# Integrating a product of three Spherical Harmonics

The following command is returned unevaluated. The answer is well known to be related to Wigner's 3j Symbol which is also a defined function in Mathematica.

   Integrate[
SphericalHarmonicY[l, -m, θ, ϕ] SphericalHarmonicY[1, 0, θ, ϕ] *
SphericalHarmonicY[l + 1, m, θ, ϕ] Sin[θ],
{ϕ, 0, 2 π}, {θ, 0, π}]

• People here generally like users to post code as Mathematica code instead of images or TeX, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. You may find this this meta Q&A helpful Feb 1, 2018 at 1:52
• Your Y_1 is undefined (for instance). Feb 1, 2018 at 1:52
• Try to add constraints for m and l, and/or substitute their definition in terms of the associate Legendre polynomials and complex exponentials... Feb 1, 2018 at 12:33
• Adding the usual constraints on m and l does not make any difference. I don't see why I should have to write SphericalHarmonics in terms of Legendre polynomials, that is something the algorithm has to do internally. Feb 1, 2018 at 18:16