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).

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Wikitext context	{{Use American English\|date = January 2019}} {{Short description\|Special function defined by an integral}} [[image:Fresnel Integrals (Unnormalised).svg\|250px\|thumb\| Plots of {{math\|{{color\|#b30000\|''S''(''x'')}}}} and {{math\|{{color\|#00b300\|''C''(''x'')}}}}. The maximum of {{math\|''C''(''x'')}} is about {{math\|0.977451424}}. If the integrands of {{mvar\|S}} and {{mvar\|C}} were defined using {{math\|π''t''<sup>2</sup>/2}} instead of {{math\|''t''<sup>2</sup>}}, then the image would be scaled vertically and horizontally (see below).]] The '''Fresnel integrals''' ''S''(''x'') and ''C''(''x'') are two [[transcendental function]]s named after [[Augustin-Jean Fresnel]] that are used in [[optics]] and are closely related to the [[error function]] (erf). They arise in the description of [[near and far field\|near-field]] [[Fresnel diffraction]] phenomena and are defined through the following [[integral]] representations: : <math>S(x) = \int_0^x \sin(t^2)\,dt, \quad C(x) = \int_0^x \cos(t^2)\,dt.</math> The simultaneous [[parametric equation\|parametric plot]] of ''S''(''x'') and ''C''(''x'') is the [[Euler spiral]] (also known as the Cornu spiral or clothoid). Recently, they have been used in the design of highways and other engineering projects.<ref name=Stewart>{{harvnb\|Stewart\|2008\|page=383}}.</ref> == Definition == [[image:Fresnel Integrals (Normalised).svg\|250px\|thumb\| Fresnel integrals with arguments {{pi}}''t''<sup>2</sup>/2 instead of ''t''<sup>2</sup> converge to 0.5.]] The Fresnel integrals admit the following [[power series expansion]]s that converge for all ''x'': :<math> \begin{align} S(x) & =\int_0^x \sin(t^2)\,dt=\sum_{n=0}^{\infin}(-1)^n\frac{x^{4n+3}}{(2n+1)!(4n+3)}. \\ C(x) & =\int_0^x \cos(t^2)\,dt=\sum_{n=0}^{\infin}(-1)^n\frac{x^{4n+1}}{(2n)!(4n+1)}. \end{align} </math> Some widely used tables<ref>{{AS ref\|7, eqn 7.3.1 – 7.3.2}}</ref><ref>{{dlmf\|id=7.2.iii\|title=Error Functions, Dawson’s and Fresnel Integrals: Properties\|first=N. M.\|last=Temme}}.</ref> use <math>\pi t^2 / 2</math> instead of <math>t^2</math> for the argument of the integrals defining ''S''(''x'') and ''C''(''x''). This changes their [[Limit of a function#Limits at infinity\|limits at infinity]] from <math>\textstyle (1/2) \cdot \sqrt{\pi/2}</math> to <math>1/2</math> and the arc length for the first spiral turn from <math>(2\pi)</math> to 2 (at <math>t=2</math>). These alternative functions are usually known as '''normalized Fresnel integrals'''. == Euler spiral == {{Main\|Euler spiral}} [[Image:Cornu Spiral.svg\|250px\|thumb\| Euler spiral <math>(x,y) = (C(t), S(t))</math>. The spiral converges to the centre of the holes in the image as <math>t</math> tends to positive or negative infinity.]] The '''Euler [[spiral]]''', also known as '''Cornu spiral''' or '''clothoid''', is the curve generated by a [[parametric plot]] of <math>S(t)</math> against <math>C(t)</math>. The Cornu spiral was created by [[Marie Alfred Cornu]] as a [[nomogram]] for diffraction computations in science and engineering. From the definitions of Fresnel integrals, the [[infinitesimal]]s <math>dx</math> and <math>dy</math> are thus: :<math>\begin{align} dx &= C'(t)\,dt = \cos(t^2)\,dt, \\ dy &= S'(t)\,dt = \sin(t^2)\,dt. \end{align}</math> Thus the length of the spiral measured from the [[Origin (mathematics)\|origin]] can be expressed as : <math>L = \int_0^{t_0} \sqrt {dx^2 + dy^2} = \int_0^{t_0} dt = t_0. </math> That is, the parameter <math>t</math> is the curve length measured from the origin <math>(0,0)</math>, and the Euler spiral has [[Infinity\|infinite]] length. The vector <math>(\cos(t^2), \sin(t^2))</math> also expresses the [[unit vector\|unit]] [[tangent vector]] along the spiral, giving <math>\theta = t^2</math>. Since ''t'' is the curve length, the curvature <math>\kappa</math> can be expressed as : <math> \kappa = \frac{1}{R} = \frac{d\theta}{dt} = 2t. </math> Thus the rate of change of curvature with respect to the curve length is : <math>\frac{d\kappa}{dt} = \frac {d^2\theta}{dt^2} = 2. </math> An Euler spiral has the property that its [[curvature]] at any point is proportional to the distance along the spiral, measured from the origin. This property makes it useful as a [[track transition curve\|transition curve]] in highway and railway engineering: If a vehicle follows the spiral at unit speed, the parameter <math>t</math> in the above derivatives also represents the time. Consequently, a vehicle following the spiral at constant speed will have a constant rate of [[angular acceleration]]. Sections from Euler spirals are commonly incorporated into the shape of roller-coaster loops to make what are known as [[clothoid loop]]s. == Properties == * ''C''(''x'') and ''S''(''x'') are [[odd function]]s of ''x''. * Asymptotics of the Fresnel integrals as <math>x \to \infty</math> are given by the formulas: :: <math> \begin{align} S(x) & =\sqrt{\frac{\pi}{2}} \left( \frac{\mbox{sign}{(x)}}{2} - \left[1+O(x^{-4}) \right] \left( \frac{\cos{(x^{2})}}{x \sqrt{2 \pi}} + \frac{\sin{(x^2)}}{ x^3 \sqrt{8 \pi}} \right) \right), \\ C(x) & =\sqrt{\frac{\pi}{2}} \left( \frac{\mbox{sign}{(x)}}{2} + \left[1+O(x^{-4}) \right] \left( \frac{\sin{(x^2)}}{x \sqrt{2 \pi}} - \frac{\cos{(x^{2})}}{ x^3 \sqrt{8 \pi}} \right) \right). \end{align} </math> [[File:Fresnel-sin.png\|right\|thumb\|300px\|Complex Fresnel integral S(z)]] * Using the power series expansions above, the Fresnel integrals can be extended to the domain of [[complex number]]s, where they become [[analytic function]]s of a complex variable. * ''C(z)'' and ''S(z)'' are [[entire function]]s of the complex variable ''z''. * The Fresnel integrals can be expressed using the [[error function]] as follows:<ref>functions.wolfram.com, [http://functions.wolfram.com/GammaBetaErf/FresnelS/27/01/ Fresnel integral S: Representations through equivalent functions] and [http://functions.wolfram.com/GammaBetaErf/FresnelC/27/01/ Fresnel integral C: Representations through equivalent functions]. Note: Wolfram uses the Abramowitz & Stegun convention, which differs from the one in this article by factors of <math>\sqrt{\pi/2}</math>.</ref> [[File:Fresnel-cos.png\|right\|thumb\|300px\|Complex Fresnel integral ''C''(''z'')]] :: <math> \begin{align} S(z) & =\sqrt{\frac{\pi}{2}} \frac{1+i}{4} \left[ \operatorname{erf}\left(\frac{1+i}{\sqrt{2}}z\right) -i \operatorname{erf}\left(\frac{1-i}{\sqrt{2}}z\right) \right], \\ C(z) & =\sqrt{\frac{\pi}{2}}\frac{1-i}{4} \left[ \operatorname{erf}\left(\frac{1+i}{\sqrt{2}}z\right) + i \operatorname{erf}\left(\frac{1-i}{\sqrt{2}}z\right) \right]. \end{align} </math> :or :: <math> \begin{align} C(z) + i S(z) & = \sqrt{\frac{\pi}{2}}\frac{1+i}{2} \operatorname{erf}\left(\frac{1-i}{\sqrt{2}}z\right), \\ S(z) + i C(z) & = \sqrt{\frac{\pi}{2}}\frac{1+i}{2} \operatorname{erf}\left(\frac{1+i}{\sqrt{2}}z\right). \end{align} </math> ===Limits as ''x'' approaches infinity=== The integrals defining ''C''(''x'') and ''S''(''x'') cannot be evaluated in the [[closed-form expression\|closed form]] in terms of [[elementary function]]s, except in special cases. The [[limit of a function\|limits]] of these functions as ''x'' goes to infinity are known: ::<math>\int_0^\infty \cos t^2\,dt = \int_0^\infty \sin t^2 \, dt = \frac{\sqrt{2\pi}}{4} = \sqrt{\frac{\pi}{8}} \approx 0.6267.</math> [[Image:Fresnel Integral Contour.svg\|right\|250px\|thumb\|The sector contour used to calculate the limits of the Fresnel integrals]] The limits of {{math\|''C(x)''}} and {{math\|''S(x)''}} as the argument ''x'' tends to infinity can be found by using several methods. One of them<ref>Another method based on [[Integration using parametric derivatives\|parametric integration]] is described for example in {{harvnb\|Zajta\|Goel\|1989}}.</ref> uses a [[contour integral]] of the function :<math>e^{-t^2}</math> around the boundary of the [[Circular sector\|sector]]-shaped region in the [[complex plane]] formed by the positive {{math\|''x''}}-axis, the bisector of the first quadrant {{math\|''y'' {{=}} ''x''}} with {{math\|''x'' ≥ 0}}, and a circular arc of radius {{math\|''R''}} centered at the origin. As {{math\|''R''}} goes to infinity, the integral along the circular arc <math>\gamma_2</math> tends to {{math\|0}} <math>\left\|\int_{\gamma_2}e^{-t^2}dt\right\| \leq \int_{\gamma_2}\|e^{-t^2}\|dt = R\int_0^{\pi/4}e^{-R^2\cos2t}dt \leq R\int_0^{\pi/4}e^{-R^2(1-\frac{4}{\pi}t)}dt = \frac{\pi}{4R}\left(1-e^{-R^2}\right),</math> where polar coordinates <math>z=Re^{it}</math> were used and [[Jordan's inequality]] was utilised for the second inequality. The integral along the real axis <math>\gamma_1</math> tends to the half [[Gaussian integral]] :<math>\int_0^\infty e^{-t^2} \, dt = \frac{\sqrt{\pi}}{2}.</math> Note too that because the integrand is an [[entire function]] on the complex plane, its integral along the whole contour is zero. Overall, we must have :<math>\int_0^\infty e^{-t^2} \, dt = \int_{\gamma_3} e^{-t^2} \, dt,</math> where <math>\gamma_3</math> denotes the bisector of the first quadrant, as in the diagram. To evaluate the right hand side, parametrize the bisector as :<math>t = re^{\pi i /4} = \frac{\sqrt{2}}{2}(1 + i)r</math> where r ranges from 0 to <math>+\infty</math>. Note that the square of this expression is just <math>+ir^2</math>. Therefore, substitution gives the right hand side as :<math>\int_0^\infty e^{-ir^2}\frac{\sqrt{2}}{2}(1 + i) \, dr.</math> Using [[Euler's formula]] to take real and imaginary parts of <math>e^{-ir^2}</math> gives this as :<math> \begin{align} & \int_0^\infty (\cos(r^2) - i\sin(r^2))\frac{\sqrt{2}}{2}(1 + i) \, dr \\ = {} & \frac{\sqrt{2}}{2} \int_0^\infty \left[\cos(r^2) + \sin(r^2) + i \left(\cos(r^2) - \sin(r^2)\right) \right] \, dr = \frac{\sqrt{\pi}}{2} + 0i, \end{align} </math> where we have written <math>0i</math> to emphasize that the original Gaussian integral's value is completely real with zero imaginary part. Letting <math>I_C = \int_0^\infty \cos(r^2) \, dr, I_S = \int_0^\infty \sin(r^2) \, dr</math> and then equating real and imaginary parts produces the following system of two equations in the two unknowns <math>I_C, I_S</math>: : <math> \begin{align} I_C + I_S & = \sqrt{\frac{\pi}{2}}, \\ I_C - I_S & = 0. \end{align} </math> Solving this for <math>I_C</math> and <math>I_S</math> gives the desired result. == Generalization == The integral : <math>\int x^m \exp(ix^n)\,dx = \int\sum_{l=0}^\infty\frac{i^lx^{m+nl}}{l!}\,dx = \sum_{l=0}^\infty \frac{i^l}{(m+nl+1)}\frac{x^{m+nl+1}}{l!}</math> is a [[confluent hypergeometric function]] and also an [[incomplete gamma function]]<ref>{{harvnb\|Mathar\|2012}}.</ref> : <math> \begin{align} \int x^m \exp(ix^n)\,dx & =\frac{x^{m+1}}{m+1}\,_1F_1\left(\begin{array}{c} \frac{m+1}{n}\\1+\frac{m+1}{n}\end{array}\mid ix^n\right) \\ & =\frac{1}{n} i^{(m+1)/n}\gamma\left(\frac{m+1}{n},-ix^n\right), \end{align} </math> which reduces to Fresnel integrals if real or imaginary parts are taken: : <math>\int x^m\sin(x^n)\,dx = \frac{x^{m+n+1}}{m+n+1} \,_1F_2\left(\begin{array}{c}\frac{1}{2}+\frac{m+1}{2n}\\ \frac{3}{2}+\frac{m+1}{2n},\frac{3}{2}\end{array}\mid -\frac{x^{2n}}{4}\right)</math>. The leading term in the asymptotic expansion is : <math> _1F_1 \left(\begin{array}{c}\frac{m+1}{n}\\1+\frac{m+1}{n} \end{array}\mid ix^n\right)\sim \frac{m+1}{n}\,\Gamma\left(\frac{m+1}{n}\right) e^{i\pi(m+1)/(2n)} x^{-m-1},</math> and therefore : <math>\int_0^\infty x^m \exp(ix^n)\,dx = \frac{1}{n} \,\Gamma\left(\frac{m+1}{n}\right)e^{i\pi(m+1)/(2n)}.</math> For ''m'' = 0, the imaginary part of this equation in particular is : <math>\int_0^\infty\sin(x^a)\,dx = \Gamma\left(1+\frac{1}{a} \right) \sin\left(\frac{\pi}{2a}\right),</math> with the left-hand side converging for ''a'' > 1 and the right-hand side being its analytical extension to the whole plane less where lie the poles of <math>\Gamma(a^{-1})</math>. The Kummer transformation of the confluent hypergeometric function is : <math> \int x^m \exp(ix^n)\,dx = V_{n,m}(x)e^{ix^n},</math> with : <math>V_{n,m} := \frac{x^{m+1}}{m+1}\,_1F_1\left(\begin{array}{c} 1 \\ 1 + \frac{m+1}{n} \end{array}\mid -ix^n\right).</math> == Numerical approximation == For computation to arbitrary precision, the power series is suitable for small argument. For large argument, asymptotic expansions converge faster.<ref>{{dlmf\|id=7.12.ii\|title=Error Functions, Dawson’s and Fresnel Integrals: Asymptotic expansions\|first=N. M.\|last=Temme}}.</ref> Continued fraction methods may also be used.<ref>{{harvnb\|Press\|Teukolsky\|Vetterling\|Flannery\|2007}}.</ref> For computation to particular target precision, other approximations have been developed. Cody<ref>{{harvnb\|Cody\|1968}}.</ref> developed a set of efficient approximations based on rational functions that give relative errors down to {{val\|2\|e=-19}}. A FORTRAN implementation of the Cody approximation that includes the values of the coefficients needed for implementation in other languages was published by van Snyder.<ref>{{harvnb\|van Snyder\|1993}}.</ref> Boersma developed an approximation with error less than {{val\|1.6\|e=-9}}.<ref>{{harvnb\|Boersma\|1960}}.</ref> == Applications == The Fresnel integrals were originally used in the calculation of the electromagnetic field intensity in an environment where light bends around opaque objects.<ref name=Beatty>{{harvnb\|Beatty\|2013}}.</ref> More recently, they have been used in the design of highways and railways, specifically their curvature transition zones, see [[track transition curve]].<ref name=Stewart /> Other applications are roller coasters<ref name=Beatty /> or calculating the transitions on a [[velodrome]] track to allow rapid entry to the bends and gradual exit.{{citation needed\|date=March 2014}} == See also == {{Portal\|Mathematics}} * [[Böhmer integral]] * [[Fresnel zone]] * [[Track transition curve]] * [[Euler spiral]] * [[Zone plate]] [[Dirichlet integral]] == Notes == {{Reflist}} == References == {{AS ref\|7\|297}} {{cite journal \| first1=Mohammad \| last1=Alazah \| title=Computing Fresnel integrals via modified trapezium rules \| year=2012 \| arxiv=1209.3451 \| doi=10.1007/s00211-014-0627-z \| volume=128 \| issue=4 \| journal=Numerische Mathematik \| pages=635–661 \| bibcode=2012arXiv1209.3451A \| s2cid=13934493 }} {{cite web \| last=Beatty \| first=Thomas \| title=How to evaluate Fresnel Integrals \| url=http://www.thomasbeatty.com/MATH%20PAGES/ARCHIVES%20-%20NOTES/Complex%20Variables/How%20to%20evaluate%20Fresnel%20Integrals.pdf \| work=FGCU Math - Summer 2013 \| year=2013 \| accessdate=27 July 2013 }} {{cite journal \| first1=J. \| last1=Boersma \| title=Computation of Fresnel Integrals \| journal=Math. Comp. \| volume=14 \| issue=72 \| year=1960 \| pages=380 \| mr=0121973 \| doi=10.1090/S0025-5718-1960-0121973-3 \| doi-access=free }} {{cite journal \| first1=Roland \| last1=Bulirsch \| title=Numerical calculation of the sine, cosine and Fresnel integrals \| year=1967 \| volume=9 \| number=5 \| pages=380–385 \| journal=Numer. Math. \| doi=10.1007/BF02162153 \| s2cid=121794086 }} {{cite journal \| last1=Cody \| first1=William J. \| title=Chebyshev approximations for the Fresnel integrals \| journal=Math. Comp. \| date=1968 \| volume=22 \| issue=102 \| pages=450–453 \| doi=10.1090/S0025-5718-68-99871-2 \| url=http://www.ams.org/journals/mcom/1968-22-102/S0025-5718-68-99871-2/S0025-5718-68-99871-2.pdf }} {{cite journal \| first1=R. J. \| last1=Hangelbroek \| title=Numerical approximation of Fresnel integrals by means of Chebyshev polynomials \| journal=J. Eng. Math. \| year=1967 \| volume=1 \| number=1 \| pages=37–50 \| doi=10.1007/BF01793638 \| bibcode = 1967JEnMa...1...37H \| s2cid=122271446 }} {{cite arxiv \| first1=R. J. \| last1=Mathar \| title=Series Expansion of Generalized Fresnel Integrals \| year=2012 \| eprint=1211.3963 \| class=math.CA }} {{cite web \| first1=R. \| last1=Nave \| url=http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/cornu.html#c1 \| title=The Cornu spiral \| year=2002 }} ''(Uses πt²/2 instead of t².)'' {{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 \| location=New York \| isbn=978-0-521-88068-8 \| chapter=Section 6.8.1. Fresnel Integrals \| chapter-url=http://apps.nrbook.com/empanel/index.html#pg=297 }} {{dlmf \| id=7 \| first=N. M. \| last=Temme \| title=Error Functions, Dawson’s and Fresnel Integrals }} {{cite journal \| last1=van Snyder \| first1=W. \| title=Algorithm 723: Fresnel integrals \| journal=ACM Trans. Math. Softw. \| year=1993 \| volume=19 \| issue=4 \| pages=452–456 \| doi=10.1145/168173.168193 \| s2cid=12346795 }} {{cite book \| first=James \| last=Stewart \| title=Calculus Early Transcendentals \| url=https://books.google.com/books?id=v4-1Rk7uaEcC&q=%22design+of+highways%22&pg=PA383 \| year=2008 \| publisher=Cengage Learning EMEA \| isbn=978-0-495-38273-7 }} {{cite book \| first1=A. \| last1=van Wijngaarden \| first2=W. L. \| last2=Scheen \| title=Table of Fresnel Integrals \| year=1949 \| series= Verhandl. Konink. Ned. Akad. Wetenschapen \| volume=19 \| number=4 }} {{cite journal \| first1=Aurel J. \| last1=Zajta \| first2=Sudhir K. \| last2=Goel \| title=Parametric Integration Techniques \| year=1989 \| journal=Mathematics Magazine \| volume=62 \| number=5 \| pages=318–322 \| doi=10.1080/0025570X.1989.11977462 }} ==External links== [http://www.netlib.org/cephes/ Cephes], [[Free and open-source software\|free/open-source]] C++/C code to compute Fresnel integrals among other special functions. Used in [[SciPy]] and [[ALGLIB]]. [http://ab-initio.mit.edu/Faddeeva Faddeeva Package], [[Free and open-source software\|free/open-source]] C++/C code to compute complex error functions (from which the Fresnel integrals can be obtained), with wrappers for Matlab, Python, and other languages. * {{springer\|title=Fresnel integrals\|id=p/f041720}} {{cite web \|url=http://fy.chalmers.se/LISEBERG/eng/loop_pe.html \|title=Roller Coaster Loop Shapes \|accessdate=2008-08-13 \|url-status=dead \|archiveurl=https://web.archive.org/web/20080923190052/http://fy.chalmers.se/LISEBERG/eng/loop_pe.html \|archivedate=September 23, 2008 }} {{mathworld\|title=Fresnel Integrals\|urlname=FresnelIntegrals}} *{{mathworld\|title=Cornu Spiral\|urlname=CornuSpiral}} {{Nonelementary Integral}} [[Category:Integral calculus]] [[Category:Spirals]] [[Category:Optics]] [[Category:Special functions]] [[Category:Special hypergeometric functions]] [[Category:Analytic functions]]
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