Problem 1670 (difficulty: 5/10)

Calculate the first 6 terms in the Laurent series of \(\displaystyle \ctg z\) and \(\displaystyle \pi\ctg (\pi z)\) on the domain \(\displaystyle 0<|z|<\pi\). What are the residues of \(\displaystyle \displaystyle\frac{\ctg z}z\), \(\displaystyle \displaystyle\frac{\ctg z}{z^2}\), ..., \(\displaystyle \displaystyle\frac{\ctg z}{z^5}\) in \(\displaystyle 0\)?

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