Acta Applicandae Mathematica

, Volume 56, Issue 1, pp 1–98

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

  • John P. Boyd


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 methods asymptotic hyperasymptotic exponential smallness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

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

Personalised recommendations