Solutions of differential equations of the form , where u is a large parameter and is holomorphic in a simply connected domain which includes the point , can be expanded in terms of Airy functions. The point , is called the turning point. There is a well-developed theory, due to Olver, concerning uniform expansions; that is, expansions that hold uniformly with respect to near the turning point. This expansion is not only valid near but, depending on , the -domain may be rather large. Olver provided bounds on the remainders in the expansion. The present paper gives a different approach to treat the asymptotic problem, although the expansion in the same as Olver’s. The author considers ”slowly varying” functions A(u,, B(u, such that
is an exact solution of the above differential equation, with Ai(z) a solution of Airy’s differential equation . The author gives a detailed analysis on the properties of the functions A(u,, B(u,; these functions are called the coefficient functions, since they provide the coefficients in the asymptotic expansion. Attention is paid to the construction of error bounds, and to a comparing the new results with Olver’s approach. There is an application of the new method to Bessel functions.