Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.1 LEGENDRE FUNCTIONS

Mutual orthogonality

In section 17.4, we noted that Legendre’s equation was of Sturm–Liouville form

withp=1−x^2 ,q=0,λ=(+1) andρ= 1, and that its natural interval

was [− 1 ,1]. Since the Legendre polynomialsP(x) are regular at the end-points

x=±1, they must be mutually orthogonal over this interval, i.e.
∫ 1

− 1

P(x)Pk(x)dx=0 if=k. (18.12)

Although this result follows from the general considerations of the previous

chapter, it may also be proved directly, as shown in the following example.

Prove directly that the Legendre polynomialsP(x)are mutually orthogonal over the

interval− 1 <x< 1.

Since theP(x) satisfy Legendre’s equation we may write
[
(1−x^2 )P′

]′

+(+1)P=0,

whereP′=dP/dx. Multiplying through byPkand integrating fromx=−1tox=1,we
obtain ∫
1

− 1

Pk

[

(1−x^2 )P′

]′

dx+

∫ 1

− 1

Pk(+1)Pdx=0.

Integrating the first term by parts and noting that the boundary contribution vanishes at
both limits because of the factor 1−x^2 , we find

−

∫ 1

− 1

Pk′(1−x^2 )P′dx+

∫ 1

− 1

Pk(+1)Pdx=0.

Now, if we reverse the roles ofandkand subtract one expression from the other, we
conclude that

[k(k+1)−(+1)]

∫ 1

− 1

PkPdx=0,

and therefore, sincek=, we must have the result (18.12). As a particular case, we note
that if we putk=0weobtain
∫ 1

− 1

P(x)dx=0 for=0.

As we discussed in the previous chapter, the mutual orthogonality (and com-

pleteness) of theP(x) means that any reasonable functionf(x) (i.e. one obeying

the Dirichlet conditions discussed at the start of chapter 12) can be expressed in

the interval|x|<1 as an infinite sum of Legendre polynomials,

f(x)=

∑∞

=0

aP(x), (18.13)

where the coefficientsaare given by

a=

2 +1

2

∫ 1

− 1

f(x)P(x)dx. (18.14)