ON THE COMPUTATION OF LIMSUPS

In the last years, several asymptotic expansion algorithms have appear ed, which have the property that they can deal with very general types of singularities, such as singularities arising in the study of algeb raic differential equations. However, attention has been restricted so far to functions with ''strongly monotonic'' asymptotic behaviour: fo rmally speaking, the functions lie in a common Hardy field, or, altern atively, they are determined by transseries. In this article, we make a first step towards the treatment of functions involving oscillatory behaviour. More precisely, let phi be an algebraic function defined on [-1, 1](q), let F-1(x),...,F-q(x) be exp-log functions at infinity in x, and let psi(x) = phi(sin(F-1(x)),...,sin(F-q(x))). We give a metho d to compute lim sup(x-->infinity) psi(x). Moreover, the techniques we use are stronger than this result might suggest, and we outline furth er applications. (C) 1997 Elsevier Science B.V.