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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Identities > Functional identities > Additional relations between contiguous functions





http://functions.wolfram.com/05.10.17.0011.01









  


  










Input Form





SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] == (Csc[\[CurlyTheta]] ((-Cos[\[CurlyTheta]]) Sqrt[(n + m + 1)/(n - m)] SphericalHarmonicY[n, m + 1, \[CurlyTheta], \[CurlyPhi]] + Sqrt[((2 n + 1)/(2 n - 1)) ((n - m - 1)/(n - m))] SphericalHarmonicY[n - 1, m + 1, \[CurlyTheta], \[CurlyPhi]]))/ E^(I \[CurlyPhi]) /; -n <= m <= n - 1










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[CurlyPhi]"]]], " ", RowBox[List["Csc", "[", "\[CurlyTheta]", "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], " ", SqrtBox[FractionBox[RowBox[List["n", "+", "m", "+", "1"]], RowBox[List["n", "-", "m"]]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", RowBox[List["m", "+", "1"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], RowBox[List[RowBox[List["2", "n"]], "-", "1"]]], FractionBox[RowBox[List["n", "-", "m", "-", "1"]], RowBox[List["n", "-", "m"]]]]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", RowBox[List["m", "+", "1"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["-", "n"]], "\[LessEqual]", "m", "\[LessEqual]", RowBox[List["n", "-", "1"]]]]]]]]










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> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> &#966; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </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> <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> 1 </mn> </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> 1 </mn> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> &#8804; </mo> <mi> m </mi> <mo> &#8804; </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </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> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> &#966; </ci> </apply> </apply> <apply> <csc /> <ci> &#977; </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 /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <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'> -1 </cn> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </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'> -1 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#977; </ci> <ci> &#966; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <cos /> <ci> &#977; </ci> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> n </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#977; </ci> <ci> &#966; </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <leq /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> m </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </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[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[CurlyPhi]"]]], " ", RowBox[List["Csc", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], " ", SqrtBox[FractionBox[RowBox[List["n", "+", "m", "+", "1"]], RowBox[List["n", "-", "m"]]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", RowBox[List["m", "+", "1"]], ",", "\[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["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]]]]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", RowBox[List["m", "+", "1"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["-", "n"]], "\[LessEqual]", "m", "\[LessEqual]", RowBox[List["n", "-", "1"]]]]]]]]]]










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





2001-10-29





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