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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Specific values > Specialized values > For fixed lambda, theta, phi





http://functions.wolfram.com/07.37.03.0012.01









  


  










Input Form





SphericalHarmonicY[\[Lambda], \[Lambda], \[CurlyTheta], \[CurlyPhi]] == (Sqrt[2 \[Lambda] + 1]/(2^(\[Lambda] + 1) Sqrt[Pi] Gamma[-\[Lambda]] Sqrt[Gamma[2 \[Lambda] + 1]])) E^(I \[Lambda] \[CurlyPhi]) (Sin[\[CurlyTheta]]^2)^(\[Lambda]/2) Beta[Sin[\[CurlyTheta]/2]^2, -\[Lambda], -\[Lambda]] /; !(Element[\[Lambda], Integers] && \[Lambda] >= 0)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Lambda]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]]], RowBox[List[SuperscriptBox["2", RowBox[List["\[Lambda]", "+", "1"]]], SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Lambda]"]], "]"]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]], "]"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Lambda]", " ", "\[CurlyPhi]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "2"], ")"]], RowBox[List["\[Lambda]", "/", "2"]]], RowBox[List["Beta", "[", RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ",", RowBox[List["-", "\[Lambda]"]], ",", RowBox[List["-", "\[Lambda]"]]]], "]"]]]]]], " ", "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List["\[Lambda]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Lambda]", "\[GreaterEqual]", "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> &#955; </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> &#955; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#955; </mi> <mo> &#8290; </mo> <mi> &#966; </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mrow> <mi> &#955; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> &#977; </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </msub> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> &#955; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> /; </mo> <mrow> <mi> &#955; </mi> <mo> &#8713; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> SphericalHarmonicY </ci> <ci> &#955; </ci> <ci> &#955; </ci> <ci> &#977; </ci> <ci> &#966; </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> &#955; </ci> <ci> &#966; </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <sin /> <ci> &#977; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> &#955; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Beta </ci> <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 /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> </apply> </apply> </apply> <apply> <notin /> <ci> &#955; </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]_", ",", "\[Lambda]_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Lambda]", " ", "\[CurlyPhi]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "2"], ")"]], RowBox[List["\[Lambda]", "/", "2"]]], " ", RowBox[List["Beta", "[", RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ",", RowBox[List["-", "\[Lambda]"]], ",", RowBox[List["-", "\[Lambda]"]]]], "]"]]]], RowBox[List[SuperscriptBox["2", RowBox[List["\[Lambda]", "+", "1"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Lambda]"]], "]"]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], "]"]]]]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["\[Lambda]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Lambda]", "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]










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





2001-10-29