3 j , 6 j , 9 j Symbols - 34.4 Definition: Symbol

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
34.4.E2 6 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ l 1 ⁢ l 2 ⁢ l 3 = Δ ⁢ ( j 1 ⁢ j 2 ⁢ j 3 ) ⁢ Δ ⁢ ( j 1 ⁢ l 2 ⁢ l 3 ) ⁢ Δ ⁢ ( l 1 ⁢ j 2 ⁢ l 3 ) ⁢ Δ ⁢ ( l 1 ⁢ l 2 ⁢ j 3 ) ⁢ ∑ s ( - 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 ) ! ⁢ 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 ) ! sixj subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑙 1 subscript 𝑙 2 subscript 𝑙 3 Δ subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 Δ subscript 𝑗 1 subscript 𝑙 2 subscript 𝑙 3 Δ subscript 𝑙 1 subscript 𝑗 2 subscript 𝑙 3 Δ subscript 𝑙 1 subscript 𝑙 2 subscript 𝑗 3 subscript 𝑠 superscript 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 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 {\displaystyle{\displaystyle\mathit{6j}{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)!}}}
\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)!}		 

Error

SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {Subscript[l, 1], Subscript[l, 2], Subscript[l, 3]}] == ((Divide[(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[j, 3])!,(Subscript[j, 1]+ Subscript[j, 2]+ Subscript[j, 3]+ 1)!])^(Divide[1,2]))*\[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]
Missing Macro Error Aborted - Skipped - Because timed out
34.4.E3 6 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ l 1 ⁢ l 2 ⁢ l 3 = ( - 1 ) j 1 + j 3 + l 1 + l 3 ⁢ Δ ⁢ ( j 1 ⁢ j 2 ⁢ j 3 ) ⁢ Δ ⁢ ( 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 ) ! Δ ⁢ ( j 1 ⁢ l 2 ⁢ l 3 ) ⁢ Δ ⁢ ( 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 ) ! ⁢ F 3 4 ⁡ ( - 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 ) sixj subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑙 1 subscript 𝑙 2 subscript 𝑙 3 superscript 1 subscript 𝑗 1 subscript 𝑗 3 subscript 𝑙 1 subscript 𝑙 3 Δ subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 Δ 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 Δ subscript 𝑗 1 subscript 𝑙 2 subscript 𝑙 3 Δ 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 Gauss-hypergeometric-pFq 4 3 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 {\displaystyle{\displaystyle\mathit{6j}{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)!}\*{{}_{4}% F_{3}}\left({-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\atop-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\right)}}
\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}		 

Error

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[((Divide[(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[j, 3])!,(Subscript[j, 1]+ Subscript[j, 2]+ Subscript[j, 3]+ 1)!])^(Divide[1,2]))*\[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]
Missing Macro Error Failure - Skipped - Because timed out
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