I'm a Nero fan so I'm glad I learned about the Mandelbrot set, but I notice that said topics are not in Brown-Churchill or 'A First Course in Complex Analysis' while they are in Coursera's 'Introduction to Complex Analysis'.

If so:

Why? Why are they not in those books?

If not:

Why? How could they be useful in some introductory complex analysis classes such as the one in Coursera?

  • $\begingroup$ @JW Thanks! ^-^ $\endgroup$ – BCLC Jan 7 '17 at 18:43
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    $\begingroup$ For what it's worth, none of the talks in the "Revitalizing Complex Analysis" session at the just-completed Joint Mathematics Meetings mention this topic. $\endgroup$ – kcrisman Jan 8 '17 at 3:24
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    $\begingroup$ A traditional complex analysis course probably contains discussion of "normal families" of functions. (Perhaps as preparation for the Riemann mapping theorem.) Then (even if it is not in the book) an instructor for the course can give the correct definitions for Julia/Fatou sets. Many of the students in the course may have previously seen the popularized definitions, and thus be interested in seeing the actual definitions. $\endgroup$ – Gerald Edgar Jan 8 '17 at 14:41
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    $\begingroup$ I devoted a day to those topics last time I taught the subject because it's interesting and not too off topic if you follow something like Gamelin's text. See youtube.com/playlist?list=PLBY4G2o7DhF0TSossUvJ-CTKSLfOhQgb6 although, I don't see my day of Mandelbrot, it may be I didn't tape it as we mostly spent the time watching the excellent videos from the numberfile on the topic. Other year I had an student give a talk on it. Given the current structure of the course (lack of analysis prereq) it does seem like an optional topic. $\endgroup$ – James S. Cook Jan 17 '17 at 11:19

No, these topics are not usually included in courses on complex analysis, for several reasons, which I will explain below. At the same time, it is easier to understand why relatively old textbooks did not include such topics: things about iteration of mappings in the complex domain is a much younger topic (as old as it is by this year) than most of the rest of what is called "complex analysis", by almost 100 years. Thus, textbooks either created 60 years ago, or emulating such, would not have mentioned such things... and still would be overflowing with important ideas.

This comes to a less artifactual point: the "traditional" topics in complex analysis are highly useful across much of mathematics and other parts of science, which is why complex analysis is considered one of the pillars of the standard curriculum, especially for graduate students in math, but also for physicists and many engineers. The dynamics of iterated maps is interesting, but less universally useful, certainly at a basic level.

I have seen a few attempts to include the dynamics of iterated maps in otherwise "standard, required" graduate complex analysis courses. There are some conflicts in this scenario, although these are partly artifactual, and not an argument against trying something similar. Namely, if the standard "required" course is to prepare grad students for prelims, and the latter don't include things about iterated maps, probably something that is included was omitted, and this disserves the students. Yet, arguably, the stuff is interesting, and lends itself to actual contemporary research far more than the old-timey (but incredibly practical-useful) complex analysis. Really, I'd argue that these are "different subjects", in the same way that basic X is typically utterly different from research-in-distant-descendent-of-X.

In a related vein, it is not easy to arrange meaningful "tests" on iterated maps, any more than it is easy to arrange appropriate "tests" on any kind of still-alive mathematics. This is not a good reason to not teach such things, but it does complicate matters in the face of the inertia of structural expectations.

Finally, as @DanielRCollins speculates, situations in which the goals are popularization and promotion (for better or for worse), colorful graphics are obviously a winner. Cool pictures that nowadays can be easily programmed or at least viewed naturally achieve broader popularity than proving theorems and such.

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    $\begingroup$ Especially given that complex variable courses have that engineering/physics aim at many institutions, I agree with you. I feel that the dynamical system stuff probably fits in better as part of a full-year analysis sequence in some way (e.g. contraction mappings) though perhaps I'm influenced too much by certain books taking this point of view. $\endgroup$ – kcrisman Jan 8 '17 at 3:16

A brief stab at an answer before someone more knowledgeable comes along: My brief experience says that: no, fractal sets are not usually a topic in complex analysis (they're not in any of the 3 textbooks that I'm familiar with).

We might take a guess and say that they're in the Coursera MOOC because it provides a fluffy, visual, easy-to-promote introduction to their mass-market class.

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    $\begingroup$ I have some feelings about your last sentence. What exactly is the problem with creating beautiful courses with wide appeal? I know Petra, and she puts an incredible amount of energy into teaching. I have not taken her course, but I have heard really good things about it from people who have. $\endgroup$ – Steven Gubkin Sep 25 '20 at 17:29

It seems to me there is a fundamental disconnect between what one might call "traditional complex analysis" and the study of fractals.

Traditional complex analysis, as used in physics and engineering, is primarily about analytic and meromorphic functions, whose most important property is that they are infinitely differentiable except at a set of isolated points (their poles).

On the other hand fractals are essentially continuous but non-differentiable functions.

Also, "fractals which happen to be easy to define using complex numbers" are a subset of fractals in general.

Therefore, transplanting an "introduction to fractals" into a different branch of math doesn't seem a very useful idea. I don't see anything wrong with the notion that dynamical systems, fractals, chaos theory, etc could and/or should be added to the "core applied math" curriculum, but they deserve to form a separate topic in their own right, and/or linked to practical applications such as wavelet transforms in signal processing, the general behaviour of nonlinear differential equations, etc.

But as others have said, pretty pictures do sometimes generate motivation!


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