Exponentially-Improved Asymptotic Solutions of Ordinary Differential Equations I: The Confluent Hypergeometric Function

There has been renewed interest in both formal and rigorous theories of exponentially-small contributions to asymptotic expansions. In particular, a generalized asymptotic expansion was obtained for the confluent hypergeometric function $U(a,a - b + 1,z)$ in which the parameters a and b are complex constants, and z is a large complex variable. This expansion is expressed in terms of generalized exponential integrals and has a larger region of validity and greater accuracy than conventional expansions of Poincaré type. The expansion was established by transformations and a re-expansion of an integral representation of $U(a,a - b + 1,z)$. In this paper it is shown how the same result can be achieved by a direct differential-equation approach, thereby laying the foundation for a rigorous theory of generalized asymptotic solutions of linear differential equations.