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Realistic error bounds for asymptotic expansions via integral representations

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ARA - Applicable resurgent asymptotics: towards a universal theory

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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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