Please write here about what was the most confusing stuff in the lecture and I will try to add some additional explanation of it in the style of "manipulating power series" video.

This is so helpful! I had a question about today's lecture, why can't the rouche theorem be extended for turns around any point z_0?

I believe it can: I will talk about argument principle instead of Rouche theorem, as it is more general.
In the formulation we had in class the number of zeroes of an analytic function in a domain bounded by curve c was equal to the number of turns that f(c) makes around point 0.
More generally, the number of solutions of the equation f(z)=w_0 in a domain bounded by a curve c is equal to the number of turns f(c) makes around w_0.

Was this the kind of extension you had in mind?

Yes indeed! Thank you for the clarification

It would also be really helpful if you could go over some more examples of finding the number of zeros of a function on a given domain.

I will post the videos from the lecture on argument principle, but right now I don't have time to add extra material. I will try to do so as soon as I can.

Can you please post solutions to our Quiz 3? Most of us don't know how to do perfectly. Thanks so much!

How much of greens theorem do we have to know, because chapter 1.6 boggles my mind...aka wave your hands convincingly about why it is correct

You have to be able to apply it (for integrals of the type f(z)dz over a closed cotour, where f(z) is not necessarily analytic).