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 secondorder 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^{n1}_{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^{2/3}zeta) 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.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 March 1987
 DOI:
 10.1098/rspa.1987.0027
 Bibcode:
 1987RSPSA.410...35B