Beta functions

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral $$ \Beta(x,y) = \int_0^1 t^{x-1}(1-t)^{y-1} dt $$

The incomplete beta function, a generalization of the beta function, is defined as

$$ \Beta(x;a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt $$

The regularized incomplete beta function defined in terms of the incomplete beta function and the complete beta function: $$ I_x(a,b) = \frac{\Beta(x;a,b)}{\Beta(a,b)} $$