Divergent Series, Summability and Resurgence I: Monodromy by Claude Mitschi, David Sauzin
By Claude Mitschi, David Sauzin
Providing an trouble-free advent to analytic continuation and monodromy, the 1st a part of this quantity applies those notions to the neighborhood and worldwide examine of complicated linear differential equations, their formal options at singular issues, their monodromy and their differential Galois teams. The Riemann-Hilbert challenge is mentioned from Bolibrukh’s perspective.
the second one half expounds 1-summability and Ecalle’s concept of resurgence lower than relatively common stipulations. It comprises a variety of examples and provides an research of the singularities within the Borel airplane through “alien calculus”, which gives a whole description of the Stokes phenomenon for linear or non-linear differential or distinction equations.
the 1st of a chain of 3, entitled Divergent sequence, Summability and Resurgence, this quantity is geared toward graduate scholars, mathematicians and theoretical physicists drawn to geometric, algebraic or neighborhood analytic homes of dynamical structures. It contains beneficial routines with recommendations. the necessities are a operating wisdom of straightforward advanced research and differential algebra.
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Extra info for Divergent Series, Summability and Resurgence I: Monodromy and Resurgence
This proves that the map s → f γs is locally constant, b hence constant on [0, 1]. In particular f γ 0 ∼ f γ1 . 11. 15. Let γ be a path in C and t 0 = 0 < t1 < . . < tn = 1, a subdivision of [0, 1]. Assume that for each i = 0, . . , n there is an open disk Vi with center γ(ti ) such that Vi ∩ Vi+1 = 0/ and γ([ti ,ti+1 ]) ⊂ Vi for i ≤ n − 1. Let ( fi )0≤i≤n be a family of functions fi holomorphic in Vi and such that fi = fi+1 on Vi ∩ Vi+1 . 8, and f 0 = fn . Fig. 4 Proof. (See Fig. 4) For each t ∈ [0, 1] we have t ∈ [ti ,ti+1 ], hence γ(t) ∈ Vi for some i.
N there is an open disk Vi with center γ(ti ) such that Vi ∩ Vi+1 = 0/ and γ([ti ,ti+1 ]) ⊂ Vi for i ≤ n − 1. Let ( fi )0≤i≤n be a family of functions fi holomorphic in Vi and such that fi = fi+1 on Vi ∩ Vi+1 . 8, and f 0 = fn . Fig. 4 Proof. (See Fig. 4) For each t ∈ [0, 1] we have t ∈ [ti ,ti+1 ], hence γ(t) ∈ Vi for some i. Let Ut be the open disk with center γ(t) and maximal radius of convergence for the series expansion of fi at γ(t), and gt the sum in Ut of this series. In particular, gti = fi in Vi for each i, and the family of pairs ((Ut , gt ))t∈[0,1] clearly defines a continuation of the pair (U 0 , f 0 ) resulting in the pair (U1 , g1 ) = (U1 , fn ).
7. Let α1 , . . , αm ∈ C∗ . Consider the differential field L = C(x, eα1 x , . . , eαm x ) endowed with the usual derivation ∂ such that ∂ x = 1, ∂ eα1 x = α1 eα1 x , . . , ∂ eαm x = αm eαm x . Show that the differential field L is a Picard-Vessiot extension (a) of C(x) , (b) of C. In each case write an explicit differential system ∂ y = Ay for which L is a PicardVessiot extension of the base-field. 8. Let (k, ∂ ) be a differential field as above, C its field of constants and (L, ∂ ) a Picard-Vessiot extension of k with respect to a linear differential system(S).