Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

in (18.3) and (18.4), which we now denote byu 1 (x)andu 2 (x), we may obtain two

linearly-independent series solutions,y 1 (x)andy 2 (x), to the associated equation

by using (18.29). From the general discussion of the convergence of power series

given in section 4.5.1, we see that bothy 1 (x)andy 2 (x) will also converge for

|x|<1. Hence the general solution to (18.28) in this range is given by

y(x)=c 1 y 1 (x)+c 2 y 2 (x).

18.2.1 Associated Legendre functions for integer

Ifandmare both integers, as is the case in many physical applications, then

the general solution to (18.28) is denoted by

y(x)=c 1 Pm(x)+c 2 Qm(x), (18.31)

wherePm(x)andQm(x) are associated Legendre functions of the first and second

kind, respectively. For non-negative values ofm, these functions are related to the

ordinary Legendre functions for integerby

Pm(x)=(1−x^2 )m/^2

dmP

dxm

,Qm(x)=(1−x^2 )m/^2

dmQ

dxm

.

(18.32)

We see immediately that, as required, the associated Legendre functions reduce

to the ordinary Legendre functions whenm= 0. Since it ism^2 that appears in

the associated Legendre equation (18.28), the associated Legendre functions for

negativemvalues must be proportional to the corresponding function for non-

negativem. The constant of proportionality is a matter of convention. For the

Pm(x) it is usual to regard the definition (18.32) as being valid also for negativem

values. Although differentiating a negative number of times is not defined, when

P(x) is expressed in terms of the Rodrigues’ formula (18.9), this problem does

not occur for−≤m≤.§In this case,

P−m(x)=(−1)m

(−m)!

(+m)!

Pm(x). (18.33)

Prove the result (18.33).

From (18.32) and the Rodrigues’ formula (18.9) for the Legendre polynomials, we have

Pm(x)=

1

2 !

(1−x^2 )m/^2

d+m

dx+m

(x^2 −1),

and, without loss of generality, we may assume thatmis non-negative. It is convenient to

§Some authors defineP−m(x)=Pm(x), and similarly for theQm(x), in which casemis replaced by

|m|in the definitions (18.32). It should be noted that, in this case, many of the results presented in

this section also requiremto be replaced by|m|.