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

:$\Beta(z_1,z_2) = \int_0^1 t^(1-t)^\,dt$
for complex number inputs
$z_1, z_2$ such that $\Re(z_1), \Re(z_2)>0$.

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 capital beta.

Properties

The beta function is symmetric, meaning that
$\Beta(z_1,z_2) = \Beta(z_2,z_1)$ for all inputs $z_1$ and $z_2$.Davis (1972) 6.2.2 p.258

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

:$\Beta(z_1,z_2)=\frac.$

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 thatDavis (1972) 6.2.1 p.258

:$\Beta(m,n) =\dfrac = \frac \Bigg/ \binom.$

Relationship to the gamma function

A simple derivation of the relation $\Beta(z_1,z_2) =\frac$ can be found in Emil Artin's book ''The Gamma Function'', page 18–19.
To derive this relation, write the product of two factorials as

:\begin
\Gamma(z_1)\Gamma(z_2) &= \int_^\infty\ e^ u^\,du \cdot\int_^\infty\ e^ v^\,dv \\ pt &=\int_^\infty\int_^\infty\ e^ u^v^\, du \,dv.
\end

Changing variables by and produces

:\begin
\Gamma(z_1)\Gamma(z_2) &= \int_^\infty\int_^1 e^ (st)^(s(1-t))^s\,dt \,ds \\ pt &= \int_^\infty e^s^ \,ds\cdot\int_^1 t^(1-t)^\,dt\\
&=\Gamma(z_1+z_2) \cdot \Beta(z_1,z_2).
\end

Dividing both sides by $\Gamma(z_1+z_2)$ gives the desired result.

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

:$\beginf(u)&:=e^ u^ 1_ \\ g(u)&:=e^ u^ 1_, \end$

one has:

:$\Gamma(z_1) \Gamma(z_2) = \int_f(u)\,du\cdot \int_ g(u) \,du = \int_(f*g)(u)\,du =\Beta(z_1,z_2)\,\Gamma(z_1+z_2).$

Derivatives

We have

:$\frac \mathrm(z_1, z_2) = \mathrm(z_1, z_2) \left( \frac - \frac \right) = \mathrm(z_1, z_2) \big(\psi(z_1) - \psi(z_1 + z_2)\big),$

:$\frac \mathrm(z_1, z_2, \dots, z_n) = \mathrm(z_1, z_2, \dots, z_n) \left(\psi(z_m) - \psi\left( \sum_^n z_k \right)\right), \quad 1\le m\le n,$

where $\psi(z)$ denotes the Polygamma function.

Approximation

Stirling's approximation gives the asymptotic formula

:$\Beta(x,y) \sim \sqrt \frac$

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

:$\Beta(x,y) \sim \Gamma(y)\,x^.$

Other identities and formulas

The integral defining the beta function may be rewritten in a variety of ways, including the following:
:
\begin
\Beta(z_1,z_2) &= 2\int_0^(\sin\theta)^(\cos\theta)^\,d\theta, \\ pt &= \int_0^\infty\frac\,dt, \\ pt &= n\int_0^1t^(1-t^n)^\,dt, \\
&= (1-a)^ \int_0^1 \fracdt \qquad \text a\in\mathbb_,
\end

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 $t = \tan^2(\theta)$.

The beta function can be written as an infinite sum
: $\Beta(x,y) = \sum_^\infty \frac$
: (where $(x)_n$ is the rising factorial)
and as an infinite product
: $\Beta(x,y) = \frac \prod_^\infty \left( 1+ \dfrac\right)^.$

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

:$\Beta(x,y) = \Beta(x, y+1) + \Beta(x+1, y)$

and a simple recurrence on one coordinate:

:$\Beta(x+1,y) = \Beta(x, y) \cdot \dfrac, \quad \Beta(x,y+1) = \Beta(x, y) \cdot \dfrac.$

The positive integer values of the beta function are also the partial derivatives of a 2D function: for all nonnegative integers $m$ and $n$,
:$\Beta(m+1, n+1) = \frac(0, 0),$
where
:$h(a, b) = \frac.$
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 $x, y \geq 1$, the beta function may be written in terms of a convolution involving the truncated power function $t \mapsto t_+^x$:
:$\Beta(x,y) \cdot\left(t \mapsto t_+^\right) = \Big(t \mapsto t_+^\Big) * \Big(t \mapsto t_+^\Big)$

Evaluations at particular points may simplify significantly; for example,
:$\Beta(1,x) = \dfrac$
and
:$\Beta(x,1-x) = \dfrac, \qquad x \not \in \mathbb$

By taking $x = \frac$ in this last formula, it follows that $\Gamma(1/2) = \sqrt$.
Generalizing this into a bivariate identity for a product of beta functions leads to:
:$\Beta(x,y) \cdot \Beta(x+y,1-y) = \frac .$

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

:$\left(1-e^\right)\left(1-e^\right)\Beta(\alpha,\beta) =\int_C t^(1-t)^ \, dt.$

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 factorials, the beta function can define a binomial coefficient after adjusting indices:
:$\binom = \frac.$

Moreover, for integer , can be factored to give a closed form interpolation function for continuous values of :
:$\binom = (-1)^n\, n! \cdot\frac.$

Reciprocal beta function

The reciprocal beta function is the function about the form

:$f(x,y)=\frac$

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

:$\int_0^\pi\sin^\theta\sin y\theta~d\theta=\frac$

:$\int_0^\pi\sin^\theta\cos y\theta~d\theta=\frac$

:$\int_0^\pi\cos^\theta\sin y\theta~d\theta=\frac$

:$\int_0^\frac\cos^\theta\cos y\theta~d\theta=\frac$

Incomplete beta function

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

:$\Beta(x;\,a,b) = \int_0^x t^\,(1-t)^\,dt.$

For , 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. For positive integer ''a'' and ''b'', the incomplete beta function will be a polynomial of degree ''a'' + ''b'' - 1 with rational coefficients.

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:

:$I_x(a,b) = \frac.$

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

:$F(k;\,n,p) = \Pr\left(X \le k\right) = I_(n-k, k+1) = 1 - I_p(k+1,n-k).$

Properties

:$\begin I_0(a,b) &= 0 \\ I_1(a,b) &= 1 \\ I_x(a,1) &= x^a\\ I_x(1,b) &= 1 - (1-x)^b \\ I_x(a,b) &= 1 - I_(b,a) \\ I_x(a+1,b) &= I_x(a,b)-\frac \\ I_x(a,b+1) &= I_x(a,b)+\frac \\ \Beta(x;a,b)&=(-1)^ \Beta\left(\frac;a,1-a-b\right) \end$

Multivariate beta function

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

:$\Beta(\alpha_1,\alpha_2,\ldots\alpha_n) = \frac .$

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:

:$\Beta(\alpha_1,\alpha_2,\ldots\alpha_n) = \Beta(\alpha_1+1,\alpha_2,\ldots\alpha_n)+\Beta(\alpha_1,\alpha_2+1,\ldots\alpha_n)+\cdots+\Beta(\alpha_1,\alpha_2,\ldots\alpha_n+1) .$

Applications

The beta function is useful in computing and representing the scattering amplitude

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