Spherical harmonics

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.

References

 [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)