Is undergraduate "complex analysis" actually kind of "complex calculus"? Please provide references.

Question

As far as I know:

1. After Calculus I, II and III, math majors have basic real analysis that covers topics including uniform continuity, Riemann-Stieltjes, Bolzano-Weierstrass and is mainly proving results in Calculus that weren't proved.

2. After that, basic real analysis is, at least in my university, a prerequisite to advanced real analysis, involving Lebesgue measure and Lebesgue integral and also a prerequisite to "complex analysis", involving Cauchy-Riemann, Cauchy's Integral Theorem and Cauchy's Integral Formula, Laurent series and Residue Theorem. (In my university, basic real analysis is called "advanced calculus" and advanced real analysis is just called "real analysis")

My question is: Is undergraduate "complex analysis" actually kind of "complex calculus"?

My take is that:

1. in Calculus, we were doing more computations than proofs. In basic real analysis, we did more proofs than computations. Whenever we did computations, we often had to prove. Naturally, advanced real analysis involved much more proofs and much less computations.

2. In "complex analysis", computations of complex integrals seem to require little proving. I found them much like calculus questions where we did u-substitutions. Even the integrals where you split a path, I find are much like Calculus II integrals where you split regions or Calculus III integrals where you split surfaces or solids.

3. On the other hand, there are a lot proofs even before you get to complex integrals such as proving a particular function is constant using Cauchy-Riemann. Later for complex sequences, the problems are indeed like the basic real analysis questions. This could be an argument against the name "complex calculus"

   • This is actually comes from a comparison made by some of my physics major friends or fellow math majors in my university between "real analysis" and "complex analysis", where some say that "real analysis" is more difficult

   • Isn't the comparison wrong because "complex analysis" should actually be compared to basic real analysis instead of advanced real analysis?

   • I have a suspicion that we can't have a "complex calculus" course that is split from "complex analysis" because there wouldn't be enough topics for a whole semester so schools just decide to do the proofs and computations at once.

   • (I actually had another thought that no one would be in need of a "complex calculus" other than math majors so math majors are going to eventually learn "complex analysis", then we might as well merge what would have been "complex calculus" with it, but then what about engineering undergraduates who do Fourier transforms?)

A follow up question to this is: Can anyone suggest a text that is a successor to "complex analysis" books such as Complex Analysis by Brown Churchill?

• My opinion is that if a successor is more on proofs than computations for the same topics, then I would see "complex analysis" as "complex calculus", but if a successor is on new topics, then "complex analysis" can be so-called without quotes.

Related questions:

What is the difference between stochastic calculus and stochastic analysis?

Are calculus and real analysis the same thing?

Answer 1

You could do "complex calculus" with very little proofs, just focusing on computational methods to obtain Taylor and Laurent series expansions, the usual holomorphic functions including the logarithm, and calculating integrals using the residue theorem. Depending on how many hours per week you spend, this can be one semester.

That would mean identity theorem, open map theorem, maximum principle etc. all go into your "complex analysis" course, with many options what else to do (Schwarz lemma, Riemann mapping theorem, convergence theorems, ...) that are of a less computational nature.

Doing both at once makes a nice integrated class where students both see beautiful proofs and powerful properties, and can spend half of their exam doing nice and easy calculus questions.

Nice books on complex analysis that do not do "calculus" that much: Serge Lang's book "Complex Analysis" and Stein-Shakarchi's book "Complex Analysis". Both have a much more theoretical focus and so don't overlap with Brown-Churchill very much.

Answer 2

Don't know about your university, but a basic complex analysis course would have things like Weierstrass-Casorati, argument principle and Rouche's theorem, local and open mapping theorem, homotopy&homology versions of Cauchy, possibly Riemann mapping theorem, building up for a course on Riemann surfaces or similar. If it just involves lots of calculating contour integrals then it is a methods course (covering e.g., calculating silly integrals, asymptotic expansion, using Fourier/Laplace transform to solve linear ODEs).

About books: Brown/Churchill is all applied. If you want a complex analysis textbook, try something like H.A. Priestley's Introduction to Complex Analysis (although it contains its fair share of typos/mistakes). If you are feeling ambitious you could jump directly to Ahlfor or Rudin.