Beta function

 ( Gamma And Related Functions ) 

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

$\backslash Beta(z\_1,z\_2)\; =\; \backslash int\_0^1\; t^\{z\_1-1\}(1-t)^\{z\_2-1\}\backslash ,dt$

for complex number inputs $z\_1,\; z\_2$ such that $\backslash operatorname\{Re\}(z\_1),\; \backslash operatorname\{Re\}(z\_2)>0$.

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

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Properties The beta function is symmetric, meaning that $\backslash Beta(z\_1,z\_2)\; =\; \backslash Beta(z\_2,z\_1)$ for all inputs $z\_1$ and $z\_2$.. Specifically, see 6.2 Beta Function.

A key property of the beta function is its close relationship to the gamma function:

$\backslash Beta(z\_1,z\_2)=\backslash frac\{\backslash Gamma(z\_1)\backslash ,\backslash Gamma(z\_2)\}\{\backslash Gamma(z\_1+z\_2)\}$

A proof is given below in .

The beta function is also closely related to binomial coefficients. When (or , by symmetry) is a positive integer, it follows from the definition of the gamma function that

$\backslash Beta(m,n)\; =\backslash frac\{(m-1)!\backslash ,(n-1)!\}\{(m+n-1)!\}\; =\; \backslash frac\{m\; +\; n\}\{mn\}\; \backslash Bigg/\; \backslash binom\{m\; +\; n\}\{m\}$

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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:

$\backslash begin\{align\}$

\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}

Changing variables by and , because and , we have that the limits of integrations for are 0 to ∞ and the limits of integration for are 0 to 1. Thus produces

$\backslash begin\{align\}$

\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}

Dividing both sides by $\backslash Gamma(z\_1+z\_2)$ gives the desired result.

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

one has:

$\backslash Gamma(z\_1)\; \backslash Gamma(z\_2)\; =\; \backslash int\_\{\backslash R\}f(u)\backslash ,du\backslash cdot\; \backslash int\_\{\backslash R\}\; g(u)\; \backslash ,du\; =\; \backslash int\_\{\backslash R\}(f*g)(u)\backslash ,du\; =\backslash Beta(z\_1,z\_2)\backslash ,\backslash Gamma(z\_1+z\_2).$

See The Gamma Function, page 18–19 for a derivation of this relation.

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Differentiation of the beta function We have

$\backslash frac\{\backslash partial\}\{\backslash partial\; z\_1\}\; \backslash mathrm\{B\}(z\_1,\; z\_2)\; =\; \backslash mathrm\{B\}(z\_1,\; z\_2)\; \backslash left(\; \backslash frac\{\backslash Gamma\text{'}(z\_1)\}\{\backslash Gamma(z\_1)\}\; -\; \backslash frac\{\backslash Gamma\text{'}(z\_1\; +\; z\_2)\}\{\backslash Gamma(z\_1\; +\; z\_2)\}\; \backslash right)\; =\; \backslash mathrm\{B\}(z\_1,\; z\_2)\; \backslash big(\backslash psi(z\_1)\; -\; \backslash psi(z\_1\; +\; z\_2)\backslash big),$

$\backslash frac\{\backslash partial\}\{\backslash partial\; z\_m\}\; \backslash mathrm\{B\}(z\_1,\; z\_2,\; \backslash dots,\; z\_n)\; =\; \backslash mathrm\{B\}(z\_1,\; z\_2,\; \backslash dots,\; z\_n)\; \backslash left(\backslash psi(z\_m)\; -\; \backslash psi\backslash left(\; \backslash sum\_\{k=1\}^n\; z\_k\; \backslash right)\backslash right),\; \backslash quad\; 1\backslash le\; m\backslash le\; n,$

where $\backslash psi(z)$ denotes the digamma function.

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Approximation Stirling's approximation gives the asymptotic formula

$\backslash Beta(x,y)\; \backslash sim\; \backslash sqrt\; \{2\backslash pi\; \}\; \backslash frac\{x^\{x\; -\; 1/2\}\; y^\{y\; -\; 1/2\}\; \}\{(\; \{x\; +\; y\}\; )^\{x\; +\; y\; -\; 1/2\}\; \}$

for large and large .

If on the other hand is large and is fixed, then

$\backslash Beta(x,y)\; \backslash sim\; \backslash Gamma(y)\backslash ,x^\{-y\}.$

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Other identities and formulas The integral defining the beta function may be rewritten in a variety of ways, including the following:

$$

\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}

where in the second-to-last identity is any positive real number. One may move from the first integral to the second one by substituting $t\; =\; \backslash tan^2(\backslash theta)$.

For values $z=z\_1=z\_2\backslash neq1$ we have:

$$

\Beta(z,z) = \frac{1}{z}\int_0^{\pi / 2}\frac{1}{(\sqrtz{\sin\theta} + \sqrtz{\cos\theta})^{2z}}\,d\theta

The beta function can be written as an infinite sum

$\backslash Beta(x,y)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(1-x)\_n\}\{(y+n)\backslash ,n!\}$

If $x$ and $y$ are equal to a number $z$ we get:

$$

\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!}

(where $(x)\_n$ is the rising factorial)

and as an infinite product

$\backslash Beta(x,y)\; =\; \backslash frac\{x+y\}\{x\; y\}\; \backslash prod\_\{n=1\}^\backslash infty\; \backslash left(\; 1+\; \backslash dfrac\{x\; y\}\{n\; (x+y+n)\}\backslash right)^\{-1\}.$

The beta function satisfies several identities analogous to corresponding identities for binomial coefficients, including a version of Pascal's identity

$\backslash Beta(x,y)\; =\; \backslash Beta(x,\; y+1)\; +\; \backslash Beta(x+1,\; y)$

and a simple recurrence on one coordinate:

$\backslash Beta(x+1,y)\; =\; \backslash Beta(x,\; y)\; \backslash cdot\; \backslash dfrac\{x\}\{x+y\},\; \backslash quad\; \backslash Beta(x,y+1)\; =\; \backslash Beta(x,\; y)\; \backslash cdot\; \backslash dfrac\{y\}\{x+y\}.$

The positive integer values of the beta function are also the partial derivatives of a 2D function: for all nonnegative integers $m$ and $n$,

$\backslash Beta(m+1,\; n+1)\; =\; \backslash frac\{\backslash partial^\{m+n\}h\}\{\backslash partial\; a^m\; \backslash ,\; \backslash partial\; b^n\}(0,\; 0),$

where

$h(a,\; b)\; =\; \backslash frac\{e^a-e^b\}\{a-b\}.$

The Pascal-like identity above implies that this function is a solution to the first-order partial differential equation

$h\; =\; h\_a+h\_b.$

For $x,\; y\; \backslash geq\; 1$, the beta function may be written in terms of a convolution involving the truncated power function $t\; \backslash mapsto\; t\_+^x$:

$\backslash Beta(x,y)\; \backslash cdot\backslash left(t\; \backslash mapsto\; t\_+^\{x+y-1\}\backslash right)\; =\; \backslash Big(t\; \backslash mapsto\; t\_+^\{x-1\}\backslash Big)\; *\; \backslash Big(t\; \backslash mapsto\; t\_+^\{y-1\}\backslash Big)$

Evaluations at particular points may simplify significantly; for example,

$\backslash Beta(1,x)\; =\; \backslash dfrac\{1\}\{x\}$

and

$\backslash Beta(x,1-x)\; =\; \backslash dfrac\{\backslash pi\}\{\backslash sin(\backslash pi\; x)\},\; \backslash qquad\; x\; \backslash not\; \backslash in\; \backslash mathbb\{Z\}$

By taking $x\; =\; \backslash frac\{1\}\{2\}$ in this last formula, it follows that $\backslash Gamma(1/2)\; =\; \backslash sqrt\{\backslash pi\}$. Generalizing this into a bivariate identity for a product of beta functions leads to:

$\backslash Beta(x,y)\; \backslash cdot\; \backslash Beta(x+y,1-y)\; =\; \backslash frac\{\backslash pi\}\{x\; \backslash sin(\backslash pi\; y)\}\; .$

Euler's integral for the beta function may be converted into an integral over the Pochhammer contour as

$\backslash left(1-e^\{2\backslash pi\; i\backslash alpha\}\backslash right)\backslash left(1-e^\{2\backslash pi\; i\backslash beta\}\backslash right)\backslash Beta(\backslash alpha,\backslash beta)\; =\backslash int\_C\; t^\{\backslash alpha-1\}(1-t)^\{\backslash beta-1\}\; \backslash ,\; dt.$

This Pochhammer contour integral converges for all values of and and so gives the analytic continuation of the beta function.

Just as the gamma function for integers describes , the beta function can define a binomial coefficient after adjusting indices:

$\backslash binom\{n\}\{k\}\; =\; \backslash frac\{1\}\{(n+1)\backslash ,\backslash Beta(n-k+1,\; k+1)\}.$

Moreover, for integer , can be factored to give a closed form interpolation function for continuous values of :

$\backslash binom\{n\}\{k\}\; =\; (-1)^n\backslash ,\; n!\; \backslash cdot\backslash frac\{\backslash sin\; (\backslash pi\; k)\}\{\backslash pi\; \backslash displaystyle\backslash prod\_\{i=0\}^n\; (k-i)\}.$

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Reciprocal beta function The reciprocal beta function is the special function about the form

$f(x,y)=\backslash frac\{1\}\{\backslash Beta(x,y)\}$

Interestingly, their integral representations closely relate as the definite integral of trigonometric functions with product of its power and multiple-angle:

$\backslash int\_0^\backslash pi\backslash sin^\{x-1\}\backslash theta\backslash sin\; y\backslash theta~d\backslash theta=\backslash frac\{\backslash pi\backslash sin\backslash frac\{y\backslash pi\}\{2\}\}\{2^\{x-1\}x\backslash Beta\backslash left(\backslash frac\{x+y+1\}\{2\},\backslash frac\{x-y+1\}\{2\}\backslash right)\}$

$\backslash int\_0^\backslash pi\backslash sin^\{x-1\}\backslash theta\backslash cos\; y\backslash theta~d\backslash theta=\backslash frac\{\backslash pi\backslash cos\backslash frac\{y\backslash pi\}\{2\}\}\{2^\{x-1\}x\backslash Beta\backslash left(\backslash frac\{x+y+1\}\{2\},\backslash frac\{x-y+1\}\{2\}\backslash right)\}$

$\backslash int\_0^\backslash pi\backslash cos^\{x-1\}\backslash theta\backslash sin\; y\backslash theta~d\backslash theta=\backslash frac\{\backslash pi\backslash cos\backslash frac\{y\backslash pi\}\{2\}\}\{2^\{x-1\}x\backslash Beta\backslash left(\backslash frac\{x+y+1\}\{2\},\backslash frac\{x-y+1\}\{2\}\backslash right)\}$

$\backslash int\_0^\backslash frac\{\backslash pi\}\{2\}\backslash cos^\{x-1\}\backslash theta\backslash cos\; y\backslash theta~d\backslash theta=\backslash frac\{\backslash pi\}\{2^xx\backslash Beta\backslash left(\backslash frac\{x+y+1\}\{2\},\backslash frac\{x-y+1\}\{2\}\backslash right)\}$

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Incomplete beta function The incomplete beta function, a generalization of the beta function, is defined as

$\backslash Beta(x;\backslash ,a,b)\; =\; \backslash int\_0^x\; t^\{a-1\}\backslash ,(1-t)^\{b-1\}\backslash ,dt.$

For , 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 $t\; =\; \backslash sin^2\backslash theta$ and $t\; =\; \backslash frac1\{1+s\}$, we can show that

$\backslash Beta(x;\backslash ,a,b)\; =\; 2\; \backslash int\_0^\{\backslash arcsin\; \backslash sqrt\; x\}\; \backslash sin^\{2a-1\backslash !\}\backslash theta\backslash cos^\{2b-1\backslash !\}\backslash theta\backslash ,\backslash mathrm\; d\backslash theta\; =\; \backslash int\_\{\backslash frac\{1-x\}x\}^\backslash infty\; \backslash frac\{s^\{b-1\}\}\{(1+s)^\{a+b\}\}\backslash ,\backslash mathrm\; ds$

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:

$I\_x(a,b)\; =\; \backslash frac\{\backslash Beta(x;\backslash ,a,b)\}\{\backslash Beta(a,b)\}.$

The regularized incomplete beta function is the cumulative distribution function of the beta distribution, and is related to the cumulative distribution function $F(k;\backslash ,n,p)$ of a random variable following a binomial distribution with probability of single success and number of Bernoulli trials :

$F(k;\backslash ,n,p)\; =\; \backslash Pr\backslash left(X\; \backslash le\; k\backslash right)\; =\; I\_\{1-p\}(n-k,\; k+1)\; =\; 1\; -\; I\_p(k+1,n-k).$

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Properties

$\backslash begin\{align\}$

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}

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Continued fraction expansion The continued fraction expansion

$\backslash Beta(x;\backslash ,a,b)\; =\; \backslash frac\{x^\{a\}\; (1\; -\; x)^\{b\}\}\{a\; \backslash left(\; 1\; +\; \backslash frac\{1\; +\}\; \backslash frac\{1\; +\}\; \backslash frac\{1\; +\}\; \backslash frac\{1\; +\}\; \backslash cdots\; \backslash right)\}$

with odd and even coefficients respectively

$\{d\}\_\{2\; m\; +\; 1\}\; =\; -\; \backslash frac\{(a\; +\; m)\; (a\; +\; b\; +\; m)\; x\}\{(a\; +\; 2\; m)\; (a\; +\; 2\; m\; +\; 1)\}$

$\{d\}\_\{2\; m\}\; =\; \backslash frac\{m\; (b\; -\; m)\; x\}\{(a\; +\; 2\; m\; -\; 1)\; (a\; +\; 2\; m)\}$

converges rapidly when $x$ is not close to 1. The $4\; m$ and $4\; m\; +\; 1$ convergents are less than $\backslash Beta(x;\backslash ,a,b)$, while the $4\; m\; +\; 2$ and $4\; m\; +\; 3$ convergents are greater than $\backslash Beta(x;\backslash ,a,b)$.

For $x\; >\; \backslash frac\{a\; +\; 1\}\{a\; +\; b\; +\; 2\}$, the function may be evaluated more efficiently using $\backslash Beta(x;\backslash ,a,b)\; =\; \backslash Beta(a,\; b)\; -\; \backslash Beta(1\; -\; x;\backslash ,b,a)$.

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

$\backslash Beta(\backslash alpha\_1,\backslash alpha\_2,\backslash ldots\backslash alpha\_n)\; =\; \backslash frac\{\backslash Gamma(\backslash alpha\_1)\backslash ,\backslash Gamma(\backslash alpha\_2)\; \backslash cdots\; \backslash Gamma(\backslash alpha\_n)\}\{\backslash Gamma(\backslash alpha\_1\; +\; \backslash alpha\_2\; +\; \backslash cdots\; +\; \backslash alpha\_n)\}\; .$

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:

$\backslash Beta(\backslash alpha\_1,\backslash alpha\_2,\backslash ldots\backslash alpha\_n)\; =\; \backslash Beta(\backslash alpha\_1+1,\backslash alpha\_2,\backslash ldots\backslash alpha\_n)+\backslash Beta(\backslash alpha\_1,\backslash alpha\_2+1,\backslash ldots\backslash alpha\_n)+\backslash cdots+\backslash Beta(\backslash alpha\_1,\backslash alpha\_2,\backslash ldots\backslash alpha\_n+1)\; .$

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Applications The beta function is useful in computing and representing the scattering amplitude for Regge trajectories. Furthermore, it was the first known S matrix 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. 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.

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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 GammaLn function (or special.gammaln in Python's SciPy package):

Value = Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b))

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 GNU Octave, it is computed using a continued fraction expansion.

The incomplete beta function has existing implementation in common languages. For instance, betainc (incomplete beta function) in MATLAB and GNU Octave, pbeta (probability of beta distribution) in R and betainc in SymPy. In SciPy, special.betainc 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 special.betainc by the result returned by the corresponding beta function. In Mathematica, Beta[x, a, b] and BetaRegularized[x, a, b] give $\backslash Beta(x;\backslash ,a,b)$ and $I\_x(a,b)$, respectively.

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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 .
• Nørlund–Rice integral
• Yule–Simon distribution

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External links

• Arbitrarily accurate values can be obtained from:
• The Wolfram functions site: Evaluate Beta Regularized incomplete beta
• danielsoper.com: Incomplete beta function calculator, Regularized incomplete beta function calculator

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