3 j , 6 j , 9 j Symbols - 34.3 Basic Properties: Symbol

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
34.3.E1 3 ⁢ j ⁢ j ⁢ j ⁢ 0 ⁢ m - m ⁢ 0 = ( - 1 ) j - m ( 2 ⁢ j + 1 ) 1 2 threej 𝑗 𝑗 0 𝑚 𝑚 0 superscript 1 𝑗 𝑚 superscript 2 𝑗 1 1 2 {\displaystyle{\displaystyle\mathit{3j}{j}{j}{0}{m}{-m}{0}=\frac{(-1)^{j-m}}{(% 2j+1)^{\frac{1}{2}}}}}
\Wignerthreejsym{j}{j}{0}{m}{-m}{0} = \frac{(-1)^{j-m}}{(2j+1)^{\frac{1}{2}}}		 

Error

ThreeJSymbol[{j, m}, {j, - m}, {m, 0}] == Divide[(- 1)^(j - m),(2*j + 1)^(Divide[1,2])]
Missing Macro Error Failure -
Failed [9 / 9]
Result: -0.16910197872576277
Test Values: {Rule[j, 1], Rule[m, 1]}


Result: 0.5773502691896258
Test Values: {Rule[j, 1], Rule[m, 2]}

... skip entries to safe data
34.3.E2 3 ⁢ j ⁢ j ⁢ j ⁢ 1 ⁢ m - m ⁢ 0 = ( - 1 ) j - m ⁢ 2 ⁢ m ( 2 ⁢ j ⁢ ( 2 ⁢ j + 1 ) ⁢ ( 2 ⁢ j + 2 ) ) 1 2 threej 𝑗 𝑗 1 𝑚 𝑚 0 superscript 1 𝑗 𝑚 2 𝑚 superscript 2 𝑗 2 𝑗 1 2 𝑗 2 1 2 {\displaystyle{\displaystyle\mathit{3j}{j}{j}{1}{m}{-m}{0}=(-1)^{j-m}\frac{2m}% {\left(2j(2j+1)(2j+2)\right)^{\frac{1}{2}}}}}
\Wignerthreejsym{j}{j}{1}{m}{-m}{0} = (-1)^{j-m}\frac{2m}{\left(2j(2j+1)(2j+2)\right)^{\frac{1}{2}}}		 j ≥ 1 2 𝑗 1 2 {\displaystyle{\displaystyle j\geq\tfrac{1}{2}}} 
Error

ThreeJSymbol[{j, m}, {j, - m}, {m, 0}] == (- 1)^(j - m)*Divide[2*m,(2*j*(2*j + 1)*(2*j + 2))^(Divide[1,2])]
Missing Macro Error Failure -
Failed [6 / 9]
Result: 0.816496580927726
Test Values: {Rule[j, 1], Rule[m, 2]}


Result: -1.224744871391589
Test Values: {Rule[j, 1], Rule[m, 3]}

... skip entries to safe data
34.3.E3 3 ⁢ j ⁢ j ⁢ j ⁢ 1 ⁢ m - m - 11 = ( - 1 ) j - m ⁢ ( 2 ⁢ ( j - m ) ⁢ ( j + m + 1 ) 2 ⁢ j ⁢ ( 2 ⁢ j + 1 ) ⁢ ( 2 ⁢ j + 2 ) ) 1 2 threej 𝑗 𝑗 1 𝑚 𝑚 11 superscript 1 𝑗 𝑚 superscript 2 𝑗 𝑚 𝑗 𝑚 1 2 𝑗 2 𝑗 1 2 𝑗 2 1 2 {\displaystyle{\displaystyle\mathit{3j}{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}}}}
\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}}		 j ≥ 1 2 𝑗 1 2 {\displaystyle{\displaystyle j\geq\tfrac{1}{2}}} 
Error

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])
Missing Macro Error Failure -
Failed [4 / 9]
Result: Complex[0.0, 0.5773502691896258]
Test Values: {Rule[j, 1], Rule[m, 2]}


Result: Complex[0.0, -0.9128709291752769]
Test Values: {Rule[j, 1], Rule[m, 3]}

... skip entries to safe data
34.3.E6 3 ⁢ j ⁢ 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 ⁢ ( ( 2 ⁢ j 1 ) ! ⁢ ( 2 ⁢ j 2 ) ! ⁢ ( j 1 + j 2 + m 1 + m 2 ) ! ⁢ ( j 1 + j 2 - m 1 - m 2 ) ! ( 2 ⁢ j 1 + 2 ⁢ j 2 + 1 ) ! ⁢ ( j 1 + m 1 ) ! ⁢ ( j 1 - m 1 ) ! ⁢ ( j 2 + m 2 ) ! ⁢ ( j 2 - m 2 ) ! ) 1 2 threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 1 subscript 𝑚 2 superscript 1 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑚 1 subscript 𝑚 2 superscript 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 1 2 {\displaystyle{\displaystyle\mathit{3j}{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}}}}
\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}}		 

Error

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])
Missing Macro Error Failure -
Failed [277 / 300]
Result: Complex[-0.009681373425206639, 0.01697152361235145]
Test Values: {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]]]}


Result: Complex[-0.0017082454309239698, -0.0034991197231335393]
Test Values: {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]]]}

... skip entries to safe data
34.3.E7 3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ j 1 - j 1 - m 3 ⁢ m 3 = ( - 1 ) j 1 - j 2 - m 3 ⁢ ( ( 2 ⁢ j 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 ) ! ) 1 2 threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑗 1 subscript 𝑗 1 subscript 𝑚 3 subscript 𝑚 3 superscript 1 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑚 3 superscript 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 1 2 {\displaystyle{\displaystyle\mathit{3j}{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}}}}
\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}}		 

Error

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])
Missing Macro Error Failure -
Failed [265 / 300]
Result: Complex[1.73830318129122, -1.18098937472798]
Test Values: {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]]]}


Result: Complex[-2.022477625790687, -8.26930711198898]
Test Values: {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]]]}

... skip entries to safe data
34.3.E8 3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 ⁢ m 3 = 3 ⁢ j ⁢ j 2 ⁢ j 3 ⁢ j 1 ⁢ m 2 ⁢ m 3 ⁢ m 1 threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 threej subscript 𝑗 2 subscript 𝑗 3 subscript 𝑗 1 subscript 𝑚 2 subscript 𝑚 3 subscript 𝑚 1 {\displaystyle{\displaystyle\mathit{3j}{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{% 3}}=\mathit{3j}{j_{2}}{j_{3}}{j_{1}}{m_{2}}{m_{3}}{m_{1}}}}
\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}}		 

Error

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]}]
Missing Macro Error Failure - Successful [Tested: 300]
34.3.E8 3 ⁢ j ⁢ j 2 ⁢ j 3 ⁢ j 1 ⁢ m 2 ⁢ m 3 ⁢ m 1 = 3 ⁢ j ⁢ j 3 ⁢ j 1 ⁢ j 2 ⁢ m 3 ⁢ m 1 ⁢ m 2 threej subscript 𝑗 2 subscript 𝑗 3 subscript 𝑗 1 subscript 𝑚 2 subscript 𝑚 3 subscript 𝑚 1 threej subscript 𝑗 3 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑚 3 subscript 𝑚 1 subscript 𝑚 2 {\displaystyle{\displaystyle\mathit{3j}{j_{2}}{j_{3}}{j_{1}}{m_{2}}{m_{3}}{m_{% 1}}=\mathit{3j}{j_{3}}{j_{1}}{j_{2}}{m_{3}}{m_{1}}{m_{2}}}}
\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}}		 

Error

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]}]
Missing Macro Error Failure - Successful [Tested: 300]
34.3.E9 3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 ⁢ m 3 = ( - 1 ) j 1 + j 2 + j 3 ⁢ 3 ⁢ j ⁢ j 2 ⁢ j 1 ⁢ j 3 ⁢ m 2 ⁢ m 1 ⁢ m 3 threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 superscript 1 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 threej subscript 𝑗 2 subscript 𝑗 1 subscript 𝑗 3 subscript 𝑚 2 subscript 𝑚 1 subscript 𝑚 3 {\displaystyle{\displaystyle\mathit{3j}{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{% 3}}=(-1)^{j_{1}+j_{2}+j_{3}}\mathit{3j}{j_{2}}{j_{1}}{j_{3}}{m_{2}}{m_{1}}{m_{% 3}}}}
\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}}		 

Error

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]}]
Missing Macro Error Failure - Successful [Tested: 300]
34.3.E10 3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 ⁢ m 3 = ( - 1 ) j 1 + j 2 + j 3 ⁢ 3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 - m 1 - m 2 - m 3 threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 superscript 1 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 {\displaystyle{\displaystyle\mathit{3j}{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{% 3}}=(-1)^{j_{1}+j_{2}+j_{3}}\mathit{3j}{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)^{j_{1}+j_{2}+j_{3}}\Wignerthreejsym{j_{1}}{j_{2}}{j_{3}}{-m_{1}}{-m_{2}}{-m_{3}}		 

Error

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]}]
Missing Macro Error Failure - Successful [Tested: 300]
34.3.E11 3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 ⁢ m 3 = 3 ⁢ j ⁢ j 1 ⁢ 1 2 ⁢ ( j 2 + j 3 + m 1 ) ⁢ 1 2 ⁢ ( j 2 + j 3 - m 1 ) ⁢ j 2 - j 3 ⁢ 1 2 ⁢ ( j 3 - j 2 + m 1 ) + m 2 ⁢ 1 2 ⁢ ( j 3 - j 2 + m 1 ) + m 3 threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 threej subscript 𝑗 1 1 2 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 1 2 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑗 2 subscript 𝑗 3 1 2 subscript 𝑗 3 subscript 𝑗 2 subscript 𝑚 1 subscript 𝑚 2 1 2 subscript 𝑗 3 subscript 𝑗 2 subscript 𝑚 1 subscript 𝑚 3 {\displaystyle{\displaystyle\mathit{3j}{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{% 3}}=\mathit{3j}{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}}}}
\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}}		 

Error

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]}]
Missing Macro Error Failure - Successful [Tested: 300]
34.3.E12 3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 ⁢ m 3 = 3 ⁢ j ⁢ 1 2 ⁢ ( j 1 + j 2 - m 3 ) ⁢ 1 2 ⁢ ( j 2 + j 3 - m 1 ) ⁢ 1 2 ⁢ ( j 1 + j 3 - m 2 ) ⁢ j 3 - 1 2 ⁢ ( j 1 + j 2 + m 3 ) ⁢ j 1 - 1 2 ⁢ ( j 2 + j 3 + m 1 ) ⁢ j 2 - 1 2 ⁢ ( j 1 + j 3 + m 2 ) threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 threej 1 2 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑚 3 1 2 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 1 2 subscript 𝑗 1 subscript 𝑗 3 subscript 𝑚 2 subscript 𝑗 3 1 2 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑚 3 subscript 𝑗 1 1 2 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑗 2 1 2 subscript 𝑗 1 subscript 𝑗 3 subscript 𝑚 2 {\displaystyle{\displaystyle\mathit{3j}{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{% 3}}=\mathit{3j}{\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})}}}
\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})}		 

Error

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])}]
Missing Macro Error Failure - Successful [Tested: 300]
34.3.E13 ( ( j 1 + j 2 + j 3 + 1 ) ⁢ ( - j 1 + j 2 + j 3 ) ) 1 2 ⁢ 3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 ⁢ m 3 = ( ( j 2 + m 2 ) ⁢ ( j 3 - m 3 ) ) 1 2 ⁢ 3 ⁢ j ⁢ j 1 ⁢ j 2 - 1 2 ⁢ j 3 - 1 2 ⁢ m 1 ⁢ m 2 - 1 2 ⁢ m 3 + 1 2 - ( ( j 2 - m 2 ) ⁢ ( j 3 + m 3 ) ) 1 2 ⁢ 3 ⁢ j ⁢ j 1 ⁢ j 2 - 1 2 ⁢ j 3 - 1 2 ⁢ m 1 ⁢ m 2 + 1 2 ⁢ m 3 - 1 2 superscript subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 1 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 1 2 threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 superscript subscript 𝑗 2 subscript 𝑚 2 subscript 𝑗 3 subscript 𝑚 3 1 2 threej subscript 𝑗 1 subscript 𝑗 2 1 2 subscript 𝑗 3 1 2 subscript 𝑚 1 subscript 𝑚 2 1 2 subscript 𝑚 3 1 2 superscript subscript 𝑗 2 subscript 𝑚 2 subscript 𝑗 3 subscript 𝑚 3 1 2 threej subscript 𝑗 1 subscript 𝑗 2 1 2 subscript 𝑗 3 1 2 subscript 𝑚 1 subscript 𝑚 2 1 2 subscript 𝑚 3 1 2 {\displaystyle{\displaystyle\left((j_{1}+j_{2}+j_{3}+1)(-j_{1}+j_{2}+j_{3})% \right)^{\frac{1}{2}}\mathit{3j}{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}}\mathit{3j}{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}}\mathit{3j}{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_{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}}		 

Error

((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]}]
Missing Macro Error Failure - Successful [Tested: 300]
34.3.E14 ( j 1 ⁢ ( j 1 + 1 ) - j 2 ⁢ ( j 2 + 1 ) - j 3 ⁢ ( j 3 + 1 ) - 2 ⁢ m 2 ⁢ m 3 ) ⁢ 3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 ⁢ m 3 = ( ( j 2 - m 2 ) ⁢ ( j 2 + m 2 + 1 ) ⁢ ( j 3 - m 3 + 1 ) ⁢ ( j 3 + m 3 ) ) 1 2 ⁢ 3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 + 1 ⁢ m 3 - 1 + ( ( j 2 - m 2 + 1 ) ⁢ ( j 2 + m 2 ) ⁢ ( j 3 - m 3 ) ⁢ ( j 3 + m 3 + 1 ) ) 1 2 ⁢ 3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 - 1 ⁢ m 3 + 1 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 threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 superscript 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 1 2 threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 1 subscript 𝑚 3 1 superscript 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 1 2 threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 1 subscript 𝑚 3 1 {\displaystyle{\displaystyle\left(j_{1}(j_{1}+1)-j_{2}(j_{2}+1)-j_{3}(j_{3}+1)% -2m_{2}m_{3}\right)\mathit{3j}{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}}% \mathit{3j}{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}}\mathit{3j}{j_{% 1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}-1}{m_{3}+1}}}
\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}		 

Error

(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}]
Missing Macro Error Failure - Successful [Tested: 300]
34.3.E15 ( 2 ⁢ j 1 + 1 ) ⁢ ( ( j 2 ⁢ ( j 2 + 1 ) - j 3 ⁢ ( j 3 + 1 ) ) ⁢ m 1 - j 1 ⁢ ( j 1 + 1 ) ⁢ ( m 3 - m 2 ) ) ⁢ 3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 ⁢ m 3 = ( j 1 + 1 ) ⁢ ( j 1 2 - ( j 2 - j 3 ) 2 ) 1 2 ⁢ ( ( j 2 + j 3 + 1 ) 2 - j 1 2 ) 1 2 ⁢ ( j 1 2 - m 1 2 ) 1 2 ⁢ 3 ⁢ j ⁢ j 1 - 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 ⁢ m 3 + j 1 ⁢ ( ( j 1 + 1 ) 2 - ( j 2 - j 3 ) 2 ) 1 2 ⁢ ( ( j 2 + j 3 + 1 ) 2 - ( j 1 + 1 ) 2 ) 1 2 ⁢ ( ( j 1 + 1 ) 2 - m 1 2 ) 1 2 ⁢ 3 ⁢ j ⁢ j 1 + 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 ⁢ m 3 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 threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 subscript 𝑗 1 1 superscript superscript subscript 𝑗 1 2 superscript subscript 𝑗 2 subscript 𝑗 3 2 1 2 superscript superscript subscript 𝑗 2 subscript 𝑗 3 1 2 superscript subscript 𝑗 1 2 1 2 superscript superscript subscript 𝑗 1 2 superscript subscript 𝑚 1 2 1 2 threej subscript 𝑗 1 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 subscript 𝑗 1 superscript superscript subscript 𝑗 1 1 2 superscript subscript 𝑗 2 subscript 𝑗 3 2 1 2 superscript superscript subscript 𝑗 2 subscript 𝑗 3 1 2 superscript subscript 𝑗 1 1 2 1 2 superscript superscript subscript 𝑗 1 1 2 superscript subscript 𝑚 1 2 1 2 threej subscript 𝑗 1 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 {\displaystyle{\displaystyle(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)\mathit{3j}{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}}\mathit{3j}{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}}\mathit{3j}{j_{1}+1}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}}}}
(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}}		 

Error

(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]}] == (Subscript[j, 1]+ 1)*((Subscript[j, 1])^(2)-(Subscript[j, 2]- Subscript[j, 3])^(2))^(Divide[1,2])*((Subscript[j, 2]+ Subscript[j, 3]+ 1)^(2)- (Subscript[j, 1])^(2))^(Divide[1,2])*((Subscript[j, 1])^(2)- (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, 2]+ Subscript[j, 3]+ 1)^(2)-(Subscript[j, 1]+ 1)^(2))^(Divide[1,2])*((Subscript[j, 1]+ 1)^(2)- (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]}]
Missing Macro Error Failure - Successful [Tested: 300]
34.3.E18 ∑ m 1 ⁢ m 2 ⁢ m 3 3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 ⁢ m 3 ⁢ 3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ m 2 ⁢ m 3 = 1 subscript subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 1 {\displaystyle{\displaystyle\sum_{m_{1}m_{2}m_{3}}\mathit{3j}{j_{1}}{j_{2}}{j_% {3}}{m_{1}}{m_{2}}{m_{3}}\mathit{3j}{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{3}}% =1}}
\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		 

Error

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
Translation Error Translation Error - -
34.3.E19 P l 1 ⁡ ( cos ⁡ θ ) ⁢ P l 2 ⁡ ( cos ⁡ θ ) = ∑ l ( 2 ⁢ l + 1 ) ⁢ 3 ⁢ j ⁢ l 1 ⁢ l 2 ⁢ l ⁢ 000 2 ⁢ P l ⁡ ( cos ⁡ θ ) shorthand-Legendre-P-first-kind subscript 𝑙 1 𝜃 shorthand-Legendre-P-first-kind subscript 𝑙 2 𝜃 subscript 𝑙 2 𝑙 1 threej subscript 𝑙 1 subscript 𝑙 2 𝑙 superscript 000 2 shorthand-Legendre-P-first-kind 𝑙 𝜃 {\displaystyle{\displaystyle P_{l_{1}}\left(\cos\theta\right)P_{l_{2}}\left(% \cos\theta\right)=\sum_{l}(2l+1)\mathit{3j}{l_{1}}{l_{2}}{l}{0}{0}{0}^{2}P_{l}% \left(\cos\theta\right)}}
\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}}		 

Error

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]
Missing Macro Error Failure - Skipped - Because timed out
34.3.E20 Y l 1 , m 1 ⁡ ( θ , ϕ ) ⁢ Y l 2 , m 2 ⁡ ( θ , ϕ ) = ∑ l , m ( ( 2 ⁢ l 1 + 1 ) ⁢ ( 2 ⁢ l 2 + 1 ) ⁢ ( 2 ⁢ l + 1 ) 4 ⁢ π ) 1 2 ⁢ 3 ⁢ j ⁢ l 1 ⁢ l 2 ⁢ l ⁢ m 1 ⁢ m 2 ⁢ m ⁢ Y l , m ⁡ ( θ , ϕ ) ¯ ⁢ 3 ⁢ j ⁢ l 1 ⁢ l 2 ⁢ l ⁢ 000 spherical-harmonic-Y subscript 𝑙 1 subscript 𝑚 1 𝜃 italic-ϕ spherical-harmonic-Y subscript 𝑙 2 subscript 𝑚 2 𝜃 italic-ϕ subscript 𝑙 𝑚 superscript 2 subscript 𝑙 1 1 2 subscript 𝑙 2 1 2 𝑙 1 4 𝜋 1 2 threej subscript 𝑙 1 subscript 𝑙 2 𝑙 subscript 𝑚 1 subscript 𝑚 2 𝑚 spherical-harmonic-Y 𝑙 𝑚 𝜃 italic-ϕ threej subscript 𝑙 1 subscript 𝑙 2 𝑙 000 {\displaystyle{\displaystyle Y_{{l_{1}},{m_{1}}}\left(\theta,\phi\right)Y_{{l_% {2}},{m_{2}}}\left(\theta,\phi\right)=\sum_{l,m}\left(\frac{(2l_{1}+1)(2l_{2}+% 1)(2l+1)}{4\pi}\right)^{\frac{1}{2}}\mathit{3j}{l_{1}}{l_{2}}{l}{m_{1}}{m_{2}}% {m}\overline{Y_{{l},{m}}\left(\theta,\phi\right)}\mathit{3j}{l_{1}}{l_{2}}{l}{% 0}{0}{0}}}
\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}		 

Error

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]
Translation Error Translation Error - -
34.3.E21 ∫ 0 π P l 1 ⁡ ( cos ⁡ θ ) ⁢ P l 2 ⁡ ( cos ⁡ θ ) ⁢ P l 3 ⁡ ( cos ⁡ θ ) ⁢ sin ⁡ θ ⁢ d θ = 2 ⁢ 3 ⁢ j ⁢ l 1 ⁢ l 2 ⁢ l 3 ⁢ 000 2 superscript subscript 0 𝜋 shorthand-Legendre-P-first-kind subscript 𝑙 1 𝜃 shorthand-Legendre-P-first-kind subscript 𝑙 2 𝜃 shorthand-Legendre-P-first-kind subscript 𝑙 3 𝜃 𝜃 𝜃 2 threej subscript 𝑙 1 subscript 𝑙 2 subscript 𝑙 3 superscript 000 2 {\displaystyle{\displaystyle\int_{0}^{\pi}P_{l_{1}}\left(\cos\theta\right)P_{l% _{2}}\left(\cos\theta\right)P_{l_{3}}\left(\cos\theta\right)\sin\theta\mathrm{% d}\theta=2\mathit{3j}{l_{1}}{l_{2}}{l_{3}}{0}{0}{0}^{2}}}
\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}		 

Error

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)
Missing Macro Error Failure - Skipped - Because timed out
34.3.E22 ∫ 0 2 ⁢ π ∫ 0 π Y l 1 , m 1 ⁡ ( θ , ϕ ) ⁢ Y l 2 , m 2 ⁡ ( θ , ϕ ) ⁢ Y l 3 , m 3 ⁡ ( θ , ϕ ) ⁢ sin ⁡ θ ⁢ d θ ⁢ d ϕ = ( ( 2 ⁢ l 1 + 1 ) ⁢ ( 2 ⁢ l 2 + 1 ) ⁢ ( 2 ⁢ l 3 + 1 ) 4 ⁢ π ) 1 2 ⁢ 3 ⁢ j ⁢ l 1 ⁢ l 2 ⁢ l 3 ⁢ 000 ⁢ 3 ⁢ j ⁢ l 1 ⁢ l 2 ⁢ l 3 ⁢ m 1 ⁢ m 2 ⁢ m 3 superscript subscript 0 2 𝜋 superscript subscript 0 𝜋 spherical-harmonic-Y subscript 𝑙 1 subscript 𝑚 1 𝜃 italic-ϕ spherical-harmonic-Y subscript 𝑙 2 subscript 𝑚 2 𝜃 italic-ϕ spherical-harmonic-Y subscript 𝑙 3 subscript 𝑚 3 𝜃 italic-ϕ 𝜃 𝜃 italic-ϕ superscript 2 subscript 𝑙 1 1 2 subscript 𝑙 2 1 2 subscript 𝑙 3 1 4 𝜋 1 2 threej subscript 𝑙 1 subscript 𝑙 2 subscript 𝑙 3 000 threej subscript 𝑙 1 subscript 𝑙 2 subscript 𝑙 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 {\displaystyle{\displaystyle\int_{0}^{2\pi}\!\int_{0}^{\pi}Y_{{l_{1}},{m_{1}}}% \left(\theta,\phi\right)Y_{{l_{2}},{m_{2}}}\left(\theta,\phi\right)Y_{{l_{3}},% {m_{3}}}\left(\theta,\phi\right)\sin\theta\mathrm{d}\theta\mathrm{d}\phi=\left% (\frac{(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi}\right)^{\frac{1}{2}}\mathit{3j}{l% _{1}}{l_{2}}{l_{3}}{0}{0}{0}\mathit{3j}{l_{1}}{l_{2}}{l_{3}}{m_{1}}{m_{2}}{m_{% 3}}}}
\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}}		 

Error

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]}]
Missing Macro Error Aborted - Skipped - Because timed out
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