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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Series representations > Generalized power series > Expansions at theta==0





http://functions.wolfram.com/05.10.06.0019.01









  


  










Input Form





SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] \[Proportional] (Sqrt[(2 n + 1)/(4 Pi)] (Sqrt[Gamma[n - m + 1]]/Sqrt[Gamma[n + m + 1]]) 2^m E^(I \[CurlyPhi] m) (1/Gamma[1 - m] + (1/12) (-((3 n (n + 1))/Gamma[2 - m]) + 1/Gamma[-m]) \[CurlyTheta]^2 + (1/1440) ((30 n (n + 1) (m + 1))/Gamma[2 - m] + (45 (n - 1) n (n + 1) (n + 2))/Gamma[3 - m] + (7 - 5 m)/Gamma[-m]) \[CurlyTheta]^4 + (1/362880) (-((63 n (n + 1) (4 + m (3 + 5 m)))/ Gamma[2 - m]) - (945 (n - 1) n (n + 1) (n + 2) (m + 2))/ Gamma[3 - m] - (945 (n - 1) n (n + 1) (n + 2) (n^2 + n - 6))/ Gamma[4 - m] + (124 + 7 m (5 m - 21))/Gamma[-m]) \[CurlyTheta]^6 + O[\[CurlyTheta]^8]))/(\[CurlyTheta]^2)^(m/2) /; (\[CurlyTheta] -> 0)










Standard Form





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MathML Form







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</ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> &#977; </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> &#977; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], RowBox[List["4", " ", "\[Pi]"]]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["n", "-", "m", "+", "1"]], "]"]]], " ", SuperscriptBox["2", "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "m"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[CurlyTheta]", "2"], ")"]], RowBox[List["-", FractionBox["m", "2"]]]], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m"]], "]"]]], "+", RowBox[List[FractionBox["1", "12"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["3", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]]]], "+", FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], " ", SuperscriptBox["\[CurlyTheta]", "2"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["30", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["45", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["3", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["7", "-", RowBox[List["5", " ", "m"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], " ", SuperscriptBox["\[CurlyTheta]", "4"]]], "1440"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["63", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["5", " ", "m"]]]], ")"]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]]]], "-", FractionBox[RowBox[List["945", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "+", "2"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["3", "-", "m"]], "]"]]], "-", FractionBox[RowBox[List["945", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["n", "2"], "+", "n", "-", "6"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["4", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["124", "+", RowBox[List["7", " ", "m", " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", "m"]], "-", "21"]], ")"]]]]]], RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], " ", SuperscriptBox["\[CurlyTheta]", "6"]]], "362880"], "+", SuperscriptBox[RowBox[List["O", "[", "\[CurlyTheta]", "]"]], "8"]]], ")"]]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["n", "+", "m", "+", "1"]], "]"]]]], "/;", RowBox[List["(", RowBox[List["\[CurlyTheta]", "\[Rule]", "0"]], ")"]]]]]]]]










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





2001-10-29