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ThreeJSymbol






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ThreeJSymbol[{j1,m1},{j2,m2},{j3,m3}] > Integral representations > Multiple integral representations > For the direct function itself





http://functions.wolfram.com/07.39.07.0007.01









  


  










Input Form





ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}] == ((-1)^(Subscript[j, 2] - Subscript[j, 1] + Subscript[m, 3])/(8 Pi^2)) ((Sqrt[(Subscript[j, 1] + Subscript[j, 2] + Subscript[j, 3] + 1)!] Sqrt[(Subscript[j, 1] + Subscript[j, 2] - Subscript[j, 3])!])/ (Sqrt[(2 Subscript[j, 1])!] Sqrt[(2 Subscript[j, 2])!])) Integrate[Sin[\[Beta]] WignerD[Subscript[m, 1], Subscript[j, 1], Subscript[j, 1], \[Alpha], \[Beta], \[Gamma]] WignerD[Subscript[m, 2], -Subscript[j, 2], Subscript[j, 2], \[Alpha], \[Beta], \[Gamma]] Conjugate[WignerD[-Subscript[m, 3], Subscript[j, 1] - Subscript[j, 2], Subscript[j, 3], \[Alpha], \[Beta], \[Gamma]]], {\[Alpha], 0, 2 Pi}, {\[Beta], 0, Pi}, {\[Gamma], 0, 2 Pi}] /; \[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\ \[ScriptI]\[ScriptC]\[ScriptA]\[ScriptL]\[ScriptCapitalQ][ {Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}]










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mo> ( </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;(&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> <mtext> &#8287; </mtext> <mtable> <mtr> <mtd> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mn> 3 </mn> </msub> </mtd> </mtr> </mtable> <mtext> &#8287; </mtext> <semantics> <mo> ) </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;)&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 3 </mn> </msub> </mrow> </msup> <mrow> <mn> 8 </mn> <mo> &#8290; 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</mo> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msubsup> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> &#960; </mi> </msubsup> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msubsup> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#946; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> D </mi> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> , </mo> <mi> &#946; </mi> <mo> , </mo> <mi> &#947; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> D </mi> <mrow> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> , </mo> <mi> &#946; </mi> <mo> , </mo> <mi> &#947; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mover> <mrow> <msubsup> <mi> D </mi> <mrow> <mrow> <mo> - </mo> <msub> <mi> m </mi> <mn> 3 </mn> </msub> </mrow> <mo> , </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> , </mo> <mi> &#946; </mi> <mo> , </mo> <mi> &#947; </mi> </mrow> <mo> ) </mo> </mrow> <mo> _ </mo> </mover> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> &#947; </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> &#946; </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> &#945; </mi> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> &#119979;&#119997;&#120014;&#120008;&#119998;&#119992;&#119990;&#8467;&#119980; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> , </mo> <msub> <mi> m </mi> <mn> 3 </mn> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> ThreeJSymbol </ci> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <factorial /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <factorial /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <factorial /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> &#947; </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </uplimit> <apply> <int /> <bvar> <ci> &#946; </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <pi /> </uplimit> <apply> <int /> <bvar> <ci> &#945; </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </uplimit> <apply> <times /> <apply> <sin /> <ci> &#946; </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> D </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> &#945; </ci> <ci> &#946; </ci> <ci> &#947; </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> D </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#945; </ci> <ci> &#946; </ci> <ci> &#947; </ci> </apply> <apply> <ci> OverBar </ci> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> D </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> </apply> <ci> &#945; </ci> <ci> &#946; </ci> <ci> &#947; </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> &#119979;&#119997;&#120014;&#120008;&#119998;&#119992;&#119990;&#8467;&#119980; </ci> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2001-12-21