Acta Applicandae Mathematica

, Volume 56, Issue 1, pp 1–98

The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series

  • John P. Boyd

DOI: 10.1023/A:1006145903624

Cite this article as:
Boyd, J.P. Acta Applicandae Mathematicae (1999) 56: 1. doi:10.1023/A:1006145903624


Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε 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 different 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.

perturbation methodsasymptotichyperasymptoticexponential smallness

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • John P. Boyd
    • 1
  1. 1.University of MichiganAnn ArborU.S.A.