Asymptotic expansions for the coefficient functions that arise in turning-point problems

Abstract

We study the uniform asymptotic expansion for a large parameter u of solutions of second-order linear differential equations in domains of the complex plane which include a turning point. The standard theory of Olver demonstrates the existence of three such expansions, corresponding to solutions of the form Ai ( u 2/3 ζ e 2/3απi ) Ʃ n s = 0 A s (ζ)/ u 2 s + u -2 d / d ζ Ai ( u 2/3 ζ e 2/3απi ) Ʃ n -1 s =0 B s (ζ)/u 2 s + ϵ ( α ) n ( u , ζ) for α = 0, 1, 2, with bounds on ϵ ( α ) n . We proceed differently, by showing that the set of all solutions of the differential equation is of the form Ai ( u 2/3 ζ) A ( u , ζ) + u -2 (d/dζ) Ai ( u 2/3 ζ) B ( u , ζ), where Ai denotes any situation of Airy's equation. The coefficent functions A ( u , ζ) and B ( u , ζ) are the focus of our attention : we show that for sufficiently large u they are holomorphic functions of ζ in a domain including the turning point, and as functions of u are described asymptotically by descending power series in u 2 , with explicit error bounds. We apply our theory to Bessel functions.