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I want to solve this integral

Integrate[
SphericalHarmonicY[l, m, o, p]*SphericalHarmonicY[l, n, o, p]*
SphericalHarmonicY[l, i, o, p]*SphericalHarmonicY[l, j, o, p], {o, 
0, Pi}, {p, 0, 2 Pi}]

but mathematica doesn't give me an answer. I also tried: (simplified in terms of Legendre polynomials and exponential)

Integrate[
Integrate[
LegendreP[l, m, o]*LegendreP[l, n, o]*LegendreP[l, i, o]*
LegendreP[l, j, o]*Exp[I*(m - n - i + j)], {o, 0, Pi}], {p, 0, 
2 Pi}]

But still no helpful answer, can anybody help me with this?

Improve this question
$\endgroup$
3
  • $\begingroup$ The command Table[Integrate[ SphericalHarmonicY[l, m, o, p]*SphericalHarmonicY[l, n, o, p]* SphericalHarmonicY[l, i, o, p]*SphericalHarmonicY[l, j, o, p], {o, 0, Pi}, {p, 0, 2 Pi}], {l, -0, 3}, {m, -0, 3}, {n, -0, 3}, {i, -0, 3}, {j, -0, 3}] performs {{{{{1/8, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, ....}}}}}. $\endgroup$
    – user64494
    Commented Oct 16, 2019 at 19:17
  • 3
    $\begingroup$ Shouldn't the integrand have an additional Sin[o] term for uniform weighting on the sphere in spherical coordinates? (see e.g. surface element in spherical coordinates) $\endgroup$
    – Thies Heidecke
    Commented Oct 17, 2019 at 5:53
  • $\begingroup$ To get an analytical expression convert a product of 2 spherical harmonics into a sum of spherical harmonics using 3j symbols (reference.wolfram.com/language/ref/ThreeJSymbol.html). See Gaunt coefficients. Then use this formula second time. $\endgroup$
    – yarchik
    Commented Oct 17, 2019 at 7:22

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