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?

Solution 1:

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.

Solution 2:

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.