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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Differential equations > Partial differential equations





http://functions.wolfram.com/05.10.13.0004.01









  


  










Input Form





(1/Sin[\[CurlyTheta]]) D[Sin[\[CurlyTheta]] D[SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]], \[CurlyTheta]], \[CurlyTheta]] + (1/Sin[\[CurlyTheta]]^2) D[SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]], {\[CurlyPhi], 2}] + n (n + 1) SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] == 0










Standard Form





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["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], ",", "\[CurlyTheta]"]], "]"]]]], ",", "\[CurlyTheta]"]], "]"]]]], "+", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "2"]], RowBox[List["D", "[", RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], ",", RowBox[List["{", RowBox[List["\[CurlyPhi]", ",", "2"]], "}"]]]], "]"]]]], "+", RowBox[List["n", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]]]], "\[Equal]", "0"]]]]










MathML Form







<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> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mo> &#8706; </mo> <mrow> <mo> &#8706; </mo> <mi> &#977; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> sin </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> n </mi> <mi> m </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> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mn> 2 </mn> </msup> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> &#966; </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </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> &#977; </ms> <ms> ) </ms> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8706; </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8706; </ms> <ms> &#977; </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> &#977; </ms> <ms> ) </ms> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#8706; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> Y </ms> <ms> n </ms> <ms> m </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#977; </ms> <ms> , </ms> <ms> &#966; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8706; </ms> <ms> &#977; </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> &#977; </ms> <ms> ) </ms> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> &#8706; </ms> <ms> 2 </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> Y </ms> <ms> n </ms> <ms> m </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#977; </ms> <ms> , </ms> <ms> &#966; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8706; </ms> <apply> <ci> SuperscriptBox </ci> <ms> &#966; </ms> <ms> 2 </ms> </apply> </list> </apply> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </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> n </ms> <ms> m </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#977; </ms> <ms> , </ms> <ms> &#966; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> &#10869; </ms> <ms> 0 </ms> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





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["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]]]]]], ")"]]]], RowBox[List["Sin", "[", "\[CurlyTheta]_", "]"]]], "+", FractionBox[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[CurlyPhi]_", ",", "2"]], "}"]]]]], RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]]]], SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]_", "]"]], "2"]], "+", RowBox[List["n_", " ", RowBox[List["(", RowBox[List["n_", "+", "1"]], ")"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", "0"]]]]










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





2001-10-29





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