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http://functions.wolfram.com/07.38.23.0025.01
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Sum[ClebschGordan[{b, \[Beta]}, {a, \[Alpha]}, {c, \[Gamma]}]
ClebschGordan[{f, \[CurlyPhi]}, {j, \[Mu]}, {c, \[Gamma]}]
ClebschGordan[{b, \[Beta]}, {g, \[Eta]}, {e, \[Epsilon]}]
ClebschGordan[{f, \[CurlyPhi]}, {d, \[Delta]}, {e, \[Epsilon]}],
{\[Beta], -b, b}, {\[Gamma], -c, c}, {\[Epsilon], -e, e},
{\[CurlyPhi], -f, f}] == (-1)^(a - b + f - j) (2 c + 1) (2 e + 1)
Sum[ClebschGordan[{g, \[Eta]}, {j, \[Mu]}, {k, \[Kappa]}]
ClebschGordan[{d, \[Delta]}, {a, \[Alpha]}, {k, \[Kappa]}]
NineJSymbol[{c, b, a}, {f, e, d}, {j, g, k}],
{k, Max[Abs[g - j], Abs[a - d]], Min[g + j, a + d]}, {\[Kappa], -k, k}]
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Beta]", "=", RowBox[List["-", "b"]]]], "b"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Gamma]", "=", RowBox[List["-", "c"]]]], "c"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Epsilon]", "=", RowBox[List["-", "e"]]]], "e"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[CurlyPhi]", "=", RowBox[List["-", "f"]]]], "f"], RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["b", ",", "\[Beta]"]], "}"]], ",", RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List["c", ",", "\[Gamma]"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["f", ",", "\[CurlyPhi]"]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "\[Mu]"]], "}"]], ",", RowBox[List["{", RowBox[List["c", ",", "\[Gamma]"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["b", ",", "\[Beta]"]], "}"]], ",", RowBox[List["{", RowBox[List["g", ",", "\[Eta]"]], "}"]], ",", RowBox[List["{", RowBox[List["e", ",", "\[Epsilon]"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["f", ",", "\[CurlyPhi]"]], "}"]], ",", RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]], ",", RowBox[List["{", RowBox[List["e", ",", "\[Epsilon]"]], "}"]]]], "]"]]]]]]]]]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["a", "-", "b", "+", "f", "-", "j"]]], RowBox[List["(", RowBox[List[RowBox[List["2", "c"]], "+", "1"]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["2", "e"]], "+", "1"]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List["g", "-", "j"]], "]"]], ",", RowBox[List["Abs", "[", RowBox[List["a", "-", "d"]], "]"]]]], "]"]]]], RowBox[List["Min", "[", RowBox[List[RowBox[List["g", "+", "j"]], ",", RowBox[List["a", "+", "d"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Kappa]", "=", RowBox[List["-", "k"]]]], "k"], RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["g", ",", "\[Eta]"]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "\[Mu]"]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "\[Kappa]"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]], ",", RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "\[Kappa]"]], "}"]]]], "]"]], RowBox[List["NineJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List["c", ",", "b", ",", "a"]], "}"]], ",", RowBox[List["{", RowBox[List["f", ",", "e", ",", "d"]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "g", ",", "k"]], "}"]]]], "]"]]]]]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> β </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mi> b </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> γ </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> c </mi> </mrow> </mrow> <mi> c </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> ϵ </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> e </mi> </mrow> </mrow> <mi> e </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> φ </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> f </mi> </mrow> </mrow> <mi> f </mi> </munderover> <mrow> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> a </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> β </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> α </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> a </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> c </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> γ </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["b", "\[MediumSpace]", "a", "\[MediumSpace]", "\[Beta]", "\[MediumSpace]", "\[Alpha]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["b", "\[MediumSpace]", "a", "\[MediumSpace]", "c", "\[MediumSpace]", "\[Gamma]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> j </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> φ </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> μ </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> j </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> c </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> γ </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["f", "\[MediumSpace]", "j", "\[MediumSpace]", "\[CurlyPhi]", "\[MediumSpace]", "\[Mu]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["f", "\[MediumSpace]", "j", "\[MediumSpace]", "c", "\[MediumSpace]", "\[Gamma]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> g </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> β </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> η </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> g </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> e </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> ϵ </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["b", "\[MediumSpace]", "g", "\[MediumSpace]", "\[Beta]", "\[MediumSpace]", "\[Eta]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["b", "\[MediumSpace]", "g", "\[MediumSpace]", "e", "\[MediumSpace]", "\[Epsilon]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> d </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> φ </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> δ </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> d </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> e </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> ϵ </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["f", "\[MediumSpace]", "d", "\[MediumSpace]", "\[CurlyPhi]", "\[MediumSpace]", "\[Delta]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["f", "\[MediumSpace]", "d", "\[MediumSpace]", "e", "\[MediumSpace]", "\[Epsilon]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mi> f </mi> <mo> - </mo> <mi> j </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> e </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> g </mi> <mo> - </mo> <mi> j </mi> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> , </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> a </mi> <mo> - </mo> <mi> d </mi> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> min </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> g </mi> <mo> + </mo> <mi> j </mi> </mrow> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> d </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> κ </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </mrow> <mi> k </mi> </munderover> <mrow> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> g </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> j </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> η </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> μ </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> g </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> j </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> k </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> κ </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["g", "\[MediumSpace]", "j", "\[MediumSpace]", "\[Eta]", "\[MediumSpace]", "\[Mu]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["g", "\[MediumSpace]", "j", "\[MediumSpace]", "k", "\[MediumSpace]", "\[Kappa]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> a </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> δ </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> α </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> a </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> k </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> κ </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["d", "\[MediumSpace]", "a", "\[MediumSpace]", "\[Delta]", "\[MediumSpace]", "\[Alpha]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["d", "\[MediumSpace]", "a", "\[MediumSpace]", "k", "\[MediumSpace]", "\[Kappa]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> { </mo> <mtable> <mtr> <mtd> <mi> c </mi> </mtd> <mtd> <mi> b </mi> </mtd> <mtd> <mi> a </mi> </mtd> </mtr> <mtr> <mtd> <mi> f </mi> </mtd> <mtd> <mi> e </mi> </mtd> <mtd> <mi> d </mi> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> <mtd> <mi> g </mi> </mtd> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> } </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> φ </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> f </ci> </apply> </lowlimit> <uplimit> <ci> f </ci> </uplimit> <apply> <sum /> <bvar> <ci> ϵ </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> </lowlimit> <uplimit> <ci> e </ci> </uplimit> <apply> <sum /> <bvar> <ci> γ </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </lowlimit> <uplimit> <ci> c </ci> </uplimit> <apply> <sum /> <bvar> <ci> β </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </lowlimit> <uplimit> <ci> b </ci> </uplimit> <apply> <times /> <apply> <ci> ClebschGordan </ci> <list> <ci> b </ci> <ci> β </ci> </list> <list> <ci> a </ci> <ci> α </ci> </list> <list> <ci> c </ci> <ci> γ </ci> </list> </apply> <apply> <ci> ClebschGordan </ci> <list> <ci> f </ci> <ci> φ </ci> </list> <list> <ci> j </ci> <ci> μ </ci> </list> <list> <ci> c </ci> <ci> γ </ci> </list> </apply> <apply> <ci> ClebschGordan </ci> <list> <ci> b </ci> <ci> β </ci> </list> <list> <ci> g </ci> <ci> η </ci> </list> <list> <ci> e </ci> <ci> ϵ </ci> </list> </apply> <apply> <ci> ClebschGordan </ci> <list> <ci> f </ci> <ci> φ </ci> </list> <list> <ci> d </ci> <ci> δ </ci> </list> <list> <ci> e </ci> <ci> ϵ </ci> </list> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> f </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <sum /> <bvar> <ci> κ </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <max /> <apply> <abs /> <apply> <plus /> <ci> g </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <abs /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <min /> <apply> <plus /> <ci> g </ci> <ci> j </ci> </apply> <apply> <plus /> <ci> a </ci> <ci> d </ci> </apply> </apply> </uplimit> <apply> <times /> <apply> <ci> ClebschGordan </ci> <list> <ci> g </ci> <ci> η </ci> </list> <list> <ci> j </ci> <ci> μ </ci> </list> <list> <ci> k </ci> <ci> κ </ci> </list> </apply> <apply> <ci> ClebschGordan </ci> <list> <ci> d </ci> <ci> δ </ci> </list> <list> <ci> a </ci> <ci> α </ci> </list> <list> <ci> k </ci> <ci> κ </ci> </list> </apply> <list> <list> <list> <ci> c </ci> <ci> b </ci> <ci> a </ci> </list> <list> <ci> f </ci> <ci> e </ci> <ci> d </ci> </list> <list> <ci> j </ci> <ci> g </ci> <ci> k </ci> </list> </list> </list> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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