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http://functions.wolfram.com/05.10.17.0004.01
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SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] ==
((2 (1 - m))/Sqrt[(n + m) (n - m + 1)]) E^(I \[CurlyPhi])
Cot[\[CurlyTheta]] SphericalHarmonicY[n, m - 1, \[CurlyTheta],
\[CurlyPhi]] + (((m - 1) (m - 2) - n (n + 1))/
Sqrt[(n + m) (n + m - 1) (n - m + 2) (n - m + 1)]) E^(2 I \[CurlyPhi])
SphericalHarmonicY[n, m - 2, \[CurlyTheta], \[CurlyPhi]] /;
Inequality[0, Less, m, LessEqual, n]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " "]], RowBox[List[" ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], RowBox[List["(", RowBox[List["n", "-", "m", "+", "1"]], ")"]]]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]"]]], " ", RowBox[List["Cot", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", RowBox[List["m", "-", "1"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "-", "2"]], ")"]]]], "-", RowBox[List["n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]]]], ")"]], " "]], RowBox[List[" ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], RowBox[List["(", RowBox[List["n", "+", "m", "-", "1"]], ")"]], RowBox[List["(", RowBox[List["n", "-", "m", "+", "2"]], ")"]], RowBox[List["(", RowBox[List["n", "-", "m", "+", "1"]], ")"]]]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["2", "\[ImaginaryI]", " ", "\[CurlyPhi]"]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", RowBox[List["m", "-", "2"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]]]]]], "/;", RowBox[List["0", "<", "m", "\[LessEqual]", "n"]]]]]]
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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> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> φ </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> φ </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mn> 0 </mn> <mo> < </mo> <mi> m </mi> <mo> ≤ </mo> <mi> n </mi> </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> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> n </ci> <ci> m </ci> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <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> φ </ci> </apply> </apply> <apply> <cot /> <ci> ϑ </ci> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <ci> ϑ </ci> <ci> φ </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> n </ci> <ci> m </ci> </apply> <apply> <plus /> <ci> n </ci> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <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 /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> φ </ci> </apply> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> <ci> ϑ </ci> <ci> φ </ci> </apply> </apply> </apply> </apply> <apply> <ci> Inequality </ci> <cn type='integer'> 0 </cn> <lt /> <ci> m </ci> <leq /> <ci> n </ci> </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[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]"]]], " ", RowBox[List["Cot", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", RowBox[List["m", "-", "1"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "m", "+", "1"]], ")"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "-", "2"]], ")"]]]], "-", RowBox[List["n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[CurlyPhi]"]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", RowBox[List["m", "-", "2"]], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "m", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "m", "+", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "m", "+", "1"]], ")"]]]]]]]], "/;", RowBox[List["0", "<", "m", "\[LessEqual]", "n"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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