Exact Asymptotics

Abstract

Singularities of linear complex differential equations is a subject with a long history. New methods, often of an algebraic nature, have kept the subject youthful and growing. In this chapter we treat the asymptotic theory of divergent solutions and the more refined theory of multisummation of those solutions. The theory of multisummation has been developed by many authors, such as W. Baiser, B.L.J. Braaksma, J. Écalle, W.B. Jurkat, D. Lutz, M. Loday-Richaud, B. Malgrange, J. Martinet, J.-R Ramis, and Y. Sibuya. Excellent bibliographies can be found in [177] and [180]. Our aim is to give a complete proof of the multisummation theorem, based on what is called “the Main Asymptotic Existence Theorem” and some sheaf cohomology. In particular, the involved analytic theory of Laplace and Borel transforms has been avoided. However, the link between the cohomology groups and the Laplace and Borel method is made transparent in examples. This way of presenting the theory is close to that of Malgrange [195].