Problem 1700 (difficulty: 5/10)

Let \(\displaystyle N_k\) be the square with vertices \(\displaystyle \pm(k+\frac12)\pm(k+\frac12)i\). What is

\(\displaystyle \frac1{2\pi i}\int_{N_k}\frac{\pi\ctg\pi z}{z^2}\dz? \)

What identity results if we let \(\displaystyle k\to\infty\)?

Give me another random problem!