Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONSwhich on collecting terms gives∞∑{(n +2)(n +1)a n+2 − [n(n +1)− l(l +1)]a n } x n =0.n=0The recurrence relation is therefore[n(n +1)− l(l +1)]a n+2 = a n , (18.2)(n +1)(n +2)for n =0, 1, 2,... . If we choose a 0 = 1 and a 1 = 0 then we obtain the solutiony 1 (x) =1− l(l +1) x2+(l − 2)l(l +1)(l +3)x4 − ··· , (18.3)2! 4!whereas on choosing a 0 = 0 and a 1 = 1 we find a second solutiony 2 (x) =x − (l − 1)(l +2) x3+(l − 3)(l − 1)(l +2)(l +4)x5 − ··· . (18.4)3! 5!By applying the ratio test to these series (see subsection 4.3.2), we find that bothseries converge for |x| < 1, and so their radius of convergence is unity, which(as expected) is the distance to the nearest singular point of the equation. Since(18.3) contains only even powers of x and (18.4) contains only odd powers, thesetwo solutions cannot be proportional to one another, and are therefore linearlyindependent. Hence, the general solution to (18.1) for |x| < 1isy(x) =c 1 y 1 (x)+c 2 y 2 (x).18.1.1 Legendre functions for integer lIn many physical applications the parameter l in Legendre’s equation (18.1) isan integer, i.e. l =0, 1, 2,... . In this case, the recurrence relation (18.2) gives[l(l +1)− l(l +1)]a l+2 = a l =0,(l +1)(l +2)i.e. the series terminates and we obtain a polynomial solution of order l. Inparticular, if l is even, then y 1 (x) in (18.3) reduces to a polynomial, whereas if l isoddthesameistrueofy 2 (x) in (18.4). These solutions (suitably normalised) arecalled the Legendre polynomials of order l; they are written P l (x) and are validfor all finite x. It is conventional to normalise P l (x) in such a way that P l (1) = 1,and as a consequence P l (−1) = (−1) l . The first few Legendre polynomials areeasily constructed and are given byP 0 (x) =1,P 2 (x) = 1 2 (3x2 − 1),P 4 (x) = 1 8 (35x4 − 30x 2 +3),P 1 (x) =x,P 3 (x) = 1 2 (5x3 − 3x),P 5 (x) = 1 8 (63x5 − 70x 3 +15x).578