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 be an algebraic function defined on , let be exp-log functions at infinity in , and let

We give a method to compute . Moreover, the techniques we use are stronger than this result might suggest, and we outline further applications.

Keywords: asymptotic expansion, exp-log function, oscillating function, Diophantine approximation, algorithm

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Acknowledgment: Matthias Aschenbrenner noted a small but annoying mistake in the way Lemma 6 was applied in the proof of Theorem 7. We fixed this mistake in the corrected version.