Asymptotic expansions and converging factors I. General theory and basic converging factors

Abstract

It is shown that by the application of Borel’s method of summation to the later terms of an asymptotic expansion, the ‘sum’ of such terms can normally be replaced by an easily calculable series involving ‘basic converging factors’. As particular consequences, [i] the remainder in a truncated asymptotic expansion can be written down once the general term in the expansion is known; [ii] the converging factor for a given asymptotic expansion can conveniently be calculated from the basic converging factors; and [iii] the Stokes phenomenon is simply expressed in terms of discontinuities in these basic quantities. Formulae and tables are given for the basic converging factors.