BetaRegularized

BetaRegularized[z,a,b]

gives the regularized incomplete beta function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For nonsingular cases, .
  • BetaRegularized[z0,z1,a,b] gives the generalized regularized incomplete beta function defined in nonsingular cases as Beta[z0,z1,a,b]/Beta[a,b].
  • Note that the arguments in BetaRegularized are arranged differently from those in GammaRegularized.
  • For certain special arguments, BetaRegularized automatically evaluates to exact values.
  • BetaRegularized can be evaluated to arbitrary numerical precision.
  • BetaRegularized automatically threads over lists.

Examples

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Basic Examples  (6)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

Scope  (34)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

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

Complex number input:

Evaluate efficiently at high precision:

Specific Values  (4)

Values of BetaRegularized at fixed points:

Values at zero:

Values at infinity:

Find a value of z for which the BetaRegularized[z,1,3]=3.5:

Visualization  (3)

Plot the BetaRegularized function for various parameters:

Plot the real part of TemplateBox[{3, a, b}, BetaRegularized]:

Plot the imaginary part of TemplateBox[{3, a, b}, BetaRegularized]:

Function Properties  (9)

TemplateBox[{z, 1, 1}, BetaRegularized] is defined for all real and complex values:

TemplateBox[{z, 1, 1}, BetaRegularized] is an odd function:

The regularized incomplete beta function TemplateBox[{x, a, 1}, BetaRegularized] is an analytic function of for positive integer :

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

For other values of , TemplateBox[{x, a, 1}, BetaRegularized] is neither analytic nor meromorphic:

TemplateBox[{x, 1, 2}, BetaRegularized] is neither non-increasing nor non-decreasing:

TemplateBox[{x, a, 1}, BetaRegularized] is injective for positive odd but not positive even :

TemplateBox[{x, a, 1}, BetaRegularized] is surjective for positive odd but not positive even :

TemplateBox[{x, a, 1}, BetaRegularized] is non-negative for positive even but indefinite for odd :

TemplateBox[{x, a, 1}, BetaRegularized] is convex for positive even :

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to z:

First derivative with respect to a:

First derivative with respect to b:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when a=2 and b=3:

Formula for the ^(th) derivative with respect to z:

Integration  (3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Series Expansions  (5)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

FourierSeries:

Find the series expansion at Infinity:

Find the series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Function Identities and Simplifications  (3)

Regularized incomplete beta function is related to the incomplete beta function:

Recurrence relationship:

BetaRegularized may reduce to a simpler form:

Generalizations & Extensions  (8)

Ordinary Regularized Incomplete Beta Function  (5)

Evaluate at integer and halfinteger arguments:

Infinite arguments give symbolic results:

BetaRegularized threads elementwise over lists:

BetaRegularized can be applied to power series:

Series expansion at infinity:

Give the result for an arbitrary symbolic direction:

Generalized Regularized Incomplete Beta Function  (3)

Evaluate at integer and halfinteger arguments:

Series expansions at generic points:

Series expansion at infinity:

Applications  (3)

Plot of the absolute value of BetaRegularized 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:

CDF of the Student t distribution:

Plot the CDF for various parameters:

Properties & Relations  (3)

Use FunctionExpand to express through Gamma and Beta functions:

Numerically find a root of a transcendental equation:

Compose with the inverse function:

Use PowerExpand to disregard multivaluedness ambiguity:

Possible Issues  (3)

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

Machinenumber inputs can give highprecision results:

Regularized beta functions are typically not generated by FullSimplify:

Wolfram Research (1991), BetaRegularized, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaRegularized.html.

Text

Wolfram Research (1991), BetaRegularized, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaRegularized.html.

BibTeX

@misc{reference.wolfram_2021_betaregularized, author="Wolfram Research", title="{BetaRegularized}", year="1991", howpublished="\url{https://reference.wolfram.com/language/ref/BetaRegularized.html}", note=[Accessed: 28-November-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_betaregularized, organization={Wolfram Research}, title={BetaRegularized}, year={1991}, url={https://reference.wolfram.com/language/ref/BetaRegularized.html}, note=[Accessed: 28-November-2021 ]}

CMS

Wolfram Language. 1991. "BetaRegularized." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BetaRegularized.html.

APA

Wolfram Language. (1991). BetaRegularized. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BetaRegularized.html