Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.3 SPHERICAL HARMONICS

be derived in a number of ways, such as using the generating function (18.40) or

by differentiation of the recurrence relations for the Legendre polynomialsP(x).

Use the recurrence relation(2n+1)Pn=Pn′+1−Pn′− 1 for Legendre polynomials to derive

the result (18.43).

Differentiating the recurrence relation for the Legendre polynomialsmtimes, we have

(2n+1)

dmPn

dxm

=

dm+1Pn+1

dxm+1

−

dm+1Pn− 1

dxm+1

.

Multiplying through by (1−x^2 )(m+1)/^2 and using the definition (18.32) immediately gives
the result (18.43).

18.3 Spherical harmonics

The associated Legendre functions discussed in the previous section occur most

commonly when obtaining solutions in spherical polar coordinates of Laplace’s

equation∇^2 u= 0 (see section 21.3.1). In particular, one finds that, for solutions

that are finite on the polar axis, the angular part of the solution is given by

Θ(θ)Φ(φ)=Pm(cosθ)(Ccosmφ+Dsinmφ),

whereandmare integers with−≤m≤. This general form is sufficiently

common that particular functions ofθandφcalledspherical harmonicsare

defined and tabulated. The spherical harmonicsYm(θ, φ) are defined by

Ym(θ, φ)=(−1)m

[

2 +1

4 π

(−m)!

(+m)!

] 1 / 2

Pm(cosθ) exp(imφ). (18.45)

Using (18.33), we note that

Y−m(θ, φ)=(−1)m

[

Ym(θ, φ)

]∗

,

where the asterisk denotes complex conjugation. The first few spherical harmonics

Ym(θ, φ)≡Ymare as follows:

Y 00 =

√

1

4 π,Y

0

1 =

√

3

4 πcosθ,

Y 1 ±^1 = ∓

√

3

8 πsinθexp(±iφ),Y

0

2 =

√

5

16 π(3 cos

(^2) θ−1),
Y 2 ±^1 = ∓
√
15
8 πsinθcosθexp(±iφ),Y
± 2
2 =
√
15
32 πsin
(^2) θexp(± 2 iφ).
Since they contain as theirθ-dependent part the solutionPmto the associated
Legendre equation, theYmare mutually orthogonal when integrated from−1to
+1 overd(cosθ). Their mutual orthogonality with respect toφ(0≤φ≤ 2 π)is
even more obvious. The numerical factor in (18.45) is chosen to make theYman