As far as I know:
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.
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:
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.
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.
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.