§7.19 Voigt Functions

For $x\in ℝ$ and $t>0$,

7.19.1		$$\mathsf{U}(x,t)=\frac{1}{\sqrt{4\pi t}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-{(x-y)}^2/(4t)}}{1+y^2}dy,$$
ⓘ Defines: $\mathsf{U}(x,t)$: Voigt function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: 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		$$\mathsf{V}(x,t)=\frac{1}{\sqrt{4\pi t}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{y{\mathrm{e}}^{-{(x-y)}^2/(4t)}}{1+y^2}dy.$$
ⓘ Defines: $\mathsf{V}(x,t)$: Voigt function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: 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		$$\mathsf{U}(x,t)+\mathrm{i}\mathsf{V}(x,t)=\sqrt{\frac{\pi }{4t}}{\mathrm{e}}^{z^2}\mathrm{erfc}z,$$
$z=(1-\mathrm{i}x)/(2\sqrt{t})$.
ⓘ Symbols: $\mathsf{U}(x,t)$: Voigt function, $\mathsf{V}(x,t)$: Voigt function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable and $z$: complex variable Referenced by: §7.19(i), §7.22(iv) Permalink: http://dlmf.nist.gov/7.19.E3 Encodings: TeX, pMML, png See also: Annotations for §7.19(i), §7.19 and Ch.7

7.19.4		$$H(a,u)=\frac{a}{\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}dt}{{(u-t)}^2+a^2}=\frac{1}{a\sqrt{\pi }}\mathsf{U}(\frac{u}{a},\frac{1}{4a^2}).$$
ⓘ Defines: $H(a,u)$: line-broadening function Symbols: $\mathsf{U}(x,t)$: Voigt function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: 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

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

7.19.5		$\underset{t\to 0}{lim}\mathsf{U}(x,t)$	$={\displaystyle \frac{1}{1+x^2}},$
		$\underset{t\to 0}{lim}\mathsf{V}(x,t)$	$={\displaystyle \frac{x}{1+x^2}}.$
ⓘ Symbols: $\mathsf{U}(x,t)$: Voigt function, $\mathsf{V}(x,t)$: 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		$\mathsf{U}(-x,t)$	$=\mathsf{U}(x,t),$
		$\mathsf{V}(-x,t)$	$=-\mathsf{V}(x,t).$
ⓘ Symbols: $\mathsf{U}(x,t)$: Voigt function, $\mathsf{V}(x,t)$: 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		$0$
		$-1$	$\le \mathsf{V}(x,t)\le 1.$
ⓘ Symbols: $\mathsf{U}(x,t)$: Voigt function, $\mathsf{V}(x,t)$: 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		$\mathsf{V}(x,t)$	$=x\mathsf{U}(x,t)+2t{\displaystyle \frac{\partial \mathsf{U}(x,t)}{\partial x}},$
ⓘ Symbols: $\mathsf{U}(x,t)$: Voigt function, $\mathsf{V}(x,t)$: Voigt function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$ and $x$: real variable Referenced by: §7.19(iii) Permalink: http://dlmf.nist.gov/7.19.E8 Encodings: TeX, pMML, png See also: Annotations for §7.19(iii), §7.19 and Ch.7
7.19.9		$\mathsf{U}(x,t)$	$=1-x\mathsf{V}(x,t)-2t{\displaystyle \frac{\partial \mathsf{V}(x,t)}{\partial x}}.$
ⓘ Symbols: $\mathsf{U}(x,t)$: Voigt function, $\mathsf{V}(x,t)$: Voigt function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$ and $x$: real variable Referenced by: §7.19(iii) Permalink: http://dlmf.nist.gov/7.19.E9 Encodings: TeX, pMML, png See also: Annotations for §7.19(iii), §7.19 and Ch.7

7.19.10		$$\mathsf{U}(\frac{u}{a},\frac{1}{4a^2})=a{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at-\frac{1}{4}{t}^2}\cos \left(ut\right)dt,$$
ⓘ Symbols: $\mathsf{U}(x,t)$: Voigt function, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm and $\int$: integral Referenced by: §7.19(iii) Permalink: http://dlmf.nist.gov/7.19.E10 Encodings: TeX, pMML, png See also: Annotations for §7.19(iv), §7.19 and Ch.7

7.19.11		$$\mathsf{V}(\frac{u}{a},\frac{1}{4a^2})=a{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at-\frac{1}{4}{t}^2}\sin \left(ut\right)dt.$$
ⓘ Symbols: $\mathsf{V}(x,t)$: Voigt function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\sin z$: sine function Referenced by: §7.19(iii) Permalink: http://dlmf.nist.gov/7.19.E11 Encodings: TeX, pMML, png See also: Annotations for §7.19(iv), §7.19 and Ch.7