Mathematical Methods for Physics and Engineering - Matematica.NET

6.6 HINTS AND ANSWERS(a) Let R be a real positive number and define K m byK m =∫ R−R(R 2 − x 2) mdx.Show, using integration by parts, that K m satisfies the recurrence relation(2m +1)K m =2mR 2 K m−1 .(b) For integer n, define I n = K n and J n = K n+1/2 . Evaluate I 0 and J 0 directlyand hence prove that(c)I n = 22n+1 (n!) 2 R 2n+1and J n =(2n +1)!A sequence of functions V n (R) is defined byV 0 (R) =1,∫ R (√ )V n (R) = V n−1 R2 − x 2−Rπ(2n +1)!R2n+22 2n+1 n!(n +1)! .dx, n ≥ 1.Prove by induction thatV 2n (R) = πn R 2n, V 2n+1 (R) = πn 2 2n+1 n!R 2n+1.n!(2n +1)!(d) For interest,(i) show that V 2n+2 (1)