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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Differential equations > Partial differential equations





http://functions.wolfram.com/07.37.13.0003.01









  


  










Input Form





E^(I \[CurlyPhi]) (D[SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]], \[CurlyTheta]] + I Cot[\[CurlyTheta]] D[SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]], \[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]]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]"]]], RowBox[List["(", RowBox[List[RowBox[List["D", "[", RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], ",", "\[CurlyTheta]"]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cot", "[", "\[CurlyTheta]", "]"]], RowBox[List["D", "[", RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], ",", "\[CurlyPhi]"]], "]"]]]]]], ")"]]]], "\[Equal]", 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]"]], "]"]]]]]], RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", RowBox[List["\[Mu]", "+", "1"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#966; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mo> &#8706; </mo> <mrow> <msubsup> <mi> Y </mi> <mi> &#955; </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <mi> &#977; </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mfrac> <mrow> <mo> &#8706; </mo> <mrow> <msubsup> <mi> Y </mi> <mi> &#955; </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <mi> &#966; </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </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> <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> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> &#966; </ci> </apply> </apply> <apply> <plus /> <apply> <partialdiff /> <bvar> <ci> &#977; </ci> </bvar> <apply> <ci> SphericalHarmonicY </ci> <ci> &#955; </ci> <ci> &#956; </ci> <ci> &#977; </ci> <ci> &#966; </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <cot /> <ci> &#977; </ci> </apply> <apply> <partialdiff /> <bvar> <ci> &#966; </ci> </bvar> <apply> <ci> SphericalHarmonicY </ci> <ci> &#955; </ci> <ci> &#956; </ci> <ci> &#977; </ci> <ci> &#966; </ci> </apply> </apply> </apply> </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> <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> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]_"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["\[CurlyTheta]_"]]], RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]_", ",", "\[Mu]_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cot", "[", "\[CurlyTheta]_", "]"]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["\[CurlyPhi]_"]]], RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]_", ",", "\[Mu]_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]]]]]]]], ")"]]]], "]"]], "\[RuleDelayed]", 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"]], "]"]]]]], ")"]], " ", 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]"]], "]"]]]]]]]]]]










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





2001-10-29





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