Question

Real Analysis Prerequisites for Complex Analysis

Answer 1

Off the top of my head, the topology of R\^n, uniform convergence of functions, convergence of series, derivatives and integrals all play an important role. It may sound dumb to say "derivatives and integrals" here, but the sophistication with which you use them needs to be at a level more like analysis and less like basic calculus.

If Rudin didn't click for you the first time round, you could try a different book. "Mathematical Analysis" by Apostol is a bit easier (but still at a good level) and covers more material than Rudin. You could also have a look at Burkill's "Second Course in Analysis", much of which is actually devoted to complex functions.

Answer 2

Derivatives and line integral in R^2 in particular. Higher dimensions are not needed. Surface integral is very rare.

Answer 3

Stokes's theorem/Green's theorem. Important near the beginning.

Familiarity with argument using continuity, connectedness and compactness. For example, if you need to show why a continuous function on a closed disk has a point with minimum absolute value it should be obvious to you.

Basic differentiation, integration, and summation techniques. Just calculus stuff, nothing too advanced.

Convergence of functions. You don't even have to worry about all the different kind of convergences, just be aware that they exist. Most of the time you only use uniform convergence, and if you encounter anything else it probably means you need to change your approach.

Laplacian, possibly important but depends on course. Because of relationship between complex analytic function and harmonic functions.

Complex analysis is a much more computational course compared to the proof-based fundamental-based real analysis. So not a lot of Real Analysis will help you. Your calculus techniques are much more important.