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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Integral representations > Multiple integral representations





http://functions.wolfram.com/07.37.07.0008.01









  


  










Input Form





SphericalHarmonicY[n, m, ϑ, φ] == E^(I*m*φ)*Sqrt[((2*n + 1)/(4*Pi))*((n + m)!/(n - m)!)]* Integrate[LegendreP[n, Subscript[t, 1]], {Subscript[t, 1], 1, Subscript[t, 2]}, {Subscript[t, 2], 1, Subscript[t, 3]}, …, {Subscript[t, m], 1, Cos[ϑ]}]/;0≤m≤n










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[CurlyPhi]"]]], SqrtBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], RowBox[List["4", "\[Pi]"]]], " ", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "!"]]]]]], " ", SuperscriptBox[RowBox[List["Csc", "[", "\[CurlyTheta]", "]"]], "m"], " ", UnderscriptBox[RowBox[List[SubsuperscriptBox["\[Integral]", "1", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]], RowBox[List["\[Ellipsis]", RowBox[List[SubsuperscriptBox["\[Integral]", "1", SubscriptBox["t", "3"]], SubsuperscriptBox["\[Integral]", "1", SubscriptBox["t", "2"]]]]]]]], UnderscriptBox["\[UnderBrace]", "m"]], RowBox[List["LegendreP", "[", RowBox[List["n", ",", SubscriptBox["t", "1"]]], "]"]], " ", RowBox[List["\[DifferentialD]", SubscriptBox["t", "1"]]], RowBox[List["\[DifferentialD]", SubscriptBox["t", "2"]]], "\[Ellipsis]", " ", RowBox[List["\[DifferentialD]", SubscriptBox["t", "m"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[LessEqual]", "n"]]]]]]]]










MathML Form







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





2001-10-29