Solution
= Solution
For $\operatorname{Re}z>0$, absolute convergence permits the double-integral calculation
$$
\Gamma(z)^2
=\int_0^\infty\int_0^\infty
e^{-(s+t)}s^{z-1}t^{z-1}\,ds\,dt.
$$
Put $r=s+t$ and $u=t/(s+t)$, so $s=r(1-u)$, $t=ru$, and the Jacobian has magnitude $r$. The first quadrant becomes $r>0$, $0<u<1$, and therefore
$$
\begin{aligned}
\Gamma(z)^2
&=\int_0^\infty e^{-r}r^{2z-1}\,dr
\int_0^1(1-u)^{z-1}u^{z-1}\,du\\
&=\Gamma(2z)B(z,z).
\end{aligned}
$$