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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Identities > Recurrence identities > Consecutive neighbors





http://functions.wolfram.com/05.10.17.0002.01









  


  










Input Form





SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] == Sqrt[((2 n - 1) (2 n + 1))/((n - m) (n + m))] Cos[\[CurlyTheta]] SphericalHarmonicY[n - 1, m, \[CurlyTheta], \[CurlyPhi]] - Sqrt[((2 n + 1) (n - m - 1) (n + m - 1))/((2 n - 3) (n - m) (n + m))] SphericalHarmonicY[n - 2, m, \[CurlyTheta], \[CurlyPhi]] /; Abs[m] < n










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "-", "1"]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "1"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]]]]]], RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "-", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "1"]], ")"]], RowBox[List["(", RowBox[List["n", "-", "m", "-", "1"]], ")"]], RowBox[List["(", RowBox[List["n", "+", "m", "-", "1"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "-", "3"]], ")"]], RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]]]]]], RowBox[List["SphericalHarmonicY", "[", RowBox[List[RowBox[List["n", "-", "2"]], ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "<", "n"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> Y </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <msubsup> <mi> Y </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> m </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mi> n </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> &#977; </ci> <ci> &#966; </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <plus /> <ci> n </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <cos /> <ci> &#977; </ci> </apply> <apply> <ci> SphericalHarmonicY </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> m </ci> <ci> &#977; </ci> <ci> &#966; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> n </ci> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -3 </cn> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <plus /> <ci> n </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> SphericalHarmonicY </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <ci> m </ci> <ci> &#977; </ci> <ci> &#966; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> m </ci> </apply> <ci> n </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]]]]]], " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "-", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "m", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "m", "-", "1"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", "3"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]]]]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[RowBox[List["n", "-", "2"]], ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "<", "n"]]]]]]]]










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





2001-10-29





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