Two new approximants for the error function

Abstract

In this paper, we present two new approximants for the Error Function. The starting point for obtaining them, is to use two alternative integral representations involving improper integrals. Both integrands include the function. Therefore, by replacing by its truncated Taylor's expansion, we obtain a rational approximant for which converges, for each x, to that function. Since these Taylor polynomials have simple roots, the improper integrals can be evaluated with the residues technique of integration in the complex plane, by using an appropriate contour of integration. By just using the roots of the polynomials, we get two new analytical expressions for the Error Function in terms of elementary functions. We show the behaviour of their corresponding errors by giving practical bounds for the absolute and relative errors, respectively.