So, I believe (but am having trouble showing) that the power series for $e^{z}$ is concentrated around indices close to $|\Re(z)|$ (or at any rate is negligible beyond those indices). More precisely, I conjecture (and would like to prove) that for any fixed $\theta \in [0,2\pi)$ and any $\epsilon>0$, we have $$\lim_{r \rightarrow \infty}\Big|\sum_{n > |r\cos(\theta)| + |r\cos(\theta)|^{1/2+\epsilon}}\frac{(re^{i\theta})^{n}}{n!}\Big|=0.$$
When $\theta=0$, this is a well-known result which follows from the fact that the Poisson distribution is asympotically normal. But I'm having trouble finding results for the case when $\Im(z)\neq 0$. Any insight would be greatly appreciated. Thanks!
I make this conjecture because it seems like $\Re(z)$ is more important than $|z|$ to the rate of convergence of $e^{z}$; for example, $e^{0+10000i}$ converges at the same rate as $e^{0}$, and much more slowly than $e^{10000}$. Any insight would be greatly appreciated. Thanks!