 Beta function

This article is about Euler beta function. For other uses, see Beta function (disambiguation).
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by
for
The beta function was studied by Euler and Legendre and was given its name by Jacques Binet; its symbol Β is a Greek capital β rather than the similar Latin capital b.
Contents
Properties
The beta function is symmetric, meaning that
When x and y are positive integers, it follows trivially from the definition of the gamma function that:
It has many other forms, including:
where is a truncated power function and the star denotes convolution. The second identity shows in particular . Some of these identities, e.g. the trigonometric formula, can be applied to deriving the volume of an nball in Cartesian coordinates.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, can be integrated to give a closed form, an interpolation function for continuous values of k:
The beta function 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.
Relationship between gamma function and beta function
To derive the integral representation of the beta function, write the product of two factorials as
Changing variables by putting u=zt, v=z(1t) shows that this is
Hence
The stated identity may be seen as a particular case of the identity for the integral of a convolution. Taking
 and , one has:
 .
Derivatives
We have
where is the digamma function.
Integrals
The Nörlund–Rice integral is a contour integral involving the beta function.
Approximation
Stirling's approximation gives the asymptotic formula
for large x and large y. If on the other hand x is large and y is fixed, then
Incomplete beta function
The incomplete beta function, a generalization of the beta function, is defined as
For x = 1, 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.
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:
Working out the integral (one can use integration by parts) for integer values of a and b, one finds:
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 X from a binomial distribution, where the "probability of success" is p and the sample size is n:
Properties
Calculation
Even if unavailable directly, the complete and incomplete Beta function values can be calculated using functions commonly included in spreadsheet or Computer algebra systems. With Excel as an example, using the GammaLn and (cumulative) Beta distribution functions, we have:
 Complete Beta Value = Exp(GammaLn(a) + GammaLn(b)  GammaLn(a + b))
and,
 Incomplete Beta Value = BetaDist(x, a, b) * Exp(GammaLn(a) + GammaLn(b)  GammaLn(a + b)).
These result from rearranging the formulae for the Beta distribution, and the incomplete beta and complete beta functions, which can also be defined as the ratio of the logs as above.
Similarly, in MATLAB and GNU Octave, betainc (Incomplete beta function) computes the regularized incomplete beta function  which is, in fact, the Cumulative Beta distribution  and so, to get the actual incomplete beta function, one must multiply the result of betainc by the result returned by the corresponding beta function.
See also
 Beta distribution
 Binomial distribution
 Jacobi sum, the analogue of the beta function over finite fields.
 Negative binomial distribution
 Yule–Simon distribution
 Uniform distribution (continuous)
 Gamma function
 Dirichlet distribution
References
 Askey, R. A.; Roy, R. (2010), "Beta function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 9780521192255, MR2723248, http://dlmf.nist.gov/5.12
 M. Zelen and N. C. Severo. in Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See §6.2, 6.6, and 26.5)
 Paris, R. B. (2010), "Incomplete beta functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 9780521192255, MR2723248, http://dlmf.nist.gov/8.17
 Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.1 Gamma Function, Beta Function, Factorials", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 9780521880688, http://apps.nrbook.com/empanel/index.html?pg=256
External links
 Evaluation of beta function using Laplace transform on PlanetMath
 Arbitrarily accurate values can be obtained from:
Categories: Gamma and related functions
 Special hypergeometric functions
Wikimedia Foundation. 2010.