# Voigt functions

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##### 1: 7.19 Voigt Functions
###### §7.19(i) Definitions Figure 7.19.1: Voigt function 𝖴 ⁡ ( x , t ) , t = 0.1 , 2.5 , 5 , 10 . Magnify Figure 7.19.2: Voigt function 𝖵 ⁡ ( x , t ) , t = 0.1 , 2.5 , 5 , 10 . Magnify
##### 2: 7.21 Physical Applications
Voigt functions $\mathsf{U}\left(x,t\right)$, $\mathsf{V}\left(x,t\right)$, can be regarded as the convolution of a Gaussian and a Lorentzian, and appear when the analysis of light (or particulate) absorption (or emission) involves thermal motion effects. …
##### 4: 7.23 Tables
• Abramowitz and Stegun (1964, Table 27.6) includes the Goodwin–Staton integral $G\left(x\right)$, $x=1(.1)3(.5)8$, 4D; also $G\left(x\right)+\ln x$, $x=0(.05)1$, 4D.

• Finn and Mugglestone (1965) includes the Voigt function $H\left(a,u\right)$, $u\in[0,22]$, $a\in[0,1]$, 6S.

• ##### 5: 7.1 Special Notation
The main functions treated in this chapter are the error function $\operatorname{erf}z$; the complementary error functions $\operatorname{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $\mathcal{F}\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the Goodwin–Staton integral $G\left(z\right)$; the repeated integrals of the complementary error function $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)$; the Voigt functions $\mathsf{U}\left(x,t\right)$ and $\mathsf{V}\left(x,t\right)$. …
##### 6: Bibliography Z
• M. R. Zaghloul and A. N. Ali (2011) Algorithm 916: computing the Faddeyeva and Voigt functions. ACM Trans. Math. Software 38 (2), pp. Art. 15, 22.
• M. R. Zaghloul (2016) Remark on “Algorithm 916: computing the Faddeyeva and Voigt functions”: efficiency improvements and Fortran translation. ACM Trans. Math. Softw. 42 (3), pp. 26:1–26:9.
• ##### 7: Bibliography
• B. H. Armstrong (1967) Spectrum line profiles: The Voigt function. J. Quant. Spectrosc. Radiat. Transfer 7, pp. 61–88.
• ##### 8: Bibliography L
• A. E. Lynas-Gray (1993) VOIGTL – A fast subroutine for Voigt function evaluation on vector processors. Comput. Phys. Comm. 75 (1-2), pp. 135–142.
• ##### 9: Bibliography W
• R. J. Wells (1999) Rapid approximation to the Voigt/Faddeeva function and its derivatives. J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.
• ##### 10: 7.25 Software
###### §7.25(iii) $\operatorname{erf}z$, $\operatorname{erfc}z$, $w\left(z\right)$, $z\in\mathbb{C}$
No research software has been found for these functions. …
###### §7.25(vi) $\mathcal{F}\left(x\right)$, $G\left(x\right)$, $\mathsf{U}\left(x,t\right)$, $\mathsf{V}\left(x,t\right)$, $x\in\mathbb{R}$
No research software has been found for these functions. …