ca.analysis and odes - what is summation in the sense of a principal value?

In one paper I saw this equality:

$$sum_{eta=-infty}^{infty}frac{z}{(z+eta)}=pi zcot(pi z)$$
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:

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

Where is my mistake?

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