Show that if $C$ is a positively oriented circle and $z_{0}$ lies outside $C$, then
$$\int_{C} \frac{dz}{z - z_{0}} = 0 $$
$\underline{Proof}$
WE have to prove $z_{0}$ lies outside $C$ if $C$ is positively oriented, i.e with $0$ to $2\pi$ limits. Let define $C$ with $z=e^{it}$
Therefore $$ \int_{0}^{2\pi} \frac{ie^{it} dt}{e^{it} -z_{0}} = \left[\log(e^{it} - z_{0})\right]\Biggr|_{0}^{2\pi}$$ $$ = \log(1 - z_{0}) - \log (1-z_{0}) = 0 $$
this indicates $z_{0}$ lies on the circle or outside the circle
Can you guys check if this is correct or not?