The asymptotic expansion of integral functions defined by Taylor series

Abstract

1.1. Considerable attention has been devoted to the behaviour of the general integral function for large values of the variable, and many important theorems have been proved in this field. On the other hand, the behaviour of a large number of particular integral functions has been studied in detail and their asymptotic expansions for certain regions of the plane obtained. There is, however, a substantial gap between the two theories. For example, much of the most interesting work on the general integral function deals with the distribution of its zeroes and other values; but many of the asymptotic expansions obtained for particular functions are not valid in the regions in which these functions have zeroes. In this paper and its sequels I propose to study several fairly wide classes of functions defined by Taylor series; from the properties of the coefficients I deduce asymptotic expansions of the function defined by the series. For the sort of functions I consider we can usually divide the whole complex plane, with the exception of certain “ barrier regions” , into a number of regions R , in each of which the function is given asymptotically by an equation of the form