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http://functions.wolfram.com/07.37.06.0028.01
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SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]] ==
((Sqrt[2 \[Lambda] + 1] Sqrt[Gamma[\[Lambda] - \[Mu] + 1]])/
(2 Pi Sqrt[Gamma[\[Lambda] + \[Mu] + 1]])) E^(I \[CurlyPhi] \[Mu])
((Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2)/(Sin[\[CurlyTheta]/2]^2)^(\[Mu]/2))
(((2^\[Lambda] (Cos[\[CurlyTheta]] - 1)^\[Lambda])/
Gamma[\[Lambda] - \[Mu] + 1]) Gamma[1/2 + \[Lambda]]
HypergeometricPFQ[{\[Mu] - \[Lambda], -\[Lambda]}, {-2 \[Lambda]},
2/(1 - Cos[\[CurlyTheta]])] +
((2^(-1 - \[Lambda]) (Cos[\[CurlyTheta]] - 1)^(-1 - \[Lambda]))/
Gamma[-\[Mu] - \[Lambda]]) Gamma[-(1/2) - \[Lambda]]
HypergeometricPFQ[{1 + \[Lambda], 1 + \[Mu] + \[Lambda]},
{2 + 2 \[Lambda]}, 2/(1 - Cos[\[CurlyTheta]])]) /;
!Element[\[CurlyTheta], Reals] && !Element[2 \[Lambda], Integers]
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</mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mfrac> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Lambda]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "2"]], HypergeometricPFQ, Rule[Editable, 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<times /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> φ </ci> <ci> μ </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <ci> μ </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> <times /> <apply> <power /> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> ϑ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> ϑ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> λ </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </list> <list> <apply> <times /> <cn type='integer'> -2 </cn> <ci> λ </ci> </apply> </list> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <ci> ϑ </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> λ </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 2 </cn> </apply> </list> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <ci> ϑ </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <notin /> <ci> ϑ </ci> <ci> ℝ </ci> </apply> <apply> <notin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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