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Question Number 145820 by Mrsof last updated on 08/Jul/21
find resideo e^((z+1)/z)
$${find}\:{resideo}\:{e}^{\left({z}+\mathrm{1}\right)/{z}} \\ $$
Commented by Mrsof last updated on 08/Jul/21
help me sir please
$${help}\:{me}\:{sir}\:{please} \\ $$
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(z) = e^((z+1)/z)  = e^(1+(1/z))  = e.e^(1/z)   The value z=0 is an essential  singularity of f.  f(z) = e(1+(1/z)+(1/(2z^2 ))+(1/(6z^3 ))+...)  The coeff. of (1/z) is e.  ⇒ Res_0 f = e
$${f}\left({z}\right)\:=\:{e}^{\frac{{z}+\mathrm{1}}{{z}}} \:=\:{e}^{\mathrm{1}+\frac{\mathrm{1}}{{z}}} \:=\:{e}.{e}^{\frac{\mathrm{1}}{{z}}} \\ $$$$\mathrm{The}\:\mathrm{value}\:{z}=\mathrm{0}\:\mathrm{is}\:\mathrm{an}\:\mathrm{essential} \\ $$$$\mathrm{singularity}\:\mathrm{of}\:{f}. \\ $$$${f}\left({z}\right)\:=\:{e}\left(\mathrm{1}+\frac{\mathrm{1}}{{z}}+\frac{\mathrm{1}}{\mathrm{2}{z}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{6}{z}^{\mathrm{3}} }+…\right) \\ $$$$\mathrm{The}\:\mathrm{coeff}.\:\mathrm{of}\:\frac{\mathrm{1}}{{z}}\:\mathrm{is}\:{e}. \\ $$$$\Rightarrow\:\mathrm{Res}_{\mathrm{0}} {f}\:=\:{e} \\ $$

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