CVR18

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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{e^{tz}}{(z+2)^2}\,

This has a pole at z=-2\, 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=-2} =\lim_{z\to -2} \frac{d}{dz}\left[(z+2)^2 f(z)\right] = \lim_{z\to -2} \frac{d}{dz}\left[e^{tz}\right] =\lim_{z\to -2}te^{tz}=-2e^{-2t},

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=-2} +\mathrm{Res}_{z=\infty} = -2e^{-2t} + \mathrm{Res}_{z=\infty}=0\,

so \mathrm{Res}_{z=\infty} = 2e^{-2t}\,


Main Page : Complex Variables : Residues

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