The function $F(t)$ is defined, for $\operatorname{Re} t>-1$, by

$$F(t)=\int_{0}^{\infty} \frac{u^{t} e^{-u}}{1+u} d u$$

and by analytic continuation elsewhere in the complex $t$-plane. By considering the integral of a suitable function round a Hankel contour, obtain the analytic continuation of $F(t)$ and hence show that singularities of $F(t)$ can occur only at $z=-1,-2,-3, \ldots$.