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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Dingle's final main rule\, Berry's transition\, an
d Howls' conjecture - GergÅ‘ Nemes (Tokyo Metropoli
tan University)
DTSTART;TZID=Europe/London:20221103T112000
DTEND;TZID=Europe/London:20221103T121000
UID:TALK185228AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/185228
DESCRIPTION:The Stokes phenomenon is the apparent discontinuou
s change in the form of the asymptotic expansion o
f a function across certain rays in the complex pl
ane\, known as Stokes lines\, as additional expans
ions\, pre-factored by exponentially small terms\,
appear in its representation. It was first observ
ed by G. G. Stokes while studying the asymptotic b
ehaviour of the Airy function. R. B. Dingle propos
ed a set of rules for locating Stokes lines and co
ntinuing asymptotic expansions across them. Includ
ed among these rules is the ``final main rule" sta
ting that half the discontinuity in form occurs on
reaching the Stokes line\, and half on leaving it
the other side. M. V. Berry demonstrated that\, i
f an asymptotic expansion is terminated just befor
e its numerically least term\, the transition betw
een two different asymptotic forms across a Stokes
line is effected smoothly and not discontinuously
as in the conventional interpretation of the Stok
es phenomenon. On a Stokes line\, in accordance wi
th Dingle's final main rule\, Berry's law predicts
a multiplier of 1/2 for the emerging small expone
ntials. In this talk\, we consider two closely rel
ated asymptotic expansions in which the multiplier
s of exponentially small contributions may no long
er obey Dingle's rule: their values can differ fro
m 1/2 on a Stokes line and can be non-zero only on
the line itself. This unusual behaviour of the mu
ltipliers is a result of a sequence of higher-orde
r Stokes phenomena. We show that these phenomena a
re rapid but smooth transitions in the remainder t
erms of a series of optimally truncated hyperasymp
totic re-expansions. To this end\, we verify a con
jecture due to C. J. Howls.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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