Is there an analytic function defined on $\Bbb C$ except for Gaussian integers where it has poles of order 1 and residue 1?

Question

I need a function defined for all complex variables $z$, except for at all the Gaussian integers, where it has poles of order 1 and residue 1. The function has to be complex-analytic.

Can anyone think of such a function?

Answer 1

The Mittag-Leffler theorem ensures that there exists such a function, and one can follow a proof of that theorem to construct such a function. For example, $$ \frac1z + z^2 \sum_{(m,n)\in\Bbb Z^2\setminus(0,0)} \frac1{(m+ni)^2(z-(m+ni))} $$ is such a function.