On the computation of limsups

Overview & Synopsis

In the last years, several asymptotic expansion algorithms have appeared, which have the property that they can deal with very general types of singularities, such as singularities arising in the study of algebraic differential equations. However, attention has been restricted so far to functions with “strongly monotonic” asymptotic behaviour: formally speaking, the functions lie in a common Hardy field, or, alternatively, they are determined by transseries. In this article, we make a first step towards the treatment of functions involving oscillatory behaviour. More precisely, let ϕ 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 ψ(x)=ϕ(sin (F_1(x)),…,sin (F_q(x))). We give a method to compute limsup_(x∞) ψ(x). Moreover, the techniques we use are stronger than this result might suggest, and we outline further applications.