# Spherical harmonics

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of degree A restriction of a homogeneous harmonic polynomial of degree in variables to the unit sphere of the Euclidean space , . In particular, when , the spherical harmonics are the classical spherical functions.

Let , , , . The basic property of spherical harmonics is the property of orthogonality: If and are spherical harmonics of degree and , respectively, with , then The simplest spherical harmonics are the zonal spherical harmonics. For any and any , a zonal spherical harmonic exists which is constant on any parallel of the sphere that is orthogonal to the vector . The zonal spherical harmonics differ from the Legendre polynomials , when , or from the ultraspherical polynomials , when , only by a constant factor: where the polynomials are defined, when , by the generating function  , , . The polynomials , are orthogonal with weight and form an orthogonal basis of the space . If is a function in with , then there is a unique set of spherical harmonics such that where the series converges in the norm of .

Expansions in spherical harmonics are largely analogous to expansions in Fourier series, of which they are essentially a generalization. The homogeneous harmonic polynomials are sometimes called spatial spherical harmonics. By virtue of the homogeneity spherical harmonics are sometimes also called surface spherical harmonics.

How to Cite This Entry:
Spherical harmonics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spherical_harmonics&oldid=12872
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article