For $f$ analytic on some domain with condition, $\sum\limits_{n\in\mathbb Z}f(n)=\sum\limits_{n\in\mathbb Z}\int_{-\infty}^\infty f(x)e^{-2\pi ixn}dx$

Question

I am stuck on this question:

Suppose that $f$ is analytic in $\{z \in \mathbb{C}: |\text{Im}(z)| < b\}$ for some $b > 0$, and there exists a constant $A > 0$ such that \begin{align*} |f(x + iy)| \leq \dfrac{A}{1 + x^2} \end{align*} for all $x \in \mathbb{R}$ and $|y| < b$. Prove that \begin{align*} \sum_{n \in \mathbb{Z}} f(n) = \sum_{n \in \mathbb{Z}}\bigg(\int_{-\infty}^\infty f(x)e^{-2\pi ixn}dx\bigg) \end{align*}

I was thinking of considering a complex function which has residues at every integer point, but I am not sure how to do it. Any help would be great. Thank you.