§14.30 Spherical and Spheroidal Harmonics

§14.30(i) Definitions

With and integers such that , and and angles such that , ,

14.30.2

are known as spherical harmonics. are known as surface harmonics of the first kind: tesseral for and sectorial for . Sometimes is denoted by ; also the definition of can differ from (14.30.1), for example, by inclusion of a factor .

and () are often referred to as the prolate spheroidal harmonics of the first and second kinds, respectively. and () are known as oblate spheroidal harmonics of the first and second kinds, respectively. Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics and which are real when and .

§14.30(ii) Basic Properties

Most mathematical properties of can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter.

¶ Orthogonality

here and elsewhere in this section the asterisk (*) denotes complex conjugate.

§14.30(iii) Sums

¶ Distributional Completeness

For a series representation of the product of two Dirac deltas in terms of products of spherical harmonics see §1.17(iii).

§14.30(iv) Applications

In general, spherical harmonics are defined as the class of homogeneous harmonic polynomials. See Andrews et al. (1999, Chapter 9). The special class of spherical harmonics , defined by (14.30.1), appear in many physical applications. As an example, Laplace’s equation in spherical coordinates (§1.5(ii)):

has solutions , which are everywhere one-valued and continuous.

In the quantization of angular momentum the spherical harmonics are normalized solutions of the eigenvalue equation

where is the reduced Planck’s constant, and is the angular momentum operator in spherical coordinates:

14.30.12

see Edmonds (1974, §2.5).

For applications in geophysics see Stacey (1977, §§4.2, 6.3, and 8.1).