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 f }

${\displaystyle f}$

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

Semantic latex: f

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Translation: f

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f

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SymPy

Translation: f

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f

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f

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Maple

Translation: f

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f

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f

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Is part of

$\eta =1.36603(f_{L}/f)-0.47719(f_{L}/f)^{2}+0.11116(f_{L}/f)^{3}$

$V_{p}(x,f)=\eta \cdot L(x,f)+(1-\eta )\cdot G(x,f)$

$f_{G}$

$\varphi _{f}(t;\sigma ,\gamma ,\mu _{\mathrm {G} },\mu _{\mathrm {L} })=e^{i(\mu _{\mathrm {G} }+\mu _{\mathrm {L} })t-\sigma ^{2}t^{2}/2-\gamma |t|}$

$f_{L}$

$\varphi _{f}(t;\sigma ,\gamma )=E(e^{ixt})=e^{-\sigma ^{2}t^{2}/2-\gamma |t|}$

$f=[f_{G}^{5}+2.69269f_{G}^{4}f_{L}+2.42843f_{G}^{3}f_{L}^{2}+4.47163f_{G}^{2}f_{L}^{3}+0.07842f_{G}f_{L}^{4}+f_{L}^{5}]^{1/5}$

$f_{\mathrm {G} }=2\sigma {\sqrt {2\ln(2)}}$

$f_{\mathrm {L} }=2\gamma$

$f_{\mathrm {V} }\approx f_{\mathrm {L} }/2+{\sqrt {f_{\mathrm {L} }^{2}/4+f_{\mathrm {G} }^{2}}}$

$f_{\mathrm {V} }\approx 0.5346f_{\mathrm {L} }+{\sqrt {0.2166f_{\mathrm {L} }^{2}+f_{\mathrm {G} }^{2}}}$

Description

parameter

Gaussian

Full width

function of Lorentz

simple formula

total FWHM

FWHM

%

mathematical definition

pseudo-Voigt profile

FWHM of the Gaussian profile

characteristic function

FWHM of the Lorentzian profile

rough approximation for the relation

product

better approximation with an accuracy

convolution

Voigt profile

width of the Voigt

Lorentzian profile

Gaussian profile

Complete translation information:

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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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