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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Differentiation > Low-order differentiation > With respect to theta





http://functions.wolfram.com/07.37.20.0011.01









  


  










Input Form





D[SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]], {\[CurlyTheta], 2}] == \[Mu] (\[Mu] Cot[\[CurlyTheta]]^2 - Csc[\[CurlyTheta]]^2) SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]] + (((Sqrt[Gamma[\[Lambda] - \[Mu] + 1]] Sqrt[Gamma[\[Lambda] + \[Mu] + 2]])/ (Sqrt[Gamma[\[Lambda] + \[Mu] + 1]] Sqrt[Gamma[\[Lambda] - \[Mu]]])) (2 \[Mu] + 1) Cot[\[CurlyTheta]] SphericalHarmonicY[\[Lambda], \[Mu] + 1, \[CurlyTheta], \[CurlyPhi]])/E^(I \[CurlyPhi]) + (((Sqrt[Gamma[\[Lambda] - \[Mu] + 1]] Sqrt[Gamma[\[Lambda] + \[Mu] + 3]])/ (Sqrt[Gamma[\[Lambda] + \[Mu] + 1]] Sqrt[Gamma[\[Lambda] - \[Mu] - 1]])) SphericalHarmonicY[\[Lambda], \[Mu] + 2, \[CurlyTheta], \[CurlyPhi]])/ E^(2 I \[CurlyPhi])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[CurlyTheta]", ",", "2"]], "}"]]], RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Mu]", " ", SuperscriptBox[RowBox[List["Cot", "[", "\[CurlyTheta]", "]"]], "2"]]], "-", SuperscriptBox[RowBox[List["Csc", "[", "\[CurlyTheta]", "]"]], "2"]]], ")"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[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]"]], "]"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", "\[Mu]"]], "+", "1"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[CurlyPhi]"]]], " ", RowBox[List["Cot", "[", "\[CurlyTheta]", "]"]], " ", 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]", "+", "3"]], "]"]]]]], RowBox[List[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "-", "1"]], "]"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], "\[ImaginaryI]", " ", "\[CurlyPhi]"]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", RowBox[List["\[Mu]", "+", "2"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]]]]]]]]










MathML Form







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</ci> <ci> &#966; </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> &#956; </ci> <apply> <plus /> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <apply> <cot /> <ci> &#977; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <csc /> <ci> &#977; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> &#955; </ci> <ci> &#956; </ci> <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; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[CurlyTheta]_", ",", "2"]], "}"]]]]], RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]_", ",", "\[Mu]_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Mu]", " ", SuperscriptBox[RowBox[List["Cot", "[", "\[CurlyTheta]", "]"]], "2"]]], "-", SuperscriptBox[RowBox[List["Csc", "[", "\[CurlyTheta]", "]"]], "2"]]], ")"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "+", 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["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[CurlyPhi]"]]], " ", RowBox[List["Cot", "[", "\[CurlyTheta]", "]"]], " ", 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]"]], "]"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "3"]], "]"]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[CurlyPhi]"]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", RowBox[List["\[Mu]", "+", "2"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], RowBox[List[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "-", "1"]], "]"]]]]]]]]]]]]










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





2001-10-29





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