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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/07.37.06.0023.01









  


  










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 + O[(\[CurlyTheta] - Pi)^2]) - (((2^\[Mu] Sin[\[Lambda] Pi] Gamma[\[Mu]])/Pi) (1 + O[(\[CurlyTheta] - Pi)^2]))/((\[CurlyTheta] - Pi)^2)^(\[Mu]/2)) /; (\[CurlyTheta] -> Pi) && !Element[\[Mu], Integers]










Standard Form





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MathML Form







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</ci> <pi /> </apply> <apply> <notin /> <ci> &#956; </ci> <ci> &#8484; </ci> </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", "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], "]"]], "2"]]], ")"]]]], 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", "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], "]"]], "2"]]], ")"]]]], "\[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





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