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

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

Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters

We consider the asymptotic behavior of the incomplete gamma functions gamma(-a,-z) and Gamma(-a,-z) as a goes to infinity. Uniform expansions are needed to describe the transition area z~a in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform equation by applying techniques used for the existing uniform expansions for gamma(a,z) and Gamma(a,z). The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given in a final section

On the use of Hadamard expansions in hyperasymptotic evaluation: differential equations of hypergeometric type

We describe how a modification of a common technique for developing asymptotic expansions of solutions of linear differential equations can be used to derive Hadamard expansions of solutions of differential equations. Hadamard expansions are convergent series that share some of the features of hyperasymptotic expansions, particularly that of having exponentially small remainders when truncated, and, as a consequence, provide a useful computational tool for evaluating special functions. The methods we discuss can be applied to linear differential equations of hypergeometric type and may have wider applicability

Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation

Using a method mixing Mellin-Barnes representation and Borel resummation we show how to obtain hyperasymptotic expansions from the (divergent) formal power series which follow from the perturbative evaluation of arbitrary "$N$-point" functions for the simple case of zero-dimensional $\phi^4$ field theory. This hyperasymptotic improvement appears from an iterative procedure, based on inverse factorial expansions, and gives birth to interwoven non-perturbative partial sums whose coefficients are related to the perturbative ones by an interesting resurgence phenomenon. It is a non-perturbative improvement in the sense that, for some optimal truncations of the partial sums, the remainder at a given hyperasymptotic level is exponentially suppressed compared to the remainder at the preceding hyperasymptotic level. The Mellin-Barnes representation allows our results to be automatically valid for a wide range of the phase of the complex coupling constant, including Stokes lines. A numerical analysis is performed to emphasize the improved accuracy that this method allows to reach compared to the usual perturbative approach, and the importance of hyperasymptotic optimal truncation schemes.Comment: v2: one reference added, one paragraph added in the conclusions, small changes in the text, corrected typos; v3: published versio

Exponentially small expansions in the asymptotics of the Wright function

We consider exponentially small expansions present in the asymptotics of the generalised hypergeometric function, or Wright function, pΨq(z) for large |z| that have not been considered in the existing theory. Our interest is principally with those functions of this class that possess either a finite algebraic expansion or no such expansion and with parameter values that produce exponentially small expansions in the neighbourhood of the negative real z axis. Numerical examples are presented to demonstrate the presence of these exponentially small expansions

Uniform Asymptotic Methods for Integrals

We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson's lemma, Laplace's method, the saddle point method, and the method of stationary phase. Certain developments in the field of asymptotic analysis will be compared with De Bruijn's book {\em Asymptotic Methods in Analysis}. The classical methods can be modified for obtaining expansions that hold uniformly with respect to additional parameters. We give an overview of examples in which special functions, such as the complementary error function, Airy functions, and Bessel functions, are used as approximations in uniform asymptotic expansions.Comment: 31 pages, 3 figure

The resurgence properties of the incomplete gamma function I

In this paper we derive new representations for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by C. J. Howls (Howls, Proc. R. Soc. Lond. A 439 (1992) 373--396). Using these representations, we obtain a number of properties of the asymptotic expansions of the incomplete gamma function with large arguments, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.Comment: 36 pages, 4 figures. arXiv admin note: text overlap with arXiv:1311.2522, arXiv:1309.2209, arXiv:1312.276