Abstract

We shall consider the problem of deriving realistic error

bounds for asymptotic expansions arising from integrals. It was demonstrated by

W. G. C. Boyd in the early 1990's that Cauchy-Heine-type representations for

remainder terms are quite suitable for obtaining such bounds. I will show that

the Borel transform can lead to a more globally valid expression for remainder

terms involving R. B. Dingle's terminant function as a kernel. We will see

through examples that such a representation is, in a sense,



optimal: it leads to error bounds that are valid in large

sectors and which cannot be improved in general. Building on the important

results of Sir M. V. Berry and C. J. Howls, I will provide analogous results

for asymptotic expansions arising from integrals with saddles.



Finally, I will show how a Cauchy-Heine-type argument can

be applied to implicit problems by outlining the recent proof of a conjecture

of F. W. J. Olver on the large negative zeros of the Airy function.