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http://functions.wolfram.com/07.37.06.0032.01
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SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]] ==
E^(I \[CurlyPhi] \[Mu]) (((-1)^(\[Mu] - \[Lambda] - 1) 2^\[Lambda]
Sqrt[2 \[Lambda] + 1] Sqrt[Gamma[\[Lambda] - \[Mu] + 1]]
Sqrt[Gamma[\[Lambda] + \[Mu] + 1]])/(Sqrt[Pi] Gamma[2 \[Lambda] + 2]
Gamma[-\[Lambda]])) (Cos[\[CurlyTheta]] - 1)^(-\[Lambda] - 1)
((Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2)/(Sin[\[CurlyTheta]/2]^2)^(\[Mu]/2))
Hypergeometric2F1[\[Lambda] + 1, \[Lambda] + \[Mu] + 1, 2 + 2 \[Lambda],
2/(1 - Cos[\[CurlyTheta]])] /; Element[2 \[Lambda] + 1, Integers] &&
2 \[Lambda] + 1 >= 0 && Element[\[Mu] - \[Lambda], Integers] &&
\[Mu] - \[Lambda] > 0 && !(Element[\[Lambda], Integers] && \[Lambda] >= 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", " ", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "\[Mu]"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Mu]", "-", "\[Lambda]", "-", "1"]]], " ", SuperscriptBox["2", "\[Lambda]"], SqrtBox[RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]]], RowBox[List[SqrtBox["\[Pi]"], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "2"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Lambda]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "1"]]], " ", 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["Hypergeometric2F1", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "1"]], ",", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], ",", RowBox[List["2", "+", RowBox[List["2", "\[Lambda]"]]]], ",", FractionBox["2", RowBox[List["1", "-", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]]]], "]"]]]]]], "/;", RowBox[List[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["\[Mu]", "-", "\[Lambda]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["\[Mu]", "-", "\[Lambda]"]], ">", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["\[Lambda]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Lambda]", "\[GreaterEqual]", "0"]]]], "]"]]]]]]]]
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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> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> - </mo> <mi> λ </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mi> λ </mi> </msup> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </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> <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> <mrow> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> λ </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <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> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </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"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "2"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["1", "-", RowBox[List["cos", "(", "\[CurlyTheta]", ")"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <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> μ </mi> <mo> - </mo> <mi> λ </mi> </mrow> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> λ </mi> <mo> ∉ </mo> <mi> ℕ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <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'> -1 </cn> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <ci> λ </ci> </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> <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> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </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> <ci> Hypergeometric2F1 </ci> <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> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 2 </cn> </apply> <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> <and /> <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> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </apply> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <notin /> <ci> λ </ci> <ci> ℕ </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[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Mu]", "-", "\[Lambda]", "-", "1"]]], " ", SuperscriptBox["2", "\[Lambda]"], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "1"]], ",", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], ",", RowBox[List["2", "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", FractionBox["2", RowBox[List["1", "-", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "2"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Lambda]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]], "/;", RowBox[List[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["\[Mu]", "-", "\[Lambda]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["\[Mu]", "-", "\[Lambda]"]], ">", "0"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["\[Lambda]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Lambda]", "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
