This question asks about the integrability of $1/z$ in a vertical line in the complex plane. The proof is: integrate in rectangles centred at the origin and increase the length of their vertical sides. Then the two integrals in horizontal sides will be very small, and the two integrals in vertical sides will sum up, so the result is $\pi i$.
The reason why they sum up is that the function $1$ is even and the function $z$ is odd.
I am wondering what we can say about the integrability of the function $e^z/z$.
Of course the line of the integrations is not the imaginary axis.
In general are there some criteria (positive and non-trivial) using which we can claim integrability of a meromorphic function in a vertical line not containing the poles ?
Reduce it to an improper integral over $\mathbb R$. With $z = \sigma + i t$, $\int_{-\infty}^\infty dt/(\sigma + i t)$ diverges, but the principal value integral exists. $\int_{-\infty}^\infty dt \,e^{\sigma + i t}/(\sigma + i t)$ conditionally converges by Dirichlet's test.