CVR9

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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

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