Symbolic asymptotics and the computation of limits (Q1354703)

Computer algebra has seen a great deal of development in the last fifteen years or so. However for the computation of limits and related areas of asymptotics, most systems have continued to use ad-hoc methods until quite recently. On the algorithmic level, attempts at a more systematic treatment began in the mid-to-late eighties, and these are now bearing fruit in implementations. The present author's approach to symbolic asymptotics has been founded on three areas. The first of these is zero equivalence, which is concerned with deciding whether a given expression involving transcendental functions is functionally equivalent to zero. The second is the theory of Hardy fields, which are ordered differential fields of function germs. The third ingredient required is a suitable theory of generalized asymptotic expansions. Power series are not generally sufficient for this purpose. The author's own nested expansions represent one possibility here; another would be Ecalle's transseries. The present paper presents a brief survey of these three areas and their use in the computation of limits.