Question Number 145820 by Mrsof last updated on 08/Jul/21
$${find}\:{resideo}\:{e}^{\left({z}+\mathrm{1}\right)/{z}} \\ $$
Commented by Mrsof last updated on 08/Jul/21
$${help}\:{me}\:{sir}\:{please} \\ $$
Commented by Mrsof last updated on 08/Jul/21
$$????? \\ $$
Commented by Mrsof last updated on 09/Jul/21
$$????????? \\ $$
Answered by Olaf_Thorendsen last updated on 09/Jul/21
$${f}\left({z}\right)\:=\:{e}^{\frac{{z}+\mathrm{1}}{{z}}} \:=\:{e}^{\mathrm{1}+\frac{\mathrm{1}}{{z}}} \:=\:{e}.{e}^{\frac{\mathrm{1}}{{z}}} \\ $$$$\mathrm{The}\:\mathrm{value}\:{z}=\mathrm{0}\:\mathrm{is}\:\mathrm{an}\:\mathrm{essential} \\ $$$$\mathrm{singularity}\:\mathrm{of}\:{f}. \\ $$$${f}\left({z}\right)\:=\:{e}\left(\mathrm{1}+\frac{\mathrm{1}}{{z}}+\frac{\mathrm{1}}{\mathrm{2}{z}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{6}{z}^{\mathrm{3}} }+…\right) \\ $$$$\mathrm{The}\:\mathrm{coeff}.\:\mathrm{of}\:\frac{\mathrm{1}}{{z}}\:\mathrm{is}\:{e}. \\ $$$$\Rightarrow\:\mathrm{Res}_{\mathrm{0}} {f}\:=\:{e} \\ $$