, 1]+ j + 1)!])^(Divide[1,2])</code> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/34.8.E2 34.8.E2] || [[Item:Q9773|<math>\cos@@{\theta} = \frac{j_{1}(j_{1}+1)+j_{2}(j_{2}+1)-j_{3}(j_{3}+1)}{2\sqrt{j_{1}(j_{1}+1)j_{2}(j_{2}+1)}}</math>]] || <code>cos(theta) = (j[1]*(j[1]+ 1)+ j[2]*(j[2]+ 1)- j[3]*(j[3]+ 1))/(2*sqrt(j[1]*(j[1]+ 1)* j[2]*(j[2]+ 1)))</code> || <code>Cos[\[Theta]] == Divide[Subscript[j, 1]*(Subscript[j, 1]+ 1)+ Subscript[j, 2]*(Subscript[j, 2]+ 1)- Subscript[j, 3]*(Subscript[j, 3]+ 1),2*Sqrt[Subscript[j, 1]*(Subscript[j, 1]+ 1)* Subscript[j, 2]*(Subscript[j, 2]+ 1)]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.2305430189-.3969495503*I <- {theta = 1/2*3^(1/2)+1/2*I, j[1] = 1/2*3^(1/2)+1/2*I, j[2] = 1/2*3^(1/2)+1/2*I, j[3] = 1/2*3^(1/2)+1/2*I}</code><br><code>-.4524696831-.2139368486*I <- {theta = 1/2*3^(1/2)+1/2*I, j[1] = 1/2*3^(1/2)+1/2*I, j[2] = 1/2*3^(1/2)+1/2*I, j[3] = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[0.23054301905722518, -0.39694955022903244] <- {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-0.452469682834994, -0.2139368483368131] <- {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, 3], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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