Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.2 ASSOCIATED LEGENDRE FUNCTIONS

Sinced^2 (1−x^2 )/dx^2 =(−1)(2)!, and noting that (−1)^2 +2m= 1, we have

Im=

1

22 (!)^2

(2)!(+m)!

(−m)!

∫ 1

− 1

(1−x^2 )dx.

We have already shown in section 18.1.2 that

K≡

∫ 1

− 1

(1−x^2 )dx=

22 +1(!)^2

(2+1)!

,

and so we obtain the final result

Im=

2

2 +1

(+m)!

(−m)!

.

The orthogonality and normalisation conditions, (18.36) and (18.37) respec-

tively, mean that the associated Legendre functionsPm(x), withmfixed, may be

used in a similar way to the Legendre polynomials to expand any reasonable

functionf(x) on the interval|x|<1 in a series of the form

f(x)=

∑∞

k=0

am+kPmm+k(x), (18.38)

where, in this case, the coefficients are given by

a=

2 +1

2

(−m)!

(+m)!

∫ 1

− 1

f(x)Pm(x)dx.

We note that the series takes the form (18.38) becausePm(x)=0form>.

Finally, it is worth noting that the associated Legendre functionsPm(x) must

also obey a second orthogonality relationship. This has to be so because one may

equally well write the associated Legendre equation (18.28) in Sturm–Liouville

form (py)′+qy+λρy= 0, withp=1−x^2 ,q=(+1),λ=−m^2 andρ=(1−x^2 )−^1 ;

once again the natural interval is [− 1 ,1]. Since the associated Legendre functions

Pm(x) are regular at the end-pointsx=±1, they must therefore be mutually

orthogonal with respect to the weight function (1−x^2 )−^1 over this interval for a

fixed value of,i.e.

∫ 1

− 1

Pm(x)Pk(x)(1−x^2 )−^1 dx=0 if|m|=|k|. (18.39)

One may also show straightforwardly that the corresponding normalisation con-

dition whenm=kis given by

∫ 1

− 1

Pm(x)Pm(x)(1−x^2 )−^1 dx=

(+m)!

m(−m)!

.

In solving physical problems, however, the orthogonality condition (18.39) is not

of any practical use.