Beta

Evaluate for complex arguments:

Evaluate for large arguments:

Evaluate to high precision:

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

Series expansion:

TraditionalForm formatting:

Evaluate symbolically in special cases:

Infinite arguments give symbolic results:

Beta threads element-wise over lists:

Beta can be applied to power series:

Series expansion at poles:

Series expansion at infinity:

Evaluate symbolically at integer and half-integer orders:

Series expansion at any point:

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 d-dimensional hypersphere:

Low-dimensional 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:

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:

Generate from integrals:

Obtain as special cases of hypergeometric functions:

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

Machine-number inputs can give high-precision 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:

Nest Beta over the complex plane:

The determinant of the matrix of reciprocals of beta functions is :