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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Integral representations > Integral representations of negative integer order





http://functions.wolfram.com/05.10.07.0009.01









  


  










Input Form





SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] == ((-1)^(m + n)/(n - m)!) r^(n + 1) Sqrt[((2 n + 1) (n - m)!)/(4 Pi (n + m)!)] Nest[Function[f, D[f, x] + I D[f, y]], D[1/r, {z, n - m}], m] /; r == Sqrt[x^2 + y^2 + z^2] && x == r Cos[\[CurlyPhi]] Sin[\[CurlyTheta]] && y == r Sin[\[CurlyPhi]] Sin[\[CurlyTheta]] && z == r Cos[\[CurlyTheta]] && 0 <= m <= n










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "n"]]], " ", SuperscriptBox["r", RowBox[List["n", "+", "1"]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "!"]]]], RowBox[List["4", " ", "\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], "!"]]]]]], " ", RowBox[List["Nest", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List["f", ",", RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["x"]]], "f"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["y"]]], "f"]]]]]]]], "]"]], ",", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z", ",", RowBox[List["n", "-", "m"]]]], "}"]]]]], FractionBox["1", "r"]]], ",", "m"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "!"]]], "/;", RowBox[List[RowBox[List["r", "\[Equal]", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", SuperscriptBox["z", "2"]]]]]], "&&", RowBox[List["x", "\[Equal]", RowBox[List["r", " ", RowBox[List["Cos", "[", "\[CurlyPhi]", "]"]], " ", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]]]], "&&", RowBox[List["y", "\[Equal]", RowBox[List["r", " ", RowBox[List["Sin", "[", "\[CurlyPhi]", "]"]], " ", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]]]], "&&", RowBox[List["z", "\[Equal]", RowBox[List["r", " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]]], "&&", RowBox[List["0", "\[LessEqual]", "m", "\[LessEqual]", "n"]]]]]]]]]]










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





2001-10-29