Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONSin (18.3) and (18.4), which we now denote by u 1 (x) andu 2 (x), we may obtain twolinearly-independent series solutions, y 1 (x) andy 2 (x), to the associated equationby using (18.29). From the general discussion of the convergence of power seriesgiven in section 4.5.1, we see that both y 1 (x) andy 2 (x) will also converge for|x| < 1. Hence the general solution to (18.28) in this range is given byy(x) =c 1 y 1 (x)+c 2 y 2 (x).18.2.1 Associated Legendre functions for integer lIf l and m are both integers, as is the case in many physical applications, thenthe general solution to (18.28) is denoted byy(x) =c 1 P m l (x)+c 2 Q m l (x), (18.31)where Pl m(x) andQm l (x) are associated Legendre functions of the first and secondkind, respectively. For non-negative values of m, these functions are related to theordinary Legendre functions for integer l byP m l (x) =(1− x 2 ) m/2 dm P ldx m ,Qm l (x) =(1− x 2 ) m/2 dm Q ldx m . (18.32)We see immediately that, as required, the associated Legendre functions reduceto the ordinary Legendre functions when m = 0. Since it is m 2 that appears inthe associated Legendre equation (18.28), the associated Legendre functions fornegative m values must be proportional to the corresponding function for nonnegativem. The constant of proportionality is a matter of convention. For thePl m (x) it is usual to regard the definition (18.32) as being valid also for negative mvalues. Although differentiating a negative number of times is not defined, whenP l (x) is expressed in terms of the Rodrigues’ formula (18.9), this problem doesnot occur for −l ≤ m ≤ l. § In this case,Pl−mm (l − m)!(x) =(−1)(l + m)! P l m (x). (18.33)◮Prove the result (18.33).From (18.32) and the Rodrigues’ formula (18.9) for the Legendre polynomials, we havePl m (x) = 12 l l! (1 − x2 m/2 dl+m)dx l+m (x2 − 1) l ,and, without loss of generality, we may assume that m is non-negative. It is convenient to§ Some authors define Pl−m (x) =Pl m(x), and similarly for the Qm l (x), in which case m is replaced by|m| in the definitions (18.32). It should be noted that, in this case, many of the results presented inthis section also require m to be replaced by |m|.588