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.


[1] P.M. Morse, H. Feshbach, "Methods of theoretical physics" , 1–2 , McGraw-Hill (1953)
[2] E.M. Stein, G. Weiss, "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971)



[a1] I.M. Gel'fand, R.A. Minlos, Z.Ya. Shapiro, "Representations of the rotation group and the Lorentz group, and their applications" , Macmillan (1963) (Translated from Russian)
[a2] N.Ya. Vilenkin, "Special functions and the theory of group representations" , Amer. Math. Soc. (1968) (Translated from Russian)
[a3] N.Ya. Vilenkin, A.U. Klimyk, "Special functions, group representations, and integral transforms" , 1 , Kluwer (1991) (Translated from Russian)
How to Cite This Entry:
Spherical harmonics. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article