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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Series representations > Generalized power series > Expansions at sin(theta/2)==0





http://functions.wolfram.com/05.10.06.0004.02









  


  










Input Form





SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] \[Proportional] (Sqrt[(2 n + 1)/(4 Pi)] (Sqrt[Gamma[n - m + 1]]/ (Gamma[1 - m] Sqrt[Gamma[n + m + 1]])) E^(I \[CurlyPhi] m) (1 + O[Sin[\[CurlyTheta]/2]^2]))/(Sin[\[CurlyTheta]/2]^2)^(m/2) /; (Sin[\[CurlyTheta]/2] -> 0) && Element[-m, Integers] && -m >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Proportional]", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], RowBox[List["4", "\[Pi]"]]]], FractionBox[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["n", "-", "m", "+", "1"]], "]"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "m"]], "]"]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["n", "+", "m", "+", "1"]], "]"]]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "m"]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List[RowBox[List["-", "m"]], "/", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "\[Rule]", "0"]], ")"]], "\[And]", RowBox[List[RowBox[List["-", "m"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "m"]], "\[GreaterEqual]", "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> <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> <mo> &#8733; </mo> <mrow> <msqrt> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mfrac> <mrow> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#966; </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mtext> </mtext> <msup> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> &#977; </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> m </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> &#977; </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> &#977; </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> &#977; </ci> <ci> &#966; </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </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> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> &#966; </ci> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> &#977; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> &#977; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <sin /> <apply> <times /> <ci> &#977; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], RowBox[List["4", " ", "\[Pi]"]]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["n", "-", "m", "+", "1"]], "]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "m"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["-", FractionBox["m", "2"]]]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "]"]], "2"]]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "m"]], "]"]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["n", "+", "m", "+", "1"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "\[Rule]", "0"]], ")"]], "&&", RowBox[List[RowBox[List["-", "m"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "m"]], "\[GreaterEqual]", "0"]]]]]]]]]]










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





2001-10-29





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