Assuming that V(x) ≈ (1 - μ) G_1(x) + μ L_1(x) is a very good approximation of the Voigt function, in this work we analytically find μ from mathematical properties of V(x). G_1(x) and L_1(x) represent a Gaussian and a Lorentzian function, respectively, with the same height and HWHM as V(x), the Voigt function, x being the distance from the function center. In this paper we extend the analysis that we have done in a previous paper, where μ is only a function of a; a being the ratio of the Lorentz width to the Gaussian width. Using one of the differential equation that V(x) satisfies, in the present paper we obtain μ as a function, not only of a, but also of x. Kielkopf first proposed μ (a, x) based on numerical arguments. We find that the Voigt function calculated with the expression μ (a, x) we have obtained in this paper, deviates from the exact value less than μ(a) does, specially for high |x| values.
Assuming that V(x) ≈ (1 - μ) G_1(x) + μ L_1(x) is a very good approximation of the Voigt function, in this work we analytically find μ from mathematical properties of V(x). G_1(x) and L_1(x) represent a Gaussian and a Lorentzian function, respectively, with the same height and HWHM as V(x), the Voigt function, x being the distance from the function center. μ is obtained as a function of a, a being the ratio of the Lorentz width to the Gaussian width. We find that, the Voigt function calculated with the expression we have obtained for μ, deviates from the exact value less than 0.5% with respect to the peak value.
In this work we present the explicit representations of the Voigt function K(a,b) (the convolution between a Gaussian and a Lorentzian function), the function N(a,b) defined as the convolution of Gaussian and dispersion distributions as well as the complex error function erf(a+ib), all in terms of the Kummer functions M(α,γ,a^2). The expansions are valid for all values of the parameter a (the relation between Lorentzian and Gaussian widths at the half maxima). Previous analytical works were known only when the parameter a≤1, or were based on numerical interpolations or empirical approximations. Also, new series and asymptotic expansions are presented.
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