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ThreeJSymbol






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ThreeJSymbol[{j1,m1},{j2,m2},{j3,m3}] > Specific values > Specialized values > Fixed j1, j2





http://functions.wolfram.com/07.39.03.0039.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", RowBox[List["-", SubscriptBox["j", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"], "+", "2"]], ",", RowBox[List[RowBox[List["-", SubscriptBox["j", "1"]]], "+", SubscriptBox["j", "2"]]]]], "}"]]]], "]"]], " ", "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], SubscriptBox["j", "1"]]], "+", RowBox[List["2", SubscriptBox["j", "2"]]]]]], FractionBox[RowBox[List[SqrtBox[SubscriptBox["j", "2"]], " ", SqrtBox[RowBox[List[RowBox[List["2", SubscriptBox["j", "2"]]], "-", "1"]]]]], RowBox[List[SqrtBox[RowBox[List[SubscriptBox["j", "1"], "+", "1"]]], SqrtBox[RowBox[List[RowBox[List["2", SubscriptBox["j", "1"]]], "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", SubscriptBox["j", "1"]]], "+", "3"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["2", SubscriptBox["j", "1"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["j", "1"], ">", "0"]], "\[And]", RowBox[List[RowBox[List["2", SubscriptBox["j", "2"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["2", SubscriptBox["j", "2"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["2", SubscriptBox["j", "2"]]], "-", "2"]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["j", "1"], ">", RowBox[List[SubscriptBox["j", "2"], "-", "1"]]]]]]]]]]










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> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 2 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mtd> <mtd> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> </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> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mfrac> <mrow> <msqrt> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <msqrt> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> </msqrt> </mrow> </mfrac> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> &gt; </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <and /> <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> j </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <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'> 2 </cn> </apply> </apply> </list> <list> <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> <cn type='integer'> 2 </cn> </apply> <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> </list> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -2 </cn> </apply> <ci> &#8469; </ci> </apply> <apply> <gt /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", RowBox[List["-", SubscriptBox["j", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"], "+", "2"]], ",", RowBox[List[RowBox[List["-", SubscriptBox["j", "1"]]], "+", SubscriptBox["j", "2"]]]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SubscriptBox["j", "1"]]], "+", RowBox[List["2", " ", SubscriptBox["j", "2"]]]]]], " ", RowBox[List["(", RowBox[List[SqrtBox[SubscriptBox["j", "2"]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "2"]]], "-", "1"]]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List[SubscriptBox["j", "1"], "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "1"]]], "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "1"]]], "+", "3"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "1"]]], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["j", "1"], ">", "0"]], "&&", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "2"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "2"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "2"]]], "-", "2"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["j", "1"], ">", RowBox[List[SubscriptBox["j", "2"], "-", "1"]]]]]]]]]]]]










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





2001-10-29





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