Spherical harmonics

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations that commonly occur in science. The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).

Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree ${\displaystyle \ell }$ in ${\displaystyle (x,y,z)}$ that obey Laplace's equation. Functions that satisfy Laplace's equation are often said to be harmonic, hence the name spherical harmonics. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of ${\displaystyle r^{\ell }}$ from the above-mentioned polynomial of degree ${\displaystyle \ell }$; the remaining factor can be regarded as a function of the spherical angular coordinates ${\displaystyle \theta }$ and ${\displaystyle \varphi }$ only, or equivalently of the orientational unit vector ${\displaystyle {\mathbf {r} }}$ specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition.

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