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 I_x(a,b) }

${\displaystyle I_{x}(a,b)}$

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

Semantic latex: \normincBetaI{x}@{a}{b}

Confidence: 0.86273327448479

Mathematica

Translation: BetaRegularized[x, a, b]

Information

Sub Equations

BetaRegularized[x, a, b]

Free variables

a

b

x

Symbol info

Regularized incomplete Beta function; Example: \normincBetaI{x}@{a}{b}

Will be translated to: BetaRegularized[$0, $1, $2] Relevant links to definitions: DLMF: http://dlmf.nist.gov/8.17#E2 Mathematica: https://reference.wolfram.com/language/ref/BetaRegularized.html

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \normincBetaI [\normincBetaI]

Tests

Symbolic

Numeric

Maple

Translation:

Information

Symbol info

(LaTeX -> Maple) No translation possible for given token: Cannot extract information from feature set: \normincBetaI [\normincBetaI]

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$x$

Is part of

$F(k;\,n,p)=\Pr \left(X\leq k\right)=I_{1-p}(n-k,k+1)=1-I_{p}(k+1,n-k)$

${\begin{aligned}I_{0}(a,b)&=0\\I_{1}(a,b)&=1\\I_{x}(a,1)&=x^{a}\\I_{x}(1,b)&=1-(1-x)^{b}\\I_{x}(a,b)&=1-I_{1-x}(b,a)\\I_{x}(a+1,b)&=I_{x}(a,b)-{\frac {x^{a}(1-x)^{b}}{a\mathrm {B} (a,b)}}\\I_{x}(a,b+1)&=I_{x}(a,b)+{\frac {x^{a}(1-x)^{b}}{b\mathrm {B} (a,b)}}\\\mathrm {B} (x;a,b)&=(-1)^{a}\mathrm {B} \left({\frac {x}{x-1}};a,1-a-b\right)\end{aligned}}$

$I_{x}(a,b)={\frac {\mathrm {B} (x;\,a,b)}{\mathrm {B} (a,b)}}$

Description

x

b

Beta

Mathematica

complete beta function

cumulative distribution function of the beta distribution

term of the incomplete beta function

incomplete beta function

cumulative distribution function

regularized incomplete beta function

regularized beta function

binomial distribution with probability

number of Bernoulli trial

single success

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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