Beta function

\(\displaystyle \mathrm {B} (x,y)=\int _{0}^{1}t^{x-1}(1-t)^{y-1}\,dt\)

for Re \(x\), Re \(y > 0\).

\(\displaystyle \mathrm {B} (x,y)=\mathrm {B} (y,x) = \frac{\Gamma (x)\,\Gamma (y)}{\Gamma (x+y)} ={\dfrac {(x-1)!\,(y-1)!}{(x+y-1)!}} \)