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Incomplete beta function: Fixed typo

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{{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 (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(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 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(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> \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 coefficient]]s. 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> \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>\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| ''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>\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\\
&=\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 [[convolution#Integration|integral of a convolution]]. Taking

:<math>\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> \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>\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>\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

:<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 ==
The integral defining the beta function may be rewritten in a variety of ways, including the following:
:<math>
\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>
\Beta(z,z) = \frac{1}{z}\int_0^{\pi / 2}\frac{1}{(\sqrt[z]{\sin\theta} + \sqrt[z]{\cos\theta})^{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>\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>
\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 [[falling and rising factorials|rising factorial]])
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> \Beta(x,y) = \Beta(x, y+1) + \Beta(x+1, y)</math>

and a simple recurrence on one coordinate:

:<math>\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><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>

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>\Beta(m+1, n+1) = \frac{\partial^{m+n}h}{\partial a^m \, \partial b^n}(0, 0),</math>
where
:<math>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>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> \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> \Beta(1,x) = \dfrac{1}{x} </math>
and
:<math> \Beta(x,1-x) = \dfrac{\pi}{\sin(\pi x)}, \qquad x \not \in \mathbb{Z} </math><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>

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> \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>

==Reciprocal beta function==
The '''reciprocal beta function''' is the [[special function|function]] about the form

:<math>f(x,y)=\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<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 = [[Abramowitz and Stegun|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>

:<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. For positive integers ''a'' and ''b'', the incomplete beta function will be a polynomial of degree ''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>\Beta(x;\,a,b) = 2 \int_0^{\arcsin \sqrt x} \sin^{2a-1\!}\theta\cos^{2b-1\!}\theta\,\mathrm d\theta = \int_{\frac{1-x}x}^\infty \frac{s^{b-1}}{(1+s)^{a+b}}\,\mathrm ds</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:

:<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===

:<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)} \\
\int \Beta(x;a,b) \mathrm{d}x &= 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 [[generalized continued fraction|continued fraction]] expansion

:<math>\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 +} \frac{{d}_{4}}{1 +} \cdots \right)}</math>

with odd and even coefficients respectively

:<math>{d}_{2 m + 1} = - \frac{(a + m) (a + b + m) x}{(a + 2 m) (a + 2 m + 1)}</math>
:<math>{d}_{2 m} = \frac{m (b - m) x}{(a + 2 m - 1) (a + 2 m)}</math>

converges rapidly when <math>x</math> is not close to 1. 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>.

For <math>x > \frac{a + 1}{a + b + 2}</math>, the function may be evaluated more efficiently using <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>\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. For example, it satisfies a similar version of Pascal's identity:

:<math>\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 [[S matrix|scattering amplitude]] in [[string theory]], first [[Veneziano amplitude|conjectured]] by [[Gabriele Veneziano]]. It also occurs in the theory of the [[preferential attachment]] process, a type of stochastic [[urn problem|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 system]]s.

In [[Microsoft Excel]], for example, the complete beta function can be computed with the <code>[[Gamma_function#Log-gamma function|GammaLn]]</code> function (or <code>special.gammaln</code> in [[Python (programming language)|Python's]] [[SciPy]] package):
:<code>Value = Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b))</code>

This result follows from the properties [[#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 (programming language)|R]] and <code>betainc</code> in [[SymPy]]. In [[SciPy]], <code>special.betainc</code> computes the [[Beta distribution#Cumulative distribution function|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 field]]s.
* [[Nørlund–Rice integral]]
* [[Yule–Simon distribution]]

{{More footnotes|date=November 2010}}

==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]

{{Authority control}}

[[Category:Gamma and related functions]]
[[Category:Special hypergeometric functions]]

Beta function - Revision history

imported>Nailujon: /* growthexperiments-addlink-summary-summary:2|1|0 */

imported>PolyconvexPoster: /* Incomplete beta function */Fixed typo