Solution
= Solution
For real $z>0$, apply <integration by parts> to the <Gamma integral> with $u=t^z$ and $dv=e^{-t}dt$:
$$
\Gamma(z+1)=\int_0^{\infty}t^ze^{-t}\,dt=[-t^ze^{-t}]_0^{\infty}+z\int_0^{\infty}t^{z-1}e^{-t}\,dt.
$$
The boundary term vanishes: $t^z\to0$ at zero, and exponential decay dominates the power at infinity. This proves the <Gamma function recurrence>
$$
\boxed{\Gamma(z+1)=z\Gamma(z).}
$$