CVR5

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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) = \frac{\sin 2z}{(z+1)^3}\,

This has poles at z=-1\, of multiplicity k=3\,.

Use the formula \mathrm{Res}_{z=z_0^k} 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=-1} =\lim_{z\to -1} \frac{d^2}{{dz}^2} (z+1)^3 f(z) = \frac{1}{2} \lim_{z\to -1} \frac{d^2}{{dz}^2} \sin 2z\,

= \frac{1}{2} \lim_{z\to -1} (-4)\sin 2z = -2 \sin(-2) = 2\sin 2\,

If f(z)\, is analytic except at isolated singular points, then the sum of all the residues of f(z)\, equals 0.

So \mathrm{Res}_{z=-1}+\mathrm{Res}_{z=\infty}=2\sin 2+\mathrm{Res}_{z=\infty}=0\, and \mathrm{Res}_{z=\infty}=-2\sin 2\,


Main Page : Complex Variables : Residues

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