Construct a meromorphic function on the complex plane whose poles are simple poles at the Gaussian integers $m+ n\iota$ with residue $1$.

Question

We know that on summing up all $ \frac{1}{z-(m+n\iota)} $ will not help as $\sum_{m,n\in\Bbb Z} \frac{1}{z-(m+n\iota)} $ is a divergent series. I think some sort of trick is to be used like done in Mittag-Leffler's theorem. If you have any other solution, please write.

This (Finding complex function) is the duplicate of the question but there is no answer to this question. Even I read comments, but they were using some Hadamard Product and Weierstrass Factor which I am completely unaware of.

Thanks!