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Asymptotic expansions for the coefficient functions that arise in turning-point problems. (English) Zbl 0614.33011
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.
Reviewer: N.M.Temme
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
34E05Asymptotic expansions (ODE)
30E10Approximation in the complex domain
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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