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Question Number 148960 by tabata last updated on 01/Aug/21

$$findtheresideof\left(z\right)=\frac{z}{z^n-1}$$

Answered by mathmax by abdo last updated on 02/Aug/21

$$\mathrm{les}\mathrm{residus}\mathrm{ici}\mathrm{sont}\mathrm{les}\mathrm{poles}\mathrm{de}\mathrm{f}\mathrm{et}\mathrm{se}\mathrm{sent}\mathrm{les}\mathrm{racines}{\mathrm{n}}^{\mathrm{eme}}\mathrm{de}\mathrm{lunite}$$$$\mathrm{cad}{\mathrm{z}}_{\mathrm{k}}={\mathrm{e}}^{\frac{\mathrm{i}2\mathrm{k}\pi }{\mathrm{n}}}\mathrm{and}\mathrm{k}\in \left[[0,\mathrm{n}-1]\right]\Rightarrow \mathrm{f}\left(\mathrm{z}\right)=\frac{\mathrm{z}}{\prod _{\mathrm{k}=0}^{\mathrm{n}-1}(\mathrm{z}-{\mathrm{z}}_{\mathrm{k}})}$$$$\mathrm{f}\left(\mathrm{z}\right)=\sum _{\mathrm{k}=0}^{\mathrm{\infty }}\frac{{\mathrm{a}}_{\mathrm{k}}}{\mathrm{z}-{\mathrm{z}}_{\mathrm{k}}}\Rightarrow \mathrm{Res}(\mathrm{f},{\mathrm{z}}_{\mathrm{k}})={\mathrm{a}}_{\mathrm{k}}=\frac{{\mathrm{z}}_{\mathrm{k}}}{{\mathrm{nz}}_{\mathrm{k}}^{\mathrm{n}-1}}=\frac{{\mathrm{z}}_{\mathrm{k}}^2}{\mathrm{n}}$$$$\mathrm{autre}\mathrm{methode}\mathrm{f}\left(\mathrm{z}\right)=\frac{\mathrm{z}}{(\mathrm{z}-{\mathrm{z}}_0)....(\mathrm{z}-{\mathrm{z}}_{\mathrm{k}-1})(\mathrm{z}-{\mathrm{z}}_{\mathrm{k}})(\mathrm{z}-{\mathrm{z}}_{\mathrm{k}+1})...(\mathrm{z}-{\mathrm{z}}_{\mathrm{n}-1}}$$$$\Rightarrow \mathrm{Res}(\mathrm{f},{\mathrm{z}}_{\mathrm{k}})=\frac{{\mathrm{z}}_{\mathrm{k}}}{({\mathrm{z}}_{\mathrm{k}}-{\mathrm{z}}_0)...({\mathrm{z}}_{\mathrm{k}}-{\mathrm{z}}_{\mathrm{k}-1})({\mathrm{z}}_{\mathrm{k}}-{\mathrm{z}}_{\mathrm{k}+1})....({\mathrm{z}}_{\mathrm{k}}-{\mathrm{z}}_{\mathrm{n}-1})}$$

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