A novel asymptotic technique for integrals involving the Hankel contour and the Bleistein asymptotic formula

TL;DR

This paper introduces a new asymptotic technique for integrals involving the Hankel contour, connecting it to Bleistein's asymptotic formula, with applications to functions like the gamma and zeta functions.

Contribution

A rigorous, all-orders asymptotic method for Hankel contour integrals, linking leading order terms to Bleistein's integral in boundary stationary point cases.

Findings

Derived explicit asymptotic series for integrals involving the Hankel contour.

Connected the leading order of these integrals to Bleistein's integral form.

Applicable to asymptotic analysis of functions like the gamma and zeta functions.

Abstract

Several important functions, including the gamma function, as well as several infinite sums, admit integral representations involving the Hankel contour. In addition, the large $t$ asymptotic analysis of several recently derived identities satisfied by the Riemann zeta function requires computing the asymptotic form of certain integrals which also involve the Hankel contour; these integrals depend on a real parameter, $\alpha$. A rigorous asymptotic technique is presented here for computing such integrals to all orders. For certain values of $\alpha$, the relevant formula, in addition to an asymptotic series of explicit terms, also contains a specific integral. It is shown that, remarkably, the leading order of this integral can be written in the form of the leading order of the Bleistein integral. The latter integral arises in the implementation of the classical steepest descent method in the case that the stationary point coincides with one of the boundary points of the integral under consideration.