I need to calculate the infinite summation below $$\sum_{n=-\infty}^\infty\frac{1}{(z_n-4)^2(z_n+2)^2}\ln\left[1+\frac{(z_n-4)(z_n+2)}{(z_n-5)(z_n+3)}\right],$$ where $z_n=\mathrm{i}\frac{2n\pi}{\beta}$ and $\beta$ is a fixed positive number.
There're singularities at $z=5,z=-3$ or the roots $z=1\pm\frac{5}{\sqrt{2}}$ of the argument inside $\ln$. The branch cut of $\ln$ prevents us from using residue theorem together with functions like $\frac{1}{\mathrm{exp}{[\beta z]}-1}$ to evaluate the summation. So this way is wrong at least for this case? I think the summation is not divergent. There must be a way to evaluate it.
Thanks in advance.