Non-trivial Asymptotic Formulas by Symbolic Computation

Giuliano Gargiulo
Saverio Salerno

Abstract
We want to study the asymptotic behaviour of the complete elliptic integral of the first kind K(m)  when m->1. This is motivated, for example, by the occurrence of  K(m)  as capacitance of a circular capacitor with slit (or similar geometries - see e.g. [9]) -, m being essentially the ratio between the slit's length and the radius of the circle. We show that the analysis of the asymptotic behaviour can be done in several ways (including series expansion and summation, symbolic integration and computation of limits) using both the numerical and symbolical capabilities of state-of-the-art Computer Algebra Systems.