ThreeJSymbol[{Subscript[j, 1], 0}, {Subscript[j, 2], 0}, {Subscript[j, 3], 0}] == ((-1)^(Subscript[j, 2] - Subscript[j, 3])/ 2^(Subscript[j, 1] + Subscript[j, 2] + Subscript[j, 3] + 1)) ((Sqrt[(Subscript[j, 1] - Subscript[j, 2] + Subscript[j, 3])!] Sqrt[(Subscript[j, 1] + Subscript[j, 2] - Subscript[j, 3])!] Sqrt[(Subscript[j, 1] + Subscript[j, 2] + Subscript[j, 3] + 1)!])/ (Subscript[j, 1]! Subscript[j, 2]! Subscript[j, 3]! Sqrt[(-Subscript[j, 1] + Subscript[j, 2] + Subscript[j, 3])!])) Integrate[(1 - t^2)^Subscript[j, 2] D[(1 - t^2)^Subscript[j, 3], {t, -Subscript[j, 1] + Subscript[j, 2] + Subscript[j, 3]}], {t, -1, 1}] /; Element[Subscript[j, 1], Integers] && Subscript[j, 1] >= 0 && Element[Subscript[j, 2], Integers] && Subscript[j, 2] >= 0 && Element[Subscript[j, 3], Integers] && Subscript[j, 3] >= 0 && Abs[Subscript[j, 1] - Subscript[j, 2]] <= Subscript[j, 3] <= Subscript[j, 1] + Subscript[j, 2]

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</mo> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 1 </mn> </msubsup> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </msup> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </msup> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> t </mi> <mrow> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; 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2001-10-29