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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Integration > Definite integration > Multiple integration





http://functions.wolfram.com/05.10.21.0010.01









  


  










Input Form





Integrate[Sin[\[CurlyTheta]] SphericalHarmonicY[Subscript[n, 1], Subscript[m, 1], \[CurlyTheta], \[CurlyPhi]] SphericalHarmonicY[Subscript[n, 2], Subscript[m, 2], \[CurlyTheta], \[CurlyPhi]] SphericalHarmonicY[Subscript[n, 3], Subscript[m, 3], \[CurlyTheta], -\[CurlyPhi]], {\[CurlyTheta], 0, Pi}, {\[CurlyPhi], 0, 2 Pi}] == Sqrt[((2 Subscript[n, 1] + 1) (2 Subscript[n, 2] + 1))/ (4 Pi (2 Subscript[n, 3] + 1))] ClebschGordan[{Subscript[n, 1], 0}, {Subscript[n, 2], 0}, {Subscript[n, 3], 0}] ClebschGordan[{Subscript[n, 1], Subscript[m, 1]}, {Subscript[n, 2], Subscript[m, 2]}, {Subscript[n, 3], Subscript[m, 3]}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[SubsuperscriptBox["\[Integral]", "0", RowBox[List["2", " ", "\[Pi]"]]], RowBox[List[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["m", "1"], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n", "2"], ",", SubscriptBox["m", "2"], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n", "3"], ",", SubscriptBox["m", "3"], ",", "\[CurlyTheta]", ",", RowBox[List["-", "\[CurlyPhi]"]]]], "]"]], RowBox[List["\[DifferentialD]", "\[CurlyPhi]"]], RowBox[List["\[DifferentialD]", "\[CurlyTheta]"]]]]]]]], "\[Equal]", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", SubscriptBox["n", "1"]]], "+", "1"]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["2", SubscriptBox["n", "2"]]], "+", "1"]], ")"]]]], RowBox[List["4", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", SubscriptBox["n", "3"]]], "+", "1"]], ")"]]]]]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["n", "1"], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["n", "2"], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["n", "3"], ",", "0"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["n", "2"], ",", SubscriptBox["m", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["n", "3"], ",", SubscriptBox["m", "3"]]], "}"]]]], "]"]]]]]]]]










MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[SubsuperscriptBox["\[Integral]", "0", RowBox[List["2", " ", "\[Pi]"]]], RowBox[List[RowBox[List[RowBox[List["Sin", "[", "\[CurlyTheta]_", "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["m_", "1"], ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n_", "2"], ",", SubscriptBox["m_", "2"], ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n_", "3"], ",", SubscriptBox["m_", "3"], ",", "\[CurlyTheta]_", ",", RowBox[List["-", "\[CurlyPhi]_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "\[CurlyPhi]_"]], RowBox[List["\[DifferentialD]", "\[CurlyTheta]_"]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SubscriptBox["nn", "1"]]], "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SubscriptBox["nn", "2"]]], "+", "1"]], ")"]]]], RowBox[List["4", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SubscriptBox["nn", "3"]]], "+", "1"]], ")"]]]]]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["nn", "1"], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["nn", "2"], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["nn", "3"], ",", "0"]], "}"]]]], "]"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["nn", "1"], ",", SubscriptBox["mm", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["nn", "2"], ",", SubscriptBox["mm", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["nn", "3"], ",", SubscriptBox["mm", "3"]]], "}"]]]], "]"]]]]]]]]










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





2001-10-29





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