Mathematical Methods for Physics and Engineering - Matematica.NET

18.1 LEGENDRE FUNCTIONSEquation (18.16) can then be written, using (18.15), ash ∑ P n h n =(1− 2xh + h 2 ) ∑ P nh ′ n ,and equating the coefficients of h n+1 we obtain the recurrence relationP n = P n+1 ′ − 2xP n ′ + P n−1. ′ (18.18)Equations (18.16) and (18.17) can be combined as(x − h) ∑ P nh ′ n = h ∑ nP n h n−1 ,from which the coefficent of h n yields a second recurrence relation,xP n ′ − P n−1 ′ = nP n ; (18.19)eliminating P n−1 ′ between (18.18) and (18.19) then gives the further result(n +1)P n = P n+1 ′ − xP n. ′ (18.20)If we now take the result (18.20) with n replaced by n − 1andaddx times (18.19) to itwe obtain(1 − x 2 )P n ′ = n(P n−1 − xP n ). (18.21)Finally, differentiating both sides with respect to x and using (18.19) again, we find(1 − x 2 )P n ′′ − 2xP n ′ = n[(P n−1 ′ − xP n) ′ − P n ]= n(−nP n − P n )=−n(n +1)P n ,andsotheP n defined by (18.15) do indeed satisfy Legendre’s equation. ◭The above example shows that the functions P n (x) defined by (18.15) satisfyLegendre’s equation with l = n (an integer) and, also from (18.15), these functionsare regular at x = ±1. Thus P n must be some multiple of the nth Legendrepolynomial. It therefore remains only to verify the normalisation. This is easilydone at x =1,whenG becomesG(1,h)=[(1− h) 2 ] −1/2 =1+h + h 2 + ··· ,and we can see that all the P n so defined have P n (1) = 1 as required, and are thusidentical to the Legendre polynomials.A particular use of the generating function (18.15) is in representing the inversedistance between two points in three-dimensional space in terms of Legendrepolynomials. If two points r and r ′ are at distances r and r ′ , respectively, fromthe origin, with r ′ r,however,585