Let $a \neq 0\in \mathbb{C}$ and consider the function $$f(z) = \frac{a^{z}}{\Gamma(z+1)}.$$
It is true that $f$ is bounded in every compact set contained in $\mathbb{C}\setminus\{-1,-2,-3,-4,\dots\}$ ?
Observation: In the calculation of $a^z$ i'm considering only the principal argument.