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.
BetaRegularized can be used with Interval and CenteredInterval objects. »

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 (36)

Numerical Evaluation (6)

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:

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 BetaRegularized function using MatrixFunction:

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 :

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 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 half‐integer 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 half‐integer arguments:

Series expansions at generic points:

Series expansion at infinity:

Applications (4)

Plot of the absolute value of BetaRegularized 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 CDF of StudentTDistribution is given in terms of BetaRegularized functions:

Plot the CDF for various parameters:

The inverse probability:

The CDF of FRatioDistribution is given in terms of BetaRegularized functions:

Plot the CDF for various values of the numerator and denominator degrees of freedom:

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:

Machine‐number inputs can give high‐precision results:

Regularized beta functions are typically not generated by FullSimplify:

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