Question

The value of the contour integral in th complex-plane $\oint \frac{z^3-2z+3}{z-2}dz$ along the contour $|z|=3$, taken counter-clockwise is:

• A. $18\pi i$
• B. $0$
• C. $14\pi i$
• D. $48\pi i$

Answer 1

The correct option is C $14\pi i$

Given contour integral is $\oint \frac{z^3-2z+3}{z-2}dz$ where $f\left(z\right)=\frac{z^3-2z+3}{z-2}$ and the countour is $|z|=3$ in counter clockwise. The pole $z=2$ lies inside the contour $|z|=3$
$Res\left(f\right(z\left)\right)={lim}_{z\to 2}(z-2)\left[\frac{z^3-2z+3}{(z-2)}\right]$
$=8-2\left(2\right)+3=7$
By Cauchy residue theorem
$\int \frac{z^3-2+3}{(z-2)}dz=2\pi i\left[Res\right(f\left(z\right)\left)\right]$
$=2\pi i\left(7\right)=14\pi i$