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e
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
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




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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.


References

1. R. C. Ackerberg and R. E. O™Malley, Jr., Boundary layer problems
exhibiting resonance, Stud. Appl. Math., 49 (1970), pp. 277“295. Clas-
sical paper illustrating the failure of standard matched asymptotics; this
can be resolved by incorporating exponentially small terms in the analysis
(MacGillivray, 1997).
2. T. C. Adamson, Jr. and G. K. Richey, Unsteady transonic ¬‚ows with shock
waves in two-dimensional channels, J. Fluid Mech., 60 (1973), pp. 363“382.
Show the key role of exponentially small terms.
3. J. R. Airey, The “converging factor” in asymptotic series and the calculation
of Bessel, Laguerre and other functions, Philosophical Magazine, 24 (1937),
pp. 521“552. Hyperasymptotic approximation to some special functions for
large |x|.
4. T. R. Akylas and R. H. J. Grimshaw, Solitary internal waves with oscil-
latory tails, J. Fluid Mech., 242 (1992), pp. 279“298. Theory agrees with
observations of Farmer and Smith (1980).
5. T. R. Akylas and T.-S. Yang, On short-scale oscillatory tails of long-wave
disturbances, Stud. Appl. Math., 94 (1995), pp. 1“20. Nonlocal solitary waves;
perturbation theory in Fourier space.
6. G. Alvarez, Coupling-constant behavior of the resonances of the cubic anhar-
monic oscillator, Phys. Rev. A, 37 (1988), pp. 4079“4083. Beyond-all-orders
perturbation theory in quantum mechanics.
7. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-
Verlag, New York, 1978. Quote about why series diverge: pg.395.
8. G. A. Arteca, F. M. Fernandez, and E. A. Castro, Large Order Per-
turbation Theory and Summation Methods in Quantum Mechanics, Springer-
Verlag, New York, 1990. 642pp.; beyond-all-orders perturbation theory.
9. G. A. Baker, Jr. and P. Graves-Morris, Pade Approximants, Cambridge
University Press, New York, 1996.
10. G. A. Baker, Jr., J. Oitmaa, and M. J. Velgakis, Series analysis of
multivalued functions, Phys. Rev. A, 38 (1988), pp. 5316“5331. Generaliza-
tion of Pad´ approximants; the power series for a function u(z), known only
e
through its series, is used to de¬ne the polynomial coe¬cients of a di¬erential
equation, whose solution is then used as an approximation to u.
11. R. Balian, G. Parisi, and A. Voros, Quartic oscillator, in Feynman Path
Integrals, S. Albeverio, P. Combe, R. Hoegh-Krohn, G. Rideau, M. Siruge-
Collin, M. Sirugue, and R. Stora, eds., no. 106 in Lecture Notes in Physics,
Springer“Verlag, New York, 1979, pp. 337“360.
12. W. Balser, A di¬erent characterization of multisummable power series,
Analysis, 12 (1992), pp. 57“65.
13. , Summation of formal power series through iterated Laplace integrals,
Math. Scand., 70 (1992), pp. 161“171.
14. , Addendum to my paper: A di¬erent characterization of multisummable
power series, Analysis, 13 (1993), pp. 317“319.
15. , From Divergent Power Series to Analytic Functions, vol. 1582 of Lec-
ture Notes in Mathematics, Springer-Verlag, New York, 1994. 100 pp.; good
presentation of Gevrey order and asymptotics and multisummability.




ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.89
90 John P. Boyd

16. W. Balser, B. L. J. Braaksma, J.-P. Ramis, and Y. Sibuya, Multi-
summability of formal power series of linear ordinary di¬erential equations,
Asymptotic Analysis, 5 (1991), pp. 27“45.
17. W. Balser and A. Tovbis, Multisummability of iterated integrals, Asymp-
totic Analysis, 7 (1992), pp. 121“127.
18. L. Benassi, V. Grecchi, E. Harrell, and B. Simon, Bender-Wu formula
and the Stark e¬ect in hydrogen, Phys. Rev. Letters, 42 (1979), pp. 704“707.
Exponentially small corrections in quantum mechanics.
19. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for
Scientists and Engineers, McGraw-Hill, New York, 1978. 594 pp.
20. C. M. Bender and T. T. Wu, Anharmonic oscillator, Phys. Rev., 184
(1969), pp. 1231“1260.
21. , Anharmonic oscillator. II. A study of perturbation theory in large order,
Phys. Rev. D, 7 (1973), pp. 1620“1636.
22. E. Benilov, R. H. Grimshaw, and E. Kuznetsova, The generation of
radiating waves in a singularly perturbed Korteweg-deVries equation, Physica
D, 69 (1993), pp. 270“276.
23. A. S. Berman, Laminar ¬‚ow in channel with porous walls, J. Appl. Phys., 24
(1953), pp. 1232“1235. Earliest paper on an ODE (Berman-Robinson-Terrill
problem) where exponentially small corrections are important.
24. M. V. Berry, Stokes™ phenomenon; smoothing a Victorian discontinuity,
Publicationes Mathematiques IHES, 68 (1989), pp. 211“221.
25. , Uniform asymptotic smoothing of Stokes™s discontinuities, Proc. Roy.
Soc. London A, 422 (1989), pp. 7“21.
26. , Waves near Stokes lines, Proc. Roy. Soc. London A, 427 (1990),
pp. 265“280.
27. , Histories of adiabatic quantum transitions, Proc. Roy. Soc. London A,
429 (1990), pp. 61“72.
28. , In¬nitely many Stokes smoothings in the Gamma function, Proc. Roy.
Soc. London A, 434 (1991), pp. 465“472.
29. , Stokes phenomenon for superfactorial asymptotic series, Proc. Roy.
Soc. London A, 435 (1991), pp. 437“444.
30. , Asymptotics, superasymptotics, hyperasymptotics, in Asymptotics
Beyond All Orders, H. Segur, S. Tanveer, and H. Levine, eds., Plenum, Ams-
terdam, 1991, pp. 1“14.
31. , Faster than Fourier, in Quantum Coherence and Reality: in Celebra-
tion of the 60th Birthday of Yakir Aharonov, J. S. Auandan and J. L. Safko,
eds., World Scienti¬c, Singapore, 1994.
32. , Evanescent and real waves in quantum billiards and Gaussian beams,
J. Phys. A, 27 (1994), pp. L391“L398.
33. , Asymptotics, singularities and the reduction of theories, in Logic,
Methodology and Philosophy of Science IX, D. Prawitz, B. Skyrms, and
D. Westerstahl, eds., Elsevier, Amsterdam, 1994, pp. 597“607.
34. , Riemann-Siegel expansion for the zeta function: high orders and
remainders, Proc. Roy. Soc. London A, 450 (1995), pp. 439“462. Beyond
all orders asymptotics.
35. M. V. Berry and C. J. Howls, Hyperasymptotics, Proc. Roy. Soc. London
A, 430 (1990), pp. 653“668.
36. , Stokes surfaces of di¬raction catastrophes with codimension three, Non-
linearity, 3 (1990), pp. 281“291.
37. , Hyperasymptotics for integrals with saddles, Proc. Roy. Soc. London
A, 434 (1991), pp. 657“675.
38. , Unfolding the high orders of asymptotic expansions with coalescing
saddles: Singularity theory, crossover and duality, Proc. Roy. Soc. London A,
443 (1993), pp. 107“126.




ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.90
91
Exponential Asymptotics

39. , In¬nity interpreted, Physics World, 6 (1993), pp. 35“39.
40. , Overlapping Stokes smoothings: survival of the error function and
canonical catastrophe integrals, Proc. Roy. Soc. London A, 444 (1994),
pp. 201“216.
41. , High orders of the Weyl expansion for quantum billiards: Resurgence
of periodic orbits, and the Stokes phenomenon, Proc. Roy. Soc. London A, 447
(1994), pp. 527“555.
42. M. V. Berry and R. Lim, Universal transition prefactors derived by supera-
diabatic renormalization, J. Phys. A, 26 (1993), pp. 4737“4747.
43. K. Bhattacharyya, Notes on polynomial perturbation problems, Chem.
Phys. Lett., 80 (1981), pp. 257“261.
44. K. Bhattacharyya and S. P. Bhattacharyya, The sign“change argument
revisited, Chem. Phys. Lett., 76 (1980), pp. 117“119. Criterion for divergence
of asymptotic series.
45. , Reply to “another attack on the sign“change argument”, Chem. Phys.
Lett., 80 (1981), pp. 604“605.
46. P. A. Boasman and J. P. Keating, Semiclassical asymptotics of perturbed
cat maps, Proc. R. Soc. London A, 449 (1995), pp. 629“653. Shows that
the optimal truncation of the semiclassical expansion is accurate to within
an error which is an exponential function of 1/¯ . Stokes phenomenon, Borel
h
resummation, and a universal approximation to the late terms are used to
beyond the superasymptotic approximation.
47. J. P. Boyd, A Chebyshev polynomial method for computing analytic solutions
to eigenvalue problems with application to the anharmonic oscillator, J. Math.
Phys., 19 (1978), pp. 1445“1456.
48. , The nonlinear equatorial Kelvin wave, J. Phys. Oceangr., 10 (1980),
pp. 1“11.
49. , The rate of convergence of Chebyshev polynomials for functions which
have asymptotic power series about one endpoint, Math. Comp., 37 (1981),
pp. 189“196.
50. , The optimization of convergence for Chebyshev polynomial methods in
an unbounded domain, J. Comput. Phys., 45 (1982), pp. 43“79. In¬nite and
semi-in¬nite intervals; guidelines for choosing the map parameter or domain
size L.
51. , A Chebyshev polynomial rate-of-convergence theorem for Stieltjes func-
tions, Math. Comp., 39 (1982), pp. 201“206. Typo: In (24), the rightmost
expression should be 1 ’ (r + )/2.
52. , The asymptotic coe¬cients of Hermite series, J. Comput. Phys., 54
(1984), pp. 382“410.
53. , Spectral methods using rational basis functions on an in¬nite interval,
J. Comput. Phys., 69 (1987), pp. 112“142.
54. , Orthogonal rational functions on a semi-in¬nite interval, J. Comput.
Phys., 70 (1987), pp. 63“88.
55. , Chebyshev and Fourier Spectral Methods, Springer-Verlag, New York,
1989. 792 pp.
56. , New directions in solitons and nonlinear periodic waves: Polycnoidal
waves, imbricated solitons, weakly non-local solitary waves and numerical
boundary value algorithms, in Advances in Applied Mechanics, T.-Y. Wu and
J. W. Hutchinson, eds., no. 27 in Advances in Applied Mechanics, Academic
Press, New York, 1989, pp. 1“82.
57. , Non-local equatorial solitary waves, in Mesoscale/Synoptic Coherent
Structures in Geophysical Turbulence: Proc. 20th Li´ge Coll. on Hydrody-
e
namics, J. C. J. Nihoul and B. M. Jamart, eds., Elsevier, Amsterdam, 1989,
pp. 103“112. Typo: In (4.1b), 0.8266 should be 1.6532.




ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.91
92 John P. Boyd

, A numerical calculation of a weakly non-local solitary wave: the φ4
58.
breather, Nonlinearity, 3 (1990), pp. 177“195. The eigenfunction calcula-
tion (5.15, etc.) has some typographical errors corrected in Chapter 12 of
Boyd(1998).
59. , The envelope of the error for Chebyshev and Fourier interpolation, J.
Sci. Comput., 5 (1990), pp. 311“363.
60. , A Chebyshev/radiation function pseudospectral method for wave scat-
tering, Computers in Physics, 4 (1990), pp. 83“85. Numerical calculation of
exponentially small re¬‚ection.
61. , A comparison of numerical and analytical methods for the reduced wave
equation with multiple spatial scales, App. Numer. Math., 7 (1991), pp. 453“
479. Study of uxx ± ux = f ( x). Typo: 2n factor should be omitted from Eq.
(4.3).
62. , Weakly non-local solitons for capillary-gravity waves: Fifth-degree
Korteweg-de Vries equation, Physica D, 48 (1991), pp. 129“146. Typo: at
the beginning of Sec. 5, ˜Newton-Kantorovich (5.1)™ should read ˜Newton-
Kantorovich (3.2)™. Also, in the caption to Fig. 12, ˜500,000™ should be ˜70,000™.
63. , Chebyshev and Legendre spectral methods in algebraic manipulation
languages, J. Symb. Comput., 16 (1993), pp. 377“399.

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