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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Summation > Infinite summation





http://functions.wolfram.com/05.10.23.0016.01









  


  










Input Form





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])) /; m >= 0 && Element[Subscript[\[CurlyTheta], k], Reals] && Element[Subscript[\[CurlyPhi], k], Reals] && Element[k, {1, 2}]










Standard Form





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MathML Form







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</mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> _ </mo> </mover> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> w </mi> </mrow> </msqrt> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mi> m </mi> </msub> <mo> ( </mo> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> &#977; </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> &#977; </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> &#977; </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> &#977; </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#966; </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> &#966; </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8805; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> &#977; </mi> <mi> k </mi> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> &#966; </mi> <mi> k </mi> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </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> &#977; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#966; </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> &#977; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#966; </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> &#977; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <sin /> <apply> <ci> Subscript </ci> <ci> &#977; </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> &#977; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <cos /> <apply> <ci> Subscript </ci> <ci> &#977; </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> &#966; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#966; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <geq /> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> &#977; </ci> <ci> k </ci> </apply> <reals /> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> &#966; </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>










Rule Form





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





2001-10-29





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