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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. 3560
Published by: Royal Society
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 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 $Ai(u^\frac{2}{3}\zeta e^{\frac{2}{3}\alpha\pi i}) \sum^n_{s=0} \frac{A_s(\zeta)}{u^{2s}} + u^{2} \frac{d}{d\zeta} Ai(u^\frac{2}{3}\zeta e^{\frac{2}{3}\alpha\pi i}) \sum^{n1}_{s=0} \frac{B_s(\zeta)}{u^{2s}} + \epsilon^{(\alpha)}_n (u,\zeta)$ 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) + u2(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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Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences © 1987 Royal Society