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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/05.10.23.0009.01









  


  










Input Form





Sum[(1/(n! Sqrt[(2 n + 1) (n - m)! (n + m)!])) SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] w^(n - m), {n, m, Infinity}] == (((-Sin[\[CurlyTheta]]) E^(I \[CurlyPhi]))^m/(2^(m + 1) Sqrt[Pi] m!^2)) Hypergeometric0F1[m + 1, w Cos[\[CurlyTheta]/2]^2] Hypergeometric0F1[m + 1, (-w) Sin[\[CurlyTheta]/2]^2] /; m >= 0 && Element[\[CurlyTheta], Reals] && Element[\[CurlyPhi], Reals] && Abs[w] < 1










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n_", "=", "m_"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], " ", SuperscriptBox["w_", RowBox[List["n_", "-", "m_"]]]]], RowBox[List[RowBox[List["n_", "!"]], " ", SqrtBox[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_"]], ")"]], "!"]]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]"]]]]], ")"]], "m"], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List[RowBox[List["m", "+", "1"]], ",", RowBox[List["w", " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"]]]]], "]"]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List[RowBox[List["m", "+", "1"]], ",", RowBox[List[RowBox[List["-", "w"]], " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"]]]]], "]"]]]], RowBox[List[SuperscriptBox["2", RowBox[List["m", "+", "1"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "!"]], ")"]], "2"]]]], "/;", RowBox[List[RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["\[CurlyTheta]", "\[Element]", "Reals"]], "&&", RowBox[List["\[CurlyPhi]", "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", "1"]]]]]]]]]]










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





2001-10-29





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