Beta function: Definition from Answers.com

This article is about the Euler beta function. There are separate articles on the Dirichlet beta function and on the beta-function (written with a hyphen) of physics.

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by

$\mathrm{\Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,dt \!$

for $\textrm{Re}(x), \textrm{Re}(y) > 0.\,$

The beta function was studied by Euler and Legendre and was given its name by Jacques Binet.

1 Properties
2 Relationship between gamma function and beta function
3 Derivatives
4 Integrals
5 Approximation
6 Incomplete beta function
- 6.1 Properties
7 See also
8 References
9 External links

Properties

The beta function is symmetric, meaning that

$\Beta(x,y) = \Beta(y,x). \!$

It has many other forms, including:

$\Beta(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!$

$\Beta(x,y) = 2\int_0^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}\,d\theta, \qquad \textrm{Re}(x)>0,\ \textrm{Re}(y)>0 \!$

$\Beta(x,y) = \int_0^\infty\dfrac{t^{x-1}}{(1+t)^{x+y}}\,dt, \qquad \textrm{Re}(x)>0,\ \textrm{Re}(y)>0 \!$

$\Beta(x,y) = \sum_{n=0}^\infty \dfrac{{n-y \choose n}} {x+n}, \!$

$\Beta(x,y) = \frac{x+y}{x y} \prod_{n=1}^\infty \left( 1+ \dfrac{x y}{n (x+y+n)}\right)^{-1}, \!$

$\Beta(x,y) \cdot \Beta(x+y,1-y) = \dfrac{\pi}{x \sin(\pi y)}, \!$

where $\Gamma\,$ is the gamma function. The second identity shows in particular $\Gamma(1/2) = \sqrt \pi$ .

Just as the gamma function for integers describes factorials, the beta function can define a binomial coefficient after adjusting indices:

${n \choose k} = \frac1{(n+1) \Beta(n-k+1, k+1)}.$

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

$\Gamma(x)\Gamma(y) = \int_0^\infty\ e^{-u} u^{x-1}\,du \int_0^\infty\ e^{-v} v^{y-1}\,dv. \!$

Now, let $u \equiv a^2$ , $v \equiv b^2$ , so

$\begin{align} \Gamma(x)\Gamma(y) & {} = 4\int_0^\infty\ e^{-a^2} a^{2x-1}\mathrm{d}a \int_0^\infty\ e^{-b^2} b^{2y-1}\,db \\ & {} = \int_{-\infty}^\infty\ \int_{-\infty}^\infty\ e^{-(a^2+b^2)} |a|^{2x-1} |b|^{2y-1} \,da \,db. \end{align} \!$

Transforming to polar coordinates with $a = r cosθ$ , $b = r sinθ$ :

$\begin{align} \Gamma(x)\Gamma(y) & {} = \int_0^{2\pi}\ \int_0^\infty\ e^{-r^2} |r\cos\theta|^{2x-1} |r\sin\theta|^{2y-1} r \, dr \,d\theta \\ & {} = \int_0^\infty\ e^{-r^2} r^{2x+2y-2} r\, dr \int_0^{2\pi}\ |(\cos\theta)^{2x-1} (\sin\theta)^{2y-1}| \, d\theta \\ & {} = \frac{1}{2}\int_0^\infty\ e^{-r^2} r^{2(x+y-1)} \, d(r^2) 4\int_0^{\pi/2}\ (\cos\theta)^{2x-1} (\sin\theta)^{2y-1} \,d\theta \\ & {} = \Gamma(x+y) 2\int_0^{\pi/2}\ (\cos\theta)^{2x-1} (\sin\theta)^{2y-1} \, d\theta \\ & {} = \Gamma(x+y) \Beta(x,y). \end{align}$

Hence, rewrite the arguments with the usual form of beta function:

$\Beta(x,y) = \frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}.$

For another derivation, notice that the stated identity also follows as a particular case of the identity for the integral of a convolution. Taking

$f(u):=e^{-u} u^{x-1} 1_{\R_+}$ and $g(u):=e^{-u} u^{y-1} 1_{\R_+}$ , it is easy to check that:

$\Gamma(x)\Gamma(y)=\left(\int_{\R}f(u)du\right)\left(\int_{\R}g(u)du\right)=\int_{\R}(f*g)(u)du=\Beta(x, y)\,\Gamma(x+y)$ .

Derivatives

The derivatives follow:

${\partial \over \partial x} \mathrm{B}(x, y) = \mathrm{B}(x, y) \left( {\Gamma'(x) \over \Gamma(x)} - {\Gamma'(x + y) \over \Gamma(x + y)} \right) = \mathrm{B}(x, y) (\psi(x) - \psi(x + y))$

where $\ \psi(x)$ 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

$\Beta(x,y) \sim \sqrt {2\pi } \frac{{x^{x - \frac{1}{2}} y^{y - \frac{1}{2}} }}{{\left( {x + y} \right)^{x + y - \frac{1}{2}} }}$

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

$\Beta(x,y) \sim \Gamma(y)\,x^{-y}.$

Incomplete beta function

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

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

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:

$I_x(a,b) = \dfrac{\Beta(x;\,a,b)}{\Beta(a,b)}. \!$

Working out the integral (one can use integration by parts to do that) for integer values of a and b, one finds:

$I_x(a,b) = \sum_{j=a}^{a+b-1} {(a+b-1)! \over j!(a+b-1-j)!} x^j (1-x)^{a+b-1-j}.$

The regularized incomplete beta function can be used to evaluate the cumulative density function of a random variable X from a binomial distribution, where the "probability of success" is p and the sample size is n:

$F(k;n,p) = \Pr(X \le k) = I_{1-p}(n-k, k+1).$

Properties

$I_0(a,b) = 0 \,$

$I_1(a,b) = 1 \,$

$I_x(a,b) = 1 - I_{1-x}(b,a) \,$

(Many other properties could be listed here.)

References

Askey, R. A.; Roy, R. (2008), "Beta function", in Boisvert, Ronald F.; Clark, Charles W.; Lozier, Daniel M. et al., Digital Library of Mathematical Functions, N.I.S.T.
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)
W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1992. Second edition. (See section 6.4)
Evaluation of beta function using Laplace transform on PlanetMath
Arbitrarily accurate values can be obtained from The Wolfram Functions Site, Evaluate Beta Regularized Incomplete beta

External links

Cephes - C and C++ language special functions math library
Beta Function Calculator
Incomplete Beta Function Calculator
Regularized Incomplete Beta Function Calculator

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Beta function

Contents

Properties

Relationship between gamma function and beta function

Derivatives

Integrals

Approximation

Incomplete beta function

Properties

See also

References

External links