. 23
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to MeV, M. C. Gutzwiller, A. Inomata, J. R. Klauder, and L. Streit, eds.,
no. 7 in Bielefeld Encounters in Physics and Mathematics, Singapore, 1986,
Bielefeld Center for Interdisciplinary Research, World Scienti¬c, pp. 173“195.

ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.104
Exponential Asymptotics

304. , Quantum resurgence, Ann. Inst. Fourier, 43 (1993), pp. 1509“1534. In
305. , Aspects of semiclassical theory in the presence of classical chaos, Prog.
Theor. Phys., 116 (1994), pp. 17“44.
306. , Exact quantization condition for anharmonic oscillators (in one dimen-
sion), J. Phys. A: Math. Gen., 27 (1994), pp. 4653“4661.
307. P. K. A. Wai, H. H. Chen, and Y. C. Lee, Radiation by “solitons” at the
zero group-dispersion wavelength of single-mode optical ¬bers, Phys. Rev. A,
41 (19), pp. 426“439. Nonlocal envelope solitons of the TNLS Eq. Their (2.1)
contains a typo and should be q (2) = ’(39/2)A2 q (0) + 21|q (0) |2 q (0) .
308. M. J. Ward, W. D. Henshaw, and J. B. Keller, Summing logarithmic
expansions for singularly perturbed eigenvalue problems, SIAM J. Appl. Math.,
53 (1993), pp. 799“828.
309. J. A. C. Weideman, Computation of the complex error function, SIAM Jour-
nal of Numerical Analysis, 31 (1994), pp. 1497“1518. [Errata: 1995, 32, 330“
331]. These series of rational functions are useful for complex-valued z.
310. , Computing integrals of the complex error function, Proceedings of Sym-
posia in Applied Mathematics, 48 (1994), pp. 403“407. Short version of Wei-
311. , Errata: computation of the complex error function, SIAM Journal of
Numerical Analysis, 32 (1995), pp. 330“331.
312. J. A. C. Weideman and A. Cloot, Spectral methods and mappings for
evolution equations on the in¬nite line, Comput. Meth. Appl. Mech. Engr.,
80 (1990), pp. 467“481. Numerical.
313. M. I. Weinstein and J. B. Keller, Hill™s equation with a large potential,
SIAM J. Appl. Math., 45 (1985), pp. 200“214.
314. , Asymptotic behavior of stability regions for Hill™s equation, SIAM J.
Appl. Math., 47 (1987), pp. 941“958.
315. E. J. Weniger, Nonlinear sequence transformations for the acceleration
of convergence and the summation of divergent series, Computer Physics
Reports, 10 (1989), pp. 189“371.
316. , On the derivation of iterated sequence transformations for the acceler-
ation of convergence and the summation of divergent series, Comput. Phys.
Commun., 64 (1991), pp. 19“45.
317. J. Wimp, The asymptotic representation of a class of G-functions for large
parameter, Math. Comp., 21 (1967), pp. 639“646.
318. , Sequence Transformations and Their Applications, Academic Press,
New York, 1981.
319. R. Wong, Asymptotic Approximation of Integrals, Academic Press, New
York, 1989.
320. A. Wood, Stokes phenomenon for high order di¬erential equations, Zeitschrist
fur Angewandte Mathematik und Mechanik, 76 (1996), pp. 45“48. Brief
321. A. D. Wood, Exponential asymptotics and spectral theory for curved optical
waveguides, in Asymptotics Beyond All Orders, H. Segur, S. Tanveer, and
H. Levine, eds., Plenum, Amsterdam, 1991, pp. 317“326.
322. A. D. Wood and R. B. Paris, On eigenvalues with exponentially small
imaginary part, in Asymptotic and Computational Analysis, R. Wong, ed.,
Marcel Dekker, New York, 1990, pp. 741“749.
323. T.-S. Yang, On traveling-wave solutions of the Kuramoto-Sivashinsky equa-
tion, Physica D, 110 (1998), pp. 25“42. Shocks with oscillations, exponentially
small in 1/ , which grow slowly in space, and thus are (very!) nonlocal. Applies
the Akylas-Yang beyond-all-orders perturbation method in wavenumber space
to compute the far ¬eld for oscillatory shocks. These are then matched to the

ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.105
106 John P. Boyd

nonlocal regular shocks to create solitary waves (that asymptote to the same
constant as x ’ ±∞); these are con¬rmed by numerical solutions.
324. T.-S. Yang and T. R. Akylas, Radiating solitary waves of a model evolution
equation in ¬‚uids of ¬nite depth, Physica D, 82 (1995), pp. 418“425. Solve
the Intermediate-Long Wave (ILW) equation for water waves with an extra
third derivative term, which makes the solitary waves weakly nonlocal. The
Yang-Akylas matched asymptotics in wavenumber is used to calculate the
exponentially small amplitude of the far ¬eld oscillations.
325. , Weakly nonlocal gravity-capillary solitary waves, Phys. Fluids, 8 (1996),
pp. 1506“1514.
326. , Finite-amplitude e¬ects on steady lee-wave patterns in subcritical strat-
i¬ed ¬‚ow over topography, J. Fluid Mech., 308 (1996), pp. 147“170.
327. , On asymmetric gravity-capillary solitary waves, J. Fluid Mech., 330
(1997), pp. 215“232. Asymptotic analysis of classical solitons of the FKdV
equation; demonstrates the coalescence of classical bions.
328. J. Zinn-Justin, Quantum ¬eld theory and critical phenomena, Oxford Uni-
versity Press, Oxford, 1989.

Address for correspondence: Department of Atmospheric, Oceanic and Space Science
and Laboratory for Scienti¬c Computation, University of Michigan, 2455 Hayward
Avenue, Ann Arbor MI 48109, jpboyd@engin.umich.edu;
http://www-personal.engin.umich.edu:/ jpboyd/

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Exponential Asymptotics

Table IV. Theory of Stokes Phenomenon and Resurgence

Description Special Functions References
fundamental theory Ecalle[123]

quantum eigenproblems anharmonic oscillator Voros[302, 301, 303, 305, 306]
generalized zeta funcs.

critical phenomena Zinn-Justin (monograph)[328]

Erfc smoothing of Dawson™s integral, Bi Berry (1989a)[24]
Stokes phenomenon Airy function Ai Berry(1989b)[25]
Various integrals Jones[155, 156, 157]
Olver [250], McLeod[200]

Hyperasymptotics Berry-Howls [35]

Di¬raction catastrophes, Berry-Howls [36]

Waves near Stokes lines Berry [26]

Adiabatic quantum transitions Berry [27]

(eigenvalue) Airy function Wood-Paris [322, 321, 259]
exponentially small

2d order ODEs Hanson[138]

Hyperasymptotics with saddles Berry-Howls [37]

In¬nitely many Stokes smoothings Gamma function Berry [28]

Superfactorial series Berry [29]

Uniform hyperasymptotics Generalized Olver [251]
with error bounds Exponential Integral

Uniform exponentially-improved Con¬‚uent Olver [252]
asymptotics with Hypergeometric functions
error bounds

Transcendentally small re¬‚ection 2d order ODEs Gingold&Hu[132]

Multisummability Martinet&Ramis[198]

Con¬‚uent Hypergeometric Olde Daalhuis [237],Olver [253]

Stokes phenomenon: Paris [256, 257]
Mellin-Barnes integral
& high-order ODEs

Exponential asymptotics Gamma function Paris-Wood [259]

Smoothing Stokes discontinuities

Coalescing saddles Berry-Howls [38]

Brief (4 pg.) review Berry-Howls [39]

Superadiabatic renormalization Berry-Lim [42]

ODEs Fifth-Order KdV Eq. Tovbis[293]

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108 John P. Boyd

Table IV. Theory of Stokes Phenomenon and Resurgence (continued)

Description Special Functions References

Steepest descent: Error bounds W. Boyd [78]

Stokes phenomenon Olde Daalhuis [238]
& hyperasymptotics
Ecalle “alien calculus” REVIEW (in French) Candelpergheret al.[90]

Overlapping Stokes smoothings Berry-Howls [40]

Quantum billiards Berry-Howls [41]
Ecalle theory REVIEW Delabaere[112]

Weyl expansion [148]

Reduction of Theories Philosophy of Science Berry [33]

Stokes phenomenon & W. Boyd [77]
& Stieltjes transforms

Coe¬cients of ODEs Olver [254]

ODEs: irregular singularities Olde Daalhuis-Olver [244, 245, 247]

Steepest descent Gamma function W. Boyd[79]

higher order ODEs Olde Daalhuis [239, 241]

Matched asymptotics Olde Daalhuis et al [243]
& Stokes phenomenon

Stokes multipliers: Olde Daalhuis-Olver(1995b) [246]
Linear ODEs

Multisummability Balser[12, 13, 14, 15, 16, 17, 81]

quantum resurgence Voros[304]

Riemann-Siegel expansion zeta function Berry [34]

ODEs Dunster[121]

Brief reviews Paris&Wood [260, 320]

Multidimensional integrals Howls[147]

Steepest descent ODEs W. Boyd[80]

Multisummability; Gevrey separation Ramis&Schafke[266]

quantum eigenproblem quartic oscillator Delabaere & Pham[113]

re-expansion of remainders integrals Byatt-Smith[87]

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Exponential Asymptotics

Table V. Asymptotics of Fourier, Chebyshev, Hermite and Other Spectral Methods. Note: β is
the spectral “exponential index of convergence” such that an ∼ exp(’constant nβ ). r is de¬ned
by r = lim supn’∞ log | bn | (n log n)

Functions Comments References

Entire Functions, Elliott[124]
Meromorphic Functions, Elliott[125]
Branch Points on [-1,1]

Entire Functions, Uniform as well as Elliott-Szekeres[126]
f(x) when Laplace large n asymptotics
Transform known

Whittaker Miller[227]
Exponential Integral, [230, 231]
Error Integral, N´meth[232, 233, 234, 235]
Con¬‚uent Hypergeometric N´meth (1992) [236]

Whittaker Asymptotics for Meier Wimp[317]
G-function [exact
Chebyshev functions]

Many (monograph) Many (complicated) Luke[187]
exact coe¬cients
and asymptotics



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