The Generalized Exponential Integral

Abstract

This paper concerns the role of the generalized exponential integral in recently-developed theories of exponentially-improved asymptotic expansions and the Stokes phenomenon. The first part describes the asymptotic behavior of the integral when both the argument and order are large in absolute value. The second part shows how to increase the accuracy of asymptotic expansions of solutions of linear differential equations of the second order by re-expanding the remainder terms in series of generalized exponential integrals.