A convergent expansion of the Airy's integral with incomplete Gamma functions

Abstract

There are two main power series for the Airy functions, namely the Maclaurin and the asymptotic expansions. The former converges for all finite values of the complex variable, $z$, but it requires a large number of terms for large values of $|z|$, and the latter is a Poincar\'{e}-type expansion which is well-suited for such large values and where optimal truncation is possible. The asymptotic series of the Airy function shows a classical example of the Stokes phenomenon where a type of discontinuity occurs for the homonymous multipliers. A new series expansion is presented here that stems from the method of steepest descents, as can the asymptotic series, but which is convergent for all values of the complex variable. It originates in the integration of uniformly convergent power series representing the integrand of the Airy's integral in different sections of the integration path. The new series expansion is not a power series and instead relies on the calculation of complete and incomplete Gamma functions. In this sense, it is related to the Hadamard expansions. It is an alternative expansion to the two main aforementioned power series that also offers some insight into the transition zone for the Stokes' multipliers due to the splitting of the integration path. Unlike the Hadamard series, it relies on only two different expansions, separated by a branch point, one of which is centered at infinity. The interest of the new series expansion is mainly a theoretical one in a twofold way. First of all, it shows how to convert an asymptotic series into a convergent one, even if the rate of convergence may be slow for small values of $|z|$. Secondly, it sheds some light on the Stokes phenomenon for the Airy function by showing the transition of the integration paths at $\arg z = \pm 2 \pi/3$.Comment: 21 pages, 23 figures. Changes in version 2: i) Footnote 10 has been added, ii) Figure 5 has been added for a deeper analysis of the results, iii) Reference 15 has been added, iv) Typo: A $\pm$ was missing in $\arg z = \pm 2 \pi/3$ (abstract), v) Some font size changes and improved labelling in the figures Changes in version 3: minor edition change