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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/07.39.03.0029.01









  


  










Input Form





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










Standard Form





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










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> </mrow> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mn> 0 </mn> </mtd> <mtd> <mn> 0 </mn> </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> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </msup> <mo> &#8290; </mo> <mfrac> <mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <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> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </msqrt> </mrow> <mrow> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <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> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </msqrt> </mrow> </mfrac> </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> 1 </mn> </msub> <mo> &#8805; </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> </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> <cn type='integer'> 0 </cn> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 0 </cn> </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> </apply> <cn type='integer'> 0 </cn> </list> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <apply> <factorial /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </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> <power /> <apply> <factorial /> <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'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <factorial /> <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> <power /> <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'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <factorial /> <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> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#8469; </ci> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8469; </ci> </apply> <apply> <geq /> <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> </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"], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"]]], ",", "0"]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["j", "1"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "!"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SubscriptBox["j", "2"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "1"]]], "-", RowBox[List["2", " ", SubscriptBox["j", "2"]]], "+", "1"]], ")"]], "!"]]]]], ")"]]]], RowBox[List[RowBox[List[SubscriptBox["j", "2"], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"]]], ")"]], "!"]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "1"]]], "-", RowBox[List["2", " ", SubscriptBox["j", "2"]]], "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "1"]]], "+", "1"]], ")"]], "!"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["j", "1"], "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["j", "2"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["j", "2"], "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["j", "1"], "\[GreaterEqual]", SubscriptBox["j", "2"]]]]]]]]]]]










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





2001-10-29