On the reversion of an asymptotic expansion and the zeros of the Airy functions.

*(English)*Zbl 1053.33003Summary: The general theories of the derivation of inverses of functions from their power series and asymptotic expansions are discussed and compared. The asymptotic theory is applied to obtain asymptotic expansions of the zeros of the Airy functions and their derivatives, and also of the associated values of the functions or derivatives. A Maple code is constructed to generate exactly the coefficients in these expansions. The only limits on the number of coefficients are those imposed by the capacity of the computer being used and the execution time that is available.

The sign patterns of the coefficients suggest open problems pertaining to error bounds for the asymptotic expansions of the zeros and stationary values of the Airy functions.

##### MSC:

33C10 | Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ |

33F05 | Numerical approximation and evaluation of special functions |

41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |

65D20 | Computation of special functions, construction of tables |