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SUMMARY:Realistic error bounds for asymptotic expansions via integral repr
esentations - Gergő Nemes (Alfréd Rényi Institute of Mathematics\,Hunga
rian Academy of Sciences)
DTSTART:20210408T150000Z
DTEND:20210408T160000Z
UID:TALK158662@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:We shall consider the problem of deriving realistic error
<
br> bounds for asymptotic expansions arising from integrals. It was demons
trated by
W. G. C. Boyd in the early 1990'\;s that Cauchy-Heine
-type representations for
remainder terms are quite suitable for o
btaining such bounds. I will show that
the Borel transform can lea
d to a more globally valid expression for remainder
terms involvin
g 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
sec
tors and which cannot be improved in general. Building on the important
results of Sir M. V. Berry and C. J. Howls\, I will provide analogou
s results
for asymptotic expansions arising from integrals with sa
ddles.
Finally\, I will show how a Cauchy-Heine-type argu
ment can
be applied to implicit problems by outlining the recent p
roof of a conjecture
of F. W. J. Olver on the large negative zeros
of the Airy function.
LOCATION:Seminar Room 1\, Newton Institute
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