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 the computation of the asymptotic form of certain integrals which also involve the Hankel contour; these integrals depend on a real parameter, 𝛼. A rigorous asymptotic technique is presented here for computing such integrals to all orders. For certain values of 𝛼, the relevant formula, in addition to an asymptotic series of explicit terms, also contains a specific integral. It is shown that, remarkably, the leading behavior 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.