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ClebschGordan






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ClebschGordan[{j1,m1},{j2,m2},{j,m}] > Integral representations > Integral representations of negative integer order





http://functions.wolfram.com/07.38.07.0010.01









  


  










Input Form





ClebschGordan[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {j, m}] == KroneckerDelta[m, Subscript[m, 1] + Subscript[m, 2]] (-1)^(Subscript[j, 1] + Subscript[j, 2] - j) (Sqrt[(Subscript[j, 1] - Subscript[j, 2] + j)!]/ (Sqrt[(Subscript[j, 1] + Subscript[j, 2] - j)!] Sqrt[(-Subscript[j, 1] + Subscript[j, 2] + j)!] Sqrt[(Subscript[j, 1] + Subscript[j, 2] + j + 1)!])) ((Sqrt[(Subscript[j, 1] + Subscript[m, 1])!] Sqrt[(Subscript[j, 1] - Subscript[m, 1])!] Sqrt[(Subscript[j, 2] + Subscript[m, 2])!] Sqrt[(j + m)!] Sqrt[2 j + 1])/ ((Subscript[j, 1] - Subscript[j, 2] + m)! Sqrt[(Subscript[j, 2] - Subscript[m, 2])!] Sqrt[(j - m)!])) Derivative[Subscript[j, 2] - Subscript[m, 2]][ Function[t, (1 - t)^(Subscript[j, 1] + Subscript[j, 2] - j) Hypergeometric2F1[Subscript[j, 1] - Subscript[j, 2] - j, -j + m, Subscript[j, 1] - Subscript[j, 2] + m + 1, t]]][0] /; \[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\[ScriptI]\[ScriptC]\ \[ScriptA]\[ScriptL]\[ScriptCapitalQ][{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {j, m}]










Standard Form





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MathML Form







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</ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ! </ms> </list> </apply> </apply> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 2 </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ! </ms> </list> </apply> </apply> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> + </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ! </ms> </list> </apply> </apply> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> j </ms> </list> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 1 </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 2 </ms> </apply> <ms> + </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ! </ms> </list> </apply> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 2 </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ! </ms> </list> </apply> </apply> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> - </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ! </ms> </list> </apply> </apply> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> &#8706; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 2 </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> t </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> j </ms> </list> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 1 </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 2 </ms> </apply> </list> </apply> </apply> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#62387; </ms> <apply> <ci> FormBox </ci> <ms> 2 </ms> <ci> TraditionalForm </ci> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> F </ms> <apply> <ci> FormBox </ci> <ms> 1 </ms> <ci> TraditionalForm </ci> </apply> </apply> </list> </apply> <ms> &#8289; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> j </ms> </list> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 1 </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 2 </ms> </apply> </list> </apply> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> j </ms> </list> </apply> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 1 </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 2 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <ms> t </ms> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ci> Hypergeometric2F1 </ci> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8706; </ms> <apply> <ci> SuperscriptBox </ci> <ms> t </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 2 </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> </list> </apply> </apply> </list> </apply> </apply> </list> </apply> </list> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> | </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> t </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <ms> &#119979;&#119997;&#120014;&#120008;&#119998;&#119992;&#119990;&#8467;&#119980; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 1 </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> } </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 2 </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> } </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> , </ms> <ms> m </ms> </list> </apply> <ms> } </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-12-21





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