Voigt functions

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1: 7.19 Voigt Functions

§7.19 Voigt Functions

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§7.19(i) Definitions

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Figure 7.19.1: Voigt function $𝖴(x,t)$, $t=0.1$, $2.5$, $5$, $10$. Magnify

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Figure 7.19.2: Voigt function $𝖵(x,t)$, $t=0.1$, $2.5$, $5$, $10$. Magnify

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§7.19(iii) Properties

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2: 7.21 Physical Applications

… ►Voigt functions $𝖴(x,t)$, $𝖵(x,t)$, 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. …

3: 7.22 Methods of Computation

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§7.22(iv) Voigt Functions

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4: 7.23 Tables

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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.

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

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5: 7.1 Special Notation

… ►The main functions treated in this chapter are the error function $\mathrm{erf}z$; the complementary error functions $\mathrm{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the Goodwin–Staton integral $G\left(z\right)$; the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$; the Voigt functions $𝖴(x,t)$ and $𝖵(x,t)$. …

6: Bibliography Z

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Algorithm 916: computing the Faddeyeva and Voigt functions. ACM Trans. Math. Software 38 (2), pp. Art. 15, 22.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

2nd item, 5th item.

Encodings:

BibTeX

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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.

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Notes:

Includes MATLAB and Fortran programs claiming 6S or 13S accuracy for single and double precision

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry

Referenced by:

3rd item, 6th item, §7.25(iii), §7.25(vi).

Encodings:

BibTeX

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7: Bibliography

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Spectrum line profiles: The Voigt function. J. Quant. Spectrosc. Radiat. Transfer 7, pp. 61–88.

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External Links:

Document

Referenced by:

§7.19(iv), §7.21.

Encodings:

BibTeX

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8: Bibliography L

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VOIGTL – A fast subroutine for Voigt function evaluation on vector processors. Comput. Phys. Comm. 75 (1-2), pp. 135–142.

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Notes:

Single-precision Fortran, minimum accuracy 13D.

External Links:

Document, Code, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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9: Bibliography W

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Rapid approximation to the Voigt/Faddeeva function and its derivatives. J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.

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Notes:

Includes Fortran code in the appendices to compute the functions to an accuracy between ${10}^{-2}$ and ${10}^{-5}$.

External Links:

ISSN 0022-4073, Document

Referenced by:

§7.21, §7.21.

Encodings:

BibTeX

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10: Software Index

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	Open Source													With Book						Commercial
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7.25(vi) $ℱ\left(x\right)$, $G\left(x\right)$, $𝖴(x,t)$, $𝖵(x,t)$, $x\in ℝ$			✓	✓	✓						✓			✓	✓	✓	✓			✓	✓	✓	✓
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …