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http://functions.wolfram.com/07.37.06.0035.01
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SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta],
\[CurlyPhi]] \[Proportional] E^(I \[CurlyPhi] \[Mu])
((2^(\[Lambda] - 1) Gamma[1/2 + \[Lambda]] Sqrt[2 \[Lambda] + 1])/
(Pi Sqrt[Gamma[\[Lambda] - \[Mu] + 1]]
Sqrt[Gamma[\[Lambda] + \[Mu] + 1]])) Cos[\[CurlyTheta]]^\[Lambda]
((Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2)/(Sin[\[CurlyTheta]/2]^2)^(\[Mu]/2))
(1 + O[1/Cos[\[CurlyTheta]]]) /; (Abs[Cos[\[CurlyTheta]]] -> Infinity) &&
Element[2 \[Lambda] + 1, Integers] && 2 \[Lambda] + 1 >= 0 &&
Element[\[Lambda] - \[Mu], Integers] && -\[Lambda] <= \[Mu] <= \[Lambda] + 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Proportional]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "\[Mu]"]]], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Lambda]", "-", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Lambda]"]], "]"]], SqrtBox[RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]]]]], RowBox[List["\[Pi]", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]]]], SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "\[Lambda]"], " ", FractionBox[SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["\[Lambda]", "-", "\[Mu]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "\[Lambda]"]], "\[LessEqual]", "\[Mu]", "\[LessEqual]", RowBox[List["\[Lambda]", "+", "1"]]]]]]]]]]
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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> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> φ </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </msup> <mo> ⁢ </mo> <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> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <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> <mo> ⁢ </mo> <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> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mi> λ </mi> </msup> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mfrac> <msup> <mrow> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> ϑ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> ϑ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> λ </mi> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mi> λ </mi> </mrow> <mo> ≤ </mo> <mi> μ </mi> <mo> ≤ </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </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> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> φ </ci> <ci> μ </ci> </apply> </apply> <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='rational'> 1 <sep /> 2 </cn> </apply> </apply> <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> <times /> <pi /> <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> <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> <power /> <apply> <cos /> <ci> ϑ </ci> </apply> <ci> λ </ci> </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 /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <abs /> <apply> <cos /> <ci> ϑ </ci> </apply> </apply> <infinity /> </apply> <apply> <in /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> ℕ </ci> </apply> <apply> <in /> <apply> <plus /> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> <integers /> </apply> <apply> <leq /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> <ci> μ </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </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]_", ",", "\[Mu]_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Lambda]", "-", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Lambda]"]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "\[Lambda]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["\[Lambda]", "-", "\[Mu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "\[Lambda]"]], "\[LessEqual]", "\[Mu]", "\[LessEqual]", RowBox[List["\[Lambda]", "+", "1"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
