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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Transformations > Products, sums, and powers of the direct function > Products involving the direct function > Clebsch-Gordan series for product of two spherical harmonics





http://functions.wolfram.com/05.10.16.0007.01









  


  










Input Form





SphericalHarmonicY[Subscript[n, 1], Subscript[m, 1], \[CurlyTheta], \[CurlyPhi]] SphericalHarmonicY[Subscript[n, 2], Subscript[m, 2], \[CurlyTheta], \[CurlyPhi]] == Sqrt[((2 Subscript[n, 1] + 1) (2 Subscript[n, 2] + 1))/(4 Pi)] Sum[(SphericalHarmonicY[k, Subscript[m, 1] + Subscript[m, 2], \[CurlyTheta], \[CurlyPhi]]/Sqrt[2 k + 1]) ClebschGordan[{Subscript[n, 1], 0}, {Subscript[n, 2], 0}, {k, 0}] ClebschGordan[{Subscript[n, 1], Subscript[m, 1]}, {Subscript[n, 2], Subscript[m, 2]}, {k, Subscript[m, 1] + Subscript[m, 2]}], {k, Max[Abs[Subscript[n, 1] - Subscript[n, 2]], Abs[Subscript[m, 1] + Subscript[m, 2]]], Subscript[n, 1] + Subscript[n, 2]}] /; Element[Subscript[n, 1], Integers] && Subscript[n, 1] >= 0 && Element[Subscript[n, 2], Integers] && Subscript[n, 2] >= 0 && Element[Subscript[m, 1], Integers] && Element[Subscript[m, 2], Integers] && Abs[Subscript[m, 1]] <= Subscript[n, 1] && Abs[Subscript[m, 2]] <= Subscript[n, 2]










Standard Form





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MathML Form







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</mo> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <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> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8804; </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8804; </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





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