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Find the residues of f(z)\, at all its isolated singular points and at infinity (if infinity is not a limit point of singular points), where f(z)\, is given by

f(z)=\cot ^{3}(z)\,

This has poles at z=n\pi ,n\in {\mathbb  {Z}}\, of multiplicity k=3\,.

Use the formula {\mathrm  {Res}}_{{z=z_{0}}}f(z)={\frac  {1}{(k-1)!}}\lim _{{z\to z_{0}}}{\frac  {d^{{k-1}}}{{dz}^{{k-1}}}}\left[(z-z_{0})^{k}f(z)\right]\,

{\mathrm  {Res}}_{{z=n\pi }}={\frac  {1}{2}}\lim _{{z\to n\pi }}{\frac  {d^{2}}{{dz}^{2}}}(z-n\pi )^{3}f(z)={\frac  {1}{2}}\lim _{{z\to n\pi }}{\frac  {d^{2}}{{dz}^{2}}}\left[{\frac  {(z-n\pi )\cos z}{\sin z}}\right]^{3}=...=-1\,

In this case infinity is a limit point of singular points, so there is no residue at infinity.

Main Page : Complex Variables : Residues