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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the beta function, also called the
Euler integral In mathematics, there are two types of Euler integral: # The ''Euler integral of the first kind'' is the beta function \mathrm(z_1,z_2) = \int_0^1t^(1-t)^\,dt = \frac # The ''Euler integral of the second kind'' is the gamma function \Gamma(z) = \i ...
of the first kind, is a
special function Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...
that is closely related to the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
and to
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s. It is defined by the
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
: \Beta(z_1,z_2) = \int_0^1 t^(1-t)^\,dt for
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
inputs z_1, z_2 such that \Re(z_1), \Re(z_2)>0. The beta function was studied by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
and
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named ...
and was given its name by Jacques Binet; its symbol is a
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
capital
beta Beta (, ; uppercase , lowercase , or cursive ; grc, βῆτα, bē̂ta or ell, βήτα, víta) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Modern Greek, it represents the voiced labiod ...
.


Properties

The beta function is
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, meaning that \Beta(z_1,z_2) = \Beta(z_2,z_1) for all inputs z_1 and z_2.Davis (1972) 6.2.2 p.258 A key property of the beta function is its close relationship to the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
: : \Beta(z_1,z_2)=\frac. A proof is given below in . The beta function is also closely related to
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s. When (or , by symmetry) is a positive integer, it follows from the definition of the gamma function thatDavis (1972) 6.2.1 p.258 : \Beta(m,n) =\dfrac = \frac \Bigg/ \binom.


Relationship to the gamma function

A simple derivation of the relation \Beta(z_1,z_2) =\frac can be found in Emil Artin's book ''The Gamma Function'', page 18–19. To derive this relation, write the product of two factorials as :\begin \Gamma(z_1)\Gamma(z_2) &= \int_^\infty\ e^ u^\,du \cdot\int_^\infty\ e^ v^\,dv \\ pt &=\int_^\infty\int_^\infty\ e^ u^v^\, du \,dv. \end Changing variables by and produces :\begin \Gamma(z_1)\Gamma(z_2) &= \int_^\infty\int_^1 e^ (st)^(s(1-t))^s\,dt \,ds \\ pt &= \int_^\infty e^s^ \,ds\cdot\int_^1 t^(1-t)^\,dt\\ &=\Gamma(z_1+z_2) \cdot \Beta(z_1,z_2). \end Dividing both sides by \Gamma(z_1+z_2) gives the desired result. The stated identity may be seen as a particular case of the identity for the integral of a convolution. Taking :\beginf(u)&:=e^ u^ 1_ \\ g(u)&:=e^ u^ 1_, \end one has: : \Gamma(z_1) \Gamma(z_2) = \int_f(u)\,du\cdot \int_ g(u) \,du = \int_(f*g)(u)\,du =\Beta(z_1,z_2)\,\Gamma(z_1+z_2).


Derivatives

We have :\frac \mathrm(z_1, z_2) = \mathrm(z_1, z_2) \left( \frac - \frac \right) = \mathrm(z_1, z_2) \big(\psi(z_1) - \psi(z_1 + z_2)\big), :\frac \mathrm(z_1, z_2, \dots, z_n) = \mathrm(z_1, z_2, \dots, z_n) \left(\psi(z_m) - \psi\left( \sum_^n z_k \right)\right), \quad 1\le m\le n, where \psi(z) denotes the
Polygamma function In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function: :\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z). Thus :\psi^(z) ...
.


Approximation

Stirling's approximation In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less p ...
gives the asymptotic formula :\Beta(x,y) \sim \sqrt \frac for large and large . If on the other hand is large and is fixed, then :\Beta(x,y) \sim \Gamma(y)\,x^.


Other identities and formulas

The integral defining the beta function may be rewritten in a variety of ways, including the following: : \begin \Beta(z_1,z_2) &= 2\int_0^(\sin\theta)^(\cos\theta)^\,d\theta, \\ pt &= \int_0^\infty\frac\,dt, \\ pt &= n\int_0^1t^(1-t^n)^\,dt, \\ &= (1-a)^ \int_0^1 \fracdt \qquad \text a\in\mathbb_, \end 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 t = \tan^2(\theta). The beta function can be written as an infinite sum : \Beta(x,y) = \sum_^\infty \frac : (where (x)_n is the
rising factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...
) and as an infinite product : \Beta(x,y) = \frac \prod_^\infty \left( 1+ \dfrac\right)^. The beta function satisfies several identities analogous to corresponding identities for binomial coefficients, including a version of
Pascal's identity In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients. It states that for positive natural numbers ''n'' and ''k'', + = , where \tbinom is a binomial coefficient; one interpretation of ...
: \Beta(x,y) = \Beta(x, y+1) + \Beta(x+1, y) and a simple recurrence on one coordinate: :\Beta(x+1,y) = \Beta(x, y) \cdot \dfrac, \quad \Beta(x,y+1) = \Beta(x, y) \cdot \dfrac. The positive integer values of the beta function are also the partial derivatives of a 2D function: for all nonnegative integers m and n, :\Beta(m+1, n+1) = \frac(0, 0), where :h(a, b) = \frac. The Pascal-like identity above implies that this function is a solution to the
first-order partial differential equation In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of ''n'' variables. The equation takes the form : F(x_1,\ldots,x_n,u,u_,\ldots u_) =0. \, ...
:h = h_a+h_b. For x, y \geq 1, the beta function may be written in terms of a
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
involving the
truncated power function In mathematics, the truncated power function with exponent n is defined as :x_+^n = \begin x^n &:\ x > 0 \\ 0 &:\ x \le 0. \end In particular, :x_+ = \begin x &:\ x > 0 \\ 0 &:\ x \le 0. \end and interpret the exponent as conventional power ...
t \mapsto t_+^x: : \Beta(x,y) \cdot\left(t \mapsto t_+^\right) = \Big(t \mapsto t_+^\Big) * \Big(t \mapsto t_+^\Big) Evaluations at particular points may simplify significantly; for example, : \Beta(1,x) = \dfrac and : \Beta(x,1-x) = \dfrac, \qquad x \not \in \mathbb By taking x = \frac in this last formula, it follows that \Gamma(1/2) = \sqrt. Generalizing this into a bivariate identity for a product of beta functions leads to: : \Beta(x,y) \cdot \Beta(x+y,1-y) = \frac . Euler's integral for the beta function may be converted into an integral over the
Pochhammer contour In mathematics, the Pochhammer contour, introduced by Jordan (1887), pp. 243–244 and , is a contour in the complex plane with two points removed, used for contour integration. If ''A'' and ''B'' are loops around the two points, both starting ...
as :\left(1-e^\right)\left(1-e^\right)\Beta(\alpha,\beta) =\int_C t^(1-t)^ \, dt. This Pochhammer contour integral converges for all values of and and so gives the
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
of the beta function. Just as the gamma function for integers describes
factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...
s, the beta function can define a
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
after adjusting indices: :\binom = \frac. Moreover, for integer , can be factored to give a closed form interpolation function for continuous values of : :\binom = (-1)^n\, n! \cdot\frac.


Reciprocal beta function

The reciprocal beta function is the
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
about the form :f(x,y)=\frac Interestingly, their integral representations closely relate as the definite integral of
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
with product of its power and multiple-angle: :\int_0^\pi\sin^\theta\sin y\theta~d\theta=\frac :\int_0^\pi\sin^\theta\cos y\theta~d\theta=\frac :\int_0^\pi\cos^\theta\sin y\theta~d\theta=\frac :\int_0^\frac\cos^\theta\cos y\theta~d\theta=\frac


Incomplete beta function

The incomplete beta function, a generalization of the beta function, is defined as : \Beta(x;\,a,b) = \int_0^x t^\,(1-t)^\,dt. For , 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. For positive integer ''a'' and ''b'', the incomplete beta function will be a polynomial of degree ''a'' + ''b'' - 1 with rational coefficients. 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) = \frac. The regularized incomplete beta function is the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
of the
beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
, and is related to the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
F(k;\,n,p) of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
following a
binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...
with probability of single success and number of Bernoulli trials : :F(k;\,n,p) = \Pr\left(X \le k\right) = I_(n-k, k+1) = 1 - I_p(k+1,n-k).


Properties

:\begin I_0(a,b) &= 0 \\ I_1(a,b) &= 1 \\ I_x(a,1) &= x^a\\ I_x(1,b) &= 1 - (1-x)^b \\ I_x(a,b) &= 1 - I_(b,a) \\ I_x(a+1,b) &= I_x(a,b)-\frac \\ I_x(a,b+1) &= I_x(a,b)+\frac \\ \Beta(x;a,b)&=(-1)^ \Beta\left(\frac;a,1-a-b\right) \end


Multivariate beta function

The beta function can be extended to a function with more than two arguments: :\Beta(\alpha_1,\alpha_2,\ldots\alpha_n) = \frac . This multivariate beta function is used in the definition of the
Dirichlet distribution In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted \operatorname(\boldsymbol\alpha), is a family of continuous multivariate probability distributions parameterized by a vector \bold ...
. Its relationship to the beta function is analogous to the relationship between
multinomial coefficient In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. Theorem For any positive integer ...
s and binomial coefficients. For example, it satisfies a similar version of Pascal's identity: :\Beta(\alpha_1,\alpha_2,\ldots\alpha_n) = \Beta(\alpha_1+1,\alpha_2,\ldots\alpha_n)+\Beta(\alpha_1,\alpha_2+1,\ldots\alpha_n)+\cdots+\Beta(\alpha_1,\alpha_2,\ldots\alpha_n+1) .


Applications

The beta function is useful in computing and representing the
scattering amplitude In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.Regge trajectories Regge may refer to * Tullio Regge (1931-2014), Italian physicist, developer of Regge calculus and Regge theory * Regge calculus, formalism for producing simplicial approximations of spacetimes * Regge theory, study of the analytic properties of ...
. Furthermore, it was the first known
scattering amplitude In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
, first conjectured by
Gabriele Veneziano Gabriele Veneziano (; ; born 7 September 1942) is an Italian theoretical physicist widely considered the father of string theory. He has conducted most of his scientific activities at CERN in Geneva, Switzerland, and held the Chair of Elementa ...
. It also occurs in the theory of the
preferential attachment A preferential attachment process is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who ...
process, a type of stochastic urn process. The beta function is also important in statistics, e.g. for the
Beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
and
Beta prime distribution In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kindJohnson et al (1995), p 248) is an absolutely continuous probability distribution. Definitions ...
. As briefly alluded to previously, the beta function is closely tied with the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
and plays an important role in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
.


Software implementation

Even if unavailable directly, the complete and incomplete beta function values can be calculated using functions commonly included in
spreadsheet A spreadsheet is a computer application for computation, organization, analysis and storage of data in tabular form. Spreadsheets were developed as computerized analogs of paper accounting worksheets. The program operates on data entered in cel ...
or
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
s. In
Microsoft Excel Microsoft Excel is a spreadsheet developed by Microsoft for Microsoft Windows, Windows, macOS, Android (operating system), Android and iOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro (comp ...
, for example, the complete beta function can be computed with the GammaLn function (or special.gammaln in Python's
SciPy SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing. SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal ...
package): :Value = Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b)) 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. I
GNU Octave
it is computed using a
continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
expansion. The incomplete beta function has existing implementation in common languages. For instance, betainc (incomplete beta function) in
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
and
GNU Octave GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a langu ...
, pbeta (probability of beta distribution) in R, or special.betainc in
SciPy SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing. SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal ...
compute 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. In Mathematica, Beta
, a, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/code> and BetaRegularized
, a, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/code> give \Beta(x;\,a,b) and I_x(a,b) , respectively.


See also

*
Beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
and
Beta prime distribution In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kindJohnson et al (1995), p 248) is an absolutely continuous probability distribution. Definitions ...
, two probability distributions related to the beta function * Jacobi sum, the analogue of the beta function over
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s. * Nörlund–Rice integral *
Yule–Simon distribution In probability and statistics, the Yule–Simon distribution is a discrete probability distribution named after Udny Yule and Herbert A. Simon. Simon originally called it the ''Yule distribution''. The probability mass function (pmf) of the Yu ...


References

* * * * *


External links

* * * Arbitrarily accurate values can be obtained from: *
The Wolfram functions siteEvaluate Beta Regularized incomplete beta
**danielsoper.com
Incomplete beta function calculatorRegularized incomplete beta function calculator
{{Authority control Gamma and related functions Special hypergeometric functions