I'm having trouble understanding the difference, other than notation, between a contour integral over an open curve and a contour integral over a closed curve. So far, it seems to me that the difference is only in the limits of integration.
More specifically, I'm working on proving this: $$\lvert\oint_\Gamma \frac{cos(z)}{z}dz\rvert \le 2e\pi$$ where the path traces the unit circle once.
I know how to prove a very similar problem, just without the closed circle:
$$\lvert\int_\gamma \frac{cos(z)}{z}dz\rvert \le 2e\pi$$
To prove this I use the theorem,
$$\lvert\int_C f(z)dz\rvert \le ML$$
and then prove that
$$L= 2\pi$$ and $$M=e.$$
Can I go about this proof with the closed circle the same way?
Yes. $ML$ estimation doesn't require the curve to be open (or closed), so you can certainly use the $ML$ estimate argument in the case where $\Gamma$ is the unit circle traversed once.