beta: Beta Functions

In specfun: Special Functions

Description

The Beta function is defined as

B(p, q) = \int_0^1 t^{p-1} (1 - t)^{q-1} dt , \qquad Re(p), Re(q) > 0

where p and q are parameters.

The incomplete Beta function is defined as

B(x, p, q) = \int_0^x t^{p-1} (1 - t)^{q-1} dt

for Re(p), Re(q) > 0 and 0 < x < 1.

Usage

sp.beta(p, q)

sp.betainc(x, p, q)

Arguments

p, q	real arguments greater 0.
x	real number between 0 and 1.

Details

The Beta function is computed through its relation to the Gamma function:

B(p, q) = \frac{Γ(p) Γ(q)}{Γ(p + q)}

The incomplete Beta function satisfies the symmetry relation

B(x, p, q) = 1 - Beta(1 - x, q, p)

.

Value

Returns the result of the (incomplete) Beta function.

Examples

sp.beta(1, 1)

specfun documentation built on May 2, 2019, 4:44 p.m.