4
$\begingroup$

I have found many references to Poincaré and Borel in relation to their work on asymptotic series, but so far, every source I can get my hands on is very old, hence hard to read (this is not true in general, but in this case, texts that predate Oh notation tend not to be clear).

Can you explain the idea behind asymptotic series, give an illuminating example, and/or suggest a good modern exposition of the theory?

$\endgroup$

3 Answers 3

4
$\begingroup$

Web pages of

Michael Berry: http://www.phy.bris.ac.uk/people/berry_mv/dingle.html

and

John Boyd: http://www-personal.umich.edu/~jpboyd/

both include helpful published and publicly available sources.

$\endgroup$
2
  • 1
    $\begingroup$ Basically what I was hoping for... Thanks! $\endgroup$ May 2, 2013 at 12:10
  • $\begingroup$ @user25199: Could you check the link to Michael Berry? The link in your ans sends me to some wrong place. $\endgroup$ Aug 28, 2019 at 20:36
4
$\begingroup$

There are many modern books, for example,

MR1317343 Balser, Werner From divergent power series to analytic functions. Theory and application of multisummable power series. Lecture Notes in Mathematics, 1582. Springer-Verlag, Berlin, 1994.

MR1250603 Candelpergher, B.; Nosmas, J.-C.; Pham, F. Approche de la résurgence. Actualités Mathématiques. Hermann, Paris, 1993.

The very basic idea is the following: You frequently obtain divergent series, a) as formal solutions of differential (or functional) equations, b) as perturbation series when you vary a linear operator.

The question is whether these series have any relation to actual solutions of the problem. It often turns out that they are asymptotic series, and moreover, that they are "Borel summable". Borel summation is a procedure using a form of Laplace transform that under certain conditions recovers the function from its formal asymptotic series.

$\endgroup$
3
$\begingroup$

In N. G. de Bruijn's Asymptotic Methods in Analysis (link), Introduction, pp. 11-19, you may find on this topic the following sections, which use the Bachmann-Landau O-notation:

  • 1.5 Asymptotic series,
  • 1.6 Elementary operations on asymptotic series, and
  • 1.7 Asymptotics and Numerical Analysis.

This is a 1981 Dover Book republication of the North-Holland Publishing Co. 1970 edition; 1st edition 1958.

The classic treatise is the 1949 book by G. H. Hardy's, Divergent Series, reprinted in 1991 by The AMS Chelsea Publishing, 2nd. edition (link).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.