Integral of t^(x-1)(1-t^z)^(y-1) dt=(1/z)B(x/z,y)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
The following formula holds for $z>0$, $\mathrm{Re}(y)>0$, and $\mathrm{Re}(x)>0$: $$\displaystyle\int_0^1 t^{x-1} (1-t^z)^{y-1} \mathrm{d}t = \dfrac{1}{z} B \left( \dfrac{x}{z}, y \right),$$ where $B$ denotes the beta function.