Spherical Harmonics | stevejtrettel.site
The spherical harmonics are the eigenfunctions of the Laplace operator $\Delta$ on the round 2-dimensional sphere. From this perspective, they are a generalization of the familiar functions $\sin(n x),\cos(nx)$ on the circle, which are eigenfunctions of the 1-dimensional Laplacian $\frac{d^2}{dx^2}$. Unlike $\sin$ and $\cos$ which are determined by a single number (their frequency), spherical harmonics are parameterized by a pair of invariants $\ell,m$. For each non-negative integer $\ell$, there is a spherical harmonic $Y_{\ell m}$ for each integral $m\in[-\ell,\ell]$.
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