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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Specific values > Specialized values > For fixed m, theta, phi





http://functions.wolfram.com/05.10.03.0020.01









  


  










Input Form





SphericalHarmonicY[5, m, \[CurlyTheta], \[CurlyPhi]] == (Sqrt[11] E^(I m \[CurlyPhi]) (945 Cos[\[CurlyTheta]]^5 - 945 m Cos[\[CurlyTheta]]^4 + 210 (2 m^2 - 5) Cos[\[CurlyTheta]]^3 - 105 m (m^2 - 7) Cos[\[CurlyTheta]]^2 + 15 (m^4 - 13 m^2 + 15) Cos[\[CurlyTheta]] - m (m^4 - 20 m^2 + 64)) (Cos[\[CurlyTheta]/2]^2)^ (m/2))/(Sin[\[CurlyTheta]/2]^2)^(m/2)/(2 Sqrt[Pi] Sqrt[Gamma[6 - m]] Sqrt[Gamma[6 + m]])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["5", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["11"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[CurlyPhi]"]]], RowBox[List["(", RowBox[List[RowBox[List["945", " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "5"]]], "-", RowBox[List["945", " ", "m", " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "4"]]], "+", RowBox[List["210", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["m", "2"]]], "-", "5"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "3"]]], "-", RowBox[List["105", " ", "m", " ", RowBox[List["(", RowBox[List[SuperscriptBox["m", "2"], "-", "7"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "2"]]], "+", RowBox[List["15", " ", RowBox[List["(", RowBox[List[SuperscriptBox["m", "4"], "-", RowBox[List["13", " ", SuperscriptBox["m", "2"]]], "+", "15"]], ")"]], " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "-", RowBox[List["m", " ", RowBox[List["(", RowBox[List[SuperscriptBox["m", "4"], "-", RowBox[List["20", " ", SuperscriptBox["m", "2"]]], "+", "64"]], ")"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["m", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List[RowBox[List["-", "m"]], "/", "2"]]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", SqrtBox["\[Pi]"], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["6", "-", "m"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["6", "+", "m"]], "]"]]]]], ")"]]]]]]]]










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> 5 </mn> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msqrt> <mn> 11 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <mi> &#966; </mi> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 945 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> cos </mi> <mn> 5 </mn> </msup> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 945 </mn> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> cos </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 210 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> cos </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 105 </mn> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> m </mi> <mn> 4 </mn> </msup> <mo> - </mo> <mrow> <mn> 13 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 15 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> m </mi> <mn> 4 </mn> </msup> <mo> - </mo> <mrow> <mn> 20 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 64 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> &#977; </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> &#977; </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> + </mo> <mi> m </mi> </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'> 5 </cn> <ci> m </ci> <ci> &#977; </ci> <ci> &#966; </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 11 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> m </ci> <ci> &#966; </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 945 </cn> <apply> <power /> <apply> <cos /> <ci> &#977; </ci> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 945 </cn> <ci> m </ci> <apply> <power /> <apply> <cos /> <ci> &#977; </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 210 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -5 </cn> </apply> <apply> <power /> <apply> <cos /> <ci> &#977; </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 105 </cn> <ci> m </ci> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -7 </cn> </apply> <apply> <power /> <apply> <cos /> <ci> &#977; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 13 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 15 </cn> </apply> <apply> <cos /> <ci> &#977; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 20 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 64 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> cos </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> &#977; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> m </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> &#977; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <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> <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'> 6 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 6 </cn> <ci> m </ci> </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["5", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox["11"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[CurlyPhi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["945", " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "5"]]], "-", RowBox[List["945", " ", "m", " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "4"]]], "+", RowBox[List["210", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["m", "2"]]], "-", "5"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "3"]]], "-", RowBox[List["105", " ", "m", " ", RowBox[List["(", RowBox[List[SuperscriptBox["m", "2"], "-", "7"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "2"]]], "+", RowBox[List["15", " ", RowBox[List["(", RowBox[List[SuperscriptBox["m", "4"], "-", RowBox[List["13", " ", SuperscriptBox["m", "2"]]], "+", "15"]], ")"]], " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "-", RowBox[List["m", " ", RowBox[List["(", RowBox[List[SuperscriptBox["m", "4"], "-", RowBox[List["20", " ", SuperscriptBox["m", "2"]]], "+", "64"]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["m", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["-", FractionBox["m", "2"]]]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["6", "-", "m"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["6", "+", "m"]], "]"]]]]]]]]]]










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





2001-10-29





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