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http://functions.wolfram.com/07.37.06.0023.01
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SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta],
\[CurlyPhi]] \[Proportional] Sqrt[(2 \[Lambda] + 1)/(4 Pi)]
(Sqrt[Gamma[\[Lambda] - \[Mu] + 1]]/Sqrt[Gamma[\[Lambda] + \[Mu] + 1]])
E^(I \[Mu] \[CurlyPhi])
((Gamma[-\[Mu]]/(2^\[Mu] (Gamma[-\[Lambda] - \[Mu]]
Gamma[\[Lambda] - \[Mu] + 1]))) ((\[CurlyTheta] - Pi)^2)^(\[Mu]/2)
(1 + O[(\[CurlyTheta] - Pi)^2]) -
(((2^\[Mu] Sin[\[Lambda] Pi] Gamma[\[Mu]])/Pi)
(1 + O[(\[CurlyTheta] - Pi)^2]))/((\[CurlyTheta] - Pi)^2)^(\[Mu]/2)) /;
(\[CurlyTheta] -> Pi) && !Element[\[Mu], Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Proportional]", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]], RowBox[List["4", "\[Pi]"]]]], FractionBox[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Mu]", " ", "\[CurlyPhi]"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Mu]"]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Mu]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[SuperscriptBox["2", "\[Mu]"], RowBox[List["Sin", "[", RowBox[List["\[Lambda]", " ", "\[Pi]"]], "]"]], RowBox[List["Gamma", "[", "\[Mu]", "]"]], " "]]]], "\[Pi]"], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"], "]"]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["\[CurlyTheta]", "\[Rule]", "\[Pi]"]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List["\[Mu]", "\[Element]", "Integers"]], "]"]]]]]]]]
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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> <mrow> <msubsup> <mi> Y </mi> <mi> λ </mi> <mi> μ </mi> </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> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mfrac> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> - </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mi> φ </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> λ </mi> </mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> - </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> - </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> - </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> μ </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> μ </mi> <mo> ) </mo> </mrow> </mrow> <mi> π </mi> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> - </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> - </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> ϑ </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mi> μ </mi> <mo> ∉ </mo> <mi> ℤ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> SphericalHarmonicY </ci> <ci> λ </ci> <ci> μ </ci> <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 /> <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> 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 /> <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> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> μ </ci> <ci> φ </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> </apply> <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> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> ϑ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </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> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> ϑ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> μ </ci> </apply> <apply> <sin /> <apply> <times /> <ci> λ </ci> <pi /> </apply> </apply> <apply> <ci> Gamma </ci> <ci> μ </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> ϑ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> ϑ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> ϑ </ci> <pi /> </apply> <apply> <notin /> <ci> μ </ci> <ci> ℤ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]_", ",", "\[Mu]_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], RowBox[List["4", " ", "\[Pi]"]]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Mu]", " ", "\[CurlyPhi]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Mu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Mu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], "]"]], "2"]]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "\[Mu]"], " ", RowBox[List["Sin", "[", RowBox[List["\[Lambda]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Mu]", "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"], ")"]], RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], "]"]], "2"]]], ")"]]]], "\[Pi]"]]], ")"]]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["\[CurlyTheta]", "\[Rule]", "\[Pi]"]], ")"]], "&&", RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
