Question Number 206550 by SANOGO last updated on 18/Apr/24
$${calcul}\:\:{Residus}\:{de}\:{f}\:{en}\:{o} \\ $$$${f}\left({z}\right)=\frac{{ze}^{{z}} }{\left({z}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$
Answered by Berbere last updated on 18/Apr/24
$${f}\:\:{n}'{a}\:{pas}\:{de}\:{poles}\:{en}\:\mathrm{0};{Res}\left({f},\mathrm{0}\right)=\mathrm{0} \\ $$$$\left.\:\mathrm{1}\:{est}\:{le}\:{seul}\right\}{Pole}\:{d}\:{ordre}\:\mathrm{2} \\ $$$${Res}\left({f},\mathrm{1}\right)=\frac{\partial}{\partial{z}}\left({z}−\mathrm{1}\right)^{\mathrm{2}} .\frac{{ze}^{{z}} }{\left({z}−\mathrm{1}\right)^{\mathrm{2}} }\mid_{{z}=\mathrm{1}} =\frac{\partial}{\partial{z}}\left({ze}^{{z}} \right)\mid_{{z}=\mathrm{1}} =\left({z}+\mathrm{1}\right){e}^{{z}} \mid_{{z}=\mathrm{1}} =\mathrm{2}{e} \\ $$