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http://functions.wolfram.com/07.37.20.0012.01
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D[SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]],
{\[CurlyTheta], 2}] == (1/2) Sqrt[(2 \[Lambda] + 1)/(4 Pi)]
(Sqrt[Gamma[\[Lambda] - \[Mu] + 1]]/Sqrt[Gamma[\[Lambda] + \[Mu] + 1]])
E^(I \[CurlyPhi] \[Mu]) Csc[\[CurlyTheta]]^2
(2 (\[Lambda] + \[Mu]) Cos[\[CurlyTheta]] LegendreP[\[Lambda] - 1, \[Mu],
2, Cos[\[CurlyTheta]]] + (2 \[Mu]^2 - 2 \[Lambda] -
2 \[Lambda]^2 Sin[\[CurlyTheta]]^2) LegendreP[\[Lambda], \[Mu], 2,
Cos[\[CurlyTheta]]])
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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[FractionBox["1", "2"], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]], RowBox[List["4", "\[Pi]"]]]], " ", FractionBox[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["Csc", "[", "\[CurlyTheta]", "]"]], "2"], RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Mu]"]], ")"]], " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Lambda]", "-", "1"]], ",", "\[Mu]", ",", "2", ",", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", SuperscriptBox["\[Mu]", "2"]]], "-", RowBox[List["2", "\[Lambda]"]], "-", RowBox[List["2", SuperscriptBox["\[Lambda]", "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "2"]]]]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "2", ",", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "]"]]]]]], ")"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mn> 2 </mn> </msup> <mrow> <msubsup> <mi> Y </mi> <mi> λ </mi> <mi> μ </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> ϑ </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mfrac> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> - </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> φ </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mrow> <mi> λ </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> μ </mi> </msubsup> <mo> ( </mo> <semantics> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["cos", "(", "\[CurlyTheta]", ")"]], HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mi> λ </mi> <mi> μ </mi> </msubsup> <mo> ( </mo> <semantics> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["cos", "(", "\[CurlyTheta]", ")"]], HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> ϑ </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> SphericalHarmonicY </ci> <ci> λ </ci> <ci> μ </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> φ </ci> <ci> μ </ci> </apply> </apply> <apply> <power /> <apply> <csc /> <ci> ϑ </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> λ </ci> <ci> μ </ci> </apply> <apply> <cos /> <ci> ϑ </ci> </apply> <apply> <ci> LegendreP </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> -1 </cn> </apply> <ci> μ </ci> <cn type='integer'> 2 </cn> <apply> <cos /> <ci> ϑ </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <sin /> <ci> ϑ </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> LegendreP </ci> <ci> λ </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> <apply> <cos /> <ci> ϑ </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| 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]", FractionBox[RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], RowBox[List["4", " ", "\[Pi]"]]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["Csc", "[", "\[CurlyTheta]", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Mu]"]], ")"]], " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Lambda]", "-", "1"]], ",", "\[Mu]", ",", "2", ",", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["\[Mu]", "2"]]], "-", RowBox[List["2", " ", "\[Lambda]"]], "-", RowBox[List["2", " ", SuperscriptBox["\[Lambda]", "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "2"]]]]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "2", ",", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
