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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Realistic error bounds for asymptotic expansions v
ia integral representations - Gergő Nemes (Alfréd
Rényi Institute of Mathematics\,Hungarian Academy
of Sciences)
DTSTART;TZID=Europe/London:20210408T160000
DTEND;TZID=Europe/London:20210408T170000
UID:TALK158662AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/158662
DESCRIPTION:We shall consider the problem of deriving realisti
c error

bounds for asymptotic expansions a
rising from integrals. It was demonstrated by

<
br> W. G. C. Boyd in the early 1990'\;s that Ca
uchy-Heine-type representations for

remain
der terms are quite suitable for obtaining such bo
unds. 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 lea
ds 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 analog
ous results

for asymptotic expansions aris
ing from integrals with saddles.

Finally\, I will show how a Cauchy-Heine-type argu
ment can

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

** of F. W. J. Olver on the large negative zeros o
f the Airy function.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:INI IT
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**