Asymptotic Laplace transforms

In this work we discuss certain aspects of the classical Laplace theory that are relevant for an entirely analytic approach to justify Heaviside\u27s operational calculus methods. The approach explored here suggests an interpretation of the Heaviside operator ${cdot}$ based on the Asymptotic Laplace Transform. The asymptotic approach presented here is based on recent work by G. Lumer and F. Neubrander on the subject. In particular, we investigate the two competing definitions of the asymptotic Laplace transform used in their works, and add a third one which we suggest is more natural and convenient than the earlier ones given. We compute the asymptotic Laplace transforms of the functions $tmapsto e^{t^n}$ for $nin N$ and we show that elements in the same asymptotic class have the same asymptotic expansion at $infty.$ In particular, we present a version of Watson\u27s Lemma for the asymptotic Laplace transforms