MATH2621: Higher Complex Analysis

2 x 20% Class Tests
60% Final Exam

Prerequisite: MATH1231 or DPST1014 or MATH1241 or MATH1251 each with a mark of at least 70; Exclusion: MATH2069

I can only really criticise this course for its annoying timetabling, and that's being *really* nitpicky.

Ridiculously good course. The exams were fair and well structured (both the exams themselves and the assessment structure overall [see assessments]), the teaching couldn't be faulted and the content was brilliant. When you talk about maths clicking in a satisfying manner, this is definitely it. To find topics that get tied up as elegantly as the ones covered in this course has been somewhat rare so far, and has been much appreciated this term. Up there with one of the best courses I've taken, full stop; pick anything in the course and you could probably find at least three things great about it. Not much else to say, except just take this course if willing and able.

2 x 2hr lectures
1 x 1hr tutorial

2.5/5

Yes

Dr Arnaud Brothier

Yes

4.5/5

None

21T3

91 HD

- 2x class tests done during a lecture time, worth 20% of your mark
- 1x final exam, worth 60% of your course mark

Formally, one of
- MATH1231
- MATH1241
- MATH1251
- DPST1014
with at least a mark of 70.

Having MATH2111/multivariable calculus in your toolbox before this course also helps in some parts - specifically, the preliminaries of continuity, limits and differentiability of complex functions as well as contour integration will be familiar material if you’ve done 2111 before. This is optional though, and there’s a nice duality as well because doing 2621 before 2111 will make some parts of 2111 easier in turn.

An interesting course, but also a bit hard this term all things considered. I’ll reiterate what other reviews have already mentioned to say that the theorems and results in this course are as technical as they are surprising, so they take some time to sink in. If you’re an integration junkie, then you’ll probably love the end of this course where you learn some neat tools to tackle hard real integrals using complex methods. The difficulty of this course during the term was pretty tame but spiked significantly at exam time, with our final being a bit of a killer.

Why this course loses 1 point in rating is because it could be improved content-wise. In terms of how engaging it is, the parts of the course preceding contour integration are fairly stock-standard and, in my opinion, not too interesting as a whole. This is in large part because integration is vital to the derivations of many results in complex analysis, so you really have to know some theory of it before being able to fully appreciate them - without that, you’re just left hanging at times. With that in mind, it’s a bit of a shame then that with how fundamental integration is, it comes into the picture quite late in week 7. I definitely think time spent covering more integration at the expense of some topics in the first bit of the course (looking at you, fractional linear transformations) would be a worthwhile trade, since time during trimesters is quite precious. So much additional content that the course used to cover in semesters is now forgotten, which is a real shame - this has probably been the most affected by trimesters in that respect out of all of the courses I’ve done so far.

2x 2 hour lectures, 1x 1 hour tutorial

4/5 just because of that exam, but the content itself is non-trivial too

Yes, and I imagine in-person offerings would be the same.

Dr. Arnaud Brothier

A decent problem set was provided as well as a typed lecture notes document, on which the lecture slides seem to be based on. You get an okay selection of past tests and papers for assessment preparation, though most of the class tests in particular don’t have any solutions, so you’re on your own there.

4/5

None formally prescribed or used, but some recommended ones are
- Wunsch’s “Complex Variables with Applications”
- Needham’s “Visual Complex Analysis” for a visually/geometrically-motivated text
- Rudin’s “Real and Complex Analysis”, a classic text in analysis, but be warned that this book is quite high level

20T3

91 HD

Prerequisites:

Overall a pretty interesting course, but a lot of theorems to remember with very specific conditions. This course mostly involved conceptual questions, but not too much difficult proof writing. Remembering the specifics in complex analysis was the most difficult part of this course, and they tested that you knew them well. Both lecturers explained things well, but I found Prof. Cowling more engaging in some respects. The course was a lot more theory-focused, so I wouldn't recommend taking this course unless you enjoy going through and understanding why theorems work and how they can be used to simplify problems. Some of the integration techniques taught are really cool, but difficult to see without being prompted, and require a lot of working to show (one specific question in the final on a single integral took up a couple of pages, though it was broken into parts). A lot of the stuff taught in the course can be linked to the several variable calculus taught in MATH2111, which made understanding the content a lot easier.

1x 2hr, 1x 1hr Lecture, 1x 1hr Tutorial

3.5/5

Yes - screen and voice recorded, however the blackboard was used often.

Dr. Alessandro Ottazzi and Prof. Michael Cowling

Lecture slides and notes all posted online, and past final exams available. Tutorial problems with very very few answers (no working either).

3.5/5

Note: I don't use textbooks and can't comment on their usefulness.

2018 S2

91 HD

2 x 45 minute quizzes (each weighted 20%), final exam weighted 60%

The formal prerequisite is a mark of 70 in one of MATH1231/MATH1241/MATH1251. However, a "lecture 0" is provided as revision and is essentially sufficient as a basis for the course.

This course serves as compulsory for two of the primary mathematics majors, and one viable choice out of two for the statistics major (the other being MATH2221). For the most part it was brilliant; everything about the maths in this course was fun. (This is also what draws students majoring in statistics to this course over MATH2221.) It is the higher counterpart of MATH2521.

This course, much like the first semester courses, is a continuation of what's been taught in MATH1231/41/51. Simply put, the first year math courses teach the algebra of complex numbers, whereas this course teaches the calculus of complex numbers. Many proofs in this course are examinable, but have the luxury in that you can figure them out on the spot, so long as you know all the basic ingredients.

The lecturers are very funny and keep you engaged decently well. In particular, Dr Michael Cowling drops hints on what might be in the exam, based off previous years. It still ended up being a bit of a bomb though with more twisted questions this year, but for the most part it is fairly relaxed. (In fact, if the final exam didn't drop the bombs, the difficulty would've only been 1.5/5)

The course really depicts how different and surprisingly beautiful the adapting of calculus to complex numbers can be. Many things that hold for real analysis are broken when taken to complex numbers, but more powerful results are derived.

Note that this course is the expansion of the former course MATH2620 (3 UoC), and was first taught in 2014.

3 x 1 hour lectures, 1 hour tutorial

3/5

Yes, but in saying that you miss out on anything drawn on the blackboard

Dr Alessandro Ottazzi, Dr Michael Cowling

4/5

Nil

2017/2

90 HD