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http://functions.wolfram.com/05.10.06.0008.01
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SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] ==
((-1)^m Sqrt[((2 n + 1) (n + m)!)/(Pi (n - m)!)] E^(I \[CurlyPhi] m)
2^(m - 1) Sum[((Pochhammer[-n, k - m] Pochhammer[n + 1, k - m])/
((k - m)! k!)) Sin[\[CurlyTheta]/2]^(2 k), {k, 0, n + m}])/
(Sin[\[CurlyTheta]]^2)^(m/2)
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Cell[BoxData[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], "!"]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "!"]]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "m"]]], SuperscriptBox["2", RowBox[List["m", "-", "1"]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "2"], ")"]], RowBox[List[RowBox[List["-", "m"]], "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "+", "m"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", RowBox[List["k", "-", "m"]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m"]], ")"]], "!"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], RowBox[List["2", "k"]]]]]]]]]]]]]
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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> <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> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> φ </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> m </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], RowBox[List["k", "-", "m"]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], RowBox[List["k", "-", "m"]]], Pochhammer] </annotation> </semantics> <mtext> </mtext> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> ϑ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> φ </ci> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <sin /> <ci> ϑ </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> ϑ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], "!"]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "!"]]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "m"]]], " ", SuperscriptBox["2", RowBox[List["m", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "2"], ")"]], RowBox[List["-", FractionBox["m", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "+", "m"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", RowBox[List["k", "-", "m"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m"]], ")"]], "!"]], " ", RowBox[List["k", "!"]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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