Hi guys, I just went through chapter 2.1 and found that the example 10 is quite confusing. I am particularly wondering, firstly, why the derivative is du/dx+i dv/dx? Why can't we take the derivative with respect to y? It says Theorem 3 implies the rationale of taking the derivative. Indeed it does, but I could not find how does Theorem 3 indicate the computation of the derivative. Furthermore, by Cauchy-Riemann equation we have du/dx=dv/dy, and du/dy=-dv/dx, if we do take the derivative with respect to y, then the derivative of log(z) would have real part of z and imaginary part of z switched in numerator. Does this cause a problem?

And secondly, it has drawn the conclusion that function log(z) is not analytic on any domain D that contains a simple closed curve that surrounds the origin. So I think log(z) on complex plane is similar as log(x) on real plane, and they both does not have derivative at 0. I have a rough idea about its reasoning but I dont know if I am on the correct track, correct me if I am wrong pls.

Btw I have highlighted the two parts that I have mentioned in my question in the picture below.