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ThreeJSymbol






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ThreeJSymbol[{j1,m1},{j2,m2},{j3,m3}] > Summation > Finite summation > Involving one 3j symbol





http://functions.wolfram.com/07.39.23.0005.01









  


  










Input Form





Sum[((2 Subscript[j, 1] + 1)/(Subscript[j, 1] (Subscript[j, 1] + 1) - n (n + 1))) ThreeJSymbol[{Subscript[j, 1], 0}, {Subscript[j, 2], 0}, {Subscript[j, 3], 0}]^2, {Subscript[j, 1], Abs[Subscript[j, 2] - Subscript[j, 3]], Subscript[j, 2] + Subscript[j, 3]}] == 0 /; Element[n, Integers] && Abs[j - Subscript[j, 2]] <= n <= j + Subscript[j, 2] && Element[(n + Subscript[j, 2] + j + 1)/2, Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "1"], "=", RowBox[List["Abs", "[", RowBox[List[SubscriptBox["j", "2"], "-", SubscriptBox["j", "3"]]], "]"]]]], RowBox[List[SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"]]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["2", SubscriptBox["j", "1"]]], "+", "1"]], RowBox[List[RowBox[List[SubscriptBox["j", "1"], RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", "1"]], ")"]]]], "-", RowBox[List["n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]]]]], SuperscriptBox[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[SubscriptBox["j", "3"], ",", "0"]], "}"]]]], "]"]], "2"]]]]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["j", "-", SubscriptBox["j", "2"]]], "]"]], "\[LessEqual]", "n", "\[LessEqual]", RowBox[List["j", "+", SubscriptBox["j", "2"]]]]], "\[And]", RowBox[List[FractionBox[RowBox[List["n", "+", SubscriptBox["j", "2"], "+", "j", "+", "1"]], "2"], "\[Element]", "Integers"]]]]]]]]










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> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mrow> </munderover> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <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> <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> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8804; </mo> <mi> n </mi> <mo> &#8804; </mo> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mfrac> <mrow> <mi> j </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <apply> <abs /> <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'> 3 </cn> </apply> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <plus /> <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> </apply> </uplimit> <apply> <times /> <apply> <times /> <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> <power /> <apply> <plus /> <apply> <times /> <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'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <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> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 0 </cn> </list> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <leq /> <apply> <abs /> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <ci> n </ci> <apply> <plus /> <ci> j </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <in /> <apply> <times /> <apply> <plus /> <ci> j </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "1"], "=", RowBox[List["Abs", "[", RowBox[List[SubscriptBox["j", "2"], "-", SubscriptBox["j", "3"]]], "]"]]]], RowBox[List[SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"]]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "1"]]], "+", "1"]], ")"]], " ", SuperscriptBox[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[SubscriptBox["j", "3"], ",", "0"]], "}"]]]], "]"]], "2"]]], RowBox[List[RowBox[List[SubscriptBox["j", "1"], " ", RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", "1"]], ")"]]]], "-", RowBox[List["n_", " ", RowBox[List["(", RowBox[List["n_", "+", "1"]], ")"]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["j", "-", SubscriptBox["j", "2"]]], "]"]], "\[LessEqual]", "n", "\[LessEqual]", RowBox[List["j", "+", SubscriptBox["j", "2"]]]]], "&&", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "+", SubscriptBox["j", "2"], "+", "j", "+", "1"]], ")"]]]], "\[Element]", "Integers"]]]]]]]]]]










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





2001-12-21





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