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http://functions.wolfram.com/07.37.03.0017.01
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SphericalHarmonicY[\[Lambda], -\[Lambda], \[CurlyTheta], \[CurlyPhi]] ==
Sqrt[(2 \[Lambda] + 1)/Pi] (Sqrt[Gamma[2 \[Lambda] + 1]]/
(E^(I \[CurlyPhi] \[Lambda]) (2^(\[Lambda] + 1) Gamma[\[Lambda] + 1])))
(Sin[\[CurlyTheta]]^2)^(\[Lambda]/2)
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Cell[BoxData[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", RowBox[List["-", "\[Lambda]"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], "\[Pi]"]], " ", FractionBox[RowBox[List[SqrtBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]], "]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[CurlyPhi]", " ", "\[Lambda]"]]]]], RowBox[List[SuperscriptBox["2", RowBox[List["\[Lambda]", "+", "1"]]], RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "1"]], "]"]]]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "2"], ")"]], RowBox[List["\[Lambda]", "/", "2"]]]]]]]]]
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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> <msubsup> <mi> Y </mi> <mi> λ </mi> <mrow> <mo> - </mo> <mi> λ </mi> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msqrt> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> π </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <mfrac> <mrow> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> φ </mi> <mo> ⁢ </mo> <mi> λ </mi> </mrow> </msup> </mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mrow> <mi> λ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> SphericalHarmonicY </ci> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> <ci> ϑ </ci> <ci> φ </ci> </apply> <apply> <times /> <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 /> <pi /> <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 /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> φ </ci> <ci> λ </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <sin /> <ci> ϑ </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> λ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]_", ",", RowBox[List["-", "\[Lambda]_"]], ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], "\[Pi]"]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], "]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[CurlyPhi]", " ", "\[Lambda]"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "2"], ")"]], RowBox[List["\[Lambda]", "/", "2"]]]]], RowBox[List[SuperscriptBox["2", RowBox[List["\[Lambda]", "+", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "1"]], "]"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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