Mathematical Methods for Physics and Engineering - Matematica.NET

18.4 CHEBYSHEV FUNCTIONS1T 0T 2T 30.5T 1−1−0.50.51−0.5−1Figure 18.3The first four Chebyshev polynomials of the first kind.From (18.56) and (18.57), we see immediately that U n (x) isapolynomial of ordern, but that W n (x) isnot a polynomial. In practice, it is usual to work entirely interms of T n (x)andU n (x), which are known, respectively, as Chebyshev polynomialsof the first and second kind. In particular, we note that the general solution toChebyshev’s equation can be written in terms of these polynomials as⎧√⎨c 1 T n (x)+c 2 1 − x2 U n−1 (x) for n =1, 2, 3,...,y(x) =⎩c 1 + c 2 sin −1 x for n =0.The n = 0 solution could also be written as d 1 + c 2 cos −1 x with d 1 = c 1 + 1 2 πc 2.The first few Chebyshev polynomials of the first kind are easily constructedand are given byT 0 (x) =1,T 1 (x) =x,T 2 (x) =2x 2 − 1, T 3 (x) =4x 3 − 3x,T 4 (x) =8x 4 − 8x 2 +1,T 5 (x) =16x 5 − 20x 3 +5x.The functions T 0 (x), T 1 (x), T 2 (x) andT 3 (x) are plotted in figure 18.3. In general,the Chebyshev polynomials T n (x) satisfy T n (−x) =(−1) n T n (x), which is easilydeduced from (18.56). Similarly, it is straightforward to deduce the following597