Spherical Surface Harmonic

Spherical surface harmonics are special functions used in studying physical phenomena in regions of space bounded by spherical surfaces and in solving physical problems exhibiting spherical symmetry.

Spherical surface harmonics are solutions of the differential equation

obtained by applying separation of variables to Laplace’s differential equation expressed in spherical coordinates r, θ, and ɸ. The general form of the solution is

Here, the a_m are constants, and the P^m_l (cos θ) are associated Legendre functions of degree l and order m defined by the equality

where the P_n are Legendre polynomials.

Spherical surface harmonics can be regarded as functions on the unit sphere. The functions

form a complete orthonormal system on the sphere. This system plays the same role in the expansion of functions on the sphere as does the trigonometric system of functions {e^imɸ} in the expansion of functions on the circle. Functions on the sphere that are independent of the coordinate ɸ can be expanded in zonal spherical harmonics:

Under rotations of the sphere, spherical surface harmonics of degree l

are linearly transformed in accordance with the equation

where q^–1M is the point into which the point M on the sphere is carried by the rotation q^–1. The coefficients are the matrix elements of an irreducible unitary representation of weight l of the group of rotations of the sphere. These matrix elements are called generalized spherical surface harmonics. They can be used, for example, to expand vector and tensor fields on the unit sphere and to solve certain problems in elasticity theory.

Equation (1) is connected with the addition theorem for zonal spherical harmonics:

Here, cos γ = cos θ cos θ’ + sin θ sin θ’ cos (ɸ—ɸ’), where γ is the spherical distance between the point (θ, ɸ) and the point (θ’, ɸ’).

Potential theory is a typical example of the many areas in which spherical harmonics may be applied to problems in mathematical physics and mechanics. Suppose σ = σ (θ, ɸ) is the surface density of the distribution of mass on a sphere of radius R with center at the origin. If σ can be expanded in a series of spherical harmonics

that converges uniformly on the surface of the sphere, the potential corresponding to this mass distribution is

at each point (r, θ, ɸ) outside the sphere and is

at each point inside the sphere. The general terms of these series are solid spherical harmonics of degree n – 1 and n, respectively.

Spherical surface harmonics were introduced by A. Legendre and P. de Laplace in the late 18th century.