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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Identities > Functional identities > Relations between contiguous functions





http://functions.wolfram.com/07.37.17.0006.01









  


  










Input Form





SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]] == (-(Tan[\[CurlyTheta]]/(2 \[Mu]))) ((((\[Lambda] (\[Lambda] + 1) - \[Mu] (\[Mu] - 1)) Sqrt[Gamma[\[Lambda] + \[Mu]]] Sqrt[Gamma[\[Lambda] - \[Mu] + 1]])/ (Sqrt[Gamma[\[Lambda] - \[Mu] + 2]] Sqrt[Gamma[\[Lambda] + \[Mu] + 1]])) E^(I \[CurlyPhi]) SphericalHarmonicY[\[Lambda], \[Mu] - 1, \[CurlyTheta], \[CurlyPhi]] + (((Sqrt[Gamma[\[Lambda] - \[Mu] + 1]] Sqrt[Gamma[\[Lambda] + \[Mu] + 2]])/ (Sqrt[Gamma[\[Lambda] + \[Mu] + 1]] Sqrt[Gamma[\[Lambda] - \[Mu]]])) SphericalHarmonicY[\[Lambda], \[Mu] + 1, \[CurlyTheta], \[CurlyPhi]])/ E^(I \[CurlyPhi]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Tan", "[", "\[CurlyTheta]", "]"]], RowBox[List["2", "\[Mu]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[Lambda]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]]]], "-", RowBox[List["\[Mu]", RowBox[List["(", RowBox[List["\[Mu]", "-", "1"]], ")"]]]]]], ")"]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]]]], RowBox[List[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "2"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]"]]], RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", RowBox[List["\[Mu]", "-", "1"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "2"]], "]"]]]]], RowBox[List[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]"]], "]"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[CurlyPhi]"]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", RowBox[List["\[Mu]", "+", "1"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]]]], ")"]]]]]]]]










MathML Form







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</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> <mfrac> <mrow> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> <mrow> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> &#966; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mi> Y </mi> <mi> &#955; </mi> <mrow> <mi> &#956; </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> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> SphericalHarmonicY </ci> <ci> &#955; </ci> <ci> &#956; </ci> <ci> &#977; </ci> <ci> &#966; </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <tan /> <ci> &#977; </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> &#955; </ci> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#956; </ci> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <ci> &#956; </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> &#966; </ci> </apply> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> &#955; </ci> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -1 </cn> </apply> <ci> &#977; </ci> <ci> &#966; </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> &#966; </ci> </apply> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> &#955; </ci> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#977; </ci> <ci> &#966; </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]_", ",", "\[Mu]_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Tan", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[Lambda]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]]]], "-", RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "1"]], ")"]]]]]], ")"]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]"]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", RowBox[List["\[Mu]", "-", "1"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], RowBox[List[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "2"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "2"]], "]"]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[CurlyPhi]"]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", RowBox[List["\[Mu]", "+", "1"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], RowBox[List[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]"]], "]"]]]]]]]], ")"]]]], RowBox[List["2", " ", "\[Mu]"]]]]]]]]]










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





2001-10-29





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