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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 double series for product of three spherical harmonics





http://functions.wolfram.com/05.10.16.0008.01









  


  










Input Form





SphericalHarmonicY[Subscript[n, 1], Subscript[m, 1], \[CurlyTheta], \[CurlyPhi]] SphericalHarmonicY[Subscript[n, 2], Subscript[m, 2], \[CurlyTheta], \[CurlyPhi]] SphericalHarmonicY[Subscript[n, 3], Subscript[m, 3], \[CurlyTheta], \[CurlyPhi]] == (Sqrt[(2 Subscript[n, 1] + 1) (2 Subscript[n, 2] + 1) (2 Subscript[n, 3] + 1)]/(4 Pi)) Sum[(SphericalHarmonicY[Subscript[k, 2], Subscript[m, 1] + Subscript[m, 2] + Subscript[m, 3], \[CurlyTheta], \[CurlyPhi]]/ Sqrt[2 Subscript[k, 2] + 1]) ClebschGordan[{Subscript[n, 1], 0}, {Subscript[n, 2], 0}, {Subscript[k, 1], 0}] ClebschGordan[{Subscript[k, 1], 0}, {Subscript[n, 3], 0}, {Subscript[k, 2], 0}] ClebschGordan[{Subscript[n, 1], Subscript[m, 1]}, {Subscript[n, 2], Subscript[m, 2]}, {Subscript[k, 1], Subscript[m, 1] + Subscript[m, 2]}] ClebschGordan[ {Subscript[k, 1], Subscript[m, 1] + Subscript[m, 2]}, {Subscript[n, 3], Subscript[m, 3]}, {Subscript[k, 2], Subscript[m, 1] + Subscript[m, 2] + Subscript[m, 3]}], {Subscript[k, 1], Max[Abs[Subscript[n, 1] - Subscript[n, 2]], Abs[Subscript[m, 1] + Subscript[m, 2]]], Subscript[n, 1] + Subscript[n, 2]}, {Subscript[k, 2], Max[Abs[Subscript[k, 1] - Subscript[n, 3]], Abs[Subscript[m, 1] + Subscript[m, 2] + Subscript[m, 3]]], Subscript[k, 1] + Subscript[n, 3]}] /; Element[Subscript[n, j], Integers] && Subscript[n, j] >= 0 && Element[Subscript[m, j], Integers] && Abs[Subscript[m, j]] <= Subscript[n, j] && Element[j, {1, 2, 3}]










Standard Form





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MathML Form







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</mo> <mrow> <mrow> <msub> <mi> n </mi> <mi> j </mi> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> m </mi> <mi> j </mi> </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> <mi> j </mi> </msub> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8804; </mo> <msub> <mi> n </mi> <mi> j </mi> </msub> </mrow> <mo> &#8743; </mo> <mrow> <mi> j </mi> <mo> &#8712; </mo> <mrow> <mo> { </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> n </mi> <mn> 3 </mn> </msub> <msub> <mi> m </mi> <mn> 3 </mn> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> n </mi> <mn> 3 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mrow> <mi> max </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> , </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mrow> <mi> max </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> n </mi> <mn> 3 </mn> </msub> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> , </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 3 </mn> </msub> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 3 </mn> </msub> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 3 </mn> </msub> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mo> &#9001; </mo> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mn> 0 </mn> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mn> 0 </mn> </mrow> <mtext> &#8287; </mtext> <mo> &#10072; </mo> <mtext> &#8287; </mtext> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mn> 0 </mn> </mrow> </mrow> <mo> &#9002; </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[LeftAngleBracket]&quot;, RowBox[List[RowBox[List[SubscriptBox[&quot;n&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;\[MediumSpace]&quot;, &quot;0&quot;, &quot;\[MediumSpace]&quot;, &quot;0&quot;]], &quot;\[MediumSpace]&quot;, &quot;\[VerticalSeparator]&quot;, &quot;\[MediumSpace]&quot;, RowBox[List[SubscriptBox[&quot;n&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;k&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, &quot;0&quot;]]]], &quot;\[RightAngleBracket]&quot;]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mo> &#9001; </mo> <mrow> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> n </mi> <mn> 3 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mn> 0 </mn> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mn> 0 </mn> </mrow> <mtext> &#8287; </mtext> <mo> &#10072; </mo> <mtext> &#8287; </mtext> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> n </mi> <mn> 3 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mn> 0 </mn> </mrow> </mrow> <mo> &#9002; </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[LeftAngleBracket]&quot;, RowBox[List[RowBox[List[SubscriptBox[&quot;k&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;n&quot;, &quot;3&quot;], &quot;\[MediumSpace]&quot;, &quot;0&quot;, &quot;\[MediumSpace]&quot;, &quot;0&quot;]], &quot;\[MediumSpace]&quot;, &quot;\[VerticalSeparator]&quot;, &quot;\[MediumSpace]&quot;, RowBox[List[SubscriptBox[&quot;k&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;n&quot;, &quot;3&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;k&quot;, &quot;2&quot;], &quot;\[MediumSpace]&quot;, &quot;0&quot;]]]], &quot;\[RightAngleBracket]&quot;]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mo> &#9001; </mo> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> <mtext> &#8287; </mtext> <mo> &#10072; </mo> <mtext> &#8287; </mtext> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> &#9002; </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[LeftAngleBracket]&quot;, RowBox[List[RowBox[List[SubscriptBox[&quot;n&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;m&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;m&quot;, &quot;2&quot;]]], &quot;\[MediumSpace]&quot;, &quot;\[VerticalSeparator]&quot;, &quot;\[MediumSpace]&quot;, RowBox[List[RowBox[List[SubscriptBox[&quot;n&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;k&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;m&quot;, &quot;1&quot;]]], &quot;+&quot;, SubscriptBox[&quot;m&quot;, &quot;2&quot;]]]]], &quot;\[RightAngleBracket]&quot;]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mo> &#9001; </mo> <mrow> <mrow> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> n </mi> <mn> 3 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> m </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mtext> &#8287; </mtext> <mo> &#10072; </mo> <mtext> &#8287; </mtext> <mrow> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> n </mi> <mn> 3 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> &#9002; </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[LeftAngleBracket]&quot;, RowBox[List[RowBox[List[RowBox[List[SubscriptBox[&quot;k&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;n&quot;, &quot;3&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;m&quot;, &quot;1&quot;]]], &quot;+&quot;, RowBox[List[SubscriptBox[&quot;m&quot;, &quot;2&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;m&quot;, &quot;3&quot;]]]]], &quot;\[MediumSpace]&quot;, &quot;\[VerticalSeparator]&quot;, &quot;\[MediumSpace]&quot;, RowBox[List[RowBox[List[SubscriptBox[&quot;k&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;n&quot;, &quot;3&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;k&quot;, &quot;2&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;m&quot;, &quot;1&quot;]]], &quot;+&quot;, SubscriptBox[&quot;m&quot;, &quot;2&quot;], &quot;+&quot;, SubscriptBox[&quot;m&quot;, &quot;3&quot;]]]]], &quot;\[RightAngleBracket]&quot;]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> /; </mo> <mrow> <mrow> <msub> <mi> n </mi> <mi> j </mi> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> m </mi> <mi> j </mi> </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> <mi> j </mi> </msub> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8804; </mo> <msub> <mi> n </mi> <mi> j </mi> </msub> </mrow> <mo> &#8743; </mo> <mrow> <mi> j </mi> <mo> &#8712; </mo> <mrow> <mo> { </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





© 1998- Wolfram Research, Inc.