Error bounds and exponential improvement for the asymptotic expansion of the Barnes GG-function

In this paper we establish new integral representations for the remainder term of the known asymptotic expansion of the logarithm of the Barnes $G$-function. Using these representations, we obtain explicit and numerically computable error bounds for the asymptotic series, which are much simpler than the ones obtained earlier by other authors. We find that along the imaginary axis, suddenly infinitely many exponentially small terms appear in the asymptotic expansion of the Barnes $G$-function. Employing one of our representations for the remainder term, we derive an exponentially improved asymptotic expansion for the logarithm of the Barnes $G$-function, which shows that the appearance of these exponentially small terms is in fact smooth, thereby proving the Berry transition property of the asymptotic series of the $G$-function.Comment: 14 pages, accepted for publication in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science

Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal

In (Boyd, Proc. R. Soc. Lond. A 447 (1994) 609--630), W. G. C. Boyd derived a resurgence representation for the gamma function, exploiting the reformulation of the method of steepest descents by M. Berry and C. Howls (Berry and Howls, Proc. R. Soc. Lond. A 434 (1991) 657--675). Using this representation, he was able to derive a number of properties of the asymptotic expansion for the gamma function, including explicit and realistic error bounds, the smooth transition of the Stokes discontinuities, and asymptotics for the late coefficients. The main aim of this paper is to modify the resurgence formula of Boyd making it suitable for deriving better error estimates for the asymptotic expansions of the gamma function and its reciprocal. We also prove the exponentially improved versions of these expansions complete with error terms. Finally, we provide new (formal) asymptotic expansions for the coefficients appearing in the asymptotic series and compare their numerical efficacy with the results of earlier authors.Comment: 22 pages, accepted for publication in Proceedings of the Royal Society of Edinburgh, Section A: Mathematical and Physical Science

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

Dingle's final main rule, Berry's transition, and Howls' conjecture

The Stokes phenomenon is the apparent discontinuous change in the form of the asymptotic expansion of a function across certain rays in the complex plane, known as Stokes lines, as additional expansions, pre-factored by exponentially small terms, appear in its representation. It was first observed by G. G. Stokes while studying the asymptotic behaviour of the Airy function. R. B. Dingle proposed a set of rules for locating Stokes lines and continuing asymptotic expansions across them. Included among these rules is the "final main rule" stating that half the discontinuity in form occurs on reaching the Stokes line, and half on leaving it the other side. M. V. Berry demonstrated that, if an asymptotic expansion is terminated just before its numerically least term, the transition between two different asymptotic forms across a Stokes line is effected smoothly and not discontinuously as in the conventional interpretation of the Stokes phenomenon. On a Stokes line, in accordance with Dingle's final main rule, Berry's law predicts a multiplier of $\frac{1}{2}$ for the emerging small exponentials. In this paper, we consider two closely related asymptotic expansions in which the multipliers of exponentially small contributions may no longer obey Dingle's rule: their values can differ from $\frac{1}{2}$ on a Stokes line and can be non-zero only on the line itself. This unusual behaviour of the multipliers is a result of a sequence of higher-order Stokes phenomena. We show that these phenomena are rapid but smooth transitions in the remainder terms of a series of optimally truncated hyperasymptotic re-expansions. To this end, we verify a conjecture due to C. J. Howls.Comment: 23 pages, 2 figures, accepted for publication in Journal of Physics A: Mathematical and Theoretical. The exposition was improved based on the referees' comment