3. Remark 2.11.1

What about meromorphic functions with infinitely many poles? Then we cannot subtract the principal parts of the poles as we did before, since 1;;the new sum may not converge. Consider:

f (z) = π \cot (π z) = \frac{π \cos (π z)}{\sin (π z)} .

This function has degree 1;; $1$ poles at every $n \in Z$ . However the series:

\sum_{n = - \infty}^{\infty} \frac{1}{z - n}

does not converge (looks like sum of two harmonic series). How do we ammend this?