Resommation des series formelles

Abstract

Resummation of formal seriesSolutions of second order linear complex ordinary differential equations near singularities Summary. Formulae for solutions of complex ordinary differential equations in the neighbourhood of irregular singularities contain almost every time divergent series. The Resummation Theory developed in the field of Analytic Functional Equations by J.P. Ramis provides us with a tool-box to perform in different ways some effective calculations and to compare their results. We take full advantage of the possibilities of Computer Algebra, especially of exact rational evaluation.Les travaux r6eents de B. Malgrange, J.P. Ramis, J. Della Dora, C. Dicrescenzo, D. Duval, et E. Tournier ont permis de "calculer" ees solutions par des algorithmes formels. I1 6tait indispensable de calculer ces solutions "exactement", cause de leur comportement 6ventuellement explosif. D'autre part, les 6tudes r6centes de J.P. Ramis, prolongeant les travaux classiques de Borel, Nevanlinna, Watson .... ont permis de passer du traitement de type "sommation au plus petit terme des s6ries asymptotiques" dans de "petits secteurs" du plan complexe /t des resommations dans des "grands secteurs" des s6ries a(x) en g6n6ral divergentes. Cette resommation est vraiment effective quand ai(x ) est une s6rie ksommable, si keQ. Ceci est toujours le cas pour les 6quations du second ordre, notamment pour les 6quations des fonctions sp6ciales. Par contre, dans le cas

Exact semiclassical expansions for one-dimensional quantum oscillators

“…Actually many formal and numerical procedure for resummation are available, see for instance Ref. 42. In this way it would be interesting to compare the numerical computations based on Padé approximants, 41,40 with the powerful hyperasymptotics methods of Refs.…”

Comparison between Borel-Padé summation and factorial series, as time integration methods

“…In fact, Sû(t) can be represented by a more general convergent series, namely by a generalised factorial series (GFS) [44,41]:…”

“…The GFS converges for Re(1/t) > 1/T and s n = nω e −iθ for any ω > ω 0 and some ω 0 > 0. ω 0 can be linked to the location of the closest singularity of P in the complex Borel plane. It is said that, numerically, the convergence of the GFS is more and more rapid as ω get close to ω 0 [41]. For linear equations, the closest singularity can be "read" from the coefficients of the equations and can be evaluated algorithmically.…”

“…Borel summation of a divergent but Gevrey seriesû consists in (a) computing the Borel transform Bû ofû which is a convergent series, (b) prolonging Bû along in a 394 AHMAD DEEB, A. HAMDOUNI AND DINA RAZAFINDRALANDY domain containing a semi-line from the origin, and (c) taking the Laplace transform to go back to the physical space [36,41]. At the end, one gets the so-called Borelsum ofû which is an holomorphic function (in some domain) and is an actual solution of the equation.…”

Summation of divergent power series by means of factorial series

“…and Ramis and Thomann [36]. Thomann [42,43] discussed the summation of formal power series with the help of generalizations of factorial series, where the Pochhammer symbols (z) n+1 = z(z + 1) . .…”