In physics, these functions are encountered as the eigenfunctions of the angular momentum operator and they consist of prefactors that depend on their l and m indices multiplied by the associated Legendre polynomial.
In the standard analysis of CMB polarization, the polarization field is decomposed into the spin-weighted variety of these mathematical objects. The method of partial-wave expansion relies on expressing the scattering amplitude as a sum of these objects. The matrix elements generated by the action of a tensor operator on one of these functions is proportional to a Clebsch-Gordan coefficient according to the Wigner-Eckhart theorem. While the Laguerre polynomials solve the radial part of the hydrogen wavefunction, these functions give the angular component of the wavefunction. In physics, these functions are encountered as the eigenfunctions of the angular momentum operator and they consist of prefactors that depend on their l and m indices multiplied by the associated Legendre polynomial. For ten points, identify these mathematical functions which solve the angular part of Laplace's equation in their namesake coordinate system.
In the standard analysis of CMB polarization, the polarization field is decomposed into the spin-weighted variety of these mathematical objects.
{spherical harmonics}