# Do you need real analysis to understand complex analysis?

I'm debating whether I should take a course, in complex analysis (using Bak as a text). I've already taken Munkres level topology and "very light" real analysis (proving the basic theorems about calculus) using the text Wade.

The complex analysis course is supposedly difficult and will even cover the Prime Number Theorem in the end. Do you think it's better to take Rudin level real analysis first?

-
It will depend on (among other things) how the course is taught in the beginning, i.e. what you are assumed comfortable with, which we don't have enough info to judge. If you are not already pretty comfortable with $\varepsilon$-$\delta$ proofs, sequences, series, and functional limits, then more primer on the basics of analysis would serve you well. Beyond mathematical maturity and these basics there isn't much needed specifically from real analysis to take on complex analysis. (In particular, the pathological cases you have to be wary of in real analysis do not appear in complex analysis.) – Jonas Meyer Jan 16 '12 at 6:46
Why don't you ask the instructor of the course? He or she should be able to tell you how much of a background in real analysis the students will be expected to have. – Robert Israel Jan 16 '12 at 6:47
Can you take both at once? – Jonas Meyer Jan 16 '12 at 6:49
As Jonas said, "It will depend on how the course is taught." There could be a lot needed from real analysis and even functional analysis - e.g. you could get the Prime Number Theorem by a route passing through Wiener's and Ingham's Tauberian theorems. – Robert Israel Jan 16 '12 at 6:58
(I looked at "Complex Analysis - Bak and Newman" for this) Since you've read Munkres, I'm assuming you are pretty mathematically mature and since you read Wade, I assume you "get" analysis. It definitely wouldn't hurt to take a Rudin-based course before this but I wouldn't say it's necessary at all, this type of complex analysis is pretty self-contained (and when it does borrow from real analysis, it's usually stuff like "definition of derivative" and done "in the obvious way"). – tomcuchta Jan 16 '12 at 8:43

If the course teaches complex analysis from a geometric perspective-emphasizing the properties of analytic maps of the plane as a "calculus of oriented angles", as I did in my undergraduate complex analysis course-then believe it or not, you'll need very little if any real analysis except for certain results (like Cauchy's theorem and convergent series). For example, a good way to think of the derivative in the complex plane as a sequence of "infinitesimal" rotations of a tangent line to a circle centered at a point in the Argand plane-whereas the sequence of rotated tangent lines converges to the point by contracting in length along increasingly smaller subcircles. Also, most of the standard transformational geometry of the Euclidean plane has very elegant reformulations in terms of the standard analytic functions of the plane, such as the complex exponential in plane polar coordinates. If the course focuses on these aspects of elementary complex analysis, you'd be better off brushing up on your basic geometry then real analysis! However, if the course develops complex analysis via a rigorous development of the complex plane as a metric or normed space and focuses on infinite series, then that's a different story and you'll need a lot more rigorous real analysis to get comfortable with it.

-

I am not familiar with the usual content of a light real analysis course but it would be really helpful if you have:

• As mentioned in the comments: Knowledge about convergence of sequences and series (with $\epsilon$-$\delta$-management), the notion of continuity and knowledge about real integrals (Riemann integral is appropriate).
• A good understanding of the derivative of a function $\mathbb{R}^2 \rightarrow \mathbb{R}^2$.
• Some theorems which allow you to interchange differentiation and integration as well as limits and integration (which are basically the same). You need these to proof the backbone theorems like Cauchy's integral formula.

Some things of a real analysis course which you probably not need are measure theory and the Lebesgue integral.

-

If you are not able to ask the instructor for some reason, I would suggest you to devote some hours looking at the book.

-