In section 6.5, the book describes using Miller's algorithm to compute integer-order Bessel functions in the region x < n. In describing how to choose the order at which to start the downward recurrence, the text says: "If you play with the asymptotic forms (6.5.3) and (6.5.5), you should be able to convince yourself that the answer is to start larger than the desired n by an addititive amount of order sqrt(constant X n), where the square root on the constant is, very roughly, the number of significant figures of accuracy."

(6.5.3) and (6.5.5) are the approximations J~(x/2)^n/n! for x < n and J~0.45/n^(1/3) for x ~ n. I'm normally pretty good at this sort of thing, but I can't for the life of me derive the asserted relationship. Can someone help me out here?