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 :

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:

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]:

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:

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:

Compute the indefinite integral using Integrate :

Verify the anti-derivative:

Definite integral:

More integrals:

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:

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

Recurrence relationship:

BetaRegularized may reduce to a simpler form:

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:

Evaluate at integer and half‐integer arguments:

Series expansions at generic points:

Series expansion at infinity: