Beta function

{{Short description|Mathematical function}} {{About|the Euler beta function}} [[File:Beta function.svg|thumb|Contour plot of the beta function]]

In mathematics, the '''beta function''', also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral

<math display="block"> \Beta(z_1,z_2) = \int_0^1 t^{z_1-1}(1-t)^{z_2-1}\,dt</math> for complex number inputs <math> z_1, z_2 </math> such that <math> \operatorname{Re}(z_1), \operatorname{Re}(z_2)>0</math>.

The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol {{math|Β}} is a Greek capital beta.

== Properties == The beta function is symmetric, meaning that <math> \Beta(z_1,z_2) = \Beta(z_2,z_1)</math> for all inputs <math>z_1</math> and <math>z_2</math>.<ref name=Davis622>{{citation | last = Davis | first = Philip J. | title = Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables | 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 | publisher = Dover Publications | location = New York | isbn = 978-0-486-61272-0 | year = 1972 | url = https://archive.org/details/handbookofmathe000abra/page/258/mode/2up?view=theater | page = 258 }}. Specifically, see 6.2 Beta Function.</ref>

A key property of the beta function is its close relationship to the gamma function:<ref name=Davis622/>

<math display="block"> \Beta(z_1,z_2)=\frac{\Gamma(z_1)\,\Gamma(z_2)}{\Gamma(z_1+z_2)}</math>

A proof is given below in {{slink||Relationship to the gamma function}}.

The beta function is also closely related to binomial coefficients. When {{mvar|m}} (or {{mvar|n}}, by symmetry) is a positive integer, it follows from the definition of the gamma function {{math|Γ}} that<ref name=Davis622/>

<math display="block"> \Beta(m,n) =\frac{(m-1)!\,(n-1)!}{(m+n-1)!} = \frac{m + n}{mn} \Bigg/ \binom{m + n}{m} </math>

== Relationship to the gamma function == To derive this relation, write the product of two factorials as integrals. Since they are integrals in two separate variables, we can combine them into an iterated integral:

<math display="block">\begin{align} \Gamma(z_1)\Gamma(z_2) &= \int_{u=0}^\infty\ e^{-u} u^{z_1-1}\,du \cdot\int_{v=0}^\infty\ e^{-v} v^{z_2-1}\,dv \\[6pt] &=\int_{v=0}^\infty\int_{u=0}^\infty\ e^{-u-v} u^{z_1-1}v^{z_2-1}\, du \,dv. \end{align}</math>

Changing variables by {{math|''u'' {{=}} ''st''}} and {{math|''v'' {{=}} ''s''(1 − ''t'')}}, because {{math|''u + v'' {{=}} ''s''}} and {{math|1=''u'' / (''u''+''v'') = ''t''}}, we have that the limits of integrations for {{math| ''s''}} are 0 to ∞ and the limits of integration for {{math| ''t''}} are 0 to 1. Thus produces

<math display="block">\begin{align} \Gamma(z_1)\Gamma(z_2) &= \int_{s=0}^\infty\int_{t=0}^1 e^{-s} (st)^{z_1-1}(s(1-t))^{z_2-1}s\,dt \,ds \\[6pt] &= \int_{s=0}^\infty e^{-s}s^{z_1+z_2-1} \,ds\cdot\int_{t=0}^1 t^{z_1-1}(1-t)^{z_2-1}\,dt\\[1ex] &=\Gamma(z_1+z_2) \cdot \Beta(z_1,z_2). \end{align}</math>

Dividing both sides by <math>\Gamma(z_1+z_2)</math> gives the desired result.

The stated identity may be seen as a particular case of the identity for the integral of a convolution. Taking

<math display="block">\begin{align}f(u)&:=e^{-u} u^{z_1-1} 1_{\R_+} \\ g(u)&:=e^{-u} u^{z_2-1} 1_{\R_+}, \end{align}</math>

one has:

<math display="block"> \Gamma(z_1) \Gamma(z_2) = \int_{\R}f(u)\,du\cdot \int_{\R} g(u) \,du = \int_{\R}(f*g)(u)\,du =\Beta(z_1,z_2)\,\Gamma(z_1+z_2).</math>

See ''The Gamma Function'', page 18–19<ref>{{citation|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> for a derivation of this relation.

== Differentiation of the beta function ==

We have

<math display="block">\frac{\partial}{\partial z_1} \mathrm{B}(z_1, z_2) = \mathrm{B}(z_1, z_2) \left( \frac{\Gamma'(z_1)}{\Gamma(z_1)} - \frac{\Gamma'(z_1 + z_2)}{\Gamma(z_1 + z_2)} \right) = \mathrm{B}(z_1, z_2) \big(\psi(z_1) - \psi(z_1 + z_2)\big),</math>

<math display="block">\frac{\partial}{\partial z_m} \mathrm{B}(z_1, z_2, \dots, z_n) = \mathrm{B}(z_1, z_2, \dots, z_n) \left(\psi(z_m) - \psi{\left( \sum_{k=1}^n z_k \right)}\right), \quad 1\le m\le n,</math>

where <math>\psi(z)</math> denotes the digamma function.

==Approximation== Stirling's approximation gives the asymptotic formula

for large {{mvar|x}} and large {{mvar|y}}.

If on the other hand {{mvar|x}} is large and {{mvar|y}} is fixed, then

<math display="block">\Beta(x,y) \sim \Gamma(y)\,x^{-y}.</math>

== Other identities and formulas == The integral defining the beta function may be rewritten in a variety of ways, including the following: <math display="block"> \begin{align} \Beta(z_1,z_2) &= 2\int_0^{\pi / 2}(\sin\theta)^{2z_1-1}(\cos\theta)^{2z_2-1}\,d\theta, \\[6pt] &= \int_0^\infty\frac{t^{z_1-1}}{(1+t)^{z_1+z_2}}\,dt, \\[6pt] &= n\int_0^1t^{nz_1-1}(1-t^n)^{z_2-1}\,dt, \\ &= (1-a)^{z_2} \int_0^1 \frac{(1-t)^{z_1-1}t^{z_2-1}}{(1-at)^{z_1+z_2}}dt \qquad \text{for any } a\in\mathbb{R}_{\leq 1}, \end{align}</math>

where in the second-to-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>.

For values <math>z=z_1=z_2\neq1</math> we have:

<math display="block"> \Beta(z,z) = \frac{1}{z}\int_0^{\pi / 2}\frac{1}{\left(\sqrt[z]{\sin\theta} + \sqrt[z]{\cos\theta}\right) ^{2z}}\,d\theta </math>

The beta function can be written as an infinite sum<ref>{{citation|url=https://functions.wolfram.com/GammaBetaErf/Beta/06/03/0001/|title = Beta function : Series representations (Formula 06.18.06.0007)}}</ref> <math display="block">\Beta(x,y) = \sum_{n=0}^\infty \frac{(1-x)_n}{(y+n)\,n!}</math> If <math>x</math> and <math>y</math> are equal to a number <math>z</math> we get: <math display="block"> \Beta(z,z) = 2\sum_{n=0}^\infty \frac{(2z+n-1)_n (-1)^n}{(z+n)n!} = \lim_{x \to 1^-}2\sum_{n=0}^\infty \frac{(-2z)_n x^n}{(z+n)n!} </math> where <math>(x)_n</math> is the rising factorial, and as an infinite product <math display="block">\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

which can be proved as follows:

<math display="block">\begin{aligned} \Beta(x, y+1) + \Beta(x+1, y) &= \frac{\Gamma(x)\Gamma(y+1)}{\Gamma(x+y+1)} + \frac{\Gamma(x+1)\Gamma(y)}{\Gamma(x+y+1)}\\ &= \frac{y\Gamma(x)\Gamma(y) + x\Gamma(x)\Gamma(y)}{(x+y)\Gamma(x+y)}\\ &= \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\\ &= \Beta(x,y). \end{aligned} </math>

The above proof also shows the simple recurrence on one coordinate<ref>{{citation|last=Mäklin|first=Tommi|year=2022|title=Probabilistic Methods for High-Resolution Metagenomics|publisher=Unigrafia|location=Helsinki|pages=27|series=Series of publications A / Department of Computer Science, University of Helsinki|issn=2814-4031|isbn=978-951-51-8695-9|url=https://helda.helsinki.fi/bitstream/handle/10138/349862/M%C3%A4klin_Tommi_dissertation_2022.pdf}}</ref>

<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>

The positive integer values of the beta function are also the partial derivatives of a 2D function: for all nonnegative integers <math>m</math> and <math>n</math>, <math display="block">\Beta(m+1, n+1) = \frac{\partial^{m+n}h}{\partial a^m \, \partial b^n}(0, 0),</math> where <math display="block">h(a, b) = \frac{e^a-e^b}{a-b}.</math> The Pascal-like identity above implies that this function is a solution to the first-order partial differential equation <math display="block">h = h_a + h_b.</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 \mapsto t_+^x</math>: <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<ref>{{citation|title=Euler's Reflection Formula - ProofWiki|url=https://proofwiki.org/wiki/Euler%27s_Reflection_Formula | access-date=2020-09-02|website=proofwiki.org}}</ref> <math display="block"> \Beta(x,1-x) = \dfrac{\pi}{\sin(\pi x)}, \qquad x \not \in \mathbb{Z} </math>

By taking <math> x = \frac{1}{2}</math> in this last formula, it follows that <math>\Gamma(1/2) = \sqrt{\pi}</math>. Generalizing this into a bivariate identity for a product of beta functions leads to: <math display="block"> \Beta(x,y) \cdot \Beta(x+y,1-y) = \frac{\pi}{x \sin(\pi y)} .</math>

Also, using Legendre duplication formula, we get <math display="block"> 2^{z-1}\Beta(z/2,z/2) = \Beta(1/2,z/2) .</math>

Euler's integral for the beta function may be converted into an integral over the Pochhammer contour {{mvar|C}} as

<math display="block">\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 factorials, the beta function can define a binomial coefficient after adjusting indices: <math display="block">\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 display="block">\binom{n}{k} = (-1)^n\, n! \cdot\frac{\sin (\pi k)}{\pi \displaystyle\prod_{i=0}^n (k-i)}.</math>

==Reciprocal beta function== The '''reciprocal beta function''' is the function about the form

Interestingly, their integral representations closely relate as the definite integral of trigonometric functions with product of its power and multiple-angle:<ref>{{dlmf|id=5.12|title=Beta Function|first=R. B. |last=Paris}}</ref>

<math display="block">\begin{align} \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)}} \\[1ex] \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)}} \\[1ex] \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)}} \\[1ex] \int_0^\frac{\pi}{2}\cos^{x-1}\theta\cos y\theta~d\theta &= \frac{\pi}{ 2^x x \Beta{\left(\frac{x+y+1}{2},\frac{x-y+1}{2}\right)}} \end{align}</math>

==Incomplete beta function== The '''incomplete beta function''', a generalization of the beta function, is defined as<ref>{{citation | last1 = Zelen | first1 = M. | last2 = Severo | first2 = N. C. | editor1-last = Abramowitz | editor1-first = Milton | editor1-link = Milton Abramowitz | editor2-last = Stegun | editor2-first = Irene A. | editor2-link = Irene Stegun | year = 1972 | title = Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables | chapter = 26. Probability functions | pages = [https://archive.org/details/handbookofmathe000abra/page/944 944] | publisher = Dover Publications | location = New York | isbn = 978-0-486-61272-0}}</ref><ref name="paris-ibf">{{dlmf|mode=cs2 | last = Paris | first = R. B. | id = 8.17 | title = Incomplete beta functions}}</ref>

For {{math|1=''x'' = 1}}, the incomplete beta function coincides with the complete beta function. For positive integers ''a'' and ''b'', the incomplete beta function will be a polynomial of degree {{math|''a'' + ''b'' − 1}} with rational coefficients.

By the substitution <math>t = \sin^2\theta</math> and <math>t = \frac1{1+s}</math>, we can show that <math display="block">\begin{align} \Beta(x;\,a,b) &= 2 \int_0^{\arcsin \sqrt x} \sin^{2a-1\!}\theta \cos^{2b-1\!}\theta \, d\theta \\[1ex] &= \int_{\frac{1-x}x}^\infty \frac{s^{b-1}}{(1+s)^{a+b}} \, ds \end{align}</math>

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:

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 display="block">\begin{align} F(k;\,n,p) &= \Pr\left(X \le k\right) \\[1ex] &= I_{1-p}(n-k, k+1) \\[1ex] &= 1 - I_p(k+1,n-k). \end{align} </math>

===Properties===  <math display="block">\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)}, \\ \int \Beta(x;a,b) \, dx &= x \Beta(x; a, b) - \Beta(x; a+1, b), \\ \Beta(x;a,b) &= (-1)^a \Beta\left(\frac{x}{x-1};a,1-a-b\right). \end{align}</math>

===Continued fraction expansion===

The continued fraction expansion is

<math display="block">\Beta(x;\,a,b) = \frac{x^{a} (1 - x)^{b}}{a \left(1 + \frac{{d}_{1}}{1 + \frac{{d}_{2}}{1 + \frac{{d}_{3}}{1 + \cdots}}}\right)},</math>

with odd and even coefficients given by

<math display="block">\begin{align} {d}_{2m + 1} &= - \frac{(a + m) (a + b + m) x}{(a + 2 m) (a + 2 m + 1)}, \\[1ex] {d}_{2m} &= \frac{m (b - m) x}{(a + 2 m - 1) (a + 2 m)}. \end{align}</math>

The <math>4 m</math> and <math>4 m + 1</math> convergents are less than <math>\Beta(x;\,a,b)</math>, while the <math>4 m + 2</math> and <math>4 m + 3</math> convergents are greater than <math>\Beta(x;\,a,b)</math>.

It converges rapidly for <math>x<(a+1)/(a+b+2)</math>. For <math>x > (a + 1)/(a + b + 2)</math> or <math>1 - x < (b + 1)/(a + b + 2)</math>, the function may be evaluated more efficiently through the relation <math>\Beta(x;\,a,b) = \Beta(a, b) - \Beta(1 - x;\,b,a)</math>.<ref name="paris-ibf"/>

==Multivariate beta function== The beta function can be extended to a function with more than two arguments:

<math display="block">\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 coefficients and binomial coefficients. For example, it satisfies a similar version of Pascal's identity:

<math display="block">\Beta(\alpha_1,\alpha_2,\ldots\alpha_n) = \Beta(\alpha_1+1,\alpha_2,\ldots\alpha_n)+\Beta(\alpha_1,\alpha_2+1,\ldots\alpha_n)+\cdots+\Beta(\alpha_1,\alpha_2,\ldots\alpha_n+1) .</math>

== Applications == The beta function is useful in computing and representing the scattering amplitude for Regge trajectories. Furthermore, it was the first known 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 process. The beta function is also important in statistics, e.g. for the beta distribution and beta prime distribution. As briefly alluded to previously, the beta function is closely tied with the gamma function and plays an important role in calculus.

==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 systems.

In Microsoft Excel, for example, the complete beta function can be computed with the <code>GammaLn</code> function (or <code>special.gammaln</code> in Python's SciPy package): :<code>Value = Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b))</code>

This result follows from the properties listed above.

The incomplete beta function cannot be directly computed using such relations and other methods must be used. In [https://www.gnu.org/software/gsl/doc/html/specfunc.html#incomplete-beta-function GNU Octave], it is computed using a continued fraction expansion.

The incomplete beta function has existing implementation in common languages. For instance, <code>betainc</code> (incomplete beta function) in MATLAB and GNU Octave, <code>pbeta</code> (probability of beta distribution) in R and <code>betainc</code> in SymPy. In SciPy, <code>special.betainc</code> computes the regularized incomplete beta function—which is, in fact, the cumulative beta distribution. To get the actual incomplete beta function, one can multiply the result of <code>special.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. * Nørlund–Rice integral * Yule–Simon distribution

==References== {{reflist}} * {{dlmf|mode=cs2|authorlink=Richard Askey|first=R. A.|last= Askey|first2= R.|last2= Roy |id=5.12 }} * {{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 | access-date=2011-08-09 | archive-date=2021-10-27 | archive-url=https://web.archive.org/web/20211027043154/http://apps.nrbook.com/empanel/index.html?pg=256 | url-status=dead }}

==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]

Category:Gamma and related functions Category:Special hypergeometric functions