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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)=\cos({\frac  {1}{z-2}}),

This has poles at z=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=2}}=\lim _{{z\to 2}}(z-2)f(z)=\lim _{{z\to 2}}(z-2)\cos({\frac  {1}{z-2}})=0,

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

Main Page : Complex Variables : Residues