Author’s abstract: “By allowing the number of terms in an asymptotic expansion to depend on the asymptotic variable, it is possible to obtain an error term that is exponentially small as the asymptotic variable tends to its limit. This procedure is called “ exponential improvement”. It is shown how to improve exponentially the well-known Poincaré expansions for the generalized exponential integral (or incomplete Gamma function) of large argument. New uniform expansions are derived in terms of elementary functions, and also in terms of the error function. Inter alia, the results supply a rigorous foundation for some of the recent work of M. V. Berry on a smooth interpretation of the Stokes phenomenon.” Note: In the words of the author the purpose of this investigation is to show how to construct expansions for a wide class of functions. The new theory goes beyond that one presented in chapter 14 of the author’s celebrated book, Introduction to asymptotics and special functions (1974) in three ways: (i) the number of functions that can be treated successfully is greatly increased, (ii) Non elementary functions may be used in constructing expansions, (iii) as a consequence of (ii), regions of validity are extended considerably. He further states that D. S. Jones
[Lect. Notes Pure Appl. Math. 124, 241-264 (1990; Zbl 0693.41034
)] has independently investigated the generalized exponential integrals or incomplete Gamma function substantially overlaping the work of the author but his work differs significantly in the method of proof and results achieved.