Beta

Beta[a,b]

gives the Euler beta function .

Beta[z,a,b]

gives the incomplete beta function .

Details

Beta is a mathematical function, suitable for both symbolic and numerical manipulation.
$TemplateBox[{a, b}, Beta]=TemplateBox[{a}, Gamma]TemplateBox[{b}, Gamma]/TemplateBox[{{a, +, b}}, Gamma]=int_0^1t^(a-1)(1-t)^(b-1)dt$ .
$TemplateBox[{z, a, b}, Beta3]=int_0^zt^(a-1)(1-t)^(b-1)dt$ .
Beta[z,a,b] has a branch cut discontinuity in the complex plane running from to .
Beta[z₀,z₁,a,b] gives the generalized incomplete beta function .
Note that the arguments in the incomplete form of Beta are arranged differently from those in the incomplete form of Gamma.
For certain special arguments, Beta automatically evaluates to exact values.
Beta can be evaluated to arbitrary numerical precision.
Beta automatically threads over lists.
In TraditionalForm, Beta is output using \[CapitalBeta].
Beta can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples (6)

Exact values:

Evaluate numerically:

Plot over a subset of the reals:

Plot the incomplete beta function over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope (42)

Numerical Evaluation (8)

Evaluate numerically:

Evaluate symbolically in special cases:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for large arguments:

Evaluate for complex arguments:

Evaluate Beta efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix Beta function using MatrixFunction:

Specific Values (4)

Values at infinity:

Find a zero of :

Evaluate the incomplete beta function symbolically at integer and half‐integer orders:

Evaluate the generalized incomplete beta symbolically:

Visualization (2)

Plot :

Contour plot of :

Function Properties (11)

Real domain of the complete Euler beta function:

Complex domain:

Permutation symmetry:

Euler beta function has the mirror property :

The complete beta function is not an analytic function:

However, it is meromorphic:

Its singularities and discontinuities are restricted to the non-positive integers:

The incomplete beta function is an analytic function of for positive integer :

Thus, any such function will have no singularities or discontinuities:

For other values of , is neither analytic nor meromorphic:

is neither non-increasing nor non-decreasing:

is injective for positive odd but not positive even :

is surjective for positive odd but not positive even :

is non-negative for positive even but indefinite for odd :

is convex for positive even :

TraditionalForm formatting:

Differentiation (2)

First derivative of the beta function:

Higher derivatives of the beta function:

Plot higher derivatives for :

Series Expansions (5)

The beta function series expansion at poles:

The first term in the beta function series expansion around :

Asymptotic expansion of the beta function:

Incomplete beta function series expansion at any point:

Beta can be applied to power series:

Function Identities and Simplifications (4)

Generalized incomplete beta function is related to incomplete beta function:

Use FullSimplify to simplify beta functions:

Recurrence relationships:

Product relation:

Function Representations (6)

Primary definition in terms of Gamma function:

Reduce the generalized incomplete beta function to incomplete beta functions:

Integral representation of the Euler beta function:

Integral representation of the incomplete beta function:

Beta can be represented in terms of MeijerG:

Beta can be represented as a DifferentialRoot:

Generalizations & Extensions (6)

Euler Beta Function (2)

Evaluate symbolically in special cases:

Beta threads elementwise over lists:

Incomplete Beta Function (2)

Evaluate symbolically at integer and half‐integer orders:

Series expansion at any point:

Generalized Incomplete Beta Function (2)

Generalized incomplete beta function is related to incomplete beta function:

Evaluate symbolically:

Applications (5)

Plot the beta function for real positive values:

Plot of the absolute value of Beta in the complex plane:

Distribution of the average distance s of all pairs of points in a d‐dimensional hypersphere:

Low‐dimensional distributions can be expressed in elementary functions:

Plot distributions:

The PDF for the beta distribution for random variable :

Plot the PDF for various parameters:

Calculate the mean:

The probability that more than (capacity) simultaneous service requests are made can be represented in terms of Gamma and Beta functions:

Properties & Relations (7)

Express the Euler beta function as a ratio of Euler gamma functions:

Reduce the generalized incomplete beta function to incomplete beta functions:

Use FullSimplify to simplify beta functions:

Numerically find a root of a transcendental equation:

Sum expressions involving Beta:

Generating function:

Obtain as special cases of hypergeometric functions:

Beta can be represented as a DifferenceRoot:

Possible Issues (4)

Large arguments can give results too small to be computed explicitly:

Machine‐number inputs can give high‐precision results:

Algorithmically generated results often use gamma and hypergeometric rather than beta functions:

The differential equation is satisfied by a sum of incomplete beta functions:

Beta functions are typically not generated by FullSimplify:

Neat Examples (2)

Nest Beta over the complex plane:

Define the beta matrix, whose entries are reciprocals of beta functions:

The determinant of the beta matrix is :

The beta matrix is symmetric positive definite, and its Cholesky decomposition has entries of the form :

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