Equivalence of Definitions of Beta Function

Theorem
The following definitions of the Beta function are equivalent:


 * $\displaystyle B \left({x, y}\right) = \int_0^1 t^{x - 1} \left({1-t}\right)^{y - 1} \ \mathrm d t$


 * $\displaystyle B \left({x, y}\right) = 2 \int_0^{\pi / 2} \left({\sin \theta}\right)^{2x - 1} \left({\cos \theta}\right)^{2y - 1} \ \mathrm d \theta$


 * $B \left({x, y}\right) = \dfrac {\Gamma \left({x}\right) \Gamma \left({y}\right)} {\Gamma \left({x + y}\right)}$

where $\operatorname{Re} \left({x}\right), \operatorname{Re} \left({y}\right) > 0$

Proof
as required.


 * $\displaystyle \Gamma \left({x}\right) \Gamma \left({y}\right) = \int_0^\infty t^{x - 1} e^{-t} \ \mathrm d t \int_0^\infty s^{y - 1} e^{-s} \ \mathrm d s$

Substitute $t = u^2$ and $s = v^2$: