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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/05.10.17.0009.01









  


  










Input Form





SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] == E^(I \[CurlyPhi]) Csc[\[CurlyTheta]] ((-Sqrt[((1 + 2 n) (-1 + n + m))/((-1 + 2 n) (n + m))]) SphericalHarmonicY[-1 + n, -1 + m, \[CurlyTheta], \[CurlyPhi]] + Sqrt[(1 + n - m)/(n + m)] Cos[\[CurlyTheta]] SphericalHarmonicY[n, -1 + m, \[CurlyTheta], \[CurlyPhi]]) /; -n + 1 <= m <= n && m != 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]"]]], " ", RowBox[List["Csc", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n", "+", "m"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]]]]]]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "+", RowBox[List[SqrtBox[FractionBox[RowBox[List["1", "+", "n", "-", "m"]], RowBox[List["n", "+", "m"]]]], " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "n"]], "+", "1"]], "\[LessEqual]", "m", "\[LessEqual]", "n"]], "\[And]", RowBox[List["m", "\[NotEqual]", "0"]]]]]]]]










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> <mi> &#8520; </mi> <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> <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> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <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> <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> </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> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> &#8804; </mo> <mi> m </mi> <mo> &#8804; </mo> <mi> n </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8800; </mo> <mn> 0 </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 /> <imaginaryi /> <ci> &#966; </ci> </apply> </apply> <apply> <csc /> <ci> &#977; </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <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> <plus /> <ci> n </ci> <ci> m </ci> </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> <ci> n </ci> <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> <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> <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'> -1 </cn> </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'> -1 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <ci> &#977; </ci> <ci> &#966; </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <leq /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> m </ci> <ci> n </ci> </apply> <apply> <neq /> <ci> m </ci> <cn type='integer'> 0 </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["\[ImaginaryI]", " ", "\[CurlyPhi]"]]], " ", RowBox[List["Csc", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n", "+", "m"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]]]]]]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "+", RowBox[List[SqrtBox[FractionBox[RowBox[List["1", "+", "n", "-", "m"]], RowBox[List["n", "+", "m"]]]], " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "n"]], "+", "1"]], "\[LessEqual]", "m", "\[LessEqual]", "n"]], "&&", RowBox[List["m", "\[NotEqual]", "0"]]]]]]]]]]










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





2001-10-29





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