Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONS◮Prove the expression (18.14) for the coefficients in the Legendre polynomial expansionof a function f(x).If we multiply (18.13) by P k (x) and integrate from x = −1 tox = 1 then we obtain∫ 1∞∑∫ 1P k (x)f(x) dx = a l P k (x)P l (x) dx−1l=0∫ 1= a k P k (x)P k (x) dx =2a k−12k +1 ,where we have used the orthogonality property (18.12) and the normalisation property(18.11). ◭Generating functionA useful device for manipulating and studying sequences of functions or quantitieslabelled by an integer variable (here, the Legendre polynomials P l (x) labelled byl) isagenerating function. The generating function has perhaps its greatest utilityin the area of probability theory (see chapter 30). However, it is also a greatconvenience in our present study.The generating function for, say, a series of functions f n (x) forn =0, 1, 2,... isa function G(x, h) containing, as well as x, a dummy variable h such that∞∑G(x, h) = f n (x)h n ,n=0i.e. f n (x) is the coefficient of h n in the expansion of G in powers of h. The utilityof the device lies in the fact that sometimes it is possible to find a closed formfor G(x, h).For our study of Legendre polynomials let us consider the functions P n (x)defined by the equation∞∑G(x, h) =(1− 2xh + h 2 ) −1/2 = P n (x)h n . (18.15)As we show below, the functions so defined are identical to the Legendre polynomialsand the function (1 − 2xh + h 2 ) −1/2 is in fact the generating function forthem. In the process we will also deduce several useful relationships between thevarious polynomials and their derivatives.◮Show that the functions P n (x) defined by (18.15) satisfy Legendre’s equationIn the following dP n (x)/dx will be denoted by P n. ′ Firstly, we differentiate the definingequation (18.15) with respect to x and geth(1 − 2xh + h 2 ) −3/2 = ∑ P nh ′ n . (18.16)Also, we differentiate (18.15) with respect to h to yield(x − h)(1 − 2xh + h 2 ) −3/2 = ∑ nP n h n−1 . (18.17)584−1n=0