Asymptotic Expansions with Oscillating CoeÆcients

In recent years, Computer Algebra has seen signi cant advances on a wide range of fronts. One of the many areas of development has been Symbolic Asymptotics. The exp{log functions are those de ned by expressions built from the rational numbers Q and the variable, x, using arithmetic operations and the functions exp and log, with the understanding that the latter is only applied to arguments which are eventually positive. Modulo diÆculties with signs of constants, algorithms exist to determine the asymptotics of exp{log functions, [11, 31, 21]. Moreover one can add integration and extraction of algebraic roots to the signature, [37], and likewise composition with functions which are given by ordinary di erential equations and which are meromorphic at the limit, [35]. Inverse functions can be handled, [28], and expansions of implicit functions can be obtained, [27, 39]. In addition there is substantial progress with Hardy{ eld solutions of di erential equations, [33, 36, 39]. Practical development has been slower to follow, but there are now implementations of the exp{log algorithm in Maple, by Dominik Gruntz [14], and in Aldor by James Beaumont. Moreover the multiseries algorithm for inverse functions has been implemented in Maple [28] and used to give new results in combinatorics.