From Divergent Power Series to Analytic Functions Theory and Applica
Multisummability is a method which, for certain formal power series with radius of convergence equal to zero, produces an analytic function having the formal series as its asymptotic expansion. This book presents the theory of multisummabi- lity, and as a
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Werner Balser
From Divergent Power Series to Analytic Functions Theory and Application of Multisurnrnable Power Series
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Werner Balser Mathematik V Universitat DIm D-89069 Ulm, Germany
Mathematics Subject Classification (1991): 34E05, 34A25, 34A34, 30B 10, 30E15, 44AI0
ISBN 3-540-58268-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-58268-1 Springer-Verlag New York Berlin Heidelberg CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Printed in Germany Typesetting: Camera ready by author SPIN: 10130205 46/3140-543210 - Printed on acid-free paper
Preface Since the second half of the last century, asymptotic expansions have been an important and very successful tool to understand the structure of solutions of ordinary and partial differential (or difference) equations. The by now classical part of this theory has been presented in many books on differential equations in the complex plane or related topics, by such distinguished authors as Wolfgang Wasow [Wa], Yasutaka Sibuya [Si], and many others. In my opinion, the most important result in this context is (in Wasow's terminology) the Main Asymptotic Existence Theorem: it states that to every formal solution of a differential equation, and every sector (in the complex plane) of sufficiently small opening, one can find a solution of the equation having the formal one as its asymptotic expansion. This solution, in general, is not uniquely determined, and the proofs given for this theorem (in various degrees of generality) do not provide a truly efficient way to compute such a solution, say, in terms of the formal solution. In fact, to prove this result, even for linear, but in particular nonlinear equations, and to determine sharp bounds for the opening of the sector (or more generally, determine size and location of all sectors for which the theorem holds, for a given equation with "generic Stokes phenomenon") is not an easy task and has kept researchers busy until very recently; see, e.g., Ramis and Sibuya's paper on Hukuhara domains [RS 1] of 1989, or Wolfgang Jurkat's discussion of Asymptotic Sectors [Ju 1]. In the general theory of asymptotic expansions, the analogue to the Main Asymptotic Existence Theorem is usually called Ritt's Theorem, and is much easier to prove: Given any formal power series and any sector of arbitrary (but finite) opening (on t
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