Is there a complete and rigorous, yet concise, definition of what an analyzable function is, along with the related notion of accelero-summation, both in the sense of Écalle? All of the definitions I can find in the literature I find to be somewhat impenetrable. I'm familiar with transseries, so it is fine to use them in the requested definitions.
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1$\begingroup$ Can you give some context (e.g., what work of Écalle)? $\endgroup$– LSpiceCommented Aug 8, 2023 at 5:05
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$\begingroup$ If you would include, or link to, definitions you find impenetrable, we'd know what not to tell you. $\endgroup$– Gerry MyersonCommented Aug 8, 2023 at 7:51
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1 Answer
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There is this...
Dorigoni, Daniele, An introduction to resurgence, trans-series and alien calculus, Ann. Phys. 409, Article ID 167914, 38 p. (2019). ZBL1425.81003.
"In these notes we give an overview of different topics in resurgence theory from a physics point of view, but with particular mathematical flavour. After a short review of the standard Borel method for the resummation of asymptotic series, we introduce the class of simple resurgent functions, explaining their importance in physical problems. We define the Stokes automorphism and the alien derivative and discuss these objects in concrete examples using the notion of trans-series expansion. With all the tools introduced, we see how resurgence and alien calculus allow us to extract non-perturbative physics from perturbation theory. To conclude, we apply Morse theory to a toy model path integral to understand why physical observables should be resurgent functions."
answered Aug 8, 2023 at 16:17