*(English)*Zbl 0614.33011

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

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,$\zeta )$, 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.

##### 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.) |