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 = \frac{1}{2}}

${\displaystyle x={\frac {1}{2}}}$

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

Semantic latex: x = \frac{1}{2}

Confidence: 0

Mathematica

Translation: x == Divide[1,2]

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Sub Equations

x = Divide[1,2]

Free variables

x

Tests

Symbolic

Test expression: (x)-(Divide[1,2])

ERROR:

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Numeric

SymPy

Translation: x == (1)/(2)

Information

Sub Equations

x = (1)/(2)

Free variables

x

Tests

Symbolic

Numeric

Maple

Translation: x = (1)/(2)

Information

Sub Equations

x = (1)/(2)

Free variables

x

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$x$

Description

beta function

evaluation at particular point

example

last formula

power function

term of a convolution

Complete translation information:

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Specify your own input

Formula input
Wikitext context	{{About\|the Euler beta function}} [[File:Beta function contour plot.png\|thumb\|[[Contour plot]] of the beta function]] In [[mathematics]], the '''beta function''', also called the [[Euler integral (disambiguation)\|Euler integral]] of the first kind, is a [[special function]] that is closely related to the [[gamma function]] and to [[binomial coefficient]]s. It is defined by the [[integral]] :<math> \Beta(x,y) = \int_0^1 t^{x-1}(1-t)^{y-1}\,dt</math> for [[complex number]] inputs {{math\|''x'', ''y''}} such that {{math\|Re ''x'' > 0, Re ''y'' > 0}}. The beta function was studied by [[Leonhard Euler\|Euler]] and [[Adrien-Marie Legendre\|Legendre]] and was given its name by [[Jacques Philippe Marie Binet\|Jacques Binet]]; its symbol {{math\|Β}} is a [[Greek alphabet\|Greek]] capital [[Beta (letter)\|beta]]. == Properties == The beta function is [[symmetric function\|symmetric]], meaning that :<math> \Beta(x,y) = \Beta(y,x)</math> for all inputs {{mvar\|x}} and {{mvar\|y}}.<ref name=Davis622>Davis (1972) 6.2.2 p.258</ref> A key property of the beta function is its close relationship to the [[gamma function]]: one has that<ref name=Davis622/> :<math> \Beta(x,y)=\frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}.</math> (A proof is given below in {{slink\|\|Relationship to the gamma function}}.) The beta function is also closely related to [[binomial coefficient]]s. When {{mvar\|x}} and {{mvar\|y}} are positive integers, it follows from the definition of the [[gamma function]] {{math\|Γ}} that<ref name=Davis621>Davis (1972) 6.2.1 p.258</ref> :<math> \Beta(x,y) =\dfrac{(x-1)!\,(y-1)!}{(x+y-1)!} = \frac{x + y}{xy} \cdot \frac{1}{\binom{x + y}{x}}. </math> == Relationship to the gamma function == A simple derivation of the relation <math> \Beta(x,y)=\frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}</math> can be found in Emil Artin's book ''The Gamma Function'', page 18–19.<ref>{{cite book\|last1=Artin\|first1=Emil\|title=The Gamma Function\|pages=18–19\|url=http://www.plouffe.fr/simon/math/Artin%20E.%20The%20Gamma%20Function%20(1931)(23s).pdf\|access-date=2016-11-11\|archive-url=https://web.archive.org/web/20161112081854/http://www.plouffe.fr/simon/math/Artin%20E.%20The%20Gamma%20Function%20(1931)(23s).pdf\|archive-date=2016-11-12\|url-status=dead}}</ref> To derive this relation, write the product of two factorials as :<math>\begin{align} \Gamma(x)\Gamma(y) &= \int_{u=0}^\infty\ e^{-u} u^{x-1}\,du \cdot\int_{v=0}^\infty\ e^{-v} v^{y-1}\,dv \\[6pt] &=\int_{v=0}^\infty\int_{u=0}^\infty\ e^{-u-v} u^{x-1}v^{y-1}\, du \,dv. \end{align}</math> Changing variables by {{math\|''u'' {{=}} ''zt''}} and {{math\|''v'' {{=}} ''z''(1 − ''t'')}} produces :<math>\begin{align} \Gamma(x)\Gamma(y) &= \int_{z=0}^\infty\int_{t=0}^1 e^{-z} (zt)^{x-1}(z(1-t))^{y-1}z\,dt \,dz \\[6pt] &= \int_{z=0}^\infty e^{-z}z^{x+y-1} \,dz\cdot\int_{t=0}^1 t^{x-1}(1-t)^{y-1}\,dt\\ &=\Gamma(x+y) \cdot \Beta(x,y). \end{align}</math> Dividing both sides by <math>\Gamma(x+y)</math> gives the desired result. The stated identity may be seen as a particular case of the identity for the [[convolution#Integration\|integral of a convolution]]. Taking :<math>\begin{align}f(u)&:=e^{-u} u^{x-1} 1_{\R_+} \\ g(u)&:=e^{-u} u^{y-1} 1_{\R_+}, \end{align}</math> one has: :<math> \Gamma(x) \Gamma(y) = \int_{\R}f(u)\,du\cdot \int_{\R} g(u) \,du = \int_{\R}(fg)(u)\,du =\Beta(x, y)\,\Gamma(x+y).</math> == Derivatives == We have :<math>\frac{\partial}{\partial x} \mathrm{B}(x, y) = \mathrm{B}(x, y) \left( \frac{\Gamma'(x)}{\Gamma(x)} - \frac{\Gamma'(x + y)}{\Gamma(x + y)} \right) = \mathrm{B}(x, y) \big(\psi(x) - \psi(x + y)\big),</math> where <math>\psi(x)</math> is the [[digamma function]]. ==Approximation== [[Stirling's approximation]] gives the asymptotic formula :<math>\Beta(x,y) \sim \sqrt {2\pi } \frac{x^{x - 1/2} y^{y - 1/2} }{( {x + y} )^{x + y - 1/2} }</math> for large {{mvar\|x}} and large {{mvar\|y}}. If on the other hand {{mvar\|x}} is large and {{mvar\|y}} is fixed, then :<math>\Beta(x,y) \sim \Gamma(y)\,x^{-y}.</math> == Other identities and formulas == {{uncited section\|date=June 2020}} The integral defining the beta function may be rewritten in a variety of ways, including the following: <math display = block> \begin{align} \Beta(x,y) &= 2\int_0^{\pi / 2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}\,d\theta, \\[6pt] &= \int_0^\infty\frac{t^{x-1}}{(1+t)^{x+y}}\,dt, \\[6pt] &= n\int_0^1t^{nx-1}(1-t^n)^{y-1}\,dt, \end{align}</math> where in the last identity {{mvar\|n}} is any positive real number. (One may move from the first integral to the second one by substituting <math>t = \tan^2(\theta)</math>.) The beta function can be written as an infinite sum : <math>\Beta(x,y) = \sum_{n=0}^\infty \frac\binom{n-y}{n} {x+n}</math>{{dubious\|date=June 2020}} <!-- :<math> \Beta(x,y) = \frac{1}{y}\sum_{n=0}^\infty(-1)^n\frac{y^{n+1}}{n!(x+n)}</math> --> and as an infinite product : <math>\Beta(x,y) = \frac{x+y}{x y} \prod_{n=1}^\infty \left( 1+ \dfrac{x y}{n (x+y+n)}\right)^{-1}.</math> The beta function satisfies several identities analogous to corresponding identities for binomial coefficients, including a version of [[Pascal's identity]] <math display = block> \Beta(x,y) = \Beta(x, y+1) + \Beta(x+1, y)</math> and a simple recurrence on one coordinate: <math display = block>\Beta(x+1,y) = \Beta(x, y) \cdot \dfrac{x}{x+y}, \quad \Beta(x,y+1) = \Beta(x, y) \cdot \dfrac{y}{x+y}.</math> For <math>x, y \geq 1</math>, the beta function may be written in terms of a [[convolution]] involving the [[truncated power function]] {{math\|''t'' ↦ ''t''{{su\|b=+\|p=''x''}}}}: <math display = block> \Beta(x,y) \cdot\left(t \mapsto t_+^{x+y-1}\right) = \Big(t \mapsto t_+^{x-1}\Big) \Big(t \mapsto t_+^{y-1}\Big)</math> Evaluations at particular points may simplify significantly; for example, <math display = block> \Beta(1,x) = \dfrac{1}{x} </math> and <math display="block"> \Beta(x,1-x) = \dfrac{\pi}{\sin(\pi x)}, \qquad x \not \in \mathbb{Z} </math><ref>{{Cite web\|title=Euler's Reflection Formula - ProofWiki\|url=https://proofwiki.org/wiki/Euler%27s_Reflection_Formula\|access-date=2020-09-02\|website=proofwiki.org}}</ref> By taking <math> x = \frac{1}{2}</math> in this last formula, one may conclude in particular that {{math\|Γ(1/2) {{=}} {{sqrt\|π}}}}. One may also generalize the last formula into a bivariate identity for a product of beta functions: <math display="block"> \Beta(x,y) \cdot \Beta(x+y,1-y) = \frac{\pi}{x \sin(\pi y)} .</math> Euler's integral for the beta function may be converted into an integral over the [[Pochhammer contour]] {{mvar\|C}} as :<math>\left(1-e^{2\pi i\alpha}\right)\left(1-e^{2\pi i\beta}\right)\Beta(\alpha,\beta) =\int_C t^{\alpha-1}(1-t)^{\beta-1} \, dt.</math> This Pochhammer contour integral converges for all values of {{mvar\|α}} and {{mvar\|β}} and so gives the [[analytic continuation]] of the beta function. Just as the gamma function for integers describes [[factorial]]s, the beta function can define a [[binomial coefficient]] after adjusting indices: :<math>\binom{n}{k} = \frac{1}{(n+1) \Beta(n-k+1, k+1)}.</math> Moreover, for integer {{mvar\|n}}, {{math\|Β}} can be factored to give a closed form interpolation function for continuous values of {{mvar\|k}}: :<math>\binom{n}{k} = (-1)^n\, n! \cdot\frac{\sin (\pi k)}{\pi \displaystyle\prod_{i=0}^n (k-i)}.</math> The beta function was the first known [[S matrix\|scattering amplitude]] in [[string theory]], first conjectured by [[Gabriele Veneziano]]. It also occurs in the theory of the [[preferential attachment]] process, a type of stochastic [[urn problem\|urn process]]. ==Reciprocal beta function== The '''reciprocal beta function''' is the [[special function\|function]] about the form :<math>f(z)=\frac{1}{\Beta(x,y)}</math> Interestingly, their integral representations closely relate as the [[definite integral]] of [[trigonometric functions]] with product of its power and [[List of trigonometric identities#Multiple-angle formulae\|multiple-angle]]:<ref>{{dlmf\|id=5.12\|title=Beta Function\|first=R. B. \|last=Paris}}</ref> :<math>\int_0^\pi\sin^{x-1}\theta\sin y\theta~d\theta=\frac{\pi\sin\frac{y\pi}{2}}{2^{x-1}x\Beta\left(\frac{x+y+1}{2},\frac{x-y+1}{2}\right)}</math> :<math>\int_0^\pi\sin^{x-1}\theta\cos y\theta~d\theta=\frac{\pi\cos\frac{y\pi}{2}}{2^{x-1}x\Beta\left(\frac{x+y+1}{2},\frac{x-y+1}{2}\right)}</math> :<math>\int_0^\pi\cos^{x-1}\theta\sin y\theta~d\theta=\frac{\pi\cos\frac{y\pi}{2}}{2^{x-1}x\Beta\left(\frac{x+y+1}{2},\frac{x-y+1}{2}\right)}</math> :<math>\int_0^\frac{\pi}{2}\cos^{x-1}\theta\cos y\theta~d\theta=\frac{\pi}{2^xx\Beta\left(\frac{x+y+1}{2},\frac{x-y+1}{2}\right)}</math> ==Incomplete beta function== The '''incomplete beta function''', a generalization of the beta function, is defined as :<math> \Beta(x;\,a,b) = \int_0^x t^{a-1}\,(1-t)^{b-1}\,dt. </math> For {{math\|''x'' {{=}} 1}}, the incomplete beta function coincides with the complete beta function. The relationship between the two functions is like that between the gamma function and its generalization the [[incomplete gamma function]]. The '''regularized incomplete beta function''' (or '''regularized beta function''' for short) is defined in terms of the incomplete beta function and the complete beta function: :<math> I_x(a,b) = \frac{\Beta(x;\,a,b)}{\Beta(a,b)}. </math> The regularized incomplete beta function is the [[cumulative distribution function]] of the [[beta distribution]], and is related to the [[cumulative distribution function]] <math>F(k;\,n,p)</math> of a [[random variable]] {{mvar\|X}} following a [[binomial distribution]] with probability of single success {{mvar\|p}} and number of Bernoulli trials {{mvar\|n}}: :<math>F(k;\,n,p) = \Pr\left(X \le k\right) = I_{1-p}(n-k, k+1) = 1 - I_p(k+1,n-k). </math> ===Properties=== <!-- (Many other properties could be listed here.)--> :<math>\begin{align} I_0(a,b) &= 0 \\ I_1(a,b) &= 1 \\ I_x(a,1) &= x^a\\ I_x(1,b) &= 1 - (1-x)^b \\ I_x(a,b) &= 1 - I_{1-x}(b,a) \\ I_x(a+1,b) &= I_x(a,b)-\frac{x^a(1-x)^b}{a \Beta(a,b)} \\ I_x(a,b+1) &= I_x(a,b)+\frac{x^a(1-x)^b}{b \Beta(a,b)} \\ \Beta(x;a,b)&=(-1)^{a} \Beta\left(\frac{x}{x-1};a,1-a-b\right) \end{align}</math> ==Multivariate beta function== The beta function can be extended to a function with more than two arguments: :<math>\Beta(\alpha_1,\alpha_2,\ldots\alpha_n) = \frac{\Gamma(\alpha_1)\,\Gamma(\alpha_2) \cdots \Gamma(\alpha_n)}{\Gamma(\alpha_1 + \alpha_2 + \cdots + \alpha_n)} .</math> This multivariate beta function is used in the definition of the [[Dirichlet distribution]]. Its relationship to the beta function is analogous to the relationship between [[multinomial coefficient]]s and binomial coefficients. ==Software implementation== Even if unavailable directly, the complete and incomplete beta function values can be calculated using functions commonly included in [[spreadsheet]] or [[computer algebra system]]s. In [[Microsoft Excel\|Excel]], for example, the complete beta value can be calculated from the <code>[[Gamma function#Approximations\|GammaLn]]</code> function: :<code>Value = Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b))</code> An incomplete beta value can be calculated as: :<code>Value = BetaDist(x, a, b) * Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b))</code>. These result follow from the properties [[#Properties\|listed above]]. Similarly, <code>betainc</code> (incomplete beta function) in [[MATLAB]] and [[GNU Octave]], <code>pbeta</code> (probability of beta distribution) in [[R (programming language)\|R]], or <code>special.betainc</code> in [[Python (programming language)\|Python's]] [[SciPy]] package compute the [[Beta distribution#Cumulative distribution function\|regularized incomplete beta function]]—which is, in fact, the cumulative beta distribution—and so, to get the actual incomplete beta function, one must multiply the result of <code>betainc</code> by the result returned by the corresponding <code>beta</code> function. In [[Mathematica]], <code>Beta[x, a, b]</code> and <code>BetaRegularized[x, a, b]</code> give <math> \Beta(x;\,a,b) </math> and <math> I_x(a,b) </math>, respectively. ==See also== * [[Beta distribution]] and [[Beta prime distribution]], two probability distributions related to the beta function * [[Jacobi sum]], the analogue of the beta function over finite fields. * [[Nörlund–Rice integral]] * [[Yule–Simon distribution]] {{More footnotes\|date=November 2010}} ==References== {{reflist}} * {{dlmf\|authorlink=Richard Askey\|first=R. A.\|last= Askey\|first2= R.\|last2= Roy \|id=5.12 }} {{citation \| first1=M. \| last1=Zelen \| first2=N. C. \| last2=Severo \| chapter=26. Probability functions \| pages=[https://archive.org/details/handbookofmathe000abra/page/925 925–995] \| editor1-last=Abramowitz \| editor1-first=Milton \| editor1-link=Milton Abramowitz \| editor2-last=Stegun \| editor2-first=Irene A. \| editor2-link=Irene Stegun \| title=[[Abramowitz and Stegun\|Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]] \| publisher=[[Dover Publications]] \| location=New York \| isbn=978-0-486-61272-0 \| year=1972 }} {{citation \| first=Philip J. \| last=Davis \| chapter=6. Gamma function and related functions \| editor1-last=Abramowitz \| editor1-first=Milton \| editor1-link=Milton Abramowitz \| editor2-last=Stegun \| editor2-first=Irene A. \| editor2-link=Irene Stegun \| title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables \| publisher=[[Dover Publications]] \| location=New York \| isbn=978-0-486-61272-0 \| year=1972 \| url=https://archive.org/details/handbookofmathe000abra }} {{dlmf\|first=R. B. \|last=Paris\|id=8.17\|title=Incomplete beta functions}} {{Citation \| last1=Press \| first1=W. H. \| last2=Teukolsky \| first2=SA \| last3=Vetterling \| first3=WT \| last4=Flannery \| first4=BP \| 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 6.1 Gamma Function, Beta Function, Factorials \| chapter-url=http://apps.nrbook.com/empanel/index.html?pg=256}} ==External links== * {{springer\|title=Beta-function\|id=p/b015960}} * {{planetmath\|evaluationofbetafunctionusinglaplacetransform\|title=Evaluation of beta function using Laplace transform}} * Arbitrarily accurate values can be obtained from: [http://functions.wolfram.com The Wolfram Functions Site]: [http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=BetaRegularized Evaluate Beta Regularized Incomplete beta] danielsoper.com: [https://web.archive.org/web/20070120151547/http://www.danielsoper.com/statcalc/calc36.aspx Incomplete Beta Function Calculator], [https://web.archive.org/web/20070120151557/http://www.danielsoper.com/statcalc/calc37.aspx Regularized Incomplete Beta Function Calculator] {{Authority control}} {{DEFAULTSORT:Beta Function}} [[Category:Gamma and related functions]] [[Category:Special hypergeometric functions]]
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