A numerical method for generalized exponential integrals

AbstractWe present a method for evaluation of the exponential integral, Es(x), generalized to an arbitrary order s > 0. The algorithm is valid whatever s > 0 and x > 0. In the region x ⩾ 1, we start from a proper initial value, obtained by asymptotic calculation, and then compute the required Es(x) by means of a suitable combination of Taylor's expansions and recurrences, whatever s > 0. When x < 1 the starting element used is Es0(x) (0 < s0 ⩽ 1), which is obtained by the means of suitable expansions. A forward recursion finally yields the required Es(x). Numerical stability and accuracy of the proposed algorithm are discussed and some results given