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http://functions.wolfram.com/07.37.07.0007.01
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Abs[SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]]]^2 ==
((2 n + 1)/(4 Pi)) Integrate[BesselJ[m, (t Sin[\[CurlyTheta]])/2]^2
BesselJ[2 n + 1, t], {t, 0, Infinity}] /;
Element[n, Integers] && n >= 0 && Element[m, Integers] && Abs[m] <= n &&
Element[\[CurlyTheta], Reals] && Element[\[CurlyPhi], Reals] &&
0 < Sin[\[CurlyTheta]] < 1
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Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Abs", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "]"]], "2"], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], RowBox[List["4", "\[Pi]"]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["BesselJ", "[", RowBox[List["m", ",", FractionBox[RowBox[List["t", " ", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]], "2"]]], "]"]], "2"], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], ",", "t"]], "]"]], " ", RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[LessEqual]", "n"]], "\[And]", RowBox[List["\[CurlyTheta]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[CurlyPhi]", "\[Element]", "Reals"]], "\[And]", RowBox[List["0", "<", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "<", "1"]]]]]]]]
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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> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <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> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </msup> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <msup> <mrow> <msub> <mi> J </mi> <mi> m </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mi> t </mi> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> m </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ≤ </mo> <mi> n </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> ϑ </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> φ </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mn> 0 </mn> <mo> < </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <power /> <apply> <abs /> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <apply> <ci> BesselJ </ci> <ci> m </ci> <apply> <times /> <apply> <times /> <ci> t </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sin /> <ci> ϑ </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> BesselJ </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> t </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <leq /> <apply> <abs /> <ci> m </ci> </apply> <ci> n </ci> </apply> <apply> <in /> <ci> ϑ </ci> <reals /> </apply> <apply> <in /> <ci> φ </ci> <reals /> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <sin /> <ci> ϑ </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["Abs", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["BesselJ", "[", RowBox[List["m", ",", RowBox[List[FractionBox["1", "2"], " ", "t", " ", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]]]], "]"]], "2"], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ",", "t"]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[LessEqual]", "n"]], "&&", RowBox[List["\[CurlyTheta]", "\[Element]", "Reals"]], "&&", RowBox[List["\[CurlyPhi]", "\[Element]", "Reals"]], "&&", RowBox[List["0", "<", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "<", "1"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
