Question

Integrtion of the complex funciton $f\left(z\right)=\frac{z^2}{z^2-1}$, in the counterclockwise direction, around $|z-1|=1$, is

• A. $\pi i$
• B. $0$
• C. $\pi i$
• D. $2\pi i$

Answer 1

The correct option is C $\pi i$

$f\left(z\right)=\frac{z^2}{z^2-1}$
$z=-1,1$ are the simple poles of f(z)
and $z=-1$ lies outside the curve $C:|z-1|=1$
and $z=1$ lies inside the curve $C:|z-1|=1$
so, ${\oint }_{c}f\left(z\right)dz=2\pi$ Res $\left[f\right(z),z=1]$
$=2\pi i\left(\frac{1}{1+1}\right)=\pi i$