Extended Hadamard Expansions for the Airy Functions

SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3537-3558, June 2024.
Abstract. A new set of Hadamard series expansions for the Airy functions, [math] and [math], is presented. Previous Hadamard expansions were defined in terms of an infinite number of integration path subdivisions. Unlike the earlier expansions, the expansions in the present work originate in the splitting of the steepest descent into a number of segments that is not only finite but very small, and these segments are defined on the basis of the location of the branch points. One of the segments reaches to infinity, and this gives rise to the presence of upper incomplete gamma functions. This is one of the most important differences from the Hadamard series as defined in the work of R.B. Paris, where all the incomplete gamma functions are of the lower type. The interest of the new series expansion is twofold. First, it shows how to convert an asymptotic series into a convergent one with a finite splitting of the steepest descent path in a process that can be named “exactification.” Second, the inverse of the phase function that is part of the Laplace-type integral is Taylor-expanded around branch points to produce Puiseux series when necessary. Regarding their computational application, these series expansions require a relatively small number of terms for each of them to reach a very high precision.