CVR2

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$f(z) = \frac{1}{z(1-z^2)}\,$

This has poles at $z=0, \pm 1\,$ of multiplicity $k=1\,$ .

Use the formula $\mathrm{Res}_{z=z_0} f(z) = \frac{1}{(k-1)!}\lim_{z\to z_0} \frac{d^{k-1}}{{dz}^{k-1}}\left[(z-z_0)^k f(z)\right]\,$

$\mathrm{Res}_{z=-1} =\lim_{z\to -1} (z+1) f(z) = \lim_{z\to -1} \frac{1}{z(1-z)} = \frac{-1}{2} \,$

$\mathrm{Res}_{z=0} =\lim_{z\to 0} z f(z) = \lim_{z\to 0} \frac{1}{(1-z^2)} = 1 \,$

$\mathrm{Res}_{z=1} =\lim_{z\to 1} (z-1) f(z) = \lim_{z\to 1} \frac{-1}{z(1+z)} = \frac{-1}{2} \,$

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

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

Main Page : Complex Variables : Residues

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CVR2

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