Question

Find the laurent series about the indicated singularity for the function e^2z/(z-1)³ at z=1

Answer 1

$f(z)=\frac {e^{2z}}{(z-1)^3}$ ,

Let the given complex function be analytic in an annulus $r<|z−1|<R$  Then $f(z)$ can be expanded into Laurent's series about $z=1$

Let $z-1 =u$ , then $z = u+1,$ Putting in $f(z)$ and expanding

$f(z)=\frac {1}{u^3}[1+2(u+1)+2(u+1)^2+...]$