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http://functions.wolfram.com/05.10.02.0004.01
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SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] ==
SphericalHarmonicY[-n - 1, m, \[CurlyTheta], \[CurlyPhi]] /;
Element[-n, Integers] && -n > 0 && Element[m, Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List["SphericalHarmonicY", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "-", "1"]], ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "n"]], ">", "0"]], "\[And]", RowBox[List["m", "\[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> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msubsup> <mi> Y </mi> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> <apply> <ci> SphericalHarmonicY </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> m </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "-", "1"]], ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "n"]], ">", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
