Complex analysis: Cauchy's theorem

This is a lecture recorded by Prof. Steven Strogatz for his undergraduate class on "Applied Complex Analysis" at Cornell University. In this lecture, Prof. Strogatz walks the students through a magnificent proof of one of the main results of complex analysis, known as Cauchy's theorem. (It's also sometimes called Cauchy's integral theorem, or the Cauchy-Goursat theorem, since the proof shown here is due to Goursat.)

[giorapeniakov]

I am a little confused about the first assumption you call "Recall". In this assumption, we say that powers of z, z^n are integrated to 0 on closed loops for any integer n that is not -1. From my understanding, this result can be directly proven with anti-derivative only on circle closed loops, where the radius R is fixed and the integral is parametrized only through an angle theta. The extension of this result to arbitrary loops uses analyticity, which we are trying to prove.
I guess it was explained in a previous lecture, but I haven't seen it. If you could upload it, I'd be grateful.

[josedomingogiliglesias]

Dear professor, could you tell me which book you follow to teach the complex analysis course? It's a pleasure to listen to you. Thank you

[dneary]

What application do you use to draw? I really like the way it can do lines, boxes, curves in addition to freeform writing!