An assessment of some closed-form expressions for the Voigt function II: Utilizing rational approximations for the Gauss function

Abstract

Rational approximations for the Gauss function can be used to construct closedform expressions of the Voigt function K(x, y) in terms of rational functions, logarithms and inverse trigonometric functions. The comparison with accurate reference values indicates a relative accuracy in the percent range for y 1, but serious problems for smaller y. Furthermore, these expressions are not competitive with other algorithms with respect to computational speed. Both accuracy and speed tests indicate that supposedly "good" approximations of the integrand do not necessarily provide good approximations of the integral, i.e. Voigt function.

Notes: An assessment of some closed-form expressions for the Voigt function III: Combinations of the Lorentz and Gauss functions

“…Our conclusions now are therefore similar to those given in Schreier [25,26]: Closed-form expressions as presented here might be desirable for certain applications, but their quality is limited. In general approximations based on modern state-of-the-art numerical methods, e.g.…”

“…Fig. 1 As discussed in Schreier [24] and in our previous assessments of simple closed-form approximations of the Voigt function [12,25,26], the range of y values encountered in (8), (15), (20) and c L defined in (21). molecular spectroscopy and atmospheric and astrophysical applications spans many orders of magnitude.…”

“…Ignoring the correction terms Eqs. ( 10), (12), and ( 17 As discussed in Schreier [24] and in our previous assessments of simple closed-form approximations of the Voigt function [12,25,26], the range of y values encountered in 8), ( 15), (20) and c L defined in (21).…”

The accuracy and precision of multi-line NO-LIF thermometry in a wide range of pressures and temperatures

“…where ν is wavenumber in unit cm −1 ,ν is the spectral position of the peak intensity obtained from LIFBASE [24], Γ G and Γ L are the Gauss and Lorentz full-width at half maximum (FWHM), respectively. Since the Voigt function cannot be expressed in finite terms of elementary functions [38], for computational purposes, various alternate approaches [39][40][41][42] have been proposed in an attempt to analytically approximate the complex function.…”

“…Schreier [38,43,44] has extensively reviewed this subject. The Voigt function is a powerful tool that has important applications in many fields of science.…”