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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.

Tom Copeland
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  • Hardy's book Divergent Series? Or is that one of the usual references? – Robert Furber May 10 '20 at 14:37
  • @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. – Tom Copeland May 10 '20 at 21:05
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    Écalle, Malgrange and Ramis work on Gevrey series may be also a good track. – Loïc Teyssier Jun 05 '20 at 08:10
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    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) http://www.jointmathematicsmeetings.org/bull/2002-39-04/S0273-0979-02-00951-5/S0273-0979-02-00951-5.pdf – Conrad Jun 05 '20 at 12:17
  • @Conrad, that's fitting. I found the Temme ref (or was reminded again of him) just last night in "Early work of N.G. (Dick) de Bruijn in analysis and some of my own" by Korevaar. – Tom Copeland Jun 05 '20 at 12:42
  • @LoïcTeyssier, I found "Fonctions multisommables" by Malgrange and Ramis (in which Watson's "A theory of asymptotic series" is reffed). Which papers by E, M, and R would be a good intro to their work? – Tom Copeland Jun 05 '20 at 13:02
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    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. – user142929 Jun 18 '20 at 12:11
  • I can read these for free with my account of JSTOR. – user142929 Jun 19 '20 at 06:23
  • Worth panning, but no time for me at the moment: "How Euler Did It" , an online MAA column, written by Ed Sandifer from 2003 to 2010. These are archived at http://eulerarchive.maa.org/hedi/. – Tom Copeland Jul 30 '20 at 17:06
  • Again, back to the source--the mathemage, Euler--in "Euler and his work on infinite series" by Varadarajan (https://www.ams.org/journals/bull/2007-44-04/S0273-0979-07-01175-5/S0273-0979-07-01175-5.pdf). See also https://mathoverflow.net/questions/19201/summation-methods-for-divergent-series and https://mathoverflow.net/questions/45811/use-of-everywhere-divergent-generating-functions/45868#45868. – Tom Copeland Jul 30 '20 at 17:44
  • And "Euler's 1760 paper on divergent series" by Barbeau and Leah https://www.sciencedirect.com/science/article/pii/0315086076900306 – Tom Copeland Jul 30 '20 at 18:11
  • "Convergence from Divergence" by Costin and Dunne https://arxiv.org/abs/1705.09687 – Tom Copeland Dec 04 '20 at 15:37
  • Also a translation by Aycock of Euler's "On divergent series" https://arxiv.org/abs/1808.02841 – Tom Copeland Mar 30 '21 at 21:38
  • "Transseries for beginners" by G. A. Edgar https://arxiv.org/abs/0801.4877 – Tom Copeland May 11 '21 at 16:01
  • On the iconic divergent power series with $a_n = n!$, see refs in https://oeis.org/A003319 and the blog posts https://qchu.wordpress.com/2015/11/03/the-answer-to-the-puzzle/, https://qchu.wordpress.com/2015/11/04/the-categorical-exponential-formula/, and https://tcjpn.wordpress.com/2014/12/14/the-hirzebruch-criterion-fo-the-todd-class/. – Tom Copeland May 11 '21 at 16:49
  • Related: https://mathoverflow.net/questions/258525/how-do-i-solve-this-displaystyle-f-ef-1 and discussion on formal inversion (multiplicative and compositional) of and composition with divergent series. – Tom Copeland May 11 '21 at 17:11
  • See also the book Continued Fractions by Jones and Thron. – Tom Copeland May 18 '21 at 23:28
  • "On Numbers, Germs, and Transseries" by Aschenbrenner, van den Dries, van der Hoeven https://arxiv.org/abs/1711.06936 and "Asymptotics and Borel Summability" by Costin – Tom Copeland May 20 '21 at 15:28
  • "On the Specialness of Special Functions (The Nonrandom Effusions of the Divine Mathematician)" by Batterman and the refs therein. – Tom Copeland May 29 '21 at 03:03
  • "Differential Equations: A Dynamical Systems Approach" by Hubbard and West. See the Appendix: Asymptotic Development. – Tom Copeland Sep 11 '21 at 23:26
  • Another book by Balser not mentioned in Wikipedia: "Formal Power Series and Linear Systems of Meromorphic Differential Equations." Wikipedia and MathWorld links are https://en.wikipedia.org/wiki/Asymptotic_analysis, https://en.wikipedia.org/wiki/Divergent_series, https://en.wikipedia.org/wiki/Category:Summability_methods, https://en.wikipedia.org/wiki/Category:Asymptotic_analysis, https://mathworld.wolfram.com/AsymptoticSeries.html, https://mathworld.wolfram.com/DivergentSeries.html – Tom Copeland Sep 28 '21 at 18:03
  • Fairly comprehensive intro: Quantum Field Theory II: Quantum Electrodynamics by Zeidler – Tom Copeland Oct 24 '21 at 02:18
  • "Polynomial expansions of Analytic Functions" by Boas and Buck – Tom Copeland Feb 17 '22 at 20:10
  • "From Resurgence to BPS States" by Marino https://member.ipmu.jp/yuji.tachikawa/stringsmirrors/2019/2_M_Marino.pdf – Tom Copeland Apr 28 '22 at 20:54
  • Despite the exhortations against divergent series / formal power series by Abel, since at least Laplace, such series have found important applications in physics. See, e.g., "Divergent series: past, present, future . . . " by Christiane Rousseau (https://arxiv.org/pdf/1312.5712.pdf). – Tom Copeland Sep 28 '22 at 14:22
  • See also applications in Operational Calculus (2nd ed.) by Bremmer and van der Pol. – Tom Copeland Nov 02 '22 at 03:46
  • Informal intro with some history: the 2023 video by Sir Michael Berry - Divergent Series: From Thomas Bayes's Bewilderment to Today's Resurgence (https://youtu.be/73CLvdrdKuU?si=FVt1Lq1-dCl1Fo9E) – Tom Copeland Jan 04 '24 at 20:23
  • 'Summability, Tauberian theorems, and Fourier series' a blog post at https://www.victorchen.org/2019/10/21/summability-tauberian-fourier-series/ – Tom Copeland Jan 24 '24 at 17:55

1 Answers1

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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.

paul garrett
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  • Nice lists of references. I'm wondering if you could contribute to the somewhat related MO-Q https://mathoverflow.net/questions/397238/renormalization-in-physics-vs-dynamical-systems. – Tom Copeland Oct 04 '21 at 15:44
  • @TomCopeland, I'll take a look... :) – paul garrett Oct 04 '21 at 15:46
  • @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. – paul garrett Oct 04 '21 at 16:59