# Real before complex analysis or vice versa?

I used to learn Real Analysis before Complex Analysis in my bachelor study, but now the order is reversed in my university.
I would like to ask which order is better to learn the subjects, and which order is better to teach the subjects?

• This depends greatly on specifics that you have not given. For example, complex variables at the Churchill level is often taken by engineering and physics undergraduates who will never take a real analysis course -- my first complex analysis course, using the 1st (1973) edition of Marsden's text, had about a $2:1$ ratio of engineering majors to mathematics majors. On the other hand, you'd probably want prior exposure to Lebesgue integration before Rudin's book. Jul 7, 2021 at 5:32
• Thanks @DaveLRenfro. We used Churchill's book in our class back then, and, it's more about technical stuff, rather than theoretical. Jul 7, 2021 at 8:34
• Wooow, you learn this Rudin's Real and Complex Analysis in your Real Analysis's bachelor study, @DaveLRenfro? Amazing. Jul 7, 2021 at 8:37
• Rudin's book is almost never used in the U.S., but it is a frequently cited book, especially for those preparing for their Ph.D. qualifying exams. I mentioned it because it's famous for being very unsuitable for first courses in either real analysis or complex analysis at the graduate level (i.e. after undergraduate study), but also often considered very instructive for surveying/reviewing the material for those who have already had such courses. Since you spoke of "bachelor study" and "university", I assumed you were comparing what in the U.S. is called undergraduate work and graduate work. Jul 7, 2021 at 15:03
• Yes, that's what I mean, undergraduate study. Thank you, @DaveLRenfro. Jul 9, 2021 at 1:45

While, yes, precise advise would depend considerably on context, I do think there are some features that should/could influence curriculum design/choices.

Basic complex analysis (Cauchy's theorem and corollaries, power series and Laurent expansions, residues, ...) functions very well, answers questions, and can feel like a fulfillment or happy continuation of the positive aspects of calculus. That is, we can compute derivatives of lots of things, and compute some integrals, in lucky cases. Amazing. And, in complex analysis, "functions" are (representable as) power series, which is about as good as it gets.

Basic real analysis can be perceived as a lot of bad news and reports of danger. Yes, "rigor" is being imparted to calculus, but can easily be perceived as expensive, without much operational benefit besides the rigor itself.

But, ok, it's good to have a bit of that basic real analysis before proceeding too much further with complex analysis, since some aspects really do depend on finicky estimates, issues of uniformity-or-not, auxiliary functions not as nice as holomorphic, as opposed to the algebraic ideas in the compact Riemann surfaces direct.

Beyond the basic real analysis stuff, for applications (to complex analysis and to other things), the kind of things that have seemed most persistently useful to me involve distribution-theory (generalized functions), Fourier series and Fourier transforms, and things sometimes labelled "functional analysis", rather than subtleties about pointwise behavior and measure theory.

Our take is Real Analysis before Complex Analysis.

Real Numbers when couldn't accommodate square root of negative number - complex numbers were introduced. If you consider, complex number as an ordered pair - basically elements of $$R^{2}$$ - can that be better conceived without understanding the properties on R.

Mathematics teaches us ways to generalize abstract structures and as such Real Analysis comes as a particular case of Complex Analysis. Without having understood the properties on real numbers how can we appreciate the generalization mechanism to higher dimensions.

There can be other schools of thought - but we prefer Real Analysis before Complex Analysis.