§7.19 Voigt Functions

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Permalink:: http://dlmf.nist.gov/7.19
See also:: Annotations for Ch.7

§7.19(i) Definitions
§7.19(ii) Graphics
§7.19(iii) Properties
§7.19(iv) Other Integral Representations

§7.19(i) Definitions

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Keywords:: Voigt functions, definition, error functions, line broadening function, relation to line broadening function, relations to other functions
Notes:: (7.19.3) follows from (7.2.3) and (7.7.2).
Permalink:: http://dlmf.nist.gov/7.19.i
See also:: Annotations for §7.19 and Ch.7

For $x\in\mathbb{R}$ and ,

7.19.1 $\mathsf{U}\left(x,t\right)=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}\frac% {e^{-(x-y)^{2}/(4t)}}{1+y^{2}}\,\mathrm{d}y,$

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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 of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and : 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 $\mathsf{V}\left(x,t\right)=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}\frac% {ye^{-(x-y)^{2}/(4t)}}{1+y^{2}}\,\mathrm{d}y.$

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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 of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and : 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 $\mathsf{U}\left(x,t\right)+i\mathsf{V}\left(x,t\right)=\sqrt{\frac{\pi}{4t}}e^% {z^{2}}\operatorname{erfc}z,$ $z=(1-ix)/(2\sqrt{t})$ .

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Symbols:: $\mathsf{U}\left(\NVar{x},\NVar{t}\right)$ : Voigt function, $\mathsf{V}\left(\NVar{x},\NVar{t}\right)$ : Voigt function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : real variable and : 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 $H\left(a,u\right)=\frac{a}{\pi}\int_{-\infty}^{\infty}\frac{e^{-t^{2}}\,% \mathrm{d}t}{(u-t)^{2}+a^{2}}=\frac{1}{a\sqrt{\pi}}\mathsf{U}\left(\frac{u}{a}% ,\frac{1}{4a^{2}}\right).$

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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 of , $\mathrm{e}$ : 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(ii) Graphics

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Notes:: These graphs were produced at NIST.
Permalink:: http://dlmf.nist.gov/7.19.ii
See also:: Annotations for §7.19 and Ch.7

Figure 7.19.1: Voigt function $\mathsf{U}\left(x,t\right)$ ,

, 2.5, 5, 10. Magnify

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Symbols:: $\mathsf{U}\left(\NVar{x},\NVar{t}\right)$ : Voigt function and : real variable
Keywords:: Voigt functions, graphs
Permalink:: http://dlmf.nist.gov/7.19.F1
Encodings:: pdf, png
See also:: Annotations for §7.19(ii), §7.19 and Ch.7

Figure 7.19.2: Voigt function $\mathsf{V}\left(x,t\right)$ ,

, 2.5, 5, 10. Magnify

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Symbols:: $\mathsf{V}\left(\NVar{x},\NVar{t}\right)$ : Voigt function and : real variable
Keywords:: Voigt functions, graphs
Permalink:: http://dlmf.nist.gov/7.19.F2
Encodings:: pdf, png
See also:: Annotations for §7.19(ii), §7.19 and Ch.7

§7.19(iii) Properties

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Keywords:: Voigt functions, properties
Notes:: (7.19.5) follows from the definitions (7.19.1), (7.19.2), together with (1.17.6) or §2.3(iii). For the first of (7.19.7) use (7.19.1) for the lower bound, and (7.19.10) for the upper bound. For the second of (7.19.7) use (7.19.11). For (7.19.8) and (7.19.9) again use the definitions (7.19.1) and (7.19.2).
Permalink:: http://dlmf.nist.gov/7.19.iii
See also:: Annotations for §7.19 and Ch.7