Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
SphericalHarmonicY




Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Series representations > Generalized power series > Expansions at theta==Pi





http://functions.wolfram.com/07.37.06.0022.01









  


  








Input Form





SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]] \[Proportional] Sqrt[(2 \[Lambda] + 1)/(4 Pi)] (Sqrt[Gamma[\[Lambda] - \[Mu] + 1]]/Sqrt[Gamma[\[Lambda] + \[Mu] + 1]]) E^(I \[Mu] \[CurlyPhi]) ((Gamma[-\[Mu]]/(2^\[Mu] (Gamma[-\[Lambda] - \[Mu]] Gamma[\[Lambda] - \[Mu] + 1]))) ((\[CurlyTheta] - Pi)^2)^(\[Mu]/2) (1 + ((\[Mu] (\[Mu] + 1) - 3 \[Lambda] (\[Lambda] + 1))/ (12 (\[Mu] + 1))) (\[CurlyTheta] - Pi)^2 + ((45 \[Lambda]^3 (\[Lambda] + 2) - 30 \[Lambda] (\[Mu]^2 + \[Mu] + 1) - 15 \[Lambda]^2 (2 \[Mu]^2 + 2 \[Mu] - 1) + \[Mu] (5 \[Mu]^3 + 22 \[Mu]^2 + 31 \[Mu] + 14))/ (1440 (\[Mu] + 1) (\[Mu] + 2))) (\[CurlyTheta] - Pi)^4 + O[(\[CurlyTheta] - Pi)^6]) - (((2^\[Mu] Sin[\[Lambda] Pi] Gamma[\[Mu]])/Pi) (1 + ((3 \[Lambda] (\[Lambda] + 1) + \[Mu] (1 - \[Mu]))/ (12 (\[Mu] - 1))) (\[CurlyTheta] - Pi)^2 + ((45 \[Lambda]^3 (\[Lambda] + 2) - \[Lambda]^2 (30 \[Mu]^2 - 30 \[Mu] - 15) - 30 \[Lambda] (\[Mu]^2 - \[Mu] + 1) + \[Mu] (5 \[Mu]^3 - 22 \[Mu]^2 + 31 \[Mu] - 14))/ (1440 (\[Mu] - 2) (\[Mu] - 1))) (\[CurlyTheta] - Pi)^4 + O[(\[CurlyTheta] - Pi)^6]))/((\[CurlyTheta] - Pi)^2)^(\[Mu]/2)) /; (\[CurlyTheta] -> Pi) && !Element[\[Mu], Integers]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Proportional]", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]], RowBox[List["4", "\[Pi]"]]]], FractionBox[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Mu]", " ", "\[CurlyPhi]"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Mu]"]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Mu]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["\[Mu]", RowBox[List["(", RowBox[List["\[Mu]", "+", "1"]], ")"]]]], "-", RowBox[List["3", " ", "\[Lambda]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]]]]]], RowBox[List["12", RowBox[List["(", RowBox[List["\[Mu]", "+", "1"]], ")"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"]]], "+", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["45", " ", SuperscriptBox["\[Lambda]", "3"], RowBox[List["(", RowBox[List["\[Lambda]", "+", "2"]], ")"]]]], "-", RowBox[List["30", " ", "\[Lambda]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Mu]", "2"], "+", "\[Mu]", "+", "1"]], ")"]]]], "-", RowBox[List["15", " ", SuperscriptBox["\[Lambda]", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["2", "\[Mu]"]], "-", "1"]], ")"]]]], "+", RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["5", SuperscriptBox["\[Mu]", "3"]]], "+", RowBox[List["22", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["31", "\[Mu]"]], "+", "14"]], ")"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["1440", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "2"]], ")"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "4"]]], " ", "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "6"], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[SuperscriptBox["2", "\[Mu]"], RowBox[List["Sin", "[", RowBox[List["\[Lambda]", " ", "\[Pi]"]], "]"]], RowBox[List["Gamma", "[", "\[Mu]", "]"]], " "]]]], "\[Pi]"], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["3", " ", "\[Lambda]", RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]]]], "+", RowBox[List["\[Mu]", RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]]]]]], RowBox[List["12", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "1"]], ")"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"]]], "+", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["45", " ", SuperscriptBox["\[Lambda]", "3"], RowBox[List["(", RowBox[List["\[Lambda]", "+", "2"]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[Lambda]", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["30", SuperscriptBox["\[Mu]", "2"]]], "-", RowBox[List["30", "\[Mu]"]], "-", "15"]], ")"]]]], "-", RowBox[List["30", " ", "\[Lambda]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Mu]", "2"], "-", "\[Mu]", "+", "1"]], ")"]]]], "+", RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["5", SuperscriptBox["\[Mu]", "3"]]], "-", RowBox[List["22", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["31", "\[Mu]"]], "-", "14"]], ")"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["1440", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "1"]], ")"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "4"]]], " ", "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "6"], "]"]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["\[CurlyTheta]", "\[Rule]", "\[Pi]"]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List["\[Mu]", "\[Element]", "Integers"]], "]"]]]]]]]]








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> &#956; </mi> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <msqrt> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mfrac> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mi> &#966; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 45 </mn> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 30 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 22 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 31 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mn> 14 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1440 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <msup> <mn> 2 </mn> <mi> &#956; </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#956; </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mi> &#960; </mi> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 45 </mn> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> &#955; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 30 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 30 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> - </mo> <mn> 15 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 30 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 22 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 31 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> - </mo> <mn> 14 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1440 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#956; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> SphericalHarmonicY </ci> <ci> &#955; </ci> <ci> &#956; </ci> <ci> &#977; </ci> <ci> &#966; </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </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> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </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> &#955; </ci> <ci> &#956; </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 /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> &#956; </ci> <ci> &#966; </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> &#977; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> &#956; </ci> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> &#955; </ci> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> &#977; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 30 </cn> <ci> &#955; </ci> <apply> <plus /> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 22 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 31 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 14 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1440 </cn> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> &#977; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> &#977; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 6 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <apply> <sin /> <apply> <times /> <ci> &#955; </ci> <pi /> </apply> </apply> <apply> <ci> Gamma </ci> <ci> &#956; </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> &#977; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> &#955; </ci> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> &#977; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 30 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 30 </cn> <ci> &#956; </ci> </apply> </apply> <cn type='integer'> -15 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 30 </cn> <ci> &#955; </ci> <apply> <plus /> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 22 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 31 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> -14 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1440 </cn> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> &#977; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> &#977; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 6 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> &#977; </ci> <pi /> </apply> <apply> <notin /> <ci> &#956; </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]_", ",", "\[Mu]_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], RowBox[List["4", " ", "\[Pi]"]]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Mu]", " ", "\[CurlyPhi]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Mu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Mu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "1"]], ")"]]]], "-", RowBox[List["3", " ", "\[Lambda]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"]]], RowBox[List["12", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "1"]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["45", " ", SuperscriptBox["\[Lambda]", "3"], " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "2"]], ")"]]]], "-", RowBox[List["30", " ", "\[Lambda]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Mu]", "2"], "+", "\[Mu]", "+", "1"]], ")"]]]], "-", RowBox[List["15", " ", SuperscriptBox["\[Lambda]", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["2", " ", "\[Mu]"]], "-", "1"]], ")"]]]], "+", RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", SuperscriptBox["\[Mu]", "3"]]], "+", RowBox[List["22", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["31", " ", "\[Mu]"]], "+", "14"]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "4"]]], RowBox[List["1440", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "2"]], ")"]]]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], "]"]], "6"]]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "\[Mu]"], " ", RowBox[List["Sin", "[", RowBox[List["\[Lambda]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Mu]", "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"], ")"]], RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Lambda]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]]]], "+", RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"]]], RowBox[List["12", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "1"]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["45", " ", SuperscriptBox["\[Lambda]", "3"], " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "2"]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[Lambda]", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["30", " ", SuperscriptBox["\[Mu]", "2"]]], "-", RowBox[List["30", " ", "\[Mu]"]], "-", "15"]], ")"]]]], "-", RowBox[List["30", " ", "\[Lambda]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Mu]", "2"], "-", "\[Mu]", "+", "1"]], ")"]]]], "+", RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", SuperscriptBox["\[Mu]", "3"]]], "-", RowBox[List["22", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["31", " ", "\[Mu]"]], "-", "14"]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "4"]]], RowBox[List["1440", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "1"]], ")"]]]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], "]"]], "6"]]], ")"]]]], "\[Pi]"]]], ")"]]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["\[CurlyTheta]", "\[Rule]", "\[Pi]"]], ")"]], "&&", RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]]]]]]]]]]


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





2001-10-29