Spherical harmonics: Series representations (formula 05.10.06.0019)

SphericalHarmonicY

$Mathematica Notation$

Polynomials

SphericalHarmonicY[n,m,theta,phi]

Series representations

Generalized power series

Expansions at theta==0

http://functions.wolfram.com/05.10.06.0019.01

Input Form

SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] \[Proportional] (Sqrt[(2 n + 1)/(4 Pi)] (Sqrt[Gamma[n - m + 1]]/Sqrt[Gamma[n + m + 1]]) 2^m E^(I \[CurlyPhi] m) (1/Gamma[1 - m] + (1/12) (-((3 n (n + 1))/Gamma[2 - m]) + 1/Gamma[-m]) \[CurlyTheta]^2 + (1/1440) ((30 n (n + 1) (m + 1))/Gamma[2 - m] + (45 (n - 1) n (n + 1) (n + 2))/Gamma[3 - m] + (7 - 5 m)/Gamma[-m]) \[CurlyTheta]^4 + (1/362880) (-((63 n (n + 1) (4 + m (3 + 5 m)))/ Gamma[2 - m]) - (945 (n - 1) n (n + 1) (n + 2) (m + 2))/ Gamma[3 - m] - (945 (n - 1) n (n + 1) (n + 2) (n^2 + n - 6))/ Gamma[4 - m] + (124 + 7 m (5 m - 21))/Gamma[-m]) \[CurlyTheta]^6 + O[\[CurlyTheta]^8]))/(\[CurlyTheta]^2)^(m/2) /; (\[CurlyTheta] -> 0)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Proportional]", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], RowBox[List["4", "\[Pi]"]]]], FractionBox[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["n", "-", "m", "+", "1"]], "]"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["n", "+", "m", "+", "1"]], "]"]]]], SuperscriptBox["2", "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "m"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[CurlyTheta]", "2"], ")"]], RowBox[List[RowBox[List["-", "m"]], "/", "2"]]], RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m"]], "]"]]], "+", RowBox[List[FractionBox["1", "12"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["3", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]]]], "+", FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], SuperscriptBox["\[CurlyTheta]", "2"]]], " ", "+", RowBox[List[FractionBox["1", "1440"], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["30", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["45", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["3", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["7", "-", RowBox[List["5", " ", "m"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], SuperscriptBox["\[CurlyTheta]", "4"]]], " ", "+", RowBox[List[FractionBox["1", "362880"], RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["63", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["5", "m"]]]], ")"]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]]]], "-", FractionBox[RowBox[List["945", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "+", "2"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["3", "-", "m"]], "]"]]], "-", FractionBox[RowBox[List["945", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["n", "2"], "+", "n", "-", "6"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["4", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["124", "+", RowBox[List["7", " ", "m", " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", "m"]], "-", "21"]], ")"]]]]]], RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], SuperscriptBox["\[CurlyTheta]", "6"]]], " ", "+", RowBox[List["O", "[", SuperscriptBox["\[CurlyTheta]", "8"], "]"]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["\[CurlyTheta]", "\[Rule]", "0"]], ")"]]]]]]

MathML Form

<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> <msqrt> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mfrac> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mi> m </mi> </msup> <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> <mrow> <mo> ( </mo> <msup> <mi> ϑ </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 12 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ϑ </mi> <mn> 2 </mn> </msup> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 1440 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 30 </mn> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 45 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 7 </mn> <mo> - </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ϑ </mi> <mn> 4 </mn> </msup> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 362880 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 63 </mn> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> + </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> - </mo> <mfrac> <mrow> <mn> 945 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 945 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 124 </mn> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> ⁢ </mo> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 21 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ϑ </mi> <mn> 6 </mn> </msup> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> ϑ </mi> <mn> 8 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> ϑ </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <apply> <ci> Gamma </ci> <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> <apply> <power /> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> φ </ci> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <power /> <ci> ϑ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 12 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> ϑ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 1440 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 30 </cn> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 7 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> ϑ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 362880 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 63 </cn> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <cn type='integer'> 4 </cn> <apply> <times /> <ci> m </ci> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 945 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 945 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> n </ci> <cn type='integer'> -6 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 124 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <ci> m </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> m </ci> </apply> <cn type='integer'> -21 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> ϑ </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> ϑ </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> ϑ </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], RowBox[List["4", " ", "\[Pi]"]]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["n", "-", "m", "+", "1"]], "]"]]], " ", SuperscriptBox["2", "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "m"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[CurlyTheta]", "2"], ")"]], RowBox[List["-", FractionBox["m", "2"]]]], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m"]], "]"]]], "+", RowBox[List[FractionBox["1", "12"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["3", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]]]], "+", FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], " ", SuperscriptBox["\[CurlyTheta]", "2"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["30", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["45", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["3", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["7", "-", RowBox[List["5", " ", "m"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], " ", SuperscriptBox["\[CurlyTheta]", "4"]]], "1440"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["63", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["5", " ", "m"]]]], ")"]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]]]], "-", FractionBox[RowBox[List["945", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "+", "2"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["3", "-", "m"]], "]"]]], "-", FractionBox[RowBox[List["945", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["n", "2"], "+", "n", "-", "6"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["4", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["124", "+", RowBox[List["7", " ", "m", " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", "m"]], "-", "21"]], ")"]]]]]], RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], " ", SuperscriptBox["\[CurlyTheta]", "6"]]], "362880"], "+", SuperscriptBox[RowBox[List["O", "[", "\[CurlyTheta]", "]"]], "8"]]], ")"]]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["n", "+", "m", "+", "1"]], "]"]]]], "/;", RowBox[List["(", RowBox[List["\[CurlyTheta]", "\[Rule]", "0"]], ")"]]]]]]]]

Date Added to functions.wolfram.com (modification date)

2001-10-29

SphericalHarmonicY[lambda,mu,theta,phi]