Problems with rotational symmetry are naturally expanded in spherical harmonics $Y_{\ell m}(\theta,\phi)$. They are eigenfunctions of the angular part of the Laplacian on the unit sphere, $$\nabla^2_{\Omega}Y_{\ell m}=-\ell(\ell+1)Y_{\ell m},$$ with $$\ell=0,1,2,\ldots,\qquad m=-\ell,\ldots,\ell.$$

They form an orthonormal basis on the sphere: $$\int Y_{\ell m}^*(\theta,\phi)Y_{\ell' m'}(\theta,\phi),d\Omega =\delta_{\ell\ell'}\delta_{mm'}.$$ Thus a sufficiently regular angular function can be expanded as $$f(\theta,\phi)=\sum_{\ell,m}a_{\ell m}Y_{\ell m}(\theta,\phi),$$ with coefficients obtained by projection.

The lowest harmonic is constant: $$Y_{00}=\frac{1}{\sqrt{4\pi}},$$ representing an isotropic angular pattern. The $\ell=1$ harmonics have dipole-like angular dependence; higher $\ell$ describe progressively finer angular structure.

Spherical harmonics arise when separation of variables is applied in spherical coordinates: electrostatic and gravitational multipoles, wave equations, diffusion, and central-potential quantum problems all use the same angular basis. In quantum mechanics, the orbital angular-momentum operator satisfies $$L^2=-\hbar^2\nabla^2_{\Omega},$$ so the same harmonics become angular-momentum eigenfunctions.

They are the spherical analogue of Fourier modes. Fourier exponentials resolve periodic structure on a line or circle; spherical harmonics resolve angular structure on a sphere.