By Michèle Loday-Richaud
Addressing the query the best way to “sum” an influence sequence in a single variable while it diverges, that's, how you can connect to it analytic capabilities, the quantity provides solutions by means of proposing and evaluating some of the theories of k-summability and multisummability. those theories practice particularly to all suggestions of standard differential equations.
The quantity comprises purposes, examples and revisits, from a cohomological viewpoint, the gang of tangent-to-identity germs of diffeomorphisms of C studied in quantity 1. to be able to employing the theories to ideas of differential equations, an in depth survey of linear traditional differential equations is supplied, together with Gevrey asymptotic expansions, Newton polygons, index theorems and Sibuya’s evidence of the meromorphic class theorem that characterizes the Stokes phenomenon for linear differential equations.
This quantity is the second one in a chain of 3, entitled Divergent sequence, Summability and Resurgence. it's aimed toward graduate scholars and researchers in arithmetic and theoretical physics who're attracted to divergent sequence, even if heavily on the topic of the opposite volumes, it may be learn independently.
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Extra resources for Divergent Series, Summability and Resurgence II: Simple and Multiple Summability
0 denote the sector =](α + 2 jπ)/r, (β + 2 jπ)/r[ × ]0, R1/r [ so that as t runs through through . 14 (Gevrey Asymptotics and Rank Reduction) The following two assertions are equivalent: (i) for = 0, . . , r − 1 the series g(t) is an s -Gevrey asymptotic series on /r (in the variable t); (ii) for j = 0, . . , r −1 the r-rank reduced series g j (x) is an s r-Gevrey asymptotic series on (in the variable x = t r ). These results allows us to limit the study of Gevrey asymptotics to small values of s (s ≤ s0 ) or to large ones (s ≥ s1 ) at convenience.
From the fact that | 2n+1 | ≥ 2n n! and | 2n | ≥ 2n (n − 1)! we know that, if the series ˜(z) (cf. Exa. 8, p. 10) is of Gevrey type then it is at least 1/2-Gevrey. From the fact that its Borel transform is convergent it is of Gevrey type and at most 1-Gevrey. Note however that its Borel transform is an entire function and consequently, ˜(z) could be less than 1-Gevrey. 4 C[[x]]s is a differential subalgebra of C[[x]] stable under composition. Proof. C[[x]]s is clearly a vector subspace of C[[x]].
Indeed, the function f (x) − a0,x0 + x 0 1+ux sion and is 1-Gevrey asymptotic to the Euler series E(x) on . Then, it must be equal to the Euler function E(x) which has the same properties. Stokes phenomenon. 1), p. 4 by dividing x and then, differentiating once. Since the equation has no singular point but 0 (and inﬁnity) the Cauchy-Lipschitz theorem allows one to analytically continue the Euler function along any path which avoids 0 and then in particular, outside of the sector −3π/2 < arg(x) < +3π/2.