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http://functions.wolfram.com/07.37.23.0016.01
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Sum[I^(n - m) BesselJ[n + 1/2, w] SphericalHarmonicY[n, m,
Subscript[\[CurlyTheta], 1], Subscript[\[CurlyPhi], 1]]
Conjugate[SphericalHarmonicY[n, m, Subscript[\[CurlyTheta], 2],
Subscript[\[CurlyPhi], 2]]], {n, m, Infinity}] ==
(Sqrt[2 w]/(4 Pi^(3/2))) BesselJ[m, w Sin[Subscript[\[CurlyTheta], 1]]
Sin[Subscript[\[CurlyTheta], 2]]]
E^(I w Cos[Subscript[\[CurlyTheta], 1]] Cos[Subscript[\[CurlyTheta], 2]])
E^(I m (Subscript[\[CurlyPhi], 1] - Subscript[\[CurlyPhi], 2])) /;
Element[m, Integers] && m >= 0 && Element[Subscript[\[CurlyTheta], k],
Reals] && Element[Subscript[\[CurlyPhi], k], Reals] && Element[k, {1, 2}]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "m"]], "\[Infinity]"], RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "-", "m"]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "w"]], "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", SubscriptBox["\[CurlyTheta]", "1"], ",", SubscriptBox["\[CurlyPhi]", "1"]]], "]"]], " ", RowBox[List["Conjugate", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", SubscriptBox["\[CurlyTheta]", "2"], ",", SubscriptBox["\[CurlyPhi]", "2"]]], "]"]], "]"]]]]]], "\[Equal]", RowBox[List[FractionBox[SqrtBox[RowBox[List["2", "w"]]], RowBox[List["4", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], " ", RowBox[List["BesselJ", "[", RowBox[List["m", ",", RowBox[List["w", " ", RowBox[List["Sin", "[", SubscriptBox["\[CurlyTheta]", "1"], "]"]], " ", RowBox[List["Sin", "[", SubscriptBox["\[CurlyTheta]", "2"], "]"]]]]]], "]"]], SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "w", " ", RowBox[List["Cos", "[", SubscriptBox["\[CurlyTheta]", "1"], "]"]], " ", RowBox[List["Cos", "[", SubscriptBox["\[CurlyTheta]", "2"], "]"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "m", " ", RowBox[List["(", RowBox[List[SubscriptBox["\[CurlyPhi]", "1"], "-", SubscriptBox["\[CurlyPhi]", "2"]]], ")"]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["\[CurlyTheta]", "k"], "\[Element]", "Reals"]], "\[And]", RowBox[List[SubscriptBox["\[CurlyPhi]", "k"], "\[Element]", "Reals"]], "\[And]", RowBox[List["k", "\[Element]", RowBox[List["{", RowBox[List["1", ",", "2"]], "}"]]]]]]]]]]
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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> <munderover> <mo> ∑ </mo> <mrow> <mi> n </mi> <mo> = </mo> <mi> m </mi> </mrow> <mi> ∞ </mi> </munderover> <mrow> <msup> <mi> ⅈ </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <msub> <mi> J </mi> <mrow> <mi> n </mi> <mo> + </mo> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </mrow> </msub> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <msub> <mi> ϑ </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> φ </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mover> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <msub> <mi> ϑ </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> φ </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> _ </mo> </mover> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> w </mi> </mrow> </msqrt> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mi> m </mi> </msub> <mo> ( </mo> <mrow> <mi> w </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> ϑ </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> ϑ </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> w </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> ϑ </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> ϑ </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> φ </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> φ </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> ϑ </mi> <mi> k </mi> </msub> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> φ </mi> <mi> k </mi> </msub> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <mrow> <mo> { </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <ci> m </ci> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <imaginaryi /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> w </ci> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> ϑ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> φ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> OverBar </ci> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> ϑ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> φ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> w </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> m </ci> <apply> <times /> <ci> w </ci> <apply> <sin /> <apply> <ci> Subscript </ci> <ci> ϑ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <sin /> <apply> <ci> Subscript </ci> <ci> ϑ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> w </ci> <apply> <cos /> <apply> <ci> Subscript </ci> <ci> ϑ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <cos /> <apply> <ci> Subscript </ci> <ci> ϑ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> m </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> φ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> φ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <ci> ℕ </ci> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> ϑ </ci> <ci> k </ci> </apply> <reals /> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> φ </ci> <ci> k </ci> </apply> <reals /> </apply> <apply> <in /> <ci> k </ci> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n_", "=", "m_"]], "\[Infinity]"], RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["n_", "-", "m_"]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["n_", "+", FractionBox["1", "2"]]], ",", "w_"]], "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", SubscriptBox["\[CurlyTheta]_", "1"], ",", SubscriptBox["\[CurlyPhi]_", "1"]]], "]"]], " ", RowBox[List["Conjugate", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", SubscriptBox["\[CurlyTheta]_", "2"], ",", SubscriptBox["\[CurlyPhi]_", "2"]]], "]"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["2", " ", "w"]]], " ", RowBox[List["BesselJ", "[", RowBox[List["m", ",", RowBox[List["w", " ", RowBox[List["Sin", "[", SubscriptBox["\[CurlyTheta]\[CurlyTheta]", "1"], "]"]], " ", RowBox[List["Sin", "[", SubscriptBox["\[CurlyTheta]\[CurlyTheta]", "2"], "]"]]]]]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "w", " ", RowBox[List["Cos", "[", SubscriptBox["\[CurlyTheta]\[CurlyTheta]", "1"], "]"]], " ", RowBox[List["Cos", "[", SubscriptBox["\[CurlyTheta]\[CurlyTheta]", "2"], "]"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "m", " ", RowBox[List["(", RowBox[List[SubscriptBox["\[CurlyPhi]\[CurlyPhi]", "1"], "-", SubscriptBox["\[CurlyPhi]\[CurlyPhi]", "2"]]], ")"]]]]]]], RowBox[List["4", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["\[CurlyTheta]", "k"], "\[Element]", "Reals"]], "&&", RowBox[List[SubscriptBox["\[CurlyPhi]", "k"], "\[Element]", "Reals"]], "&&", RowBox[List["k", "\[Element]", RowBox[List["{", RowBox[List["1", ",", "2"]], "}"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
