I am trying to think on a function that has sequence of singularity points at $z_0=\pm1$ and also a sequence of simple poles (whatever point I would like)
I thought on the function $$f(z)= \frac{1}{sin(\frac{1}{z-1})sin(\frac{1}{z+1})}$$
so I have $z_k= \frac{1}{2\pi k} \pm1$
But I can't think on the part with the poles.