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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Summation > Infinite summation





http://functions.wolfram.com/05.10.23.0014.01









  


  










Input Form





Sum[I^(n - m) Sqrt[((2 n + 1) (n + m)!)/(n - m)!] BesselJ[n + 1/2, w] SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]], {n, m, Infinity, 2}] == ((Sqrt[w] ((-w) Sin[\[CurlyTheta]] E^(I \[CurlyPhi]))^m)/(Sqrt[2] Pi)) Cos[w Cos[\[CurlyTheta]]] /; m >= 0 && Element[\[CurlyTheta], Reals] && Element[\[CurlyPhi], Reals]










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sum", "[", RowBox[List[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["n_", "-", "m_"]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n_"]], "+", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n_", "+", "m_"]], ")"]], "!"]]]], RowBox[List[RowBox[List["(", RowBox[List["n_", "-", "m_"]], ")"]], "!"]]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["n_", "+", FractionBox["1", "2"]]], ",", "w_"]], "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]]]], ",", RowBox[List["{", RowBox[List["n_", ",", "m_", ",", "\[Infinity]", ",", "2"]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["w"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "w"]], " ", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]"]]]]], ")"]], "m"]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["w", " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "]"]]]], RowBox[List[SqrtBox["2"], " ", "\[Pi]"]]], "/;", RowBox[List[RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["\[CurlyTheta]", "\[Element]", "Reals"]], "&&", RowBox[List["\[CurlyPhi]", "\[Element]", "Reals"]]]]]]]]]]










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





2001-10-29





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