Integrate[Csc[\[CurlyTheta]] SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] Conjugate[SphericalHarmonicY[n, Subscript[m, 1], \[CurlyTheta], \[CurlyPhi]]], {\[CurlyTheta], 0, Pi}] == ((2 n + 1)/(4 Pi m)) KroneckerDelta[m, Subscript[m, 1]] /; n > 0 && Element[Subscript[m, 1], Integers] && Subscript[m, 1] >= 0 && 0 <= m <= n && Subscript[m, 1] <= 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> &#8747; </mo> <mn> 0 </mn> <mi> &#960; </mi> </msubsup> <mrow> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mover> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> _ </mo> </mover> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> &#977; </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> </mfrac> <mo> &#8290; </mo> <msub> <mi> &#948; </mi> <mrow> <mi> m </mi> <mo> , </mo> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mrow> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> m </mi> <mo> &#8804; </mo> <mi> n </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> &#8804; </mo> <mi> n </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> &#977; </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <pi /> </uplimit> <apply> <times /> <apply> <csc /> <ci> &#977; </ci> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> &#977; </ci> <ci> &#966; </ci> </apply> <apply> <ci> OverBar </ci> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#977; </ci> <ci> &#966; </ci> </apply> </apply> </apply> </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 /> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> &#948; </ci> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#8469; </ci> </apply> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> m </ci> <ci> n </ci> </apply> <apply> <leq /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <ci> n </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[RowBox[List[RowBox[List["Csc", "[", "\[CurlyTheta]_", "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], " ", RowBox[List["Conjugate", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", SubscriptBox["m_", "1"], ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]]]], RowBox[List["\[DifferentialD]", "\[CurlyTheta]_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", ",", SubscriptBox["mm", "1"]]], "]"]]]], RowBox[List["4", " ", "\[Pi]", " ", "m"]]], "/;", RowBox[List[RowBox[List["n", ">", "0"]], "&&", RowBox[List[SubscriptBox["mm", "1"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["mm", "1"], "\[GreaterEqual]", "0"]], "&&", RowBox[List["0", "\[LessEqual]", "m", "\[LessEqual]", "n"]], "&&", RowBox[List[SubscriptBox["mm", "1"], "\[LessEqual]", "n"]]]]]]]]]]

2001-10-29