Result of complex integral

Question

If $f(z)=z^{-1}$ is integrated on a circle not containing the origin, what is the result?

My guess is that since $f$ doesnt have a primitive on $\Bbb C\setminus\{0\}$ the result cannot be $0$.

Answer 1

As the circle doesn't contain the origin, therefore $\frac{1}{z}$ is analytic on the circle(closed curve), therefore the integral is $0$.

Answer 2

By homotopy, you integration path can be reduced to any point in the plane (except 0 of course) and thus you integral is indeed $0$.

Answer 3

The integral of $z^{-1}$ along any circle such that $0$ does not belong to the closed disk whose boundary is that circle is $0$.