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Zbl 0614.33011
Boyd, W.G.C.
Asymptotic expansions for the coefficient functions that arise in turning-point problems. (English)
[J] Proc. R. Soc. Lond., Ser. A 410, 35-60 (1987). ISSN 0080-4630

Solutions of differential equations of the form $w''(\zeta)=[u\sp 2\zeta +\Psi (\zeta)]w(\zeta)$, where u is a large parameter and $\Psi$ is holomorphic in a simply connected domain $\Delta$ which includes the point $\zeta =0$, can be expanded in terms of Airy functions. The point $\zeta =0$, is called the turning point. There is a well-developed theory, due to Olver, concerning uniform expansions; that is, expansions that hold uniformly with respect to $\zeta$ near the turning point. This expansion is not only valid near $\zeta =0$ but, depending on $\Psi$, the $\zeta$-domain may be rather large. Olver provided bounds on the remainders in the expansion. The present paper gives a different approach to treat the asymptotic problem, although the expansion in the same as Olver's. The author considers "slowly varying" functions A(u,$\zeta)$, B(u,$\zeta)$ such that $$ Ai(u\sp{2/3}\zeta)A(u,\zeta)+u\sp{- 4/3}Ai'(u\sp{2/3}\zeta)B(u,\zeta) $$ is an exact solution of the above differential equation, with Ai(z) a solution of Airy's differential equation $y''(z)=zy(z)$. The author gives a detailed analysis on the properties of the functions A(u,$\zeta)$, B(u,$\zeta)$; these functions are called the coefficient functions, since they provide the coefficients in the asymptotic expansion. Attention is paid to the construction of error bounds, and to a comparing the new results with Olver's approach. There is an application of the new method to Bessel functions.
[N.M.Temme]

MSC 2000:: *33C10 Cylinder functions, etc.
34E05 Asymptotic expansions (ODE)
30E10 Approximation in the complex domain
41A60 Asymptotic problems in approximation

Keywords: asymptotic expansions of differential equations; Airy-type expansions; Airy functions; turning point; error bounds; Bessel functions

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