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ThreeJSymbol






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ThreeJSymbol[{j1,m1},{j2,m2},{j3,m3}] > Identities > Recurrence identities > Consecutive neighbors





http://functions.wolfram.com/07.39.17.0003.01









  


  










Input Form





ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}] == (-((Sqrt[Subscript[j, 1] - Subscript[m, 1]] Sqrt[Subscript[j, 1] + Subscript[m, 1] + 1])/ (Sqrt[Subscript[j, 3] + Subscript[m, 3]] Sqrt[Subscript[j, 3] - Subscript[m, 3] + 1]))) ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1] + 1}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3] - 1}] - ((Sqrt[Subscript[j, 2] - Subscript[m, 2]] Sqrt[Subscript[j, 2] + Subscript[m, 2] + 1])/ (Sqrt[Subscript[j, 3] + Subscript[m, 3]] Sqrt[Subscript[j, 3] - Subscript[m, 3] + 1])) ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2] + 1}, {Subscript[j, 3], Subscript[m, 3] - 1}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "3"], ",", SubscriptBox["m", "3"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox[RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["m", "1"]]]], " ", SqrtBox[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["m", "1"], "+", "1"]]]]], RowBox[List[SqrtBox[RowBox[List[SubscriptBox["j", "3"], "+", SubscriptBox["m", "3"]]]], " ", SqrtBox[RowBox[List[SubscriptBox["j", "3"], "-", SubscriptBox["m", "3"], "+", "1"]]]]]]]], " ", RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", RowBox[List[SubscriptBox["m", "1"], "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "3"], ",", RowBox[List[SubscriptBox["m", "3"], "-", "1"]]]], "}"]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List[SubscriptBox["j", "2"], "-", SubscriptBox["m", "2"]]]], " ", SqrtBox[RowBox[List[SubscriptBox["j", "2"], "+", SubscriptBox["m", "2"], "+", "1"]]]]], RowBox[List[SqrtBox[RowBox[List[SubscriptBox["j", "3"], "+", SubscriptBox["m", "3"]]]], " ", SqrtBox[RowBox[List[SubscriptBox["j", "3"], "-", SubscriptBox["m", "3"], "+", "1"]]]]]], " ", RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", RowBox[List[SubscriptBox["m", "2"], "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "3"], ",", RowBox[List[SubscriptBox["m", "3"], "-", "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> <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> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msqrt> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <msqrt> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 3 </mn> </msub> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <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> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mtd> <mtd> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mtd> <mtd> <mrow> <msub> <mi> m </mi> <mn> 3 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </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> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <msqrt> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 3 </mn> </msub> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <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> <mrow> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mtd> <mtd> <mrow> <msub> <mi> m </mi> <mn> 3 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </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> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <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> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <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> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <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> <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'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> ThreeJSymbol </ci> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <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> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </list> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <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> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <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> <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'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <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> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </list> </apply> </apply> </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["m_", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m_", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "3"], ",", SubscriptBox["m_", "3"]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["mm", "1"]]]], " ", SqrtBox[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["mm", "1"], "+", "1"]]]]], ")"]], " ", RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", RowBox[List[SubscriptBox["mm", "1"], "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["mm", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "3"], ",", RowBox[List[SubscriptBox["mm", "3"], "-", "1"]]]], "}"]]]], "]"]]]], RowBox[List[SqrtBox[RowBox[List[SubscriptBox["j", "3"], "+", SubscriptBox["mm", "3"]]]], " ", SqrtBox[RowBox[List[SubscriptBox["j", "3"], "-", SubscriptBox["mm", "3"], "+", "1"]]]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[SubscriptBox["j", "2"], "-", SubscriptBox["mm", "2"]]]], " ", SqrtBox[RowBox[List[SubscriptBox["j", "2"], "+", SubscriptBox["mm", "2"], "+", "1"]]]]], ")"]], " ", RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["mm", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", RowBox[List[SubscriptBox["mm", "2"], "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "3"], ",", RowBox[List[SubscriptBox["mm", "3"], "-", "1"]]]], "}"]]]], "]"]]]], RowBox[List[SqrtBox[RowBox[List[SubscriptBox["j", "3"], "+", SubscriptBox["mm", "3"]]]], " ", SqrtBox[RowBox[List[SubscriptBox["j", "3"], "-", SubscriptBox["mm", "3"], "+", "1"]]]]]]]]]]]]










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





2001-10-29





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