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http://functions.wolfram.com/05.10.21.0004.01
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Integrate[Sin[\[CurlyTheta]] SphericalHarmonicY[n, m, \[CurlyTheta],
\[CurlyPhi]]^2, {\[CurlyTheta], 0, Pi}] ==
E^(2 I m \[CurlyPhi])/(2 Pi) /; 0 <= m <= n
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], SuperscriptBox[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "2"], RowBox[List["\[DifferentialD]", "\[CurlyTheta]"]]]]]], "\[Equal]", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "m", " ", "\[CurlyPhi]"]]], RowBox[List["2", " ", "\[Pi]"]]]]], "/;", RowBox[List["0", "\[LessEqual]", "m", "\[LessEqual]", "n"]]]]]]
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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> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> ϑ </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> m </mi> <mo> ⁢ </mo> <mi> φ </mi> </mrow> </msup> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mn> 0 </mn> <mo> ≤ </mo> <mi> m </mi> <mo> ≤ </mo> <mi> n </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> ϑ </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <pi /> </uplimit> <apply> <times /> <apply> <sin /> <ci> ϑ </ci> </apply> <apply> <power /> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> m </ci> <ci> φ </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> m </ci> <ci> n </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[RowBox[List[RowBox[List["Sin", "[", "\[CurlyTheta]_", "]"]], " ", SuperscriptBox[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "2"]]], RowBox[List["\[DifferentialD]", "\[CurlyTheta]_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "m", " ", "\[CurlyPhi]"]]], RowBox[List["2", " ", "\[Pi]"]]], "/;", RowBox[List["0", "\[LessEqual]", "m", "\[LessEqual]", "n"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
