Spherical harmonics

Summary

Spherical harmonics are the restrictions to the unit sphere of homogeneous polynomials that are harmonic. Those of degree $\ell$ form a finite-dimensional space, and these spaces give an orthogonal decomposition of the square-integrable functions on the sphere. They are the eigenfunctions of the Laplace–Beltrami operator of the sphere, and they play for the sphere the role that $e^{in\theta}$ plays for the circle. On the sphere in $\R^3$ they are built from the associated Legendre polynomials.