# Complex Variable vs Real Analysis 1

I took Real Analysis 1 last semester, and it was challenging, but not as bad as I thought it would be. I am considering taking Function of a Complex Variable this semester, but I am torn. I don't know much about the complex number system, and I am afraid I'll struggle. Generally, should I expect Complex Variable to be more rigorous than Analysis?

Here are the course descriptions:

Real Analysis 1: Properties of the real number system; point set theory for the real line including the Bolzano-Weierstrass theorem; sequences, functions of one variable: limits and continuity, differentiability, Riemann integrability

Function of a Complex Variable: Complex number system. Functions of a complex variable, their derivatives and integrals. Taylor and Laurent series expansions. Residue theory and applications, elementary functions, conformal mapping, and applications to physical problems.

• It is hard to tell without detail, but the complex variable course as described is likely to be very different from the real variable course. In some sense it will be much more familiar in shape, fairly heavily computational, formula-dense. – André Nicolas Jan 8 '13 at 2:21
• Frankly, while I am somewhat comfortable with basic real analysis, I have developed relatively little intuition for complex analysis (despite a control and electronics design background). The point you should take from this is that they are quite different. – copper.hat Jan 8 '13 at 3:48

I think this depends on what you're worried about, and what elements of the subject you think you might have trouble with. You mention that you thought real analysis was challenging, but you also ask specifically whether complex analysis is more rigorous than real analysis. So there are two possible aspects of the course one can look at: the level of mathematical rigor, which can differ depending on the style of the professor and/or textbook, and the difficulty of the mathematical concepts.

In terms of mathematical rigor, I would say it's rare that one would teach complex analysis more rigorously than one would teach real analysis, if it's being taught by the same professor (which, of course, is not given). That is not to say that complex analysis requires less rigor, but simply that because real analysis is in some sense more fundamental, in teaching it one often pays more attention to a lot of details.

In terms of mathematical concepts, I think it is typical for a student to say that complex analysis hearkens back to more familiar concepts learned in previous courses, e.g. basic calculus and geometry. In many respects, real analysis is a subject that has to deal with pathological issues: things like spaces that are path-connected but not connected, functions that are continuous everywhere but differentiable nowhere, metric spaces that are not complete, etc. This often makes it hard to visualize some of the concepts. Complex analysis, on the other hand, deals only with a nice space ($\mathbb{C} \cong \mathbb{R}^2$) and complex analytic functions, which turn out to be basically the nicest functions on the planet. You learn many geometric connections involving complex analytic functions, and often these connections can be visualized. I don't claim that complex analysis is an easier subject, but certainly the theory is much tighter and in many ways more elegant.

I hope this helps you in making your decision.