Shifted rectangular quadrature rule approximations to Dawson's integral \(F(x)\) (Q1298502)

Dawson's integral \(F(x)= e^{-x^2}\int^x_0 e^{t^2}dt\) is encountered in physical problems, such as the calculation of absorption line profiles in astrophysics [cf. \textit{D. G. Hummer}, Mon. Not. R. Astron. Soc. 125, 21-37 (1962; Zbl 0111.46302)]. \(F(x)\) is an analytic, odd function that vanishes at \(x= 0\) with Maclaurin series expansion. Since \textit{H. G. Dawson} [Lond. M. S. Proc. 29, 519-522 (1898; JFM 29.0245.02)], several additional methods have been developed for accurate numerical computation of \(F(x)\). The techniques based on approximation theory and analytic integration have also been used to express \(F(x)\) as a rapidly convergent series of exponential functions. In this paper, the author considers two well-known exponential series approaches: (i) The series approach based on Dawson's original work which obtains the approximation \[ F(x)\approx \pi^{-1/2} \Biggl[2hxe^{-x^2}+ \sum_{n\text{ even}}n^{-1} e^{-(x- nh)^2}\Biggr].\tag{1} \] (ii) A more recent series approach based on Sinc approximation: \textit{Rybicki} [Comput. Phys. 3, 85-87 (1989)] obtained the approximation \[ F(x)\approx \pi^{-1/2} \sum_{n\text{ odd}}n^{-1} e^{-(x- nh)^2}.\tag{2} \] It is shown how Rybicki's approximation (2) and Dawson's approximation (1) can both be easily obtained by approximating a certain improper integral with a primitive quadrature rule. This unifying observation is then used to derive rigorous error bounds for (1) and (2) which turn out to be identical.