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ClebschGordan






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ClebschGordan[{j1,m1},{j2,m2},{j,m}] > Integral representations > Multiple integral representations > For the direct function itself





http://functions.wolfram.com/07.38.07.0006.01









  


  










Input Form





ClebschGordan[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {j, m}] == ((2 I^(Subscript[j, 1] + Subscript[j, 2] - j) Sqrt[Pi] ((j + Subscript[j, 1] - Subscript[j, 2])/2)! ((j - Subscript[j, 1] + Subscript[j, 2])/2)! ((Subscript[j, 1] + Subscript[j, 2] - j)/2)! Sqrt[(Subscript[j, 1] + Subscript[j, 2] + j + 1)!])/ (Sqrt[(j + Subscript[j, 1] - Subscript[j, 2])!] Sqrt[(j - Subscript[j, 1] + Subscript[j, 2])!] Sqrt[(Subscript[j, 1] + Subscript[j, 2] - j)!] ((j + Subscript[j, 1] + Subscript[j, 2])/2)! Sqrt[2 Subscript[j, 1] + 1] Sqrt[2 Subscript[j, 2] + 1])) Integrate[Sin[\[CurlyTheta]] SphericalHarmonicY[Subscript[j, 1], Subscript[m, 1], \[CurlyTheta], \[CurlyPhi]] SphericalHarmonicY[Subscript[j, 2], Subscript[m, 2], \[CurlyTheta], \[CurlyPhi]] Conjugate[SphericalHarmonicY[j, m, \[CurlyTheta], \[CurlyPhi]]], {\[CurlyTheta], 0, Pi}, {\[CurlyPhi], 0, 2 Pi}] /; Element[(Subscript[j, 1] + Subscript[j, 2] + j)/2, Integers] && Element[m, Integers] && Element[Subscript[m, 1], Integers] && Element[Subscript[m, 2], Integers] && \[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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<ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <integers /> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <integers /> </apply> <apply> <ci> &#119979;&#119997;&#120014;&#120008;&#119998;&#119992;&#119990;&#8467;&#119980; </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> <ci> j </ci> <ci> m </ci> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29