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This series is divergent; therefore, we may be able to do something with it. -- Oliver Heaviside

[Edit (1/14/21) from the answer by Count Iblis to a recent MO-Q on math vids: An enthusiastic intro is that to the set of lectures by Carl Bender "Perturbation and Asymptotic Series." ]

Other than the usual references given in Wikipedia and Mathworld, which resources have you found helpful as intros to the topic and for advanced exploration?

I'll prime the pump with

  1. "Divergent series:taming the tails" by M. V. Berry and C. J. Howls (cf. also refs in this MO-Q)

  2. Sporadic examples in Heaviside's publications, see Heaviside's Operational Calculus, a post by Ron Doerfler.

  3. A Singular Mathematical Promenade by Etienne Guys

  4. Sum Divergent Series by the user mnoonan, a series of posts at The Everything Seminar

  5. "Euler's constant: Euler's work and modern developments" by Jeffrey Lagarias

  6. "Uniform asymptotic methods for integrals" by Nico Temme

  7. "On the Specialness of Special Functions (The Nonrandom Effusions of the Divine Mathematician)" by Robert W. Batterman

For one example of the importance of such series, see the relation between the Harer-Zagier formula and the asymptotic expansion of the digamma function in Chapter 5 "The Euler characteristic of the moduli space of curves" of the course notes "Mathematical ideas and notions of quantum field theory" by Etingof.

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  • $\begingroup$ Hardy's book Divergent Series? Or is that one of the usual references? $\endgroup$
    – Robert Furber
    May 10, 2020 at 14:37
  • $\begingroup$ @Robert Furber, yep, under asymptotic series in Wiki. (Divergent Series has been google hijacked by Hollywood). Years ago when I had an excellent home library, I saved some early paper by Hardy in which he expressed what I have called The Hardy Heuristic. Goes something like: Apply two operations consecutively in one order then reverse the order. If one order is convergent and the other divergent, you have a summation method. If you can find that ref, would be a good one. Lost my library and have no access to a good University lib myself. $\endgroup$
    – Tom Copeland
    May 10, 2020 at 21:05
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    $\begingroup$ Écalle, Malgrange and Ramis work on Gevrey series may be also a good track. $\endgroup$
    – Loïc Teyssier
    Jun 5, 2020 at 8:10
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    $\begingroup$ I would add the tauberian theory bible of J Korevaar; there is also a survey of complex Tauberian theory by the author here (pdf linked) jointmathematicsmeetings.org/bull/2002-39-04/… $\endgroup$
    – Conrad
    Jun 5, 2020 at 12:17
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    $\begingroup$ As was expressed the tauberian theorems are very important. On the side of divergent series I know from an informative point of view Kolmogorov studied divergent series (see History from the Wikipedia Carleson's theorem). Myself attempt of learning is to search and study concise articles that I can understand about divergent series, for example R. P. Agnew, A Slowly Divergent Series, The American Mathematical Monthly, Vol. 54, No. 5 (May, 1947) pp. 273-274, or T. S. Nanjundiah, Extensions of Olivier's Theorem, The American Mathematical Monthly, Vol. 76, No. 6 (Jul, 1969) pp. 666-667. $\endgroup$
    – user142929
    Jun 18, 2020 at 12:11

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As far as on-line-available things go, I've attempted to modernize some arguments and give examples of asymptotics of integrals (both Watson's Lemma and easy Laplace/saddle-point examples), as well as asymptotics for ordinary differential equations, both regular and certain irregular singular points. On-line, as well as a chapter in my Cambridge Univ Press book of 2018 (http://www.math.umn.edu/~garrett/m/v/current_version.pdf). For earlier, separate treatments, see http://www.math.umn.edu/~garrett/m/mfms/notes_2019-20/05e_asymptotics_of_integrals.pdf, http://www.math.umn.edu/~garrett/m/mfms/notes_2013-14/11b_reg_sing_pt.pdf, and http://www.math.umn.edu/~garrett/m/mfms/notes_2013-14/11c_irreg_sing_pt.pdf. Those notes (and the book, on-line or not) have substantial bibliographic/historical references.

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  • $\begingroup$ Nice lists of references. I'm wondering if you could contribute to the somewhat related MO-Q mathoverflow.net/questions/397238/…. $\endgroup$
    – Tom Copeland
    Oct 4, 2021 at 15:44
  • $\begingroup$ @TomCopeland, I'll take a look... :) $\endgroup$
    – paul garrett
    Oct 4, 2021 at 15:46
  • $\begingroup$ @TomCopeland, although rigorous notions of "asymptotic expansion" might put that other question and its answers into a broader context, it doesn't seem that that's the main issue in that question. $\endgroup$
    – paul garrett
    Oct 4, 2021 at 16:59

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