Introduction to 1-summability and resurgence
By D. Sauzin. 2014
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This text is about the mathematical use of certain divergent power series. The rst part is an introduction to 1-summability. The denitions rely on the formal Borel transform and the Laplace transform along an arbitrary direction of the complex plane. Given an arc of directions, if a power series is 1-summable in that arc, then one can attach to it a Borel-Laplace sum, i.e. a holomorphic function dened in a large enough sector and asymptotic to that power series in Gevrey sense. The second part is an introduction to Ecalle’s resurgence theory. A power series is said to be resurgent when its Borel transform is convergent and has good analytic continuation properties: there may be singularities but they must be isolated. The analysis of these singularities, through the so-called alien calculus, allows one to compare the various BorelLaplace sums attached to the same resurgent 1-summable series. In the context of analytic dierence-or-dier ential equations, this sheds light on the Stokes phenomenon. A few elementary or classical examples are given a thorough treatment (the Euler series, the Stirling series, a less known example by Poincar e). Special attention is devoted to non-linear operations: 1-summable series as well as resurgent series are shown to form algebras which are stable by composition. As an application, the resurgent approach to the classication of tangent-to-identity germs of holomorphic dieomorphisms in the simplest case is included. An example of a class of non-linear dierential equations giving rise to resurgent solutions is also presented. The exposition is as self-contained as can be, requiring only some familiarity with holomorphic functions of one complex variable.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/f5992ba828ad0b33fb210266ddf9b906f8135a39
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- A primer on resurgent transseries and their asymptotics (2018, background): “, [196, 197, 83, 208, 177, 209, 93, 84, 104, 178, 105, 106, 189, 168, 85, 205, 42, 35]; see also [110] for a review specifically on transseries.”