Sum[Conjugate[SphericalHarmonicY[n, m, Subscript[\[CurlyTheta], 1], Subscript[\[CurlyPhi], 1]]] SphericalHarmonicY[n, m, Subscript[\[CurlyTheta], 2], Subscript[\[CurlyPhi], 2]], {n, 0, Infinity}, {m, -n, n}] == DiracDelta[Subscript[\[CurlyPhi], 1] - Subscript[\[CurlyPhi], 2]] DiracDelta[Cos[Subscript[\[CurlyTheta], 1]] - Cos[Subscript[\[CurlyTheta], 2]]] /; Element[Subscript[\[CurlyTheta], k], Reals] && Element[Subscript[\[CurlyPhi], k], Reals] && Element[k, {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> &#8721; </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mi> n </mi> </munderover> <mrow> <mover> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> &#966; </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> _ </mo> </mover> <mo> &#8290; </mo> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> &#966; </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <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> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> &#977; </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> &#977; </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <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> m </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> OverBar </ci> <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> <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> <ci> DiracDelta </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> <ci> DiracDelta </ci> <apply> <plus /> <apply> <cos /> <apply> <ci> Subscript </ci> <ci> &#977; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <ci> Subscript </ci> <ci> &#977; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <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>

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2001-10-29