Spherical harmonic
= Spherical harmonic
{title2=$Y_\ell^m$}
{wiki}
A <spherical harmonic> of degree $\ell$ is an <eigenfunction> of the angular <Laplacian> on the unit <sphere>. In three dimensions, with the sign convention inherited from $\Delta=\sum_j\partial_j^2$,
$$
\Delta_{S^2}Y_\ell^m=-\ell(\ell+1)Y_\ell^m,\qquad m=-\ell,\ldots,\ell.
$$
One chooses an <orthonormal basis> for each $(2\ell+1)$-dimensional eigenspace. With surface measure $d\Omega$, examples are $Y_0^0=1/\sqrt{4\pi}$ and $Y_1^0=\sqrt{3/(4\pi)}\cos\theta$.
The degree-zero mode is an <angular monopole>, the degree-one modes form an <angular dipole>, and higher degrees resolve finer angular structure. <Separation of variables> in the <Laplace equation> pairs these angular modes with the <solid harmonics> $r^\ell Y_\ell^m$ and $r^{-\ell-1}Y_\ell^m$. This is why the same basis organizes <gravitational potential> expansions, <atomic orbitals> and sky-map multipoles.