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http://functions.wolfram.com/05.10.20.0003.01
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D[SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]],
{\[CurlyTheta], 2}] == m (m Cot[\[CurlyTheta]]^2 - Csc[\[CurlyTheta]]^2)
SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] +
((2 m + 1) Sqrt[(n - m) (n + m + 1)] Cot[\[CurlyTheta]]
SphericalHarmonicY[n, m + 1, \[CurlyTheta], \[CurlyPhi]])/
E^(I \[CurlyPhi]) + (Sqrt[(n - m) (n - m - 1) (n + m + 2) (n + m + 1)]
SphericalHarmonicY[n, m + 2, \[CurlyTheta], \[CurlyPhi]])/
E^(2 I \[CurlyPhi])
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Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[CurlyTheta]", ",", "2"]], "}"]]], RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["m", " ", SuperscriptBox[RowBox[List["Cot", "[", "\[CurlyTheta]", "]"]], "2"]]], "-", SuperscriptBox[RowBox[List["Csc", "[", "\[CurlyTheta]", "]"]], "2"]]], ")"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "+", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "m"]], "+", "1"]], ")"]], SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], RowBox[List["(", RowBox[List["n", "+", "m", "+", "1"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[CurlyPhi]"]]], " ", RowBox[List["Cot", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", RowBox[List["m", "+", "1"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], RowBox[List["(", RowBox[List["n", "-", "m", "-", "1"]], ")"]], RowBox[List["(", RowBox[List["n", "+", "m", "+", "2"]], ")"]], RowBox[List["(", RowBox[List["n", "+", "m", "+", "1"]], ")"]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], "\[ImaginaryI]", " ", "\[CurlyPhi]"]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", RowBox[List["m", "+", "2"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]]]]]]]]
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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> n </mi> <mi> m </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> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> cot </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> φ </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> φ </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> </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> n </ci> <ci> m </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <cot /> <ci> ϑ </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <csc /> <ci> ϑ </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> φ </ci> </apply> </apply> <apply> <cot /> <ci> ϑ </ci> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <ci> ϑ </ci> <ci> φ </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <imaginaryi /> <ci> φ </ci> </apply> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <ci> ϑ </ci> <ci> φ </ci> </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["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["m", " ", SuperscriptBox[RowBox[List["Cot", "[", "\[CurlyTheta]", "]"]], "2"]]], "-", SuperscriptBox[RowBox[List["Csc", "[", "\[CurlyTheta]", "]"]], "2"]]], ")"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m"]], "+", "1"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "m", "+", "1"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[CurlyPhi]"]]], " ", RowBox[List["Cot", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", RowBox[List["m", "+", "1"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "m", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "m", "+", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "m", "+", "1"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[CurlyPhi]"]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", RowBox[List["m", "+", "2"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
