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logarithmic corrections [308]. The end product neglects higher powers
of and the necessary numerics is the full solution to a PDE, but still,
their work is real progress after two decades of no advance at all.
It seems likely that such hybrid numerical-asympotic methods will
¬‚ourish in the next few years for following reasons:

’ Analytical perturbation theory has enjoyed at least a century of
development, and it is hard to grow good ideas in such old soil.




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83
Exponential Asymptotics

’ Mathematics departments, even in the non-computational areas,
are becoming more computer-friendly.

’ Hyrid algorithms have successfully attacked a number of problems
already.

’ There are broad areas where hybrids have not yet been tried.



21. History

Improving upon the minimum error of an asymptotic series has a long
history; Stieltjes himself discussed the possibility in his 1888 doctoral
thesis. Oppenheimer™s calculation of the exponentially large decay time
in the quantum Stark e¬ect and the independent discovery of quantum
tunnelling by Gamow and by Condon and Gurney all happened in 1928.
The Euler acceleration of the Stieltjes function series was ¬rst analyzed
by Rosser [273] in 1951.
One can distinguish several parallel lines of development. The ¬rst is
the calculation of “converging factors” or terminants for the asymptotic
expansions of special functions, beginning with Airey in 1937 [3] and
reaching a high degree of sophistication in the books of Dingle (1973)
and Olver(1974) [118, 249], who also give good histories of earlier work.
Another was quantum mechanics, beginning with discovery of tun-
nelling in 1928, continuing with the Pokrovskii-Khalatnikov solution for
“above-the-barrier” quantum scattering, and continued to the present
with studies of high order perturbation theory. The books written by
Arteca, Fernandez and Castro [8] and edited by LeGuillou and Zinn-
Justin[181] and Braaksma[83] are good testaments, as is a special issue
of International Journal of Quantum Chemistry[269].
A third area is KAM theory and dynamical systems theory in gener-
al. Under perturbations, integrable dynamical systems become chaot-
ic, but the chaos is con¬ned to exponentially thin regions around the
separatrices [136] for small . Through “Arnold di¬usion”, dynamical
systems can move great distances in phase space (on exponentially long
time scales) even when the perturbation is very weak.
A fourth area is “weakly nonlocal solitary waves”, that is, nonlin-
ear coherent structures that would be immortal were it not for weak
radiation away from the core of the structure. These seem to be as
ubiquitous as classical, decaying-to-zero solitary waves. Nonlocal soli-
tary waves arise in ¬ber optics, hydrodynamics, plasmas and a wide
variety of other applications. Meiss and Horton (1983) [201] seem to
have done the earliest explicit calculations. However, the existence of




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84 John P. Boyd

slowly radiating solitary waves in particle physics (“ φ4 breathers”) and
oceanography (Gulf Stream Rings) was known from observations and
initial value computations a decade earlier. The subsequent eruption of
activity is catalogued in the book by Boyd [72].
A ¬fth area is crystal formation and solidi¬cation. The 1985 work
of Kruskal and Segur [171, 172] resolved a long-standing roadblock in
the theory of dendritic ¬ngers on melt interfaces, and touched o¬ a
great plume of activity. There was rapid cross-fertilization with nonlo-
cal solitary waves because Segur and Kruskal applied their new PKKS
method to the “φ4 breather” of particle physics, contributing to the
rapid growth of exponential asymptotics for nonlinear waves.
A sixth area is ¬‚uid mechanics. The Berman-Terrill-Robinson prob-
lem [135] in ¬‚ows with suction, the radiative decay of free oscillations
bound to islands [185] and Kelvin wave instability in oceanography
and atmospheric dynamics [74, 75] were all examples in which expo-
nential smallness had been calculated in the seventies or early eighties.
Somehow, these problems remained isolated. However, boundary layer
theory always involves divergent power series and exponential smallness
as showed by example above. Fluids is an area where hyperasymptotic
technology is likely to have a vigorous future.
A seventh line of research is that pursued by Richard E. Mey-
er and his students. This began with studies of adiabatic invariants
[202, 203, 204, 219, 205, 207]. He also devised an independent solu-
tion to “above the barrier” quantum scattering: recasting the problem
as an integral equation so that the re¬‚ection coe¬cient appears as
the dominant contribution instead of as an exponentially small cor-
rection [206, 208]. This led to further studies of exponential smallness
in water waves trapped around an island [185, 209, 220], connection
across WKB turning points and wave dynamics and quantum tun-
nelling [221, 211, 222, 223, 224, 212, 214, 215, 216, 225, 218, 226].
Meyer has also written four reviews [210, 213, 217, 218].
An eighth line of development is the abstract theory of resurgence
´
and multisummability. This began with Ecalle[123] and continued with
important contributions from Pham, Ramis, Delabaere, Braaksma and
others too numerous to mention as reviewed in [285, 83, 15].
Lastly, a ninth area is the development of resurgence and Stokes
phenomenon by physicists and applied mathematicians. This grew out
´
of the abstract theory of Ecalle, which Berry learned during a visit to
France, but took resurgence in a direction that was less rigorous but
much more pragmatic and applied. The trigger was Berry™s 1989 real-
ization that the discontinuity in the numerical value of an asymptotic
expansion at a Stokes™ line could be smoothed. (The change in topology
of the steepest descent path at a Stokes lines is unavoidable, howev-




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85
Exponential Asymptotics

´
er.) Building on the books of Dingle, Olver and Ecalle, Berry, Howls,
Olver, Olde Daalhuis, Paris, Wood, W. G. C. Boyd and others have
developed smoothed, high order hyperasymptotic approximations for
many species of special functions, for the WKB method and for other
schemes for di¬erential equations. A selection is given in Table IV.
Dingle™s ideas of generic forms for the late terms in asymptotic series
and universal terminants now seem as important to the rise of exponen-
tial asymptotics as the comet-crash(?) that put an end to the dinosaurs
was in biology. Only a year later, Olver™s book developed similar ideas,
with error bounds, for ordinary di¬erential equations. And yet, though
these books were widely bought and read, their net e¬ect at the time
was as quiet as a sandcastle washed away by a rising tide. Bothered by
long-term illness, Dingle never published again.
In recent years, however, the analysis of exponentially small terms
has exploded. A special program of study at the Newton Institute at
Cambridge has brought together researchers from a wide range of ¬elds
for a workshop lasting the whole ¬rst half of 1995. The books by Boyd
[72] and Segur, Tanveer and Levine (eds.) [279] are good introductions
to the vigour and diversity of this interest.
Why was this revolution in asymptotics so slow, so long delayed?
Perhaps the most important factor is that alterations in scienti¬c world-
view, like atom bombs, require assembling a critical mass. Part of this
critical mass was provided by the parallel threads of slow develop-
ment outlined above; when ideas began to cross disciplinary boundaries,
exponential asymptotics exponentiated. Another trigger was the pop-
ularization of algebraic manipulation languages, which made it easier
to compute many terms of an asymptotic series. Lastly, applied math-
ematics is subject to fads and enthusiasms.
I myself read both the Dingle and Olver books when they ¬rst
appeared while I was still in graduate school, but was unimpressed.
First, my was not very small. Second, a string of messy hyperasymp-
totic corrections seemed a poor alternative to numerical algorithms,
which were fast and e¬cient even a quarter century ago. Modern expo-
nential asymptotics still shares these limitations, but there is now a
cadre of enthusiasts who are unbothered as there was not in Dingle™s
time.6
Still, with the emergence of exponential asymptotics as a sub¬eld
of its own with ideas shared widely from physics to ¬‚uids to nonlin-
ear optics, hyperasymptotics has been very useful, at least as the low-
est hyperasymptotic order, in a wide variety of practical applications.

6
In a language of Papua New Guinea, the word “mokita” is used to denote
“things we all know but agree not to talk about”.




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86 John P. Boyd

When the parallel threads ceased to be parallel and converged, the
ancient topic of asymptotics suddenly became very interesting again.




22. Books and Review Articles


The theme of extending asymptotic series through Borel summation
and other methods of re-expanding remainder integrals is treated in the
classic books of Dingle[118] and Olver[249]. Jones™ 1997 book is very
short (160 pages), a primer of steepest descent and hyperasymptotics
that is perhaps closest in style and spirit to the Dingle and Olver books,
but at a somewhat more elementary level. It includes a short appendix
on non-standard analysis as well as exercises at the end of each chapter.
´
Ecalle™s 1981 three-volume treatise greatly extended and generalized
earlier ideas on hyperasymptotics. Unfortunately, his work has not been
translated from French. However, Sternin and Shatalov is a recent pre-
sentation of the abstract theory of resurgence[285]. The collection of
articles edited by Braaksma[83] gives a broader but less coherent state
of the abstract resurgence work. Balser[15] is only one hundred pages
long, but is very readable, based on a course taught by the author.
Kowalenko et al.[169] is a short monograph devoted entirely to the
hyperasymptotics of a fairly narrow class of integrals. Maslov[199] is a
broad treatment of the WKB method.
Segur, Tanveer and Levine[279] is a collection of articles from a
NATO Workshop that displays the remarkable breadth of application
of beyond-all-orders asymptotics that existed even in 1991. Arteca, Fer-
nandez and Castro[8] and LeGuillou and Zinn-Justin [181] describe the
calculation of exponentially small terms in quantum mechanics through
large order perturbation theory and summation methods. Boyd[72] is
focused particularly on nonlocal solitary waves, but it includes a chap-
ter on general applications of hyperasymptotics and several chapters
on numerical methods.
Curiously, review articles seem rarer than books. Berry and Howls[39],
Paris and Wood[260] and Wood[320] have written short, semi-popular
reviews. Delabaere[112] has written (in English) an introduction to
´
Ecalle alient calculus. Olde Daalhuis and Olver[248] describe hyper-
asymptotics (and numerical methods) for linear di¬erential equations.
Byatt-Smith[87],based on an unpublished but widely circulated manuscript
of seven years earlier, is not technically a review, but it nonetheless is
one of the most readable treatments of re-expansion of remainder inte-
grals and the error function smoothing of Stokes phenomenon.




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Exponential Asymptotics

This profusion of books and reviews is helpful, but there are still
some large gaps. This present article was written to ¬ll in some of
these holes and point the reader to other summaries of progress.


23. Summary

“What they [engineers] want from applied mathematics . . . is infor-
mation that illuminates.”
” Richard E. Meyer (1992) [218][pg. 43]

Key concepts:

’ Divergence is a disease caused by a perturbative approximation
which is true for only part of the interval of integration or part of
the Fourier spectrum.

’ A power series is asymptotic when the perturbative assumption is
bad only for a part of the spectrum or integrand that makes an
exponentially small contribution.

’ When a factorially divergent series is truncated at its smallest
term, this “optimal truncation” gives an error which is typically
an exponential function of 1/ . The usual Poincar´ de¬nition of
e
asymptoticity, which refers only to powers of , is therefore rather
misleading. The neologism “superasymptotic” was therefore coined
by Berry and Howls to describe the error in an optimally-truncated
asymptotic series.

’ By appending one or more terms of a second asymptotic series
(with a di¬erent rationale) to the optimal truncation of a divergent
series, one can reduce the error below that of the superasymptot-
ic approximation to obtain a “hyperasymptotic” approximation.
This, too, is divergent, but with a minimum error far smaller than
the best “superasymptotic” approximation. (This rescale-and-add-
another-series step can be repeated for further error reduction.)

’ There are many di¬erent species of hyperasymptotic methods includ-
ing:

1. sequence acceleration schemes such as the Euler, Pad´ and
e
Hermite-Pad´ (Shafer) approximations.

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