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http://functions.wolfram.com/07.37.21.0003.01
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Integrate[((\[Lambda] - \[Kappa]) (\[Lambda] + \[Kappa] + 1) -
(\[Mu]^2 - \[Nu]^2)/Sin[\[CurlyTheta]]^2) SphericalHarmonicY[\[Lambda],
\[Mu], \[CurlyTheta], \[CurlyPhi]] SphericalHarmonicY[\[Kappa], \[Nu],
\[CurlyTheta], \[CurlyPhi]] Sin[\[CurlyTheta]], \[CurlyTheta]] ==
(\[Nu] - \[Mu]) Cos[\[CurlyTheta]] SphericalHarmonicY[\[Lambda], \[Mu],
\[CurlyTheta], \[CurlyPhi]] SphericalHarmonicY[\[Kappa], \[Nu],
\[CurlyTheta], \[CurlyPhi]] +
(Sin[\[CurlyTheta]] (((Sqrt[Gamma[\[Kappa] - \[Nu] + 1]]
Sqrt[Gamma[\[Kappa] + \[Nu] + 2]])/(Sqrt[Gamma[\[Kappa] + \[Nu] + 1]]
Sqrt[Gamma[\[Kappa] - \[Nu]]])) SphericalHarmonicY[\[Lambda], \[Mu],
\[CurlyTheta], \[CurlyPhi]] SphericalHarmonicY[\[Kappa], \[Nu] + 1,
\[CurlyTheta], \[CurlyPhi]] -
((Sqrt[Gamma[\[Lambda] - \[Mu] + 1]] Sqrt[Gamma[\[Lambda] + \[Mu] + 2]])/
(Sqrt[Gamma[\[Lambda] + \[Mu] + 1]] Sqrt[Gamma[\[Lambda] - \[Mu]]]))
SphericalHarmonicY[\[Lambda], \[Mu] + 1, \[CurlyTheta], \[CurlyPhi]]
SphericalHarmonicY[\[Kappa], \[Nu], \[CurlyTheta], \[CurlyPhi]]))/
E^(I \[CurlyPhi])
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){kind=small}
