Equivalence of Definitions of Beta Function

Theorem
For $\map \Re x, \map \Re y > 0$:

Definition 2 is equivalent to Definition 3
By definition of Gamma function:
 * $\displaystyle \map \Gamma x \, \map \Gamma y = \int_0^\infty t^{x - 1} e^{-t} \rd t \int_0^\infty s^{y - 1} e^{-s} \rd s$

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