Solution
= Solution
For $\operatorname{Re}z>0$, integration by parts gives
$$
I(z+1)
=\int_0^\infty t^ze^{-t}\,dt
=\left[-t^ze^{-t}\right]_0^\infty
+z\int_0^\infty t^{z-1}e^{-t}\,dt.
$$
The boundary term vanishes, so the <Gamma function recurrence> is
$$
\boxed{I(z+1)=zI(z)}.
$$