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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 ${w}^{\text{'}\text{'}}\left(\zeta \right)=\left[{u}^{2}\zeta +{\Psi }\left(\zeta \right)\right]w\left(\zeta \right)$, 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,$\zeta \right)$, B(u,$\zeta \right)$ such that

$Ai\left({u}^{2/3}\zeta \right)A\left(u,\zeta \right)+{u}^{-4/3}A{i}^{\text{'}}\left({u}^{2/3}\zeta \right)B\left(u,\zeta \right)$

is an exact solution of the above differential equation, with Ai(z) a solution of Airy’s differential equation ${y}^{\text{'}\text{'}}\left(z\right)=zy\left(z\right)$. The author gives a detailed analysis on the properties of the functions A(u,$\zeta \right)$, B(u,$\zeta \right)$; 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
##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 34E05 Asymptotic expansions (ODE) 30E10 Approximation in the complex domain 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)