Spherical harmonic function: Specific values (formula 07.37.03.0033)

SphericalHarmonicY

$Mathematica Notation$

Hypergeometric Functions

SphericalHarmonicY[lambda,mu,theta,phi]

Specific values

Specialized values

For fixed mu,theta,phi

http://functions.wolfram.com/07.37.03.0033.01

Input Form

SphericalHarmonicY[6, \[Mu], \[CurlyTheta], \[CurlyPhi]] == (Sqrt[13] E^(I \[Mu] \[CurlyPhi]) (-225 + 259 \[Mu]^2 - 35 \[Mu]^4 + \[Mu]^6 - 21 \[Mu] (99 - 25 \[Mu]^2 + \[Mu]^4) Cos[\[CurlyTheta]] + 105 (45 - 32 \[Mu]^2 + 2 \[Mu]^4) Cos[\[CurlyTheta]]^2 - 630 \[Mu] (-17 + 2 \[Mu]^2) Cos[\[CurlyTheta]]^3 + 4725 (-3 + \[Mu]^2) Cos[\[CurlyTheta]]^4 - 10395 \[Mu] Cos[\[CurlyTheta]]^5 + 10395 Cos[\[CurlyTheta]]^6) (Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2))/(Sin[\[CurlyTheta]/2]^2)^(\[Mu]/2)/ (2 Sqrt[Pi] Sqrt[Gamma[7 - \[Mu]]] Sqrt[Gamma[7 + \[Mu]]])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["6", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["13"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Mu]", " ", "\[CurlyPhi]"]]], RowBox[List["(", RowBox[List[RowBox[List["-", "225"]], "+", RowBox[List["259", " ", SuperscriptBox["\[Mu]", "2"]]], "-", RowBox[List["35", " ", SuperscriptBox["\[Mu]", "4"]]], "+", SuperscriptBox["\[Mu]", "6"], "-", RowBox[List["21", " ", "\[Mu]", " ", RowBox[List["(", RowBox[List["99", "-", RowBox[List["25", " ", SuperscriptBox["\[Mu]", "2"]]], "+", SuperscriptBox["\[Mu]", "4"]]], ")"]], " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "+", RowBox[List["105", " ", RowBox[List["(", RowBox[List["45", "-", RowBox[List["32", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "4"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "2"]]], "-", RowBox[List["630", " ", "\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "17"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "3"]]], "+", RowBox[List["4725", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", SuperscriptBox["\[Mu]", "2"]]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "4"]]], "-", RowBox[List["10395", " ", "\[Mu]", " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "5"]]], "+", RowBox[List["10395", " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "6"]]]]], ")"]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", SqrtBox["\[Pi]"], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["7", "-", "\[Mu]"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["7", "+", "\[Mu]"]], "]"]]]]], ")"]]]]]]]]

MathML Form

<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> <mn> 6 </mn> <mi> μ </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msqrt> <mn> 13 </mn> </msqrt> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mi> φ </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 10395 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mn> 6 </mn> </msup> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 10395 </mn> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mn> 5 </mn> </msup> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4725 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 630 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 17 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 105 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 32 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 45 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 21 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> <mo> - </mo> <mrow> <mn> 25 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 99 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mi> μ </mi> <mn> 6 </mn> </msup> <mo> - </mo> <mrow> <mn> 35 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 259 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 225 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> ϑ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> ϑ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> μ </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 7 </mn> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> SphericalHarmonicY </ci> <cn type='integer'> 6 </cn> <ci> μ </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 13 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> μ </ci> <ci> φ </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 10395 </cn> <apply> <power /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10395 </cn> <ci> μ </ci> <apply> <power /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4725 </cn> <apply> <plus /> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -3 </cn> </apply> <apply> <power /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 630 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -17 </cn> </apply> <ci> μ </ci> <apply> <power /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 105 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 45 </cn> </apply> <apply> <power /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 21 </cn> <apply> <plus /> <apply> <power /> <ci> μ </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 25 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 99 </cn> </apply> <ci> μ </ci> <apply> <cos /> <ci> ϑ </ci> </apply> </apply> </apply> <apply> <power /> <ci> μ </ci> <cn type='integer'> 6 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 35 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 259 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -225 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> ϑ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> ϑ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 7 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> μ </ci> <cn type='integer'> 7 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["6", ",", "\[Mu]_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox["13"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Mu]", " ", "\[CurlyPhi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "225"]], "+", RowBox[List["259", " ", SuperscriptBox["\[Mu]", "2"]]], "-", RowBox[List["35", " ", SuperscriptBox["\[Mu]", "4"]]], "+", SuperscriptBox["\[Mu]", "6"], "-", RowBox[List["21", " ", "\[Mu]", " ", RowBox[List["(", RowBox[List["99", "-", RowBox[List["25", " ", SuperscriptBox["\[Mu]", "2"]]], "+", SuperscriptBox["\[Mu]", "4"]]], ")"]], " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "+", RowBox[List["105", " ", RowBox[List["(", RowBox[List["45", "-", RowBox[List["32", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "4"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "2"]]], "-", RowBox[List["630", " ", "\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "17"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "3"]]], "+", RowBox[List["4725", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", SuperscriptBox["\[Mu]", "2"]]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "4"]]], "-", RowBox[List["10395", " ", "\[Mu]", " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "5"]]], "+", RowBox[List["10395", " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "6"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["-", FractionBox["\[Mu]", "2"]]]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["7", "-", "\[Mu]"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["7", "+", "\[Mu]"]], "]"]]]]]]]]]]

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

2001-10-29

SphericalHarmonicY[n,m,theta,phi]