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SUMMARY:Dingle's final main rule\, Berry's transition\, and Howls' conject
 ure - Gergő Nemes (Tokyo Metropolitan University)
DTSTART:20221103T112000Z
DTEND:20221103T121000Z
UID:TALK185228@talks.cam.ac.uk
DESCRIPTION:The Stokes phenomenon is the apparent discontinuous change in 
 the form of the asymptotic expansion of a function across certain rays in 
 the complex plane\, known as Stokes lines\, as additional expansions\, pre
 -factored by exponentially small terms\, appear in its representation. It 
 was first observed by G. G. Stokes while studying the asymptotic behaviour
  of the Airy function. R. B. Dingle proposed a set of rules for locating S
 tokes lines and continuing asymptotic expansions across them. Included amo
 ng these rules is the ``final main rule" stating that half the discontinui
 ty in form occurs on reaching the Stokes line\, and half on leaving it the
  other side. M. V. Berry demonstrated that\, if an asymptotic expansion is
  terminated just before its numerically least term\, the transition betwee
 n 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 with Dingle's final main r
 ule\, Berry's law predicts a multiplier of 1/2 for the emerging small expo
 nentials. In this talk\, we consider two closely related asymptotic expans
 ions in which the multipliers of exponentially small contributions may no 
 longer obey Dingle's rule: their values can differ from 1/2 on a Stokes li
 ne and can be non-zero only on the line itself. This unusual behaviour of 
 the multipliers is a result of a sequence of higher-order Stokes phenomena
 . We show that these phenomena are rapid but smooth transitions in the rem
 ainder terms of a series of optimally truncated hyperasymptotic re-expansi
 ons. To this end\, we verify a conjecture due to C. J. Howls.
LOCATION:Seminar Room 1\, Newton Institute
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