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
such that.
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
for all inputs and.
A key property of the beta function is its close relationship to the gamma function:
A proof is given below in.
The beta function is also closely related to binomial coefficients. When is a positive integer, it follows from the definition of the gamma function that

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:
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
Dividing both sides by 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:
See The Gamma Function, page 18–19 for a derivation of this relation.

Differentiation of the beta function

We have
where denotes the digamma function.

Approximation

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

Other identities and formulas

The integral defining the beta function may be rewritten in a variety of ways, including the following:
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.
For values we have:
The beta function can be written as an infinite sum
If and are equal to a number we get:
where is the rising factorial,
and as an infinite product
The beta function satisfies several identities analogous to corresponding identities for binomial coefficients, including a version of Pascal's identity
one can easily prove this:
similarly,

and a simple recurrence on one coordinate:
The positive integer values of the beta function are also the partial derivatives of a 2D function: for all nonnegative integers and,
where
The Pascal-like identity above implies that this function is a solution to the first-order partial differential equation
For, the beta function may be written in terms of a convolution involving the truncated power function :
Evaluations at particular points may simplify significantly; for example,
and
By taking in this last formula, it follows that.
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
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:
Moreover, for integer, can be factored to give a closed form interpolation function for continuous values of :

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:

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 and, we can show that
The regularized incomplete beta function 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 of a random variable following a binomial distribution with probability of single success and number of Bernoulli trials :

Properties

Continued fraction expansion

The continued fraction expansion is
with odd and even coefficients given by
The and convergents are less than, while the and convergents are greater than.
It converges rapidly for. For or, the function may be evaluated more efficiently through the relation.

Multivariate beta function

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

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 GammaLn function :
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 , it is computed using a continued fraction expansion.
The incomplete beta function has existing implementation in common languages. For instance, betainc in MATLAB and GNU Octave, pbeta 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 and BetaRegularized give and, respectively.