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 x \ge 4}

${\displaystyle x\geq 4}$

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

Semantic latex: x \ge 4

Confidence: 0

Mathematica

Translation: x >= 4

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Sub Equations

x >= 4

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x

Tests

Symbolic

Test expression: x\geq4

ERROR:

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Numeric

SymPy

Translation: x >= 4

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x >= 4

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x

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Translation: x >= 4

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Sub Equations

x >= 4

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x

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Symbolic

Numeric

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$x$

Description

Padé

auxiliary function

error

formula

integral

rational function

Rowe et al.

Complete translation information:

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Wikitext context	{{for\|simple integrals of trigonometric functions\|List of integrals of trigonometric functions}} {{Use American English\|date = January 2019}} {{Short description\|Special function defined by an integral}} [[Image:sine cosine integral.svg\|right\|thumb\|Si(x) (blue) and Ci(x) (green) plotted on the same plot.]] In [[mathematics]], the '''trigonometric integrals''' are a [[indexed family\|family]] of [[integral]]s involving [[trigonometric function]]s. ==Sine integral== [[Image:Sine integral.svg\|thumb\|right\|Plot of '''Si(''x'')''' for {{math\|0 ≤ ''x'' ≤ 8 ''π''}}.]] The different [[sine]] integral definitions are ::<math>\operatorname{Si}(x) = \int_0^x\frac{\sin t}{t}\,dt</math> ::<math>\operatorname{si}(x) = -\int_x^\infty\frac{\sin t}{t}\,dt~.</math> Note that the integrand {{frac\|{{math\|sin ''x''}} \| {{math\|''x''}} }} is the [[sinc function]], and also the zeroth [[Bessel function#Spherical Bessel functions: jn.2C yn\|spherical Bessel function]]. Since {{math\| sinc}} is an [[even function\|even]] [[entire function]] ([[holomorphic]] over the entire complex plane), {{math\| Si}} is entire, odd, and the integral in its definition can be taken along [[Cauchy's integral theorem\|any path]] connecting the endpoints. By definition, {{math\| Si(''x'')}} is the [[antiderivative]] of {{math\|sin ''x'' / ''x''}} whose value is zero at {{math\|''x'' {{=}} 0}}, and {{math\|si(''x'')}} is the antiderivative whose value is zero at {{math\|''x'' {{=}} ∞}}. Their difference is given by the [[Dirichlet integral]], ::<math>\operatorname{Si}(x) - \operatorname{si}(x) = \int_0^\infty\frac{\sin t}{t}\,dt = \frac{\pi}{2} \quad \text{ or } \quad \operatorname{Si}(x) = \frac{\pi}{2} + \operatorname{si}(x) ~.</math> In [[signal processing]], the oscillations of the sine integral cause [[overshoot (signal)\|overshoot]] and [[ringing artifacts]] when using the [[sinc filter]], and [[frequency domain]] ringing if using a truncated sinc filter as a [[low-pass filter]]. Related is the [[Gibbs phenomenon]]: If the sine integral is considered as the [[convolution]] of the sinc function with the [[heaviside step function]], this corresponds to truncating the [[Fourier series]], which is the cause of the Gibbs phenomenon. ==Cosine integral== [[Image:Cosine integral.svg\|thumb\|right\|Plot of {{math\|Ci(''x'')}} for {{nowrap\|0 < {{mvar\|x}} ≤ 8{{mvar\|π}} .}}]] The different [[cosine]] integral definitions are ::<math>\operatorname{Cin}(x) = \int_0^x \frac{1 - \cos t}{t}\operatorname{d}t~,</math> ::<math>\operatorname{Ci}(x) = -\int_x^\infty \frac{\cos t}{t}\operatorname{d}t = \gamma + \ln x - \int_0^x \frac{1 - \cos t}{t}\operatorname{d}t \qquad ~\text{ for } ~\left\|\operatorname{Arg}(x)\right\| < \pi~,</math> where {{mvar\|γ}} ≈ 0.57721566 ... is the [[Euler–Mascheroni constant]]. Some texts use {{math\|ci}} instead of {{math\|Ci}}. {{math\| Ci(''x'')}} is the antiderivative of {{math\|cos ''x'' / ''x''}} (which vanishes as <math>x \to \infty</math>). The two definitions are related by ::<math>\operatorname{Ci}(x) = \gamma + \ln x - \operatorname{Cin}(x)~.</math> {{math\|Cin}} is an [[even and odd functions\|even]], [[entire function]]. For that reason, some texts treat {{math\|Cin}} as the primary function, and derive {{math\|Ci}} in terms of {{math\|Cin}}. ==Hyperbolic sine integral== The [[hyperbolic sine]] integral is defined as ::<math>\operatorname{Shi}(x) =\int_0^x \frac {\sinh (t)}{t}\,dt.</math> It is related to the ordinary sine integral by ::<math>\operatorname{Si}(ix) = i\operatorname{Shi}(x).</math> ==Hyperbolic cosine integral== The [[hyperbolic cosine]] integral is :<math>\operatorname{Chi}(x) = \gamma+\ln x + \int_0^x\frac{\;\cosh t-1\;}{t}\operatorname{d}t \qquad ~ \text{ for } ~ \left\| \operatorname{Arg}(x) \right\| < \pi~,</math> where <math>\gamma</math> is the [[Euler–Mascheroni constant]]. It has the series expansion :<math>\operatorname{Chi}(x) = \gamma + \ln(x) + \frac {x^2}{4} + \frac {x^4}{96} + \frac {x^6}{4320} + \frac {x^8}{322560} + \frac{x^{10}}{36288000} + O(x^{12}).</math> ==Auxiliary functions== Trigonometric integrals can be understood in terms of the so-called "auxiliary functions" :<math> \begin{array}{rcl} f(x) &\equiv& \int_0^\infty \frac{\sin(t)}{t+x} \mathrm{d}t &=& \int_0^\infty \frac{e^{-x t}}{t^2 + 1} \mathrm{d}t &=& \quad \operatorname{Ci}(x) \sin(x) + \left[\frac{\pi}{2} - \operatorname{Si}(x) \right] \cos(x)~, \qquad \text{ and } \\ g(x) &\equiv& \int_0^\infty \frac{\cos(t)}{t+x} \mathrm{d}t &=& \int_0^\infty \frac{t e^{-x t}}{t^2 + 1} \mathrm{d}t &=& -\operatorname{Ci}(x) \cos(x) + \left[\frac{\pi}{2} - \operatorname{Si}(x) \right] \sin(x)~. \end{array} </math> Using these functions, the trigonometric integrals may be re-expressed as (cf. Abramowitz & Stegun, [http://people.math.sfu.ca/~cbm/aands/page_232.htm p. 232]) :<math> \begin{array}{rcl} \frac{\pi}{2} - \operatorname{Si}(x) = -\operatorname{si}(x) &=& f(x) \cos(x) + g(x) \sin(x)~, \qquad \text{ and } \\ \operatorname{Ci}(x) &=& f(x) \sin(x) - g(x) \cos(x)~. \\ \end{array} </math> ==Nielsen's spiral== [[Image:Nielsen's spiral.png\|thumb\|right\|Nielsen's spiral.]] The [[spiral]] formed by parametric plot of {{math\|si , ci}} is known as Nielsen's spiral. :<math>x(t) = a \times \operatorname{ci}(t)</math><br /> <math>y(t) = a \times \operatorname{si}(t)</math> The spiral is closely related to the [[Fresnel integral]]s and the [[Euler spiral]]. Nielsen's spiral has applications in vision processing, road and track construction and other areas.{{Citation needed\|date=March 2019\|reason=Need reference to back up claim.}} ==Expansion== Various expansions can be used for evaluation of trigonometric integrals, depending on the range of the argument. ===Asymptotic series (for large argument)=== :<math>\operatorname{Si}(x) \sim \frac{\pi}{2} - \frac{\cos x}{x}\left(1-\frac{2!}{x^2}+\frac{4!}{x^4}-\frac{6!}{x^6}\cdots\right) - \frac{\sin x}{x}\left(\frac{1}{x}-\frac{3!}{x^3}+\frac{5!}{x^5}-\frac{7!}{x^7}\cdots\right)</math> :<math>\operatorname{Ci}(x) \sim \frac{\sin x}{x}\left(1-\frac{2!}{x^2}+\frac{4!}{x^4}-\frac{6!}{x^6}\cdots\right) -\frac{\cos x}{x}\left(\frac{1}{x}-\frac{3!}{x^{3}}+\frac{5!}{x^5}-\frac{7!}{x^7}\cdots\right) ~.</math> These series are [[Asymptotic series\|asymptotic]] and divergent, although can be used for estimates and even precise evaluation at {{math\|ℜ(''x'') ≫ 1}}. ===Convergent series=== :<math>\operatorname{Si}(x)= \sum_{n=0}^\infty \frac{(-1)^{n}x^{2n+1}}{(2n+1)(2n+1)!}=x-\frac{x^3}{3!\cdot3}+\frac{x^5}{5!\cdot5}-\frac{x^7}{7! \cdot7}\pm\cdots</math> :<math>\operatorname{Ci}(x)= \gamma+\ln x+\sum_{n=1}^{\infty}\frac{(-1)^{n}x^{2n}}{2n(2n)!}=\gamma+\ln x-\frac{x^2}{2!\cdot2}+\frac{x^4}{4! \cdot4}\mp\cdots</math> These series are convergent at any complex {{mvar\|x}}, although for {{math\|{{mabs\|''x''}} ≫ 1}}, the series will converge slowly initially, requiring many terms for high precision. === Derivation of Series Expansion === <math>\sin\,x = x - \frac{x^3}{3!}+\frac{x^5}{5!}- \frac{x^7}{7!}+\frac{x^9}{9!}-\frac{x^{11}}{11!}+\,... </math>(Maclaurin Series Expansion) <math>\frac{\sin\,x}{x} = 1 - \frac{x^2}{3!}+\frac{x^4}{5!}- \frac{x^6}{7!}+\frac{x^8}{9!}-\frac{x^{10}}{11!}+\,... </math> <math>\therefore\int \frac{\sin\,x}{x}dx = x - \frac{x^3}{3!\cdot3}+\frac{x^5}{5!\cdot5}- \frac{x^7}{7!\cdot7}+\frac{x^9}{9!\cdot9}-\frac{x^{11}}{11!\cdot11}+\,... </math> ==Relation with the exponential integral of imaginary argument== The function : <math display="block"> \operatorname{E}_1(z) = \int_1^\infty \frac{\exp(-zt)}{t}\,dt \qquad~\text{ for }~ \Re(z) \ge 0 </math> is called the [[exponential integral]]. It is closely related to {{math\|Si}} and {{math\|Ci}}, :<math display="block"> \operatorname{E}_1(i x) = i\left(-\frac{\pi}{2} + \operatorname{Si}(x)\right)-\operatorname{Ci}(x) = i \operatorname{si}(x) - \operatorname{ci}(x) \qquad ~\text{ for }~ x > 0 ~. </math> As each respective function is analytic except for the cut at negative values of the argument, the area of validity of the relation should be extended to (Outside this range, additional terms which are integer factors of {{math\|''π''}} appear in the expression.) Cases of imaginary argument of the generalized integro-exponential function are : <math display="block"> \int_1^\infty \cos(ax)\frac{\ln x}{x} \, dx = -\frac{\pi^2}{24}+\gamma\left(\frac{\gamma}{2}+\ln a\right)+\frac{\ln^2a}{2} +\sum_{n\ge 1} \frac{(-a^2)^n}{(2n)!(2n)^2} ~, </math> which is the real part of : <math display="block"> \int_1^\infty e^{iax}\frac{\ln x}{x} \, \operatorname{d}x = -\frac{\pi^2}{24} + \gamma\left(\frac{\gamma}{2}+\ln a\right)+\frac{\ln^2 a}{2} -\frac{\pi}{2}i\left(\gamma+\ln a\right) + \sum_{n\ge 1}\frac{(ia)^n}{n!n^2} ~. </math> Similarly : <math display="block"> \int_1^\infty e^{iax}\frac{\ln x}{x^2}\, \operatorname{d}x = 1 + ia\left[ -\frac{\;\pi^2}{24} + \gamma \left( \frac{\gamma}{2} + \ln a - 1 \right) + \frac{\ln^2 a}{2} - \ln a + 1 \right] + \frac{\pi a}{2} \Bigl( \gamma+\ln a - 1 \Bigr) + \sum_{n\ge 1}\frac{(ia)^{n+1}}{(n+1)!n^2}~. </math> ==Efficient evaluation== [[Padé approximant]]s of the convergent Taylor series provide an efficient way to evaluate the functions for small arguments. The following formulae, given by Rowe et al. (2015),<ref name=RoweEtAl2015>{{cite journal \|last1=Rowe \|first1=B. \|display-authors=etal \|title=GALSIM: The modular galaxy image simulation toolkit \|journal=Astronomy and Computing \|year=2015 \|volume=10 \|page=121 \|doi=10.1016/j.ascom.2015.02.002 \|arxiv=1407.7676 \|bibcode=2015A&C....10..121R}}</ref> are accurate to better than {{math\|10<sup>−16</sup>}} for {{math\|0 ≤ ''x'' ≤ 4}}, :<math> \begin{array}{rcl} \operatorname{Si}(x) &\approx & x \cdot \left( \frac{ \begin{array}{l} 1 -4.54393409816329991\cdot 10^{-2} \cdot x^2 + 1.15457225751016682\cdot 10^{-3} \cdot x^4 - 1.41018536821330254\cdot 10^{-5} \cdot x^6 \\ ~~~ + 9.43280809438713025 \cdot 10^{-8} \cdot x^8 - 3.53201978997168357 \cdot 10^{-10} \cdot x^{10} + 7.08240282274875911 \cdot 10^{-13} \cdot x^{12} \\ ~~~ - 6.05338212010422477 \cdot 10^{-16} \cdot x^{14} \end{array} } { \begin{array}{l} 1 + 1.01162145739225565 \cdot 10^{-2} \cdot x^2 + 4.99175116169755106 \cdot 10^{-5} \cdot x^4 + 1.55654986308745614 \cdot 10^{-7} \cdot x^6 \\ ~~~ + 3.28067571055789734 \cdot 10^{-10} \cdot x^8 + 4.5049097575386581 \cdot 10^{-13} \cdot x^{10} + 3.21107051193712168 \cdot 10^{-16} \cdot x^{12} \end{array} } \right)\\ &~&\\ \operatorname{Ci}(x) &\approx & \gamma + \ln(x) +\\ && x^2 \cdot \left( \frac{ \begin{array}{l} -0.25 + 7.51851524438898291 \cdot 10^{-3} \cdot x^2 - 1.27528342240267686 \cdot 10^{-4} \cdot x^4 + 1.05297363846239184 \cdot 10^{-6} \cdot x^6 \\ ~~~ -4.68889508144848019 \cdot 10^{-9} \cdot x^8 + 1.06480802891189243 \cdot 10^{-11} \cdot x^{10} - 9.93728488857585407 \cdot 10^{-15} \cdot x^{12} \\ \end{array} } { \begin{array}{l} 1 + 1.1592605689110735 \cdot 10^{-2} \cdot x^2 + 6.72126800814254432 \cdot 10^{-5} \cdot x^4 + 2.55533277086129636 \cdot 10^{-7} \cdot x^6 \\ ~~~ + 6.97071295760958946 \cdot 10^{-10} \cdot x^8 + 1.38536352772778619 \cdot 10^{-12} \cdot x^{10} + 1.89106054713059759 \cdot 10^{-15} \cdot x^{12} \\ ~~~ + 1.39759616731376855 \cdot 10^{-18} \cdot x^{14} \\ \end{array} } \right) \end{array} </math> The integrals may be evaluated indirectly via auxiliary functions <math>f(x)</math> and <math>g(x)</math>, which are defined by {\| \|<math display="block">\operatorname{Si}(x)=\frac{\pi}{2}-f(x)\cos(x)-g(x)\sin(x)</math> \|     \|<math display="block">\operatorname{Ci}(x)=f(x)\sin(x)-g(x)\cos(x) </math> \|- \|colspan="3" align="center"\| <small>''or equivalently''</small> \|- \|<math display="block">f(x) \equiv \left[\frac{\pi}{2} - \operatorname{Si}(x)\right] \cos(x) + \operatorname{Ci}(x) \sin(x)</math> \|     \|<math display="block">g(x) \equiv \left[\frac{\pi}{2} - \operatorname{Si}(x)\right] \sin(x) - \operatorname{Ci}(x) \cos(x)</math> \|} For <math>x \ge 4</math> the [[Padé approximant\|Padé rational functions]] given below approximate <math>f(x)</math> and <math>g(x)</math> with error less than 10<sup>−16</sup>:<ref name=RoweEtAl2015/> :<math> \begin{array}{rcl} f(x) &\approx & \dfrac{1}{x} \cdot \left(\frac{ \begin{array}{l} 1 + 7.44437068161936700618 \cdot 10^2 \cdot x^{-2} + 1.96396372895146869801 \cdot 10^5 \cdot x^{-4} + 2.37750310125431834034 \cdot 10^7 \cdot x^{-6} \\ ~~~ + 1.43073403821274636888 \cdot 10^9 \cdot x^{-8} + 4.33736238870432522765 \cdot 10^{10} \cdot x^{-10} + 6.40533830574022022911 \cdot 10^{11} \cdot x^{-12} \\ ~~~ + 4.20968180571076940208 \cdot 10^{12} \cdot x^{-14} + 1.00795182980368574617 \cdot 10^{13} \cdot x^{-16} + 4.94816688199951963482 \cdot 10^{12} \cdot x^{-18} \\ ~~~ - 4.94701168645415959931 \cdot 10^{11} \cdot x^{-20} \end{array} }{ \begin{array}{l} 1 + 7.46437068161927678031 \cdot 10^2 \cdot x^{-2} + 1.97865247031583951450 \cdot 10^5 \cdot x^{-4} + 2.41535670165126845144 \cdot 10^7 \cdot x^{-6} \\ ~~~ + 1.47478952192985464958 \cdot 10^9 \cdot x^{-8} + 4.58595115847765779830 \cdot 10^{10} \cdot x^{-10} + 7.08501308149515401563 \cdot 10^{11} \cdot x^{-12} \\ ~~~ + 5.06084464593475076774 \cdot 10^{12} \cdot x^{-14} + 1.43468549171581016479 \cdot 10^{13} \cdot x^{-16} + 1.11535493509914254097 \cdot 10^{13} \cdot x^{-18} \end{array} } \right) \\ & &\\ g(x) &\approx & \dfrac{1}{x^2} \cdot \left(\frac{ \begin{array}{l} 1 + 8.1359520115168615 \cdot 10^2 \cdot x^{-2} + 2.35239181626478200 \cdot 10^5 \cdot x^{-4} +3.12557570795778731 \cdot 10^7 \cdot x^{-6} \\ ~~~ + 2.06297595146763354 \cdot 10^9 \cdot x^{-8} + 6.83052205423625007 \cdot 10^{10} \cdot x^{-10} + 1.09049528450362786 \cdot 10^{12} \cdot x^{-12} \\ ~~~ + 7.57664583257834349 \cdot 10^{12} \cdot x^{-14} + 1.81004487464664575 \cdot 10^{13} \cdot x^{-16} + 6.43291613143049485 \cdot 10^{12} \cdot x^{-18} \\ ~~~ - 1.36517137670871689 \cdot 10^{12} \cdot x^{-20} \end{array} }{ \begin{array}{l} 1 + 8.19595201151451564 \cdot 10^2 \cdot x^{-2} + 2.40036752835578777 \cdot 10^5 \cdot x^{-4} + 3.26026661647090822 \cdot 10^7 \cdot x^{-6} \\ ~~~ + 2.23355543278099360 \cdot 10^9 \cdot x^{-8} + 7.87465017341829930 \cdot 10^{10} \cdot x^{-10} + 1.39866710696414565 \cdot 10^{12} \cdot x^{-12} \\ ~~~ + 1.17164723371736605 \cdot 10^{13} \cdot x^{-14} + 4.01839087307656620 \cdot 10^{13} \cdot x^{-16} + 3.99653257887490811 \cdot 10^{13} \cdot x^{-18} \end{array} } \right) \\ \end{array} </math> ==See also== * [[Logarithmic integral]] [[Tanc function]] [[Tanhc function]] [[Sinhc function]] [[Coshc function]] == References == {{reflist\|25em}} * {{AS ref\|5\|231}} == Further reading == * {{cite arXiv \|first1=R.J. \|last1=Mathar \|eprint=0912.3844 \|title=Numerical evaluation of the oscillatory integral over exp(''i{{pi}}x'')·''x''<sup>1/''x''</sup> between 1 and ∞ \|year=2009 \|class=math.CA \|at=Appendix B}} * {{cite book \|last1=Press \|first1=W.H. \|last2=Teukolsky \|first2=S.A. \|last3=Vetterling \|first3=W.T. \|last4=Flannery \|first4=B.P. \|year=2007 \|title=Numerical Recipes: The Art of Scientific Computing \|edition=3rd \|publisher=Cambridge University Press \|publication-place=New York \|isbn=978-0-521-88068-8 \|chapter=Section 6.8.2 – Cosine and Sine Integrals \|chapter-url=http://apps.nrbook.com/empanel/index.html#pg=300}} * {{cite web \|first=Dan \|last=Sloughter \|url=http://de2de.synechism.org/c5/sec58.pdf \|title=Sine Integral Taylor series proof \|website=Difference Equations to Differential Equations}} * {{dlmf \|id=6 \|title=Exponential, Logarithmic, Sine, and Cosine Integrals \|first=N.M. \|last=Temme}} ==External links== * http://mathworld.wolfram.com/SineIntegral.html * {{springer\|title=Integral sine\|id=p/i051650}} * {{springer\|title=Integral cosine\|id=p/i051370}} {{Nonelementary Integral}} {{DEFAULTSORT:Trigonometric Integral}} [[Category:Trigonometry]] [[Category:Special functions]] [[Category:Special hypergeometric functions]] [[Category:Integrals]]
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