Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

orthonormal set, i.e.

∫ 1

− 1

∫ 2 π

0

[

Ym(θ, φ)

]∗

Ym

′

′(θ, φ)dφ d(cosθ)=δ′δmm′. (18.46)

In addition, the spherical harmonics form a complete set in that any reasonable

function (i.e. one that is likely to be met in a physical situation) ofθandφcan

be expanded as a sum of such functions,

f(θ, φ)=

∑∞

=0

∑

m=−

amYm(θ, φ), (18.47)

the constantsambeing given by

am=

∫ 1

− 1

∫ 2 π

0

[

Ym(θ, φ)

]∗

f(θ, φ)dφ d(cosθ). (18.48)

This is in exact analogy with a Fourier series and is a particular example of the

general property of Sturm–Liouville solutions.

Aside from the orthonormality condition (18.46), the most important relation-

ship obeyed by theYmis thespherical harmonic addition theorem. This reads

P(cosγ)=

4 π

2 +1

∑

m=−

Ym(θ, φ)[Ym(θ′,φ′)]∗, (18.49)

where (θ, φ)and(θ′,φ′) denote two different directions in our spherical polar coor-

dinate system that are separated by an angleγ. In general, spherical trigonometry

(or vector methods) shows that these angles obey the identity

cosγ=cosθcosθ′+sinθsinθ′cos(φ−φ′). (18.50)

Prove the spherical harmonic addition theorem (18.49).

For the sake of brevity, it will be useful to denote the directions (θ, φ)and(θ′,φ′)by
ΩandΩ′, respectively. We will also denote the element of solid angle on the sphere by
dΩ=dφ d(cosθ). We begin by deriving the form of the closure relationship obeyed by the
spherical harmonics. Using (18.47) and (18.48), and reversing the order of the summation
and integration, we may write

f(Ω) =

∫

4 π

dΩ′f(Ω′)

∑

m

Ym∗(Ω′)Ym(Ω),

where

∑

mis a convenient shorthand for the double summation in (18.47). Thus we may
write the closure relationship for the spherical harmonics as
∑

m

Ym(Ω)Ym∗(Ω′)=δ(Ω−Ω′), (18.51)

whereδ(Ω−Ω′) is a Dirac delta function with the properties thatδ(Ω−Ω′)=0ifΩ=Ω′
and

∫

4 πδ(Ω)dΩ=1.