Beta function

Summary

The beta function $B(x, y)$ is the integral $\int_0^1 t^{x-1}(1 - t)^{y-1}\, dt$ for positive $x$ and $y$. It is expressed through the gamma function by $B(x, y) = \Gamma(x)\Gamma(y)/\Gamma(x + y)$. It turns many integrals over intervals and over angles into products of gamma values. It also gives the normalising constants of the beta distributions in probability. At positive integers it is the reciprocal of a multiple of a binomial coefficient.