Beta function - WikiHQ

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

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 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>.

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

If on the other hand is large and 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:

\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_2dt \qquad \text{for any } a\in\mathbb{R}_{\leq 1},

\end{align}</math>

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 <math>t = \tan^2(\theta)</math>.

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

\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

<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:

\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

<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

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

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

<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:

Also, using Legendre duplication formula, we get

Euler's integral for the beta function may be converted into an integral over the Pochhammer contour 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 and 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 , can be factored to give a closed form interpolation function for continuous values of :

<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:

<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

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 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 following a binomial distribution with probability of single success and number of Bernoulli trials :

<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{1 + \frac{1 + \frac{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>.