1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

5.9 Spherical Coordinates; Legendre Polynomials 339

Since the differential equation (5) is easily put into self-adjoint form,

(

( 1 −x^2 )y′

)′

+μ^2 y= 0 , − 1 <x< 1 ,

it is routine to show that the Legendre polynomials satisfy the orthogonality
relation
∫ 1

− 1

Pn(x)Pm(x)dx= 0 , n=m.

By direct calculation, it can be shown that

∫ 1

− 1

P^2 n(x)dx=

2

2 n+ 1. (6)

A compact way of representing the Legendre polynomials is by means of Ro-
drigues’ formula,

Pn(x)=

1

n!2n

dn

dxn

[

(x^2 − 1 )n

]

. (7)

Elementary algebra and calculus show that thenth derivative of(x^2 − 1 )nis
apolynomialofdegreen. Substituting this polynomial into the differential
equation (5), withμ^2 =n(n+ 1 ), shows that it is a solution — bounded, of
course. Therefore, it is a multiple of the Legendre polynomialPn(x).Through
Rodrigues’ formula or otherwise, it is possible to prove the following two for-
mulas, which relate three consecutive Legendre polynomials:

( 2 n+ 1 )Pn(x)=Pn′+ 1 (x)−Pn′− 1 (x), (8)

(n+ 1 )Pn+ 1 (x)+nPn− 1 (x)=( 2 n+ 1 )xPn(x). (9)

In order to use Legendre polynomials in boundary value problems, we need
to be able to express a given functionf(x)in the form of a Legendre series,

f(x)=

∑∞

n= 0

bnPn(x), − 1 <x< 1.

From the orthogonality relation and the integral, Eq. (6), it follows that the
coefficient in the series must be

bn=^2 n+^1

2

∫ 1

− 1

f(x)Pn(x)dx. (10)

The convergence theorem for Legendre series is analogous to the one for
Fourier series in Chapter 1.