Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.2 ASSOCIATED LEGENDRE FUNCTIONS

write (x^2 −1) = (x+1)(x−1) and use Leibnitz’ theorem to evaluate the derivative, which

yields

Pm(x)=

1

2 !

(1−x^2 )m/^2

∑+m

r=0

(+m)!

r!(+m−r)!

dr(x+1)

dxr

d+m−r(x−1)

dx+m−r

.

Considering the two derivative factors in a term in the summation, we note that the first

is non-zero only forr≤and the second is non-zero for+m−r≤. Combining these

conditions yieldsm≤r≤. Performing the derivatives, we thus obtain

Pm(x)=

1

2 !

(1−x^2 )m/^2

∑

r=m

(+m)!

r!(+m−r)!

!(x+1)−r

(−r)!

!(x−1)r−m

(r−m)!

=(−1)m/^2

!(+m)!

2 

∑

r=m

(x+1)−r+

m 2

(x−1)r−

m 2

r!(+m−r)!(−r)!(r−m)!

. (18.34)

Repeating the above calculation forP−m(x) and identifying once more those terms in

the sum that are non-zero, we find

P−m(x)=(−1)−m/^2

!(−m)!

2 

∑−m

r=0

(x+1)−r−

m 2

(x−1)r+

m 2

r!(−m−r)!(−r)!(r+m)!

=(−1)−m/^2

!(−m)!

2 

∑

̄r=m

(x+1)− ̄r+

m 2

(x−1) ̄r−

m 2

( ̄r−m)!(− ̄r)!(+m− ̄r)! ̄r!

, (18.35)

where, in the second equality, we have rewritten the summation in terms of the new index
̄r=r+m. Comparing (18.34) and (18.35), we immediately arrive at the required result
(18.33).

SinceP(x) is a polynomial of order, we havePm(x)=0for|m|>.From

its definition, it is clear thatPm(x) is also a polynomial of orderifmis even,

but contains the factor (1−x^2 ) to a fractional power ifmis odd. In either case,

Pm(x) is regular atx=±1. The first few associated Legendre functions of the

first kind are easily constructed and are given by (omitting them= 0 cases)

P 11 (x)=(1−x^2 )^1 /^2 , P 21 (x)=3x(1−x^2 )^1 /^2 ,

P 22 (x)=3(1−x^2 ), P 31 (x)=^32 (5x^2 −1)(1−x^2 )^1 /^2 ,

P 32 (x)=15x(1−x^2 ), P 33 (x) = 15(1−x^2 )^3 /^2.

Finally, we note that the associated Legendre functions of the second kindQm(x),

likeQ(x), are singular atx=±1.

18.2.2 Properties of associated Legendre functionsPm(x)

When encountered in physical problems, the variablexin the associated Legendre

equation (as in the ordinary Legendre equation) is usually the cosine of the polar

angleθin spherical polar coordinates, and we then require the solutiony(x)to

be regular atx=±1 (corresponding toθ=0orθ=π). For this to occur, we

requireto be an integer and the coefficientc 2 of the functionQm(x) in (18.31)