
In
mathematics, the beta function, also called the
Euler integral 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 defin ...
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 th ...
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, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
:
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 for ...
inputs
such that
.
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 nam ...
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 labi ...
.
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 definit ...
, meaning that
for all inputs
and
.
[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 th ...
:
[
:
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 that[Davis (1972) 6.2.1 p.258]
:
Relationship to the gamma function
A simple derivation of the relation 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
:
Changing variables by and produces
:
Dividing both sides by gives the desired result.
The stated identity may be seen as a particular case of the identity for the integral of a convolution. Taking
:
one has:
:
Derivatives
We have
:
:
where denotes the Polygamma function.
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
:
for large and large . If on the other hand is large and is fixed, then
:
Other identities and formulas
The integral defining the beta function may be rewritten in a variety of ways, including the following:
:
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 .
The beta function can be written as an infinite sum
:
: (where 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) \,.
\ ...
)
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 and ,
:
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
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution' ...
involving the truncated power function :
:
Evaluations at particular points may simplify significantly; for example,
:
and
:
By taking in this last formula, it follows that .
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 as
:
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 n ...
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) ...
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:
:
Moreover, for integer , can be factored to give a closed form interpolation function for continuous values of :
:
Reciprocal beta function
The reciprocal beta function is the function about the form
:
Interestingly, their integral representations closely relate as the definite integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
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 ...
with product of its power and multiple-angle:
:
:
:
:
Incomplete beta function
The incomplete beta function, a generalization of the beta function, is defined as
:
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:
:
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, 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 ...
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 p ...
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 qu ...
with probability of single success and number of Bernoulli trials :
:
Properties
:
Multivariate beta function
The beta function can be extended to a function with more than two arguments:
:
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 \bolds ...
. 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:
:
Applications
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
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 Elementar ...
. It also occurs in the theory of the preferential attachment process, a type of stochastic urn process. The beta function is also important in statistics, e.g. for the Beta distribution and Beta prime distribution. 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 th ...
and plays an important role in calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
.
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 ce ...
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 ...
s.
In Microsoft Excel
Microsoft Excel is a spreadsheet developed by Microsoft for Windows, macOS, Android and iOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro programming language called Visual Basic for ...
, for example, the complete beta function can be computed with the GammaLn
function (or special.gammaln
in Python's SciPy 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 integ ...
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, implementa ...
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 lang ...
, pbeta
(probability of beta distribution) in R, or special.betainc
in SciPy 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
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimi ...
, Beta , a, b/code> and BetaRegularized , a, b/code> give and , respectively.
See also
* Beta distribution and Beta prime distribution, 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, subt ...
s.
* Nörlund–Rice integral
* Yule–Simon distribution
References
*
*
*
*
*
External links
*
*
* Arbitrarily accurate values can be obtained from:
*
The Wolfram functions site
Evaluate Beta Regularized incomplete beta
**danielsoper.com
Incomplete beta function calculator
Regularized incomplete beta function calculator
{{Authority control
Gamma and related functions
Special hypergeometric functions