Does the following function have any poles or zeros?

Question

$F(x)=\int_1^\infty t^xe^{-t}dt$

It doesn't look like the function has any zeros or poles on the complex plane, but I'm curious if there are.

Answer 1

Your integral is the upper incomplete Gamma function $\Gamma(x+1,1)$. This is an entire function, so no poles. There are zeros, presumably infinitely many, of which the closest to $0$ are approximately $1.772122006 \pm 4.398958636 i$.

Here is a plot of the curves $\text{Re}(\Gamma(z+1,1) = 0$ (blue) and $\text{Im}(\Gamma(z+1,1) = 0$ (red) in part of the complex plane. The zeros are the intersections of red and blue curves.