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].

Examples

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Basic Examples  (6)

Exact values:

Evaluate numerically:

Plot TemplateBox[{{1, /, 2}, b}, Beta] 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  (34)

Numerical Evaluation  (7)

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 threads elementwise over lists:

Specific Values  (4)

Values at infinity:

Find a zero of TemplateBox[{x, {-, {1, /, 2}}}, Beta]=0:

Evaluate the incomplete beta function symbolically at integer and halfinteger orders:

Evaluate the generalized incomplete beta symbolically:

Visualization  (2)

Plot TemplateBox[{{1, /, 2}, b}, Beta]:

Contour plot of TemplateBox[{a, b}, Beta]:

Function Properties  (3)

Real domain of the Euler beta function:

Complex domain:

Permutation symmetry:

Euler beta function has the mirror property TemplateBox[{TemplateBox[{a}, Conjugate, SyntaxForm -> SuperscriptBox], TemplateBox[{b}, Conjugate, SyntaxForm -> SuperscriptBox]}, Beta]=TemplateBox[{TemplateBox[{a, b}, Beta]}, Conjugate]:

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  (7)

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:

TraditionalForm formatting:

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 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 :

Introduced in 1988
 (1.0)
 |
Updated in 1996
 (3.0)