# §7.19 Voigt Functions

## §7.19(i) Definitions

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

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

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

## §7.19(iii) Properties

 7.19.5 $\displaystyle\lim_{t\to 0}\mathop{\mathsf{U}\/}\nolimits\!\left(x,t\right)$ $\displaystyle=\frac{1}{1+x^{2}},$ $\displaystyle\lim_{t\to 0}\mathop{\mathsf{V}\/}\nolimits\!\left(x,t\right)$ $\displaystyle=\frac{x}{1+x^{2}}.$ Symbols: $\mathop{\mathsf{U}\/}\nolimits\!\left(\NVar{x},\NVar{t}\right)$: Voigt function, $\mathop{\mathsf{V}\/}\nolimits\!\left(\NVar{x},\NVar{t}\right)$: 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)
 7.19.6 $\displaystyle\mathop{\mathsf{U}\/}\nolimits\!\left(-x,t\right)$ $\displaystyle=\mathop{\mathsf{U}\/}\nolimits\!\left(x,t\right),$ $\displaystyle\mathop{\mathsf{V}\/}\nolimits\!\left(-x,t\right)$ $\displaystyle=-\mathop{\mathsf{V}\/}\nolimits\!\left(x,t\right).$
 7.19.7 $\displaystyle 0$ $\displaystyle<\mathop{\mathsf{U}\/}\nolimits\!\left(x,t\right)\leq 1,$ $\displaystyle-1$ $\displaystyle\leq\mathop{\mathsf{V}\/}\nolimits\!\left(x,t\right)\leq 1.$ Symbols: $\mathop{\mathsf{U}\/}\nolimits\!\left(\NVar{x},\NVar{t}\right)$: Voigt function, $\mathop{\mathsf{V}\/}\nolimits\!\left(\NVar{x},\NVar{t}\right)$: 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)
 7.19.8 $\displaystyle\mathop{\mathsf{V}\/}\nolimits\!\left(x,t\right)$ $\displaystyle=x\mathop{\mathsf{U}\/}\nolimits\!\left(x,t\right)+2t\frac{% \partial\mathop{\mathsf{U}\/}\nolimits\!\left(x,t\right)}{\partial x},$ 7.19.9 $\displaystyle\mathop{\mathsf{U}\/}\nolimits\!\left(x,t\right)$ $\displaystyle=1-x\mathop{\mathsf{V}\/}\nolimits\!\left(x,t\right)-2t\frac{% \partial\mathop{\mathsf{V}\/}\nolimits\!\left(x,t\right)}{\partial x}.$

## §7.19(iv) Other Integral Representations

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