In one paper I saw this equality:


$$\sum\_{\eta=-\infty}^{\infty}\frac{z}{(z+\eta)}=\pi z\cot(\pi z)$$
which is the same as
$$\sum\_{\eta=-\infty}^{\infty}\frac{1}{(z+\eta)}=\pi \cot(\pi z)$$
where summation is understood in the sense of a principal value. What does it mean?


In another paper I found the next expression:


$$\frac{\exp(2\pi iaz)}{\exp(2\pi iz)-1}=\frac{1}{2\pi i}\sum\_{n=-\infty}^{\infty}\frac{\exp(2\pi ina)}{z-n}$$
for $a=0$ it is equivalent to
$$\frac{1}{\exp(2\pi iz)-1}=\frac{1}{2\pi i}\sum\_{n=-\infty}^{\infty}\frac{1}{z+n}$$
which is not exactly the same expression like in the first case.
$$\sum\_{n=-\infty}^{\infty}\frac{1}{z+n}=\pi Cot[\pi z]-i\pi$$


Where is my mistake? 


If the second formula is wrong, what is the correct formula for the second case?
$$\sum\_{n=-\infty}^{\infty}\frac{\exp(2\pi ina)}{z+n}=?$$



what is summation in the sense of a principal value?
A principal-value sum (or integral) is usually one in which unconditional summation (or integration) does not converge, so one needs to sum in a particular way to achieve convergence. I suspect that, in this case, the necessary summation is symmetric, so that we consider $\lim\_{N \to \infty} \sum\_{n = -N}^{n = N} f(n)$ instead of $\sum\_{n = 1}^\infty f(-n) + \sum\_{n = 0}^\infty f(n)$.


It's not quite clear to me what your issue is with the two formulæ you mention. Since you are summing different functions ($1/(z + n)$ versus $z/(z + n)$), it is no surprise that the answers are different. What am I missing? (Sorry, I did not notice that you had already factored out the $z$.)


