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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 ^{2}(z)\,

This has poles at z=k\pi ,k\in {\mathbb  {Z}}\, of multiplicity k=2\,.

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

=\lim _{{z\to n\pi }}2\left[{\frac  {(z-n\pi )\cos z}{\sin z}}\right]\left[{\frac  {\sin z\left[(z-n\pi )(-\sin z)+\cos z\right]-(z-n\pi )\cos ^{2}z}{\sin ^{2}(z)}}\right]=0\,

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

Main Page : Complex Variables : Residues