LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle L(x;\gamma)}

${\displaystyle L(x;\gamma )}$

... is translated to the CAS output ...

Semantic latex: L(x;\gamma)

Confidence: 0

Mathematica

Translation: L[x ; \[Gamma]]

Information

Sub Equations

L[x ; \[Gamma]]

Free variables

\[Gamma]

x

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Could be the Euler-Mascheroni constant.

But it is also a Greek letter. Be aware, that this program translated the letter as a normal Greek letter and not as a constant! Use the DLMF-Macro \EulerConstant to translate \gamma as a constant.

Tests

Symbolic

Numeric

SymPy

Translation: L(x ; Symbol('gamma'))

Information

Sub Equations

L(x ; Symbol('gamma'))

Free variables

Symbol('gamma')

x

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Could be the Euler-Mascheroni constant.

But it is also a Greek letter. Be aware, that this program translated the letter as a normal Greek letter and not as a constant! Use the DLMF-Macro \EulerConstant to translate \gamma as a constant.

Tests

Symbolic

Numeric

Maple

Translation: L(x ; gamma)

Information

Sub Equations

L(x ; gamma)

Free variables

gamma

x

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Could be the Euler-Mascheroni constant.

But it is also a Greek letter. Be aware, that this program translated the letter as a normal Greek letter and not as a constant! Use the DLMF-Macro \EulerConstant to translate \gamma as a constant.

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$x$

$G(x;\sigma )$

Is part of

$V(x;\sigma ,\gamma )\equiv \int _{-\infty }^{\infty }G(x';\sigma )L(x-x';\gamma )\,dx'$

$G(x;\sigma )$

$G(x;\sigma )\equiv {\frac {e^{-x^{2}/(2\sigma ^{2})}}{\sigma {\sqrt {2\pi }}}}$

$L(x;\gamma )\equiv {\frac {\gamma }{\pi (x^{2}+\gamma ^{2})}}$

Description

Gaussian profile

Lorentzian profile

Voigt profile

shift from the line center

case

Complete translation information:

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  "formula" : "L(x;\\gamma)",
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          "\\gamma" : "Could be the Euler-Mascheroni constant.\nBut it is also a Greek letter. Be aware, that this program translated the letter as a normal Greek letter and not as a constant!\nUse the DLMF-Macro \\EulerConstant to translate \\gamma as a constant.\n"
        }
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  "includes" : [ "x", "G(x;\\sigma)" ],
  "isPartOf" : [ "V(x;\\sigma,\\gamma) \\equiv \\int_{-\\infty}^\\infty G(x';\\sigma)L(x-x';\\gamma)\\, dx'", "G(x;\\sigma)", "G(x;\\sigma) \\equiv \\frac{e^{-x^2/(2\\sigma^2)}}{\\sigma \\sqrt{2\\pi}}", "L(x;\\gamma) \\equiv \\frac{\\gamma}{\\pi(x^2+\\gamma^2)}" ],
  "definiens" : [ {
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    "score" : 0.7125985104912714
  }, {
    "definition" : "Lorentzian profile",
    "score" : 0.7125985104912714
  }, {
    "definition" : "Voigt profile",
    "score" : 0.7125985104912714
  }, {
    "definition" : "shift from the line center",
    "score" : 0.6859086196238077
  }, {
    "definition" : "case",
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  } ]
}

Specify your own input

Formula input
Wikitext context	{{Probability distribution\| name =(Centered) Voigt \| type =density\| pdf_image =[[Image:VoigtPDF.svg\|325px\|Plot of the centered Voigt profile for four cases]]<br /><small>Plot of the centered Voigt profile for four cases. Each case has a full width at half-maximum of very nearly 3.6. The black and red profiles are the limiting cases of the Gaussian (γ =0) and the Lorentzian (σ =0) profiles respectively. </small>\| cdf_image =[[Image:VoigtCDF.svg\|325px\|Centered Voigt CDF.]]\| parameters =<math>\gamma,\sigma>0</math>\| support =<math>x\in(-\infty,\infty)</math>\| pdf =<math>\frac{\Re[w(z)]}{\sigma\sqrt{2\pi}}, ~~~z=\frac{x+i\gamma}{\sigma\sqrt{2}}</math>\| cdf =(complicated - see text)\| mean =(not defined)\| median =<math>0</math>\| mode =<math>0</math>\| variance =(not defined)\| skewness =(not defined)\| kurtosis =(not defined)\| entropy =\| mgf =(not defined)\| char =<math>e^{-\gamma\|t\|-\sigma^2 t^2/2}</math> }} The '''Voigt profile''' (named after [[Woldemar Voigt]]) is a [[probability distribution]] given by a [[convolution]] of a [[Cauchy distribution\|Cauchy-Lorentz distribution]] and a [[Normal distribution\|Gaussian distribution]]. It is often used in analyzing data from [[spectroscopy]] or [[diffraction]]. ==Definition== Without loss of generality, we can consider only centered profiles, which peak at zero. The Voigt profile is then :<math> V(x;\sigma,\gamma) \equiv \int_{-\infty}^\infty G(x';\sigma)L(x-x';\gamma)\, dx', </math> where ''x'' is the shift from the line center, <math>G(x;\sigma)</math> is the centered Gaussian profile: :<math> G(x;\sigma) \equiv \frac{e^{-x^2/(2\sigma^2)}}{\sigma \sqrt{2\pi}}, </math> and <math>L(x;\gamma)</math> is the centered Lorentzian profile: :<math> L(x;\gamma) \equiv \frac{\gamma}{\pi(x^2+\gamma^2)}. </math> The defining integral can be evaluated as: :<math> V(x;\sigma,\gamma)=\frac{\operatorname{Re}[w(z)]}{\sigma\sqrt{2 \pi}}, </math> where Re[''w''(''z'')] is the real part of the [[Faddeeva function]] evaluated for :<math> z=\frac{x+i\gamma}{\sigma\sqrt{2}}. </math> In the limiting cases of <math>\sigma=0</math> and <math> \gamma =0 </math> then <math> V(x;\sigma,\gamma) </math> simplifies to <math>L(x;\gamma)</math> and <math>G(x;\sigma)</math>, respectively. ==History and applications== In spectroscopy, a Voigt profile results from the convolution of two broadening mechanisms, one of which alone would produce a Gaussian profile (usually, as a result of the [[Doppler broadening]]), and the other would produce a Lorentzian profile. Voigt profiles are common in many branches of spectroscopy and [[Powder diffraction\|diffraction]]. Due to the expense of computing the [[Faddeeva function]], the Voigt profile is often approximated using a pseudo-Voigt profile. ==Properties== The Voigt profile is normalized: :<math> \int_{-\infty}^\infty V(x;\sigma,\gamma)\,dx = 1, </math> since it is a convolution of normalized profiles. The Lorentzian profile has no moments (other than the zeroth), and so the [[moment-generating function]] for the [[Cauchy distribution]] is not defined. It follows that the Voigt profile will not have a moment-generating function either, but the [[characteristic function (probability theory)\|characteristic function]] for the [[Cauchy distribution]] is well defined, as is the characteristic function for the [[normal distribution]]. The [[characteristic function (probability theory)\|characteristic function]] for the (centered) Voigt profile will then be the product of the two: :<math> \varphi_f(t;\sigma,\gamma) = E(e^{ixt}) = e^{-\sigma^2t^2/2 - \gamma \|t\|}. </math> Since normal distributions and Cauchy distributions are [[stable distributions]], they are each closed under [[convolution]] (up to change of scale), and it follows that the Voigt distributions are also closed under convolution. === Cumulative distribution function === Using the above definition for ''z'' , the cumulative distribution function (CDF) can be found as follows: :<math>F(x_0;\mu,\sigma) =\int_{-\infty}^{x_0} \frac{\operatorname{Re}(w(z))}{\sigma\sqrt{2\pi}}\,dx =\operatorname{Re}\left(\frac{1}{\sqrt{\pi}}\int_{z(-\infty)}^{z(x_0)} w(z)\,dz\right). </math> Substituting the definition of the [[Faddeeva function]] (scaled complex [[error function]]) yields for the indefinite integral: :<math> \frac{1}{\sqrt{\pi}}\int w(z)\,dz =\frac{1}{\sqrt{\pi}} \int e^{-z^2}\left[1-\operatorname{erf}(-iz)\right]\,dz, </math> which may be solved to yield :<math> \frac{1}{\sqrt{\pi}}\int w(z)\,dz = \frac{\operatorname{erf}(z)}{2} +\frac{iz^2}{\pi}\,_2F_2\left(1,1;\frac{3}{2},2;-z^2\right), </math> where <math>{}_2F_2</math> is a [[hypergeometric function]]. In order for the function to approach zero as ''x'' approaches negative infinity (as the CDF must do), an integration constant of 1/2 must be added. This gives for the CDF of Voigt: :<math>F(x;\mu,\sigma)=\operatorname{Re}\left[\frac{1}{2}+ \frac{\operatorname{erf}(z)}{2} +\frac{iz^2}{\pi}\,_2F_2\left(1,1;\frac{3}{2},2;-z^2\right)\right]. </math> === The uncentered Voigt profile === If the Gaussian profile is centered at <math>\mu_G</math> and the Lorentzian profile is centered at <math>\mu_L</math>, the convolution is centered at <math>\mu_G+\mu_L</math> and the characteristic function is :<math> \varphi_f(t;\sigma,\gamma,\mu_\mathrm{G},\mu_\mathrm{L})= e^{i(\mu_\mathrm{G}+\mu_\mathrm{L})t-\sigma^2t^2/2 - \gamma \|t\|}. </math> The mode and median are both located at <math>\mu_G+\mu_L</math>. === Derivative profile === The first and second derivative profiles can be expressed in terms of the [[Faddeeva function]] as follows: :<math> \frac{\partial}{\partial x} V(x;\sigma,\gamma)= -\frac{\operatorname{Re}[z ~w(z)]}{\sigma^2\sqrt{\pi}} = -\frac{x}{\sigma^2} \frac{\operatorname{Re}[w(z)]}{\sigma\sqrt{2 \pi}}+\frac{\gamma}{\sigma^2} \frac{\operatorname{Im}[w(z)]}{\sigma\sqrt{2 \pi}}; </math> :<math> \frac{\partial^2}{\partial x^2} V(x;\sigma,\gamma) = \frac{x^2-\gamma^2-\sigma^2}{\sigma^4} \frac{\operatorname{Re}[w(z)]}{\sigma\sqrt{2 \pi}} -\frac{2 x \gamma}{\sigma^4} \frac{\operatorname{Im}[w(z)]}{\sigma\sqrt{2 \pi}} +\frac{\gamma}{\sigma^4}\frac{1}{\pi}, </math> using the above definition for ''z''. ==Voigt functions== The '''Voigt functions'''<ref>{{dlmf\|id=7.19\|first=N. M. \|last=Temme\|title=Voigt function}}</ref> ''U'', ''V'', and ''H'' (sometimes called the '''line broadening function''') are defined by :<math>U(x,t)+iV(x,t) = \sqrt \frac{\pi}{4t} e^{z^2} \operatorname{erfc}(z) = \sqrt \frac{\pi}{4t} w(iz),</math> :<math>H(a,u) = \frac{U(u/a,1/4 a^2)}{a\sqrt \pi},</math> where :<math>z=(1-ix)/2\sqrt t,</math> erfc is the [[complementary error function]], and ''w''(''z'') is the [[Faddeeva function]]. === Relation to Voigt profile === :<math> V(x;\sigma,\gamma) = H(a,u) / (\sqrt 2 \sqrt \pi \sigma), </math> with :<math> a = \gamma / (\sqrt 2 \sigma) </math> and :<math> u = x / (\sqrt 2 \sigma). </math> == Numeric approximations == === Pseudo-Voigt approximation === The '''pseudo-Voigt profile''' (or '''pseudo-Voigt function''') is an approximation of the Voigt profile ''V''(''x'') using a [[linear combination]] of a [[Gaussian function\|Gaussian curve]] ''G''(''x'') and a [[Cauchy distribution\|Lorentzian curve]] ''L''(''x'') instead of their [[convolution]]. The pseudo-Voigt function is often used for calculations of experimental [[spectral line shape]]s. The mathematical definition of the normalized pseudo-Voigt profile is given by :<math> V_p(x,f) = \eta \cdot L(x,f) + (1 - \eta) \cdot G(x,f) </math> with <math> 0 < \eta < 1 </math>. <math> \eta </math> is a function of [[full width at half maximum]] (FWHM) parameter. There are several possible choices for the <math> \eta </math> parameter.<ref>{{Cite journal \| title = Determination of the Gaussian and Lorentzian content of experimental line shapes \| vauthors = Wertheim GK, Butler MA, West KW, Buchanan DN \| journal = Review of Scientific Instruments \| volume = 45 \| issue = 11 \| pages = 1369–1371 \| date = 1974 \| doi = 10.1063/1.1686503 \| bibcode = 1974RScI...45.1369W }}</ref><ref>{{Cite journal \| title = The Use of the Pseudo-Voigt Function in the Variance Method of X-ray Line-Broadening Analysis \| last = Sánchez-Bajo \| first = F. \| author2 = F. L. Cumbrera \| journal = Journal of Applied Crystallography \| doi = 10.1107/S0021889896015464 \| volume = 30 \| issue = 4 \| pages = 427–430 \| date = August 1997 \| doi-access = free }}</ref><ref>{{Cite journal \| title = Simple empirical analytical approximation to the Voigt profile \| vauthors = Liu Y, Lin J, Huang G, Guo Y, Duan C \| journal = JOSA B \| volume = 18 \| issue = 5 \| pages = 666–672 \| date = 2001 \| doi=10.1364/josab.18.000666 \| bibcode = 2001JOSAB..18..666L }}</ref><ref>{{Cite journal \| title = The Voigt Profile as a Sum of a Gaussian and a Lorentzian Functions, when the Weight Coefficient Depends Only on the Widths Ratio \| vauthors =Di Rocco HO, Cruzado A \|name-list-style=amp \| journal = Acta Physica Polonica A \| volume = 122 \| issue = 4 \| pages = 666–669 \| date = 2012 \|issn=0587-4246 \|doi=10.12693/APhysPolA.122.666 \| doi-access = free }}</ref> A simple formula, accurate to 1%, is<ref>{{Cite journal \| title = Extended pseudo-Voigt function for approximating the Voigt profile \| vauthors = Ida T, Ando M, Toraya H \| journal = Journal of Applied Crystallography \| volume = 33 \| issue = 6 \| pages = 1311–1316 \| date = 2000 \| doi=10.1107/s0021889800010219 \| s2cid = 55372305 \| url = https://semanticscholar.org/paper/c081d395f2e40c960d140d9b891ccabfcfd0664f \| doi-access = free }}</ref><ref>{{Cite journal \| title = Rietveld refinement of Debye-Scherrer synchrotron X-ray data from Al<sub>2</sub>O<sub>3</sub> \| last = P. Thompson, D. E. Cox and J. B. Hastings \| journal = Journal of Applied Crystallography \| volume = 20 \| issue = 2 \| pages = 79–83 \| date = 1987 \| doi=10.1107/S0021889887087090 \| doi-access = free }}</ref> :<math> \eta = 1.36603 (f_L/f) - 0.47719 (f_L/f)^2 + 0.11116(f_L/f)^3, </math> where now, <math> \eta </math> is a function of Lorentz (<math> f_L </math>), Gaussian (<math> f_G </math>) and total (<math> f </math>) [[Full width at half maximum]] (FWHM) parameters. The total FWHM (<math> f </math>) parameter is described by: :<math> f = [f_G^5 + 2.69269 f_G^4 f_L + 2.42843 f_G^3 f_L^2 + 4.47163 f_G^2 f_L^3 + 0.07842 f_G f_L^4 + f_L^5]^{1/5}. </math> === The width of the Voigt profile === The [[full width at half maximum]] (FWHM) of the Voigt profile can be found from the widths of the associated Gaussian and Lorentzian widths. The FWHM of the Gaussian profile is :<math>f_\mathrm{G}=2\sigma\sqrt{2\ln(2)}.</math> The FWHM of the Lorentzian profile is :<math>f_\mathrm{L}=2\gamma.</math> A rough approximation for the relation between the widths of the Voigt, Gaussian, and Lorentzian profiles is: :<math>f_\mathrm{V}\approx f_\mathrm{L}/2+\sqrt{f_\mathrm{L}^2/4+f_\mathrm{G}^2}.</math> This approximation is exactly correct for a pure Gaussian. A better approximation with an accuracy of 0.02% is given by<ref>{{Cite journal \| doi = 10.1016/0022-4073(77)90161-3 \| issn = 0022-4073 \| volume = 17 \| issue = 2 \| pages = 233–236 \| last = Olivero \| first = J. J. \| author2 = R. L. Longbothum \| title = Empirical fits to the Voigt line width: A brief review \| journal = Journal of Quantitative Spectroscopy and Radiative Transfer \| date = February 1977 \| bibcode = 1977JQSRT..17..233O }}</ref> :<math>f_\mathrm{V}\approx 0.5346 f_\mathrm{L}+\sqrt{0.2166f_\mathrm{L}^2+f_\mathrm{G}^2}.</math> This approximation is exactly correct for a pure Gaussian, but has an error of about 0.000305% for a pure Lorentzian profile. ==References== {{Reflist}} ==External links== * http://jugit.fz-juelich.de/mlz/libcerf, numeric C library for complex error functions, provides a function ''voigt(x, sigma, gamma)'' with approximately 13–14 digits precision. *The original article is : Voigt, Woldemar, 1912, <nowiki>''</nowiki>Das Gesetz der Intensitätsverteilung innerhalb der Linien eines Gasspektrums<nowiki>''</nowiki>, Sitzungsbericht der Bayerischen Akademie der Wissenschaften, 25, 603 (see also: <nowiki>http://publikationen.badw.de/de/003395768</nowiki>) {{ProbDistributions\|continuous-infinite}} [[Category:Continuous distributions]] [[Category:Spectroscopy]] [[Category:Special functions]] [[Category:Probability distributions with non-finite variance]]
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LaTeX to CAS translator

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