Asymptotic Expansions for the Coefficient Functions that Arise in Turning- Point Problems

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^2/3zeta e^{2/3απ i} ) sum^n_s=0 A_s(zeta)/u^2s + u^-2 d/dzeta Ai(u^2/3zeta e^{2/3απ i}) sum^n-1_s=0 B_s(zeta)/u^2s + ɛ^(α)_n (u,zeta) for α = 0, 1, 2, with bounds on ɛ^(α)_n. We proceed differently, by showing that the set of all solutions of the differential equation is of the form Ai(u^2/3zeta) A (u, zeta) + u^-2(d/dzeta) Ai (u^2/3zeta) B (u, zeta), where Ai denotes any solution of Airy's equation. The coefficient functions A (u, zeta) and B (u, zeta) are the focus of our attention: we show that for sufficiently large u they are holomorphic functions of zeta 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.