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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 multiple series for product of several spherical harmonics





http://functions.wolfram.com/05.10.16.0009.01









  


  








Input Form





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








Standard Form





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MathML Form







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</mo> <msub> <mi> n </mi> <mi> k </mi> </msub> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> &#10869; </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> M </mi> <mn> 0 </mn> </msub> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> M </mi> <mi> j </mi> </msub> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> j </mi> </munderover> <msub> <mi> m </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> n </mi> <mi> j </mi> </msub> <msub> <mi> m </mi> <mi> j </mi> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> n </mi> <mi> j </mi> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <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> <msub> <mi> M </mi> <mn> 2 </mn> </msub> <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> <msub> <mi> M </mi> <mn> 3 </mn> </msub> <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> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </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> <mrow> <mi> p </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> - </mo> <msub> <mi> n </mi> <mi> p </mi> </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> <msub> <mi> M </mi> <mi> p </mi> </msub> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <msub> <mi> k </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> n </mi> <mi> p </mi> </msub> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> k </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> k </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> M </mi> <mi> p </mi> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> &#9001; </mo> <mrow> <mrow> <msub> <mi> k </mi> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> n </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </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> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> n </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> k </mi> <mi> j </mi> </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;, RowBox[List[&quot;j&quot;, &quot;-&quot;, &quot;1&quot;]]], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;n&quot;, RowBox[List[&quot;j&quot;, &quot;+&quot;, &quot;1&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;, RowBox[List[&quot;j&quot;, &quot;-&quot;, &quot;1&quot;]]], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;n&quot;, RowBox[List[&quot;j&quot;, &quot;+&quot;, &quot;1&quot;]]], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;k&quot;, &quot;j&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> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> n </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> M </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> m </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mtext> &#8287; </mtext> <mo> &#10072; </mo> <mtext> &#8287; </mtext> <mrow> <msub> <mi> k </mi> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> n </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> k </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> M </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> </mrow> <mo> &#9002; </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[LeftAngleBracket]&quot;, RowBox[List[RowBox[List[SubscriptBox[&quot;k&quot;, RowBox[List[&quot;j&quot;, &quot;-&quot;, &quot;1&quot;]]], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;n&quot;, RowBox[List[&quot;j&quot;, &quot;+&quot;, &quot;1&quot;]]], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;M&quot;, &quot;j&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;m&quot;, RowBox[List[&quot;j&quot;, &quot;+&quot;, &quot;1&quot;]]]]], &quot;\[MediumSpace]&quot;, &quot;\[VerticalSeparator]&quot;, &quot;\[MediumSpace]&quot;, RowBox[List[SubscriptBox[&quot;k&quot;, RowBox[List[&quot;j&quot;, &quot;-&quot;, &quot;1&quot;]]], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;n&quot;, RowBox[List[&quot;j&quot;, &quot;+&quot;, &quot;1&quot;]]], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;k&quot;, &quot;j&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;M&quot;, RowBox[List[&quot;j&quot;, &quot;+&quot;, &quot;1&quot;]]]]]]], &quot;\[RightAngleBracket]&quot;]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mi> k </mi> </msub> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> m </mi> <mi> k </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> k </mi> </msub> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8804; </mo> <msub> <mi> n </mi> <mi> k </mi> </msub> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> &#10869; </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> M </mi> <mn> 0 </mn> </msub> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> M </mi> <mi> j </mi> </msub> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> j </mi> </munderover> <msub> <mi> m </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j_", "=", "1"]], "p_"], RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n_", "j_"], ",", SubscriptBox["m_", "j_"], ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", SubscriptBox["nn", "j"]]], "+", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List[SubscriptBox["nn", "1"], "-", SubscriptBox["nn", "2"]]], "]"]], ",", RowBox[List["Abs", "[", SubscriptBox["M", "2"], "]"]]]], "]"]]]], RowBox[List[SubscriptBox["nn", "1"], "+", SubscriptBox["nn", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List[SubscriptBox["k", "1"], "-", SubscriptBox["nn", "3"]]], "]"]], ",", RowBox[List["Abs", "[", SubscriptBox["M", "3"], "]"]]]], "]"]]]], RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["nn", "3"]]]], RowBox[List["\[Ellipsis]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", RowBox[List["p", "-", "1"]]], "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List[SubscriptBox["k", RowBox[List["p", "-", "2"]]], "-", SubscriptBox["nn", "p"]]], "]"]], ",", RowBox[List["Abs", "[", SubscriptBox["M", "p"], "]"]]]], "]"]]]], RowBox[List[SubscriptBox["k", RowBox[List["p", "-", "2"]]], "+", SubscriptBox["nn", "p"]]]], FractionBox[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["k", RowBox[List["p", "-", "1"]]], ",", SubscriptBox["M", "p"], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List["p", "-", "1"]]], RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["k", RowBox[List["j", "-", "1"]]], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["n", RowBox[List["j", "+", "1"]]], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["k", "j"], ",", "0"]], "}"]]]], "]"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["k", RowBox[List["j", "-", "1"]]], ",", SubscriptBox["M", "j"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["n", RowBox[List["j", "+", "1"]]], ",", SubscriptBox["m", RowBox[List["j", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["k", "j"], ",", SubscriptBox["M", RowBox[List["j", "+", "1"]]]]], "}"]]]], "]"]]]]]]]], SqrtBox[RowBox[List[RowBox[List["2", " ", SubscriptBox["k", RowBox[List["p", "-", "1"]]]]], "+", "1"]]]]]]]]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["4", " ", "\[Pi]"]], ")"]], FractionBox[RowBox[List["p", "-", "1"]], "2"]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", ">", "1"]], "&&", RowBox[List[SubscriptBox["n", "k"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["n", "k"], "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["m", "k"], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["m", "k"], "]"]], "\[LessEqual]", SubscriptBox["n", "k"]]], "&&", RowBox[List[SubscriptBox["k", "0"], "\[Equal]", SubscriptBox["nn", "1"]]], "&&", RowBox[List[SubscriptBox["M", "0"], "\[Equal]", "0"]], "&&", RowBox[List[SubscriptBox["M", "j"], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "j"], SubscriptBox["m", "k"]]]]]]]]]]]]]


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





2001-10-29