I have a doubt on how to cancel certain poles around a given domain in an integral. Consider
$$ \int_{\partial D_R} \frac{e^{\pi i(z-1/2)^2}}{1-e^{2\pi iz}}dz, $$
where $\partial D_R$ is the parallelogram with vertices $\pm1/2\pm(1+i)R$. Use the residue theorem to show that the integral is $(1+i)/\sqrt{2}$.
The result $(1+i)/\sqrt{2}$ is precisely the integral's result at the pole $z=0$. Since $R$ is arbitrary, I must consider the other poles $z=\pm k$ ($k$ integer). The problem is that I do not know how to cancel out these other bunch of poles.