Shifted rectangular quadrature rule approximations to Dawson’s integral \(F(x)\). (English) Zbl 0933.65020
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. 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 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: 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.
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: 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.
Reviewer: C.L.Parihar (Indore)
MSC:
| 65D20 | Computation of special functions and constants, construction of tables |
| 33B20 | Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) |
Keywords:
rectangular quadrature rule; exponential series approximation; Dawson’s integral; Maclaurin series expansion; JFM 29.0245.02; exponential series; Sinc approximation; error boundsSoftware:
CephesReferences:
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