Computation of exponential integrals of a complex argument

Abstract

Previous work on exponential integrals of a real argument has produced a computational algorithm which implements backward recurrence on a three-term recurrence relation (Miller algorithm). The process on which the algorithm is based involves the truncation of an infinite sequence with a corresponding analysis of the truncation error. This error is estimated by means of (asymptotic) formulas, which are not only applicable to the real line, but also extendible to the complex plane. This fact makes the algorithm extendible to the complex plane also. However, the rate of convergence decreases sharply when the complex argument is close to the negative real axis. In order to make the algorithm more efficient, analytic continuation into a strip about the negative real axis is carried out by limited use of power series. The analytic details needed for a portable computational algorithm are presented.