# §7.19 Voigt Functions

## §7.19(i) Definitions

For $x\in\mathbb{R}$ and $t>0$,

 7.19.1 $\mathsf{U}\left(x,t\right)=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}\frac% {e^{-(x-y)^{2}/(4t)}}{1+y^{2}}\mathrm{d}y,$ ⓘ Defines: $\mathsf{U}\left(\NVar{x},\NVar{t}\right)$: Voigt function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable Referenced by: §7.19(iii) Permalink: http://dlmf.nist.gov/7.19.E1 Encodings: TeX, pMML, png See also: Annotations for §7.19(i), §7.19 and Ch.7
 7.19.2 $\mathsf{V}\left(x,t\right)=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}\frac% {ye^{-(x-y)^{2}/(4t)}}{1+y^{2}}\mathrm{d}y.$ ⓘ Defines: $\mathsf{V}\left(\NVar{x},\NVar{t}\right)$: Voigt function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable Referenced by: §7.19(iii) Permalink: http://dlmf.nist.gov/7.19.E2 Encodings: TeX, pMML, png See also: Annotations for §7.19(i), §7.19 and Ch.7
 7.19.3 $\mathsf{U}\left(x,t\right)+i\mathsf{V}\left(x,t\right)=\sqrt{\frac{\pi}{4t}}e^% {z^{2}}\operatorname{erfc}z,$ $z=(1-ix)/(2\sqrt{t})$.
 7.19.4 $H\left(a,u\right)=\frac{a}{\pi}\int_{-\infty}^{\infty}\frac{e^{-t^{2}}\mathrm{% d}t}{(u-t)^{2}+a^{2}}=\frac{1}{a\sqrt{\pi}}\mathsf{U}\left(\frac{u}{a},\frac{1% }{4a^{2}}\right).$ ⓘ Defines: $H\left(\NVar{a},\NVar{u}\right)$: line-broadening function Symbols: $\mathsf{U}\left(\NVar{x},\NVar{t}\right)$: Voigt function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm and $\int$: integral Permalink: http://dlmf.nist.gov/7.19.E4 Encodings: TeX, pMML, png See also: Annotations for §7.19(i), §7.19 and Ch.7

$H\left(a,u\right)$ is sometimes called the line broadening function; see, for example, Finn and Mugglestone (1965).

## §7.19(iii) Properties

 7.19.5 $\displaystyle\lim_{t\to 0}\mathsf{U}\left(x,t\right)$ $\displaystyle=\frac{1}{1+x^{2}},$ $\displaystyle\lim_{t\to 0}\mathsf{V}\left(x,t\right)$ $\displaystyle=\frac{x}{1+x^{2}}.$ ⓘ Symbols: $\mathsf{U}\left(\NVar{x},\NVar{t}\right)$: Voigt function, $\mathsf{V}\left(\NVar{x},\NVar{t}\right)$: Voigt function and $x$: real variable Referenced by: §7.19(iii) Permalink: http://dlmf.nist.gov/7.19.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.19(iii), §7.19 and Ch.7
 7.19.6 $\displaystyle\mathsf{U}\left(-x,t\right)$ $\displaystyle=\mathsf{U}\left(x,t\right),$ $\displaystyle\mathsf{V}\left(-x,t\right)$ $\displaystyle=-\mathsf{V}\left(x,t\right).$ ⓘ Symbols: $\mathsf{U}\left(\NVar{x},\NVar{t}\right)$: Voigt function, $\mathsf{V}\left(\NVar{x},\NVar{t}\right)$: Voigt function and $x$: real variable Permalink: http://dlmf.nist.gov/7.19.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.19(iii), §7.19 and Ch.7
 7.19.7 $\displaystyle 0$ $\displaystyle<\mathsf{U}\left(x,t\right)\leq 1,$ $\displaystyle-1$ $\displaystyle\leq\mathsf{V}\left(x,t\right)\leq 1.$ ⓘ Symbols: $\mathsf{U}\left(\NVar{x},\NVar{t}\right)$: Voigt function, $\mathsf{V}\left(\NVar{x},\NVar{t}\right)$: Voigt function and $x$: real variable Referenced by: §7.19(iii) Permalink: http://dlmf.nist.gov/7.19.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.19(iii), §7.19 and Ch.7
 7.19.8 $\displaystyle\mathsf{V}\left(x,t\right)$ $\displaystyle=x\mathsf{U}\left(x,t\right)+2t\frac{\partial\mathsf{U}\left(x,t% \right)}{\partial x},$ 7.19.9 $\displaystyle\mathsf{U}\left(x,t\right)$ $\displaystyle=1-x\mathsf{V}\left(x,t\right)-2t\frac{\partial\mathsf{V}\left(x,% t\right)}{\partial x}.$

## §7.19(iv) Other Integral Representations

 7.19.10 $\mathsf{U}\left(\frac{u}{a},\frac{1}{4a^{2}}\right)=a\int_{0}^{\infty}e^{-at-% \frac{1}{4}t^{2}}\cos\left(ut\right)\mathrm{d}t,$
 7.19.11 $\mathsf{V}\left(\frac{u}{a},\frac{1}{4a^{2}}\right)=a\int_{0}^{\infty}e^{-at-% \frac{1}{4}t^{2}}\sin\left(ut\right)\mathrm{d}t.$