2. complex-plane matched asymptotics (the Pokrovskii-Khalatnikov-

Krusal-Segur method)

ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.87

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

e

special class of Stieljtes functions (Chebyshev: [49, 51] and Pad´: [19]),

e

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

obscure.

Acknowledgments

This work was supported by the National Science Foundation through

grant OCE9119459 and by the Department of Energy through contract

ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.88

89

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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