I learned this years ago in university, but I seldom used and whenever I needed it, I had to look it up again and again. I don't know why, but this topic just doesn't stick in my head.
Analytic at a boundary point would probably imply differentiable on a neighborhood of that boundary point that extends beyond the closure of D. Maybe you can prove that's always doable given the first conclusion, but that might be a distraction from this. I haven't given it much thought.
Knowing that you can safely pull the derivative inside the integral sign requires knowing a fair amount about analytic functions, probably about as much as proving the derivative version of Cauchy's Integral Formula.
Yeah I prefer to prove the Cauchy Integral identity (the one with higher derivative) using the Cauchy integral formula and expanding it to get the Taylor series expansion. Solving for the coefficients naturally gives the result.
In the last warm up problem can we split C curve into two closed simple curves? And since they have the same orientation and both contain z = 1 will the answer then be just a double usual integral? Thanks in advance.
8:12 you still need to show that the integral over the circle of radius epsilon vanishes as epsilon goes to zero (granted, that's easy, but the proof is incomplete at this point)
my understanding of it is that it’s bc C has additional structure to it that R does not, and the “niceness” is an emergent property of that added structure
More specifically, Cauchy Schwartz implies that the derivative of complex differentiable functions are just rotations and scalings of the complex plane, while it could be any arbitrary linear transformation for general real differentiable functions
