Limits of Liouvillian Functions

Abstract

We are concerned with the calculation of limits of real-valued functions, and more generally with their asymptotic growth. One source of difficulties is that, for example, two large functions may cancel to give a much smaller function. We present a general method for handling such cancellation problems. As is normal in this area, we have to assume the existence of an oracle which determines the sign of any constants that we meet. On that basis we show how to compute the asymptotic forms of real Liouvillian functions. That is to say, elements of a field given by a tower of extensions of the basic constants by integrals, exponentials and real algebraic functions. Our method centres on the concept of an asymptotic field, in which difficulties caused by cancellation can be resolved. We show how to pass from a given asymptotic field to an elementary extension. The asymptotic expressions we derive will, in general, contain a finite number of arbitrary constants of integration. In addition, we provide different asymptotic expressions corresponding to different branches of algebraic functions. We do not determine the arbitrary constants or branches by specifying values at finite points.