Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONSUsing (18.65) and setting x =cosθ immediately gives a rearrangement of the requiredresult (18.69). Similarly, adding the plus and minus cases of result (18.68) givessin(n +1)θ +sin(n − 1)θ =2sinnθ cos θ.Dividing through on both sides by sin θ and using (18.66) yields (18.70). ◭The recurrence relations (18.69) and (18.70) are extremely useful in the practicalcomputation of Chebyshev polynomials. For example, given the values of T 0 (x)and T 1 (x) at some point x, the result (18.69) may be used iteratively to obtainthe value of any T n (x) at that point; similarly, (18.70) may be used to calculatethe value of any U n (x) at some point x, given the values of U 0 (x) andU 1 (x) atthat point.Further recurrence relations satisfied by the Chebyshev polynomials areT n (x) =U n (x) − xU n−1 (x), (18.71)(1 − x 2 )U n (x) =xT n+1 (x) − T n+2 (x), (18.72)which establish useful relationships between the two sets of polynomials T n (x)and U n (x). The relation (18.71) follows immediately from (18.68), whereas (18.72)follows from (18.67), with n replaced by n + 1, on noting that sin 2 θ =1− x 2 .Additional useful results concerning the derivatives of Chebyshev polynomialsmay be obtained from (18.65) and (18.66), as illustrated in the following example.◮Show thatT n(x) ′ =nU n−1 (x),(1 − x 2 )U n(x) ′ =xU n (x) − (n +1)T n+1 (x).These results are most easily derived from the expressions (18.65) and (18.66) by notingthat d/dx =(−1/ sin θ) d/dθ. Thus,T n(x) ′ =− 1 d(cos nθ) n sin nθ= = nU n−1 (x).sin θ dθ sin θSimilarly, we findU n(x) ′ =− 1 [d sin(n +1)θsin θ dθ sin θwhich rearranges immediately to yield the stated result. ◭]sin(n +1)θ cos θ (n +1)cos(n +1)θ= −sin 3 θsin 2 θ= xU n(x)1 − x − (n +1)T n+1(x),2 1 − x 2Bessel’s equation has the form18.5 Bessel functionsx 2 y ′′ + xy ′ +(x 2 − ν 2 )y =0, (18.73)which has a regular singular point at x = 0 and an essential singularity at x = ∞.The parameter ν is a given number, which we may take as ≥ 0 with no loss of602