The function $I(z)$, defined by

$$I(z)=\int_{0}^{\infty} t^{z-1} e^{-t} d t$$

is analytic for $\operatorname{Re} z>0$.

(i) Show that $I(z+1)=z I(z)$.

(ii) Use part (i) to construct an analytic continuation of $I(z)$ into Re $z \leqslant 0$, except at isolated singular points, which you need to identify.