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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Integration > Definite integration > Multiple integration





http://functions.wolfram.com/07.37.21.0013.01









  


  










Input Form





Integrate[Sin[\[CurlyTheta]] Conjugate[SphericalHarmonicY[Subscript[n, 1], Subscript[m, 1], \[CurlyTheta], \[CurlyPhi]]] SphericalHarmonicY[Subscript[n, 2], Subscript[m, 2], \[CurlyTheta], \[CurlyPhi]], {\[CurlyTheta], 0, Pi}, {\[CurlyPhi], 0, 2 Pi}] == KroneckerDelta[Subscript[n, 1], Subscript[n, 2]] KroneckerDelta[Subscript[m, 1], Subscript[m, 2]] /; Element[Subscript[n, 1], Integers] && Element[Subscript[n, 2], Integers] && Element[Subscript[m, 1], Integers] && Element[Subscript[m, 2], Integers]










Standard Form





Cell[BoxData[RowBox[List[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["Conjugate", "[", 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["\[DifferentialD]", "\[CurlyPhi]"]], " ", RowBox[List["\[DifferentialD]", "\[CurlyTheta]"]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"]]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["m", "1"], ",", SubscriptBox["m", "2"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["n", "1"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["n", "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["m", "1"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["m", "2"], "\[Element]", "Integers"]]]]]]]]










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["Conjugate", "[", 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["\[DifferentialD]", "\[CurlyPhi]_"]], RowBox[List["\[DifferentialD]", "\[CurlyTheta]_"]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["nn", "1"], ",", SubscriptBox["nn", "2"]]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["mm", "1"], ",", SubscriptBox["mm", "2"]]], "]"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["nn", "1"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["nn", "2"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["mm", "1"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["mm", "2"], "\[Element]", "Integers"]]]]]]]]]]










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





2001-10-29





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