I'm working on some past exam papers and I wanted to see if I'm thinking of the following in the correct way;
Given a contour integral $\int_{\gamma}\frac{1}{z}dz$ I want to find all of it's possible values given that $\gamma$ is the semi-circle starting at $1$ and going to $-1$
My thinking is that as the radius of the semicircle is $1$ then $|z|=1$ and so $z=e^{i\theta}$.
We can also parametrise $\gamma$ as $e^{i\theta}, \theta \in [-\pi,\pi]$
This means we can express our integral as;
$\int_{\gamma}\frac{1}{z}dz=\int_{\pi}^{-\pi}e^{-i\theta}ie^{i\theta}d\theta=\int_{\pi}^{-\pi}id\theta=0$
So what does the question mean by find all the values of $\int_{\gamma}\frac{1}{z}$ ?
It seems to me there is only one ?