Back to Introduction to Complex Analysis

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1,040 ratings

This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. We will start by introducing the complex plane, along with the algebra and geometry of complex numbers, and then we will make our way via differentiation, integration, complex dynamics, power series representation and Laurent series into territories at the edge of what is known today. Each module consists of five video lectures with embedded quizzes, followed by an electronically graded homework assignment. Additionally, modules 1, 3, and 5 also contain a peer assessment.
The homework assignments will require time to think through and practice the concepts discussed in the lectures. In fact, a significant amount of your learning will happen while completing the homework assignments. These assignments are not meant to be completed quickly; rather you'll need paper and pen with you to work through the questions. In total, we expect that the course will take 6-12 hours of work per module, depending on your background....

RK

Apr 5, 2018

The lectures were very easy to follow and the exercises fitted these lectures well. This course was not always very rigorous, but a great introduction to complex analysis nevertheless. Thank you!

RK

Aug 9, 2022

Very useful course that can be followed regardless of your mathematical background - whether you have even heard of imaginary numbers or only have done pre-calculus work. A gem of a course.

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By Supakorn R

•Sep 11, 2017

I know I'm giving this 3 stars, but I wanted to point out that this is not a rigorous introduction to complex variables. For those getting started - it's an excellent introduction, but for those familiar with proofs and prefer more, resources like University of Alberta's Math 411 - Honors Complex Variables at https://www.math.ualberta.ca/~bowman/m411/m411.pdf is a much more proof-based. I hope (and I really do hope) that someone will offer a honours-level (which rigorous proofs and reasoning) on Coursera one day, and I would like to take that course.

By Hrvoje B

•Mar 23, 2018

The course covers beautiful corner of mathematics and is a wonderful introduction to the topic of Complex Analysis. I would highly recommend this course for undergraduate students. In my opinion maybe it`s too tough for highschool graduates.

Material was well motivated and clearly presented. The quizes and homework assignments generally may took longer than the stated time, but that`s just fine. Feedback on quizzes was excellent and useful.

In the end, Prof. Petra Bonfert-Taylor, does an excellent job at explaining concepts simply. The main reason for that are her extraordinary pedagogic skills. She made a well organized and concise course. I hope there will be an opportunity to take another course with her again Overall, I`m very satisfied and grateful!

By James B

•May 15, 2019

Lots of neat math! It's challenging and requires a substantial time commitment. Though in CS, my love of math was re-awakened. If you are in CS and only want to know the basics of how to rotate objects with complex numbers and polar coordinates, then this course is way more than you need. If you enjoy math, have had little or no complex analysis and are curious about what's up with the complex plane, I recommend this course.

By Rens K

•Apr 6, 2018

The lectures were very easy to follow and the exercises fitted these lectures well. This course was not always very rigorous, but a great introduction to complex analysis nevertheless. Thank you!

By Natasha S

•Jun 25, 2018

The prof makes it easy to understand yet fascinating. I enjoyed video checkpoints, quizzes and peer reviewed assignments. This course encourages you to think and discover new things.

By Evgeniy K

•Feb 21, 2021

The course is not very in-depth, but it helps to recall the Complex Analysis if you have studied it before. To the disadvantages of the course, I would attribute the fact that the Laurent series and singular points of analytic functions are not considered in sufficient detail. This is a very difficult topic. Series expansions are mainly considered for rational functions, and other cases are considered very briefly. As a result, by the final exam, I felt rather unconfident with the Laurent series. Also, it seems like the author intentionally avoids the use of some well-established terms, such as holomorphic and meromorphic functions, and I don’t think it is a good decision. Many concepts are intuitive and seem clear during the lectures, but are difficult when it comes time to quizzes. For me personally, the most interesting was the second week devoted to fractals. These topics are often not covered in the standard course of Complex Analysis, and almost all this information was new to me. The elements of number theory (Riemann's hypothesis and Euler's results) also made me happy. The overall impression of the course is positive, but some hard topics are unjustifiably superficially presented. It might be better to extend the course for one or two weeks.

By Arpon P

•Feb 25, 2017

This course is fairly appropriate for those who have completed high school eduaction and opt to pursue higher studies in the field of science and engineering. This is an introductory course on complex analysis and does not cover advanced topics like zeta function, Manderbolt set etc in great detail. If you are doing major on Mathematics and looking for a graduate level course, this is not for you.This course offers an introduction to complex numbers, then discusses briefly on function iteration. In this course, you will learn on complex functions, complex derivative and integration, Cauchy-Riemann equation, Residues. The topics required to pursue undergraduate science education are covered nicely in this course.It is essential to do the quizzes and assignments to get hold of this topic. I thought the exercises in the course are not enough. I would recommend to do more exercises from any standard textbook on Complex Analysis. This course does not specifically follow any textbook. You will get some suggestions for textbook in the discussion forum.To get a deeper understanding, it is a great idea to follow the discussion forum. If you have any difficulty in understanding any topic, you can share in the discussion forum. Also following other people's questions helps you to develop an insight on this topic. Discussion forum is an integral part of this course. Use it wisely.To conclude, any high school graduate can take the course without any difficulty.

By Gregory V

•Sep 22, 2020

This is by far the best online course I have ever taken. Many years ago I took "Functions of a Complex Variable" as an undergraduate in Engineering. Now as a practicing applied mathematician in the quantitative finance space, I take a course from time to time in an area that interests me. Not only did I learn something new about complex analysis in this course, but also I actually enjoyed the experience and was sorry to get to the end! Professor B-T explains each concept very clearly and works through illustrative examples step by step. She also has an engaging manner and a very pleasant voice with only the smallest trace of a pan-European accent. Anyone wanting a solid introduction to complex analysis should take this course. For those who are familiar with complex analysis and its applications, the course is an entertaining experience which ties lots of things together from many specialized fields. Those with knowledge of fluid mechanics and classical electrodynamics will recognize where this foundational material can lead. Professor B-T has set a standard against which all MOOCs can be judged.

By Assaf B

•Feb 22, 2018

This is a beautiful area, delivered in a brilliant manner here. It succeeds in delivering most of the material in a near-uncompromising way, even though it also targets those who remember single variable calculus and nothing more.

This does not mean that the course is easy, especially if you just have the basic prerequisites, or are separated years from the last time you learned math like myself, when trying to complete tasks, you will find yourself getting back to the material and reviewing slides and videos again, and this challenge really adds to the understanding of the material. Also, as an added bonus, if you are distant many years from the time you last learnt mathematics, it will remove a little of the rust.

Brilliantly done.

By Deleted A

•Mar 21, 2017

With this wonderful complex analysis course under your belt you will be ready for the joys of Digital Signal Processing, solving Partial Differential Equations and Quantum Mechanics.

By Ryan L

•Aug 3, 2021

This course contains an overview of manipulation in complex numbers at a second-year undergraduate level. Various techniques, such as visualizing mappings, solving integrals, and manipulating series in complex numbers are introduced. The course also goes through some of the application of a few crucial theorems, such as Cauchy-Riemann Equations, Cauchy’s Theorem and Cauchy Integral Formula.

8 weeks ago, I started this course with a knowledge of some very basic real analysis (the epsilon-delta definition of convergence, differentiation, and Riemann integration). As a major in mathematics, I found this course refreshing and a pleasure to take. Dr Taylor’s voice was engaging, and the lecture notes were self-contained and well-explained.

One remarkable feature that I want to highlight is the coursework component. It consisted of graded quizzes with multiple choices, tick boxes, and some short fill in the blanks. There were also graded peer assignments once per 2 weeks, this made sure that our answers produced are readable by other people. I found the difficulty of the coursework very appropriate: it was deliberately not straightforward, and you must be careful while applying different theorems and concepts that were taught in the lessons. The peer-graded assignment also emphasized communicating mathematics carefully, with clearly given guidance and appropriate suggestions from other learners, it was a very well thought out part of the course.

The course was very enjoyable on its own. But certain features could be more well-polished. Overall, this feels like a course for applied mathematicians. Heavy focus is put on and applying the results. However, with a little bit of generalization and more discussions of proofs and their logic behind them, it can benefit more pure mathematicians that are interested in the subject. Similarly, I think the assignments (especially the peer-graded ones) can be more proof-centric, with more videos explaining the sketch of proof in more detail. Therefore, the course can aim for a balance between applied and pure content, which would in turn benefit more learners.

Moreover, even with the addition of Residue Calculus, I still think that more content can be added to the course. It currently contains topics such as Solving Real-valued Integrals, Understanding the Mandelbrot set, etc. But more applied topics, such as introducing the idea of the harmonic equation in liquid flow or heat flow using polar coordinate might be more inspiring and make the course more fulfilling.

Lastly, I found Week 5 (The first week that introduces complex integration) particularly challenging. I think I spent twice the time revising it compared to other units in this course. Therefore I think it will be better to add new content and let the whole course be 12-week long or so.

Overall, I enjoy this course a lot, and I will recommend anyone who has an interest in Complex numbers and have 1-2 years of experience in university STEM subjects.

By Marcin B

•Apr 7, 2020

Petra Bonfert-Taylor did a great job constructing this course. Her video lectures are clear and easy to follow. I took this course because I had lousy instructors at the university and this course filled the gap. There are a lot of examples and exercises where you actually have to do a lot of counting and playing with mathematical expressions. Much less focus is put on proofs, although some simple proofs are presented and sometimes the instructor outlines the main ideas behind other formal proofs. It may be beneficial to follow a standard textbook on complex analysis along with the online course. Last thing, the course is not easy. Unless you already have substantial experience with complex analysis prepare to study intensively. My experience is that it is important to re-do on your own all material covered during lectures. And indeed, without a good background on calculus and some general mathematical fluency, this course would be hard to complete.

By Carlene S d F

•May 26, 2020

This is a really nice course if you want to revise your knowledge in complex analysis. That was my case. I think for non-mathematicians or physicists is a nice course without too rigorous proofs and details, but if you want to spread your wings a nice book about the subject will be very suitable to complement the lectures and the PDF materials. For me, that took a complex analysis course at university a few years ago, it was nice to remember some concepts and finally learn a little about how to implement residues theorem to solve improper integrals, crucial for my field of expertise. The topics about complex dynamics and conformal mappings (this last one hardly properly given in a course for physicists) are also covered in a nice way. I definitely recommended this course!

By Andrew M

•Feb 12, 2021

This is one of the best online classes I've ever taken. Make no mistake, this course is rigorous and can be challenging, but the assignment questions never seemed absurdly unfair or unrelated to the topic(s) being covered. Prof. Bonfert-Taylor does an excellent job of both explaining AND providing concrete examples of the formulas and theorems. I actually keep coming back to some of the lectures from time to time, and learn something new or understand something in a new light each and every time.

By Colin Y

•Dec 22, 2018

The course covers quite a wide range of topics but are very approachable. Nice lecturer, and very clear pdf slides available for read at any time. One can follow the lecture contents either by watching the videos or by walking through the reading resources provided, either way serves my need. The assessment questions are well-designed to test the learner's understandings. Good course!

By Victor P

•Sep 21, 2016

This course has been very exciting and powerfull. The lectures are very clear and the professor use many didactic tools for improving our learning. The content of this course is completed with some other advanced topics in mathematics (for example: topology), and the course is well enriched. I recommend this course for all learners who want to learn something of advanced mathematics.

By Samopriya B

•Jun 12, 2016

Wonderful. I have been waiting for this course for some time. It begins somewhat basic, and progresses really fast, to cover some deep topics in complex analysis theory. Also, the instructor, Prof. Petra Bonfert-Taylor, does a very good job at explaining concepts simply, by not being too pedantic (e.g., with the topological preliminaries). Overall, really satisfied and grateful!

By 魏寅生

•May 3, 2020

This course is great! I tried to read books on complex analysis but failed on understanding that many definitions and proofs. Prof. Bonfert approached this subject from its applications, and put little emphasis on proofs, making it easy for us non-mathematicians to understand fancy subjects like Mandelbrot set. Thanks for providing such a great course.

By Harish M

•Apr 4, 2020

The course is excellent and Dr. Petra Bonfert-Taylor is an excellent instructor. This course is not rigorous but it touches on a lot of topics which are quintessential in Physics and to a smaller extent, Engineering. I would also recommend this to anyone who is interested in Math or is pursuing an undergraduate degree in Math.

By Arjun D

•Aug 9, 2020

Great course. One thing I would suggest is to have a list of prerequisites on the course page so people have an idea of what they should already know before joining this course. Other than that, the videos are of good quality, the homework is challenging yet stable, and the quizzes are at the right difficulty.

By Yep Y m

•Jan 24, 2021

Derivations are generally clear and easy to follow, some are abit less intuitive but Dr Petra Bonfert-Taylor makes the effort to explain it in a way that is easy for me to understand.

By Penkun H

•Apr 2, 2018

It's the first time I join a class like this, I think it's a fantastic experience for me, Although I was busy after I came back to school, it's the course itself that encourage me to finish all kinds of work in my spare time, and I do learn something about complex analysis, which I will learn in detail in my follow college years. Thanks to the course, I could have a overview about my class. Last but not least, I think the way that let us check homework for each other is really nice, I spend a few hours to finish the first one because I really want to leave an good impression for my classmates, who are really nice because they gave me warm and nice judgement. I think I would highly recommend this course for whoever want to have a overview and application of complex analysis.

By James C

•Jul 30, 2021

Very high quality course. Uncompromising and wide-ranging. Very suitable for those who already know complex analysis and are revising for e.g. postgraduate study. As an intro to complex analysis, some background in multivariate calculus (including path integrals) is needed, as is some background in university level real analysis. If you've done complex analysis before, the final exam will be straightforward, if you're learning it, it requires more thought and the high pass grade threshold insists on that. The assesments require real engagement and this isn't a course that you can approach lightly but as a summary review of the subject, you couldn't ask for more.

By Gary U

•Oct 2, 2017

Excellent course for an introduction to complex analysis. Beginning from basic concepts, the instructor develops the basis of complex analysis. The first two weeks having to do with Julia sets and Mandelbrot sets are colorful lessons, the real analysis starts on week 3. The instructor is excellent, providing step by step instructions in the presentation and also some proofs of the theorems. Week 4 lessons 4 and 5 deal with the Riemann Zeta function and the Prime Number Theorem, very interesting and addition to the course. There is much more that could be added, perhaps a further course can be developed for the MOOC.

By Meir S

•Jan 17, 2017

Excellent entry into the world of complex analysis. Dr. Petra Bonfert-Taylor carefully constructs the foundations for complex functions while constantly providing enriching examples. Complications in advanced proofs are sometimes obviated (she will mention what she chooses to skip). If you only need to learn to use complex analysis Dr. Petra Bonfert-Taylor provides more than enough guidance. Since I enjoy understanding mathematics from the axioms up, I found myself turning to outside resources to fill in the nuanced complications. Consider doing the same if you are like me.