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

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

The beta function is symmetric, meaning that

The beta function is also closely related to binomial coefficients.

When m (or n, by symmetry) is a positive integer, it follows from the definition of the gamma function Γ that[1] To derive this relation, write the product of two factorials as integrals.

Taking one has: See The Gamma Function, page 18–19[2] for a derivation of this relation.

If on the other hand x is large and y is fixed, then The integral defining the beta function may be rewritten in a variety of ways, including the following: where in the second-to-last identity n is any positive real number.

we have: The beta function can be written as an infinite sum[3] If

we get: 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 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

, 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

Generalizing this into a bivariate identity for a product of beta functions leads to: Euler's integral for the beta function may be converted into an integral over the Pochhammer contour C as This Pochhammer contour integral converges for all values of α and β 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 n, Β can be factored to give a closed form interpolation function for continuous values of k: 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:[6] The incomplete beta function, a generalization of the beta function, is defined as[7][8] For x = 1, the incomplete beta function coincides with the complete beta function.

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

of a random variable X following a binomial distribution with probability of single success p and number of Bernoulli trials n: The continued fraction expansion with odd and even coefficients respectively converges rapidly when

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: 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.

As briefly alluded to previously, the beta function is closely tied with the gamma function and plays an important role in calculus.

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): 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