Spherical harmonic

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 $\Delta ={\sum }_j{\partial }_j^2$,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.