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http://functions.wolfram.com/07.37.03.0023.01
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SphericalHarmonicY[\[Lambda], \[Lambda] + m, \[CurlyTheta], \[CurlyPhi]] ==
ComplexInfinity /; Element[m, Integers] && m > 0 &&
!Element[\[Lambda], Integers] &&
!(Element[-((m + 2 \[Lambda] + 1)/2), Integers] &&
-((m + 2 \[Lambda] + 1)/2) >= 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", RowBox[List["\[Lambda]", "+", "m"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", "ComplexInfinity"]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["\[Lambda]", "\[Element]", "Integers"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["m", "+", RowBox[List["2", "\[Lambda]"]], "+", "1"]], "2"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["m", "+", RowBox[List["2", "\[Lambda]"]], "+", "1"]], "2"]]], "\[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> <mrow> <mi> λ </mi> <mo> + </mo> <mi> m </mi> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mover> <mi> ∞ </mi> <mo> ~ </mo> </mover> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> λ </mi> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> m </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> <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> <apply> <plus /> <ci> λ </ci> <ci> m </ci> </apply> <ci> ϑ </ci> <ci> φ </ci> </apply> <apply> <ci> OverTilde </ci> <infinity /> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <notin /> <ci> λ </ci> <integers /> </apply> <apply> <notin /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <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]_", ",", RowBox[List["\[Lambda]_", "+", "m_"]], ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["ComplexInfinity", "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["!", RowBox[List["\[Lambda]", "\[Element]", "Integers"]]]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["m", "+", RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], ")"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["m", "+", RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], ")"]]]], "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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