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http://functions.wolfram.com/07.37.13.0004.01
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(1/Sin[\[CurlyTheta]]) D[Sin[\[CurlyTheta]]
D[SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]],
\[CurlyTheta]], \[CurlyTheta]] + (1/Sin[\[CurlyTheta]]^2)
D[SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]],
{\[CurlyPhi], 2}] + \[Lambda] (\[Lambda] + 1)
SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]] == 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[FractionBox["1", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]], RowBox[List["D", "[", RowBox[List[RowBox[List[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], RowBox[List["D", "[", RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], ",", "\[CurlyTheta]"]], "]"]]]], ",", "\[CurlyTheta]"]], "]"]]]], "+", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "2"]], RowBox[List["D", "[", RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], ",", RowBox[List["{", RowBox[List["\[CurlyPhi]", ",", "2"]], "}"]]]], "]"]]]], "+", RowBox[List["\[Lambda]", RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]], RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]]]], "\[Equal]", "0"]]]]
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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> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <mo> ∂ </mo> <mrow> <mo> ∂ </mo> <mi> ϑ </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mfrac> <mrow> <mo> ∂ </mo> <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> <mi> ϑ </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <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> φ </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <mi> λ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <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> <mn> 0 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <ms> ϑ </ms> <ms> ) </ms> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> ErrorBox </ci> <ms> ∂ </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ∂ </ms> <ms> ϑ </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <ms> ϑ </ms> <ms> ) </ms> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ∂ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> Y </ms> <ms> λ </ms> <ms> μ </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> ϑ </ms> <ms> , </ms> <ms> φ </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ∂ </ms> <ms> ϑ </ms> </list> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sin </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> ϑ </ms> <ms> ) </ms> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> ∂ </ms> <ms> 2 </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> Y </ms> <ms> λ </ms> <ms> μ </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> ϑ </ms> <ms> , </ms> <ms> φ </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ∂ </ms> <apply> <ci> SuperscriptBox </ci> <ms> φ </ms> <ms> 2 </ms> </apply> </list> </apply> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> λ </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> λ </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> Y </ms> <ms> λ </ms> <ms> μ </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> ϑ </ms> <ms> , </ms> <ms> φ </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ⩵ </ms> <ms> 0 </ms> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[FractionBox[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["\[CurlyTheta]_"]]], RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", "\[CurlyTheta]_", "]"]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["\[CurlyTheta]_"]]], RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]_", ",", "\[Mu]_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]]]]]], ")"]]]], RowBox[List["Sin", "[", "\[CurlyTheta]_", "]"]]], "+", FractionBox[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[CurlyPhi]_", ",", "2"]], "}"]]]]], RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]_", ",", "\[Mu]_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]]]], SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]_", "]"]], "2"]], "+", RowBox[List["\[Lambda]_", " ", RowBox[List["(", RowBox[List["\[Lambda]_", "+", "1"]], ")"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]_", ",", "\[Mu]_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", "0"]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
