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14 Legendre and Related FunctionsApplications

§14.30 Spherical and Spheroidal Harmonics

  1. §14.30(i) Definitions
  2. §14.30(ii) Basic Properties
  3. §14.30(iii) Sums
  4. §14.30(iv) Applications

§14.30(i) Definitions

With l and m integers such that |m|l, and θ and ϕ angles such that 0θπ, 0ϕ2π,

14.30.1 Yl,m(θ,ϕ)=((lm)!(2l+1)4π(l+m)!)1/2eimϕ𝖯lm(cosθ),
14.30.2 Ylm(θ,ϕ)=cos(mϕ)𝖯lm(cosθ) or sin(mϕ)𝖯lm(cosθ).

Yl,m(θ,ϕ) are known as spherical harmonics. Ylm(θ,ϕ) are known as surface harmonics of the first kind: tesseral for |m|<l and sectorial for |m|=l. Sometimes Yl,m(θ,ϕ) is denoted by il𝔇lm(θ,ϕ); also the definition of Yl,m(θ,ϕ) can differ from (14.30.1), for example, by inclusion of a factor (1)m.

Pnm(x) and Qnm(x) (x>1) are often referred to as the prolate spheroidal harmonics of the first and second kinds, respectively. Pnm(ix) and Qnm(ix) (x>0) are known as oblate spheroidal harmonics of the first and second kinds, respectively. Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics Rnm(x)=eiπn/2Pnm(ix) and Tnm(x)=ieiπn/2Qnm(ix) which are real when x>0 and n=0,1,2,.

§14.30(ii) Basic Properties

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

Explicit Representation

14.30.3 Yl,m(θ,ϕ)=(1)l+m2ll!((lm)!(2l+1)4π(l+m)!)1/2eimϕ×(sinθ)m(dd(cosθ))l+m(sinθ)2l.

Special Values

14.30.4 Yl,m(0,ϕ)={(2l+14π)1/2,m=0,0,|m|=1,2,3,,
14.30.5 Yl,m(12π,ϕ)={(1)(l+m)/2eimϕ2l(12l12m)!(12l+12m)!((lm)!(l+m)!(2l+1)4π)1/2,12l+12m,0,12l+12m.


14.30.6 Yl,m(θ,ϕ)=(1)mYl,m(θ,ϕ)¯.

Parity Operation

14.30.7 Yl,m(πθ,ϕ+π)=(1)lYl,m(θ,ϕ).


14.30.8 02π0πYl1,m1(θ,ϕ)¯Yl2,m2(θ,ϕ)sinθdθdϕ=δl1,l2δm1,m2.

See also (34.3.22), and for further related integrals see Askey et al. (1986).

Herglotz generating function

The following is the Herglotz generating function

14.30.8_5 et𝐚𝐱=4πn=0m=nntnrnλmYn,m(θ,ϕ)(2n+1)(n+m)!(nm)!,

where 𝐚=(12λλ2,i2λiλ2,1) and 𝐱=(rsinθcosϕ,rsinθsinϕ,rcosθ).

§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).

Addition Theorem

14.30.9 𝖯l(cosθ1cosθ2+sinθ1sinθ2cos(ϕ1ϕ2))=4π2l+1m=llYl,m(θ1,ϕ1)¯Yl,m(θ2,ϕ2).

See also (18.18.9) and (34.3.19).

§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 Yl,m(θ,ϕ), defined by (14.30.1), appear in many physical applications. As an example, Laplace’s equation 2W=0 in spherical coordinates (§1.5(ii)):

14.30.10 1ρ2ρ(ρ2Wρ)+1ρ2sinθθ(sinθWθ)+1ρ2sin2θ2Wϕ2=0,

has solutions W(ρ,θ,ϕ)=ρlYl,m(θ,ϕ), which are everywhere one-valued and continuous.

In the quantization of angular momentum the spherical harmonics Yl,m(θ,ϕ) are normalized solutions of the eigenvalue equations

14.30.11 L2Yl,m=2l(l+1)Yl,m,


14.30.11_5 LzYl,m=mYl,m,

where is the reduced Planck’s constant. Here, in spherical coordinates, L2 is the squared angular momentum operator:

14.30.12 L2=2(1sinθθ(sinθθ)+1sin2θ2ϕ2),

and Lz is the z component of the angular momentum operator

14.30.13 Lz=iϕ;

see Edmonds (1974, §2.5).

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