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.
• .
• .
• 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:

Beta can be used with Interval and CenteredInterval objects:

Specific Values(4)

Values at infinity:

Find a zero of :

Evaluate the incomplete beta function symbolically at integer and halfinteger 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 :

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:

Incomplete Beta Function(2)

Evaluate symbolically at integer and halfinteger orders:

Series expansion at any point:

Generalized Incomplete Beta Function(2)

Generalized incomplete beta function is related to incomplete beta function:

Evaluate symbolically:

Applications(4)

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 ddimensional hypersphere:

Lowdimensional 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:

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:

Machinenumber inputs can give highprecision 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:

The determinant of the × matrix of reciprocals of beta functions is :

Wolfram Research (1988), Beta, Wolfram Language function, https://reference.wolfram.com/language/ref/Beta.html (updated 2022).

Text

Wolfram Research (1988), Beta, Wolfram Language function, https://reference.wolfram.com/language/ref/Beta.html (updated 2022).

CMS

Wolfram Language. 1988. "Beta." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Beta.html.

APA

Wolfram Language. (1988). Beta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Beta.html

BibTeX

@misc{reference.wolfram_2022_beta, author="Wolfram Research", title="{Beta}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Beta.html}", note=[Accessed: 20-March-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_beta, organization={Wolfram Research}, title={Beta}, year={2022}, url={https://reference.wolfram.com/language/ref/Beta.html}, note=[Accessed: 20-March-2023 ]}