NationMaster - Encyclopedia: Beta function

In mathematics, the beta function (occasionally written as Beta function), also called the Euler integral of the first kind, is a special function defined by Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, there are two types of Euler integral: Euler integral of the first kind: the Beta function Euler integral of the second kind: the Gamma function For positive integers m and n See also Leonhard Euler Factorial ... In mathematics, several functions are important enough to deserve their own name. ...

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

for Re(x), Re(y) > 0.

The beta function was studied by Euler and Legendre and was given its name by Jacques Binet. Leonhard Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Adrien-Marie Legendre (September 18, 1752 â€“ January 10, 1833) was a French mathematician. ... Jacques Philippe Marie Binet was a catholic mathematician. ...

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 In mathematics, the theory of symmetric functions is part of the theory of polynomial equations, and also a substantial chapter of combinatorics. ...

$mathrm{Beta}(x,y) = mathrm{Beta}(y,x). !$

It has many other forms, including:

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

$mathrm{Beta}(x,y) = 2int_0^{pi/2}sin^{2x-1}thetacos^{2y-1}theta,dtheta, qquad Re(x)>0, Re(y)>0 !$

$mathrm{Beta}(x,y) = int_0^inftydfrac{t^{x-1}}{(1+t)^{x+y}},dt, qquad Re(x)>0, Re(y)>0 !$

$mathrm{Beta}(x,y) = dfrac{1}{y}sum_{n=0}^infty(-1)^ndfrac{(y)_{n+1}}{n!(x+n)} !$

where $Γ(x)$ is the gamma function and (x)_n is the falling factorial; i.e., $x (x - 1)(x - 2)...(x - n + 1)$ . The second identity shows in particular $Gamma(1/2) = sqrt pi$ . The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ... In mathematics, the Pochhammer symbol is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial The empty product (x)0 is defined to be 1 in both cases. ...

Just as the gamma function for integers describes factorials, the beta function can define a binomial coefficient after adjusting indices: For factorial rings in mathematics, see unique factorisation domain. ... In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is the number of combinations that exist. ...

${n choose k} = frac1{(n+1) mathrm{B}(n-k+1, k+1)}$

The beta function was the first known scattering amplitude in string theory, first conjectured by Gabriele Veneziano. The S-matrix is the matrix in quantum mechanics or quantum field theory that relates the final state in the infinite future and the initial state in the infinite past. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects called strings, rather than the zero-dimensional point... Gabriele Veneziano (b. ...

Relationship between Gamma function and Beta function

To derive the integral representation of the beta function, write the product of two factorials as

$mathrm{Gamma(x)Gamma(y)} = int_0^infty e^{-u} u^{x-1}mathrm{d}u int_0^infty e^{-v} v^{y-1}mathrm{d}v. !$

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

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

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

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

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

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

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 $ψ(x)$ is the digamma function. In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions. ...

Integrals

The Nörlund-Rice integral is a contour integral involving the beta function. In mathematics, the NÃ¶rlund-Rice integral relates the nth forward difference of a function to a path integral on the complex plane. ...

Approximation

Stirling's approximation gives the asymptotic formula The relative difference between (ln x!) and (x ln x - x) approaches zero as x increases. ...

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

Incomplete beta function

The incomplete beta function is a generalization of the beta function that replaces the definite integral of the beta function with an indefinite integral. The situation is analogous to the incomplete gamma function being a generalization of the gamma function. This article deals with the concept of an integral in calculus. ... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... In mathematics, the gamma function is defined by a definite integral. ...

The incomplete beta function is defined as

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

For x = 1, the incomplete beta function coincides with the complete beta 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 for integer values of a and b, one finds:

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

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

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

Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

External links

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

Categories: Gamma and related functions | Special hypergeometric functions

Results from FactBites:

Beta function - Wikipedia, the free encyclopedia (308 words)
	The Nörlund-Rice integral is a contour integral involving the beta function.
	The incomplete beta function is a generalization of the beta function that replaces the definite integral of the beta function with an indefinite integral.
	The situation is analogous to the incomplete gamma function being a generalization of the gamma function.

Alpha-Beta Search (1809 words)
	Beta is the worst-case scenario for the opponent.
	If a move results in a score that is greater than or equal to beta, this whole node is trash, since the opponent is not going to let the side to move achieve this position, because there is some choice the opponent can make that will avoid it.
	This caused the function to change its perspective with each recursion, to reflect the fact that the two players are in contention, and have opposite goals.

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Contents

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Relationship between Gamma function and Beta function

Derivatives

Integrals

Approximation

Incomplete beta function

Properties

See also

References

External links

Properties