# 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 :

Plot the imaginary part of :

### Function Properties(9)

is defined for all real and complex values:

is an odd function:

The regularized 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(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 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 :

Find the series expansion at :

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: 05-August-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: 05-August-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