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2. complex-plane matched asymptotics (the Pokrovskii-Khalatnikov-
Krusal-Segur method)

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88 John P. Boyd

3. Resurgence schemes
4. isolation of exponential smallness
5. special numerical algorithms, usually employing Chebyshev or
Fourier spectral methods or Gaussian quadrature
’ The history of exponential asymptotics stretches back at least a
century with several parallel lines of slow development that reached
a critical mass only within the last six years, culminating in an
explosion of both applications and theory that will touch almost
every ¬eld of science and engineering as well as mathematics
The list of open problems is large. One is a rigorous numerical test of
many-term, high order hyperasymptotic expansions versus competing
methods, such as Chebyshev series, for special function software. (The
arguments presented above suggest that the results are likely to be
unfavorable to hyperasymptotics).
Another is to create an expanded theory for the connection between
the rate of growth of power series coe¬cients or other properties of
functions with divergent power series and the rate of convergence of
Chebyshev series and Pad´ approximants. Some theorems exist for the
special class of Stieljtes functions (Chebyshev: [49, 51] and Pad´: [19]),
but little else.
An important issue is whether the Dingle terminant formalism can
be extended to weakly nonlocal solitary waves. The radiation coe¬cient
±, which is proportional to the function exp(’µ/ ) for some constant
µ, has only the trivial power series 0 + 0 · + 0 · 2 + . . .. Does ±
somehow in¬‚uence the coe¬cients of the power series subtly so that
terminants can be applied, or is the radiation condition truly a ghost,
forever invisible to methods that look only at the asymptotic form of
the power series coe¬cients?
A fourth domain of future study is to apply exponential asymptotics
to new realms. We have shown above that the theory of numerical
algorithms contains hidden beyond-all-orders terms, but this aspect of
numerical analysis is largely terra incognita.
Although applications and fundamental research on exponential-
ly small terms will doubtless continue for many years, we have tried
to show that the underlying principles are neither complicated nor


This work was supported by the National Science Foundation through
grant OCE9119459 and by the Department of Energy through contract

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Exponential Asymptotics

KC070101. I thank Richard Meyer, Michael Ward and Robert O™Malley
for helpful correspondence or conversations, and others too numerous
to mention for supplying reprints and references. I am grateful to the
three referees for their extremely careful reading of this long paper.


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