ABSTRACT
Because the approximation for Mk,,(x) is uniformly valid only in compact x intervals, the implied constant in the 0 term in (7.05) grows with n. More powerful approximations for the zeros could be constructed from an expansion for Mk,,(4kx) analogous to that of Exercise 4.6. Ex. 7.1 Let n be restricted by y...,, < ulalV2. Show that for all sufficiently large u, uniformly with respect to n, there is exactly one zero of W,,,(u,C) in the interval (7.04). Ex. 7.2 (Second approximarions ro zeros) Assume the notation of Theorem 4.1 and also that or =-a and the conditions of Exercise 4.2 are satisfied. Show that when n 1 the rth negative zeros of WZ. + 3 (u, 0 and Wznt I .~ (u , 0are given by
8.1 We now consider the extension of the approximation theorems of 553 and 4 to complex values of the variable [and parameter u. Although extensions to complex v are feasible, we shall continue to suppose that v is real and nonnegative, a condition commonly satisfied in applications. Since Iv(z) and Kv(z) are expressible in terms of Jv(iz) and Yv(iz), and vice versa, either modified or unmodified Bessel functions may be used as approximants when z is complex. But because Jv(z) and Yv(z) do not comprise a numerically satisfactory pair when z is large and complex, we shall work in terms of the modified functions. We begin with a restatement of relevant properties of Iv(z) and Kv(z) established in Chapters 2 and 7 .
