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The Devil™s Invention: Asymptotic, Superasymptotic
and Hyperasymptotic Series—

John P. Boyd †
University of Michigan

Abstract. Singular perturbation methods, such as the method of multiple scales
and the method of matched asymptotic expansions, give series in a small parameter
which are asymptotic but (usually) divergent. In this survey, we use a plethora of
examples to illustrate the cause of the divergence, and explain how this knowledge
can be exploited to generate a ”hyperasymptotic” approximation. This adds a second
asymptotic expansion, with di¬erent scaling assumptions about the size of various
terms in the problem, to achieve a minimum error much smaller than the best
possible with the original asymptotic series. (This rescale-and-add process can be
repeated further.) Weakly nonlocal solitary waves are used as an illustration.
Key words: Perturbation methods, asymptotic, hyperasymptotic, exponential small-

AMS: 34E05, 40G99, 41A60, 65B10

“Divergent series are the invention of the devil, and it is shameful
to base on them any demonstration whatsoever.”
” Niels Hendrik Abel, 1828

1. Introduction

2. The Necessity of Computing Exponentially Small Terms

3. De¬nitions and Heuristics

4. Optimal Truncation and Superasymptotics for the Stieltjes Func-

5. Hyperasymptotics for the Stieltjes Function

6. A Linear Di¬erential Equation

7. Weakly Nonlocal Solitary Waves

8. Overview of Hyperasymptotic Methods

9. Isolation of Exponential Smallness

This work was supported by the National Science Foundation through grant
OCE9119459 and by the Department of Energy through KC070101.

2 John P. Boyd

10. Darboux™s Principle and Resurgence

11. Steepest Descents

12. Stokes Phenomenon

13. Smoothing Stokes Phenomenon: Asymptotics of the Terminant

14. Matched Asymptotic Expansions in the Complex Plane: The PKKS

15. Snares and Worries: Remote but Dominant Saddle Points, Ghosts,
Interval-Extension and Sensitivity

16. Asymptotics as Hyperasymptotics for Chebyshev, Fourier and Oth-
er Spectral Methods

17. Numerical Methods for Exponential Smallness or: Poltergeist-Hunting
by the Numbers, I: Chebyshev & Fourier Spectral Methods

18. Numerical Methods, II: Sequence Acceleration and Pad´ and Hermite-
Pad´ Approximants

19. High Order Hyperasymptotics versus Chebyshev and Hermite-Pad´

20. Hybridizing Asymptotics with Numerics

21. History

22. Books and Review Articles

23. Summary

1. Introduction

Divergent asymptotic series are important in almost all branch-
es of physical science and engineering. Feynman diagrams (particle
physics), Rayleigh-Schroedinger perturbation series (quantum chem-
istry), boundary layer theory and the derivation of soliton equations
(¬‚uid mechanics) and even numerical algorithms like the “Nonlinear
Galerkin” method [66, 196] are examples. Unfortunately, classic texts
like van Dyke [297], Nayfeh [229] and Bender and Orszag [19], which
are very good on the mechanics of divergent series, largely ignore two
important questions. First, why do some series diverge for all non“zero

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Exponential Asymptotics

Table I. Non-Soliton Exponentially Small Phenomena

Phenomena Field References

Dendritic Crystal Growth Condensed Matter Kessler, Koplik & Levine [163]
Kruskal&Segur[171, 172]

Viscous Fingering Fluid Dynamics Shraiman [276]
(Sa¬man-Taylor Problem) Combescot et al.[103]
Hong & Langer [146]
Tanveer [288, 289]

Di¬usion& Merger Reaction-Di¬usion Carr [92], Hale [137],
of Fronts Systems Carr & Pego [93]
on an Exponentially Fusco & Hale [130]
Long Time Scale Laforgue& O™Malley
[173, 174, 175, 176]

Superoscillations in Applied Mathematics, Berry [31, 32]
Fourier Integrals, Quantum Mechanics,
Quantum Billiards, Electromagnetic Waves
Gaussian Beams

Rapidly-Forced Classical Chang [94]
Pendulum Physics Scheurle et al. [275]

Resonant Sloshing Fluid Mechanics Byatt-Smith & Davie [88, 89]
in a Tank

Laminar Flow Fluid Mechanics, Berman [23], Robinson [272],
in a Porous Pipe Space Plasmas Terrill [290, 291],
Terrill & Thomas [292],
Grundy & Allen [135]

Je¬rey-Hamel ¬‚ow Fluid Mechanics Bulakh [85]
Stagnation points Boundary Layer

Shocks in Nozzle Fluid Mechanics Adamson & Richey [2]

Slow Viscous Flow Past Fluid Mechanics Proudman & Pearson [264],
Circle, Sphere (Log & Power Series) Chester & Breach [98]
Skinner [283]

Log-and-Power Series Fluids, Electrostatic Ward,Henshaw

Log-and-power series Elliptic PDE on Lange&Weinitschke[179]
Domains with Small Holes

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4 John P. Boyd

Table I. Non-Soliton Exponentially Small Phenomena (continued)

Phenomena Field References

Equatorial Kelvin Wave Meteorology, Boyd & Christidis [74, 75]
Instability Oceanography Boyd&Natarov[76]

Error: Midpoint Rule Numerical Analysis Hildebrand [143]

Radiation Leakage from a Nonlinear Optics Kath & Kriegsmann [162],
Fiber Optics Waveguide Paris & Wood [258]

Particle Channeling Condensed Matter Dumas [119, 120]
in Crystals Physics

Island-Trapped Oceanography Lozano&Meyer [185],
Water Waves Meyer [210]

Chaos Onset: Physics Holmes, Marsden
Hamiltonian Systems & Scheurle [145]

Separation of Separatrices Dynamical Systems Hakim & Mallick [136]

Slow Manifold Meteorology Lorenz & Krishnamurthy [184],
in Geophysical Fluids Oceanography Boyd [65, 66]

Nonlinear Oscillators Physics Hu[149]

ODE Resonances Various Ackerberg&O™Malley[1]

French ducks (“canards”) Various MacGillivray&Liu

where is the perturbation parameter? And how can one break the
“Error Barrier” when the error of an optimally-truncated series is too
large to be useful?
This review o¬ers answers. The roots of hyperasymptotic theory
go back a century, and the particular example of the Stieltjes function
has been well understood for many decades as described in the books of
Olver [249] and Dingle [118]. Unfortunately, these ideas have percolated
only slowly into the community of derivers and users of asymptotic
I myself am a sinner. I have happily applied the method of multiple
scales for twenty years [67]. Nevertheless, I no more understood the
reason why some series diverge than why my son is lefthanded.

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Exponential Asymptotics

Table II. Selected Examples of Exponentially Small Quantum Phenomena

Phenomena References

Energy of a Quantum Fr¨man, [128]
ˇ´zek et al.[100]
Double Well (H2 , etc.) Ciˇ
Harrell[141, 142, 140]

Imaginary Part of Eigenvalue Oppenheimer [255],
of a Metastable Reinhardt [269] ,
Quantum Species: Hinton & Shaw [144],
Stark E¬ect Benassi et al.[18]
(External Electric Field)

Im(E): Cubic Anharmonicity Alvarez [6]
ˇ iˇ
C´zek & Vrscay [101]
Im(E): Quadratic Zeeman E¬ect
(External Magnetic Field)

Transition Probability, Berry & Lim [42]
Two-State Quantum System
(Exponentially Small in
Speed of Variations)

Width of Stability Bands Weinstein & Keller
for Hill™s Equation [313, 314]

Above-the-Barrier Pokrovskii
Scattering & Khalatnikov [262]
Hu&Kruskal[152, 150, 151]

Anosov-perturbed cat map: semiclassical asymptotics Boasman&Keating[46]

In this review, we shall concentrate on teaching by examples. To
make the arguments accessible to a wide readership, we shall omit
proofs. Instead, we will discuss the key ideas using the same tools of

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