Intuition on Complex Analysis

Question

I'm currently studying complex analysis via Stein's book, but I feel that, despite I understand the proofs, I'm still lacking of intuition on why the results are true. For example, it's still not natural to me why a function be holomorphic or analytic or have a primitive are all equivalent concepts, and this lack of analytic and geometric meaning makes me forget the proofs and the important results. I mean, what is behind complex function's regularity? Can anyone help me?

Answer 1

I think the brutal workhorse of complex analysis is that every holomorphic function has a local representation as a convergent power series. This is the reason why all the nice things happen, because it lets us pretend we're Euler and all the limits and integrals and derivatives commute and converge and whatnot.

If you want to get somewhat philosophical, you can say this is because absolutely convergent power series are more or less like polynomials, and so a lot of the nice things about polynomials continue being true for holomorphic functions.

Then, later, you can decide these are not acceptable things to say to people at parties, and come up with complex differentiability and the Cauchy-Riemann equations and other things that are the true definition of holomorphic functions.