DO NOT TAKE THIS CLASS WITH LETOURNEAU. it seems like a really cool subject, and maybe letourneau would have been good at teaching something else, or with a few more years under his belt; but for now, do not take functions of a complex variable with letourneau. i got an A in the class, so i say this without any spite: this was the worst-taught class i have ever taken. before i give my opinion on why the class did not suit me, i can offer 2 concrete reasons why it is a class that you should avoid at all costs:
1. the class does not cover all of the material that you need to learn in the subject, and the topics that were left out were the most important ones in the course (fourier, laplace, and schwarz-christoffel). the bottom line is, you will either need to go learn these topics on your own or take another class in complex analysis if you actually want to use any of the most important applications of the subject.
2. the professor will not match the amount of work you put in (unless, of course, you put in little to no work). letourneau was completely unprepared in lecture. he had very few office hours and frequently cancelled or rescheduled them. there were significant errors on every posted homework assignment, and they often went uncorrected until just 2 or 3 days before the assignment was due. there were frequent errors on the posted solutions. homeworks were not returned until over 2 weeks after they were handed in, and there were frequent grading errors. there were errors on the exams as well, and there were often crucial errors in the lecture notes.
on a more subjective note, i found it unacceptable the way he handled the lectures. for one, the lectures didn't prepare you for the exams. in fact, in lectures, he deliberately misled you about the exams when asked specific questions about their content. the exams were hard, but they would have been much more manageable with the exact same questions if we had been prepared in class for those problem types. even if the exams had been closely related to what was in the lecture notes, i still would have been very upset with the way the lectures were conducted. i'm sure there are plenty of professors who don't care enough to prepare thoroughly, but i would hope that they at least follow the book, or use the notes of a previous instructor. i don't know if it was foolishness, arrogance, or complete incompetence, but letourneau seemed to think he could grab a few theorems from book sections and then structure a 2.5 hour lecture around them, on the fly, from his own subject knowledge. he would often start an example, spend a few minutes struggling through it, and then say: "oh wait, this actually won't work," and then cross it all out and think of a new example. another laughable element was his "proof sketch" method, which consisted of a bunch of disjointed mathematical phrases, with inconsistent notation, scattered across 3 or 4 blackboards and connected by a tangle of long, squiggly arrows intended to reorganize his out-of-order thought train. it was as if we were the guinea pigs for a mock, live-action improv textbook that will never be written. in retrospect, i highly regret attending the lectures.
also, the ta (jeff) was even less prepared than the professor. i went to 2 of his office hours over the course of the semester, one at the beginning and one near the end. both times, every question that was asked, directly from the homework, struck jeff as the first time he had ever seen, heard, or considered it, and the students sat around while he tried (and often failed), for an inordinate amount of time, to figure out how to approach the problem. the only difference between the 2 sessions i attended was that the first one, which was the first or second meeting of the semester, was well-attended, whereas the second one was almost empty. the only other people there by the end of the semester were a few students who clearly went every week just to exchange answers and work through difficult problems together (without the ta's help).
this class was an abomination, and i'm horribly disappointed that columbia allows this kind of thing to happen. while it stood apart as the worst class i've ever taken, it wasn't alone in the realm of terribly conducted columbia math courses, and this pattern clearly reflects a lack of oversight on the department's part. at least when it comes to math, there seem to be no resources whatsoever devoted to ensuring teacher quality, and especially ta quality. it is understandable for an elite university to place greater importance on the research contributions of faculty than on their teaching contributions, but not all professors have to teach! isn't that what adjuncts and tenures are for? why do they give assistant professors and professorship fellows any classes at all? they've got plenty of veterans who've been doing this for ages, and plenty of money to fill in the gaps by bringing in adjuncts who can be hired specifically for their teaching ability. i'm not saying columbia should get rid of letourneau, because i'm sure whatever scholarly contributions he makes are worth his salary and office space, but that doesn't mean the university needs to plague students with his horrible teaching and neglectful class management. poor form, columbia, poor form.