Advanced Calculus fi..

I516 Advanced Calculus, Fifth EditionThe Laguerrepolynornials. For each cr > - 1 the system of polynomials (L~'(x)]is defined by the equations:They are orthogonal over the injinite interval 0 5 x < oo with respect to the weightfunction xffe-I.For the significance of the infinite interval and further information on thesefunctions we refer to the book of Jackson listed at the end of the chapter.All of these functions arise in a natural way in physical problems. This is illustratedin Chapter 10, where it will be shown that each of a large class of physicalproblems automatically provides a complete system of orthogonal functions.PROBLEMS1. Graph Po(x), Pl(x), . . . , P~(x).2. Prove the following parts of Theorem 17:(i) part (c), (ii) part (d), (iii) part (e). Each of these can be deduced from (7.63) alone.3. Prove part (f) of Theorem 17 by induction, with the aid of the recursion formula (e).4. Prove part (g) of Theorem 17 by squaring (a) and (b) of (7.63) and eliminating the termsin P,-, from the equations obtained.5. Prove part (h) of Theorem 17. [Hint: Show by induction that (g) gives the chain ofincquahtles:I6. Prove part (i) of Theorem 17 as a consequence of part (h).7. Prove the orthogonality condition (j). [Hint: For every very smooth function f (x) for- 15 x 5 1, let f *(x) = [(I- x2) f '(x)]'. Prove by integration by parts that (f*, g) -(f, g*) = 0. Take f = P,,(x), g = Pm(x) and use the differential equation (d) to replacef * by -n(n + 1) f and g* by -m(m + l)g, and conclude that (f, g)[m(m + 1) -n(n + I)] = 0.18. Prove part (k) of Theorem 17. [Hint: Use the recursion formula (e) to express Pn interms of and and use the orthogonality condition to show that (P,, Pn) =[(2n- l)/n](x P,, Pn-,). Apply the recursion formula to x PI, and use orthogonality toshow that (P,. PI,) = P,,-1)[(2n - 1)/(2n + I)]. Now prove by induction that(Pn. Pn)=2/(2n+ 11.19. Prove part (1) of Theorem 17 by induction, using the fact that P,,(x) is a polynomial ofnth degree.10. (a). . . (p). Prove the corresponding parts of Theorem 22 with the aid of the following1 hints: For (a) and (b), use (7.72). For (c) and (d), use (a) and (b). For (e), use (a) through(d). For (f), use (e). For (g), let u(x) = fiJ,,(crx), v(x) = &Jnl(px). Take c = cr andIII1then c = p in (f) to obtain equations u"(x) + . . . = 0, vU(x) + . . . = 0. Multiply the firstequation by -v, multiply the second by u, and add. Then integrate the result from b to 1,noting that uv" - uu" = (uv'- vu')'. Replace u and v by thcir expressions in terms of Jmand finally let b -+ 0 +. For (h), use (g) and (d). For (i), divide both sides of (h) by cr -,6and then, with cr fixed, let P -+ a. For (j), note that if Jm(a) = 0 and JA(cr) = 0, then by(g) [d x Jm(ax) Jm(~x)dx = 0 for all p #a. Let ,6 + cr to conclude that J,,,(crx) = 0