Let and be positive real numbers. Then Jp(x) and Jp(x) satisfy d dx x d dx [Jp(x)] + 2x

Chapter 11, Problem 20

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Let and be positive real numbers. Then Jp(x) and Jp(x) satisfy d dx x d dx [Jp(x)] + 2x p2 x Jp(x) = 0, (11.6.37) d dx x d dx [Jp(x)] + 2x p2 x Jp(x) = 0, (11.6.38) respectively. (a) Show that for = , 1 0 x Jp(x)Jp(x)dx = Jp()J p() Jp()J p() 2 2 . (11.6.39) [Hint: Multiply (11.6.37) by Jp(x), (11.6.38) by Jp(x), subtract the resulting equations and integrate over (0, 1).] If and are distinct zeros of Jp(x), what does your result imply? (b) In order to compute 1 0 x[Jp(x)] 2dx, we take the limit as in (11.6.39). Use LHopitals rule to compute this limit and thereby show that 1 0 x[Jp(x)] 2dx = [J p()] 2 Jp()J p() Jp()J p () 2 . (11.6.40)Substituting from Bessels equation for Jp (),show that (11.6.40) can be written as 10x[Jp(x)]2dx= 12[Jp()]2 +1 p22[Jp()]2.(c) In the case when is a zero of Jp(x), use(11.6.26) to show that your result in (b) can bewritten as 10x[Jp(x)]2dx = 12[Jp+1()]2.1