Equivalence of Definitions of Beta Function

Theorem
The following definitions of the Beta function are equivalent.

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

Definition 2 is equivalent to Definition 3
By definition of Gamma function:
 * $\displaystyle \Gamma \left({x}\right) \Gamma \left({y}\right) = \int_0^\infty t^{x - 1} e^{-t} \ \mathrm d t \int_0^\infty s^{y - 1} e^{-s} \ \mathrm d s$

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