If $f(z)$ is analytic in a simply connected domain $D$, its integral over any closed contour is $0$. I don't quite understand how the idea of contracting the contour that encloses a pole follows from that. Specifically, why are we able to say the following integral has the same value over any closed contour that encloses $z_0$, a pole of order $m$,
where $C$ is the circle of radius $\rho$ centered at $z_0$. In other words, what allows us to be able to contract an arbitrary contour (that encloses $z_0$) to the circle $C$?