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http://functions.wolfram.com/05.10.21.0013.01
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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]
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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"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> π </mi> </msubsup> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msubsup> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mtext> </mtext> <mover> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </msubsup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> _ </mo> </mover> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> φ </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> ϑ </mi> </mrow> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msub> <mi> δ </mi> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> </msub> <mo> ⁢ </mo> <msub> <mi> δ </mi> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> π </mi> </msubsup> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msubsup> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mtext> </mtext> <mover> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </msubsup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> _ </mo> </mover> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> φ </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> ϑ </mi> </mrow> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msub> <mi> δ </mi> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> </msub> <mo> ⁢ </mo> <msub> <mi> δ </mi> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| 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"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
