Calculating residua of $f(z)=\frac{1}{z^3(1-z\operatorname{ctg}z)}$. Lets define the solutions of $z=\operatorname{tg}z$ as $\lambda_i\:,i\in \mathbb{Z}$. Its easy to see that poles of $f$ are $0$ and $\lambda_i$. Now my question is how can I evaluate residua of $f$ at $\lambda_i$. I think $\lambda_i$ are poles of order $1$. From wikipedia residuum is equal to: $$\lim_{z\to\lambda_i}\frac{z-\lambda_i}{z^3(1-z\operatorname{ctg}z)}$$
So I guess my question comes to calculating this limit. Should I try to expand denominator in Laurent series around $\lambda_i$? At $0$ it is easy, because expansion is known.