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

W. G. C. Boyd
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 410, No. 1838 (Mar. 9, 1987), pp. 35-60
Stable URL: http://www.jstor.org/stable/2398279
Page Count: 26
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## 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 $Ai(u^\frac{2}{3}\zeta e^{\frac{2}{3}\alpha\pi i}) \sum^n_{s=0} \frac{A_s(\zeta)}{u^{2s}} + u^{-2} \frac{d}{d\zeta} Ai(u^\frac{2}{3}\zeta e^{\frac{2}{3}\alpha\pi i}) \sum^{n-1}_{s=0} \frac{B_s(\zeta)}{u^{2s}} + \epsilon^{(\alpha)}_n (u,\zeta)$ 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(u2/3ζ) A (u, ζ) + u-2(d/dζ) Ai (u2/3ζ) B (u, ζ), where Ai denotes any solution of Airy's equation. The coefficient 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 u2, with explicit error bounds. We apply our theory to Bessel functions.

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