ABSTRACT
Theorem 3.1 may also furnish uniform asymptotic representations when a parameter enters the differential equation in other ways. A simple illustration is provided by Exercise 3.2 on letting the parameter K tend to zero. '4.3 As an interesting application, consider the equation
obtained from the Whittaker equation (11.01) of Chapter 7 on replacing k by ik and z by iz. We indicate briefly how to obtain asymptotic solutions for positive real values of k and z, and large positive m. Making the substitutions x = -z/mt and 1 = k/m, we obtain
d W/dx2 = {m2f (x) + g (x)} W, where
Application of Theorem 3.1 produces solutions of the form
(4.03) where C is defined by
according as x E [x,, 0) or x E (- a , x,]. The error terms are bounded by (3.10) and (3.1 l), with u = m, a = - a , b = 0, and H given by (3.08).
