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Once in a while, I can't help it to ask very broad questions. I have read (a portion of) Hardy's Divergent Series. Back then, I think besides in mathematics, divergent series and the need to assign values to them hadn't arisen. But nowadays, for example, almost all explanations of the 26 dimensions of Bosonic string theory deals with such an assignment.

Also, the internet literally exploded on Youtube explaining these kinds of issues (they conveniently left out all the proofs). Questions:

  • Are there (more modern) references for the mathematical theory of the summability of divergent series, and its philosophy (I.e. usage of such things in physics) ?
  • I have seen divergent series of real numbers sum to a complex number with non-zero imaginary part. Is there something more to it (for example, are there known divergent series that lead to a quaternion) ?

I am tagging this with meta, biglist and analysis, for obvious reasons.

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As far as the use in physics is concerned, you may want to look into these references:

The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series. This book by John Boyd gives a nice non-technical overview of the modern theory of asymptotic series. You can then of course look in the references given in this book to study the rigorous details of any particualr topic.

Igor Suslov has written a review article on divergent series as it is used in quantum field theory. Like the book by Boyd, this is also quite accessible to the general public. It contains a lot of technical details of some of the popular resummation methods used in QFT.

Variational perturbation theory (also mentioned in Suslov's article), is a method to extract the asymptotic behavior of f(g) for large g (the so-called string coupling series) when some terms of its divergent perturbation series in powers of g can be computed (the so-called weak coupling series). Here you do need the underlying theory that generates the series, it's not a simple resummation method that only operates on a given divergent series. This method was developed by Hagen Kleinert, he has written this review article on this topic.

Zinn-Justin's order dependent mapping method is also a well known resummation method. You can also use this method to derive the strong coupling series from a given weak coupling series. It amounts to applying a conformal transform to the series, the conformal transform has one or more free parameters in it and you choose these parameters such that the last few terms of the series become zero. It can be argued that this yields optimal resummations (e.g. you can interpret is as optimal truncation, but instead of summing to the term of least magnitude you are making the last known terms equal to zero). The conformal mapping is thus order dependent as it depends on the order of the series.

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