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Back to Introduction to Complex Analysis

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1,006 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 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 shouvik d

•Oct 2, 2016

A great course for beginners and engineers looking to just apply concepts of Complex Analysis without going too much into mathematical rigour. This course does not require any knowledge of "Real Analysis" or "Multi-variable/Vector" calculus as a prerequisite which is one of it's greatest strengths.

This has especially helped in my preparation for a graduate-entrance exam called GATE that we have here in India.

Although, I believe the course could include some more number of problems and solutions.

By Daniel S

•Sep 26, 2016

This is an outstanding course; one of the gems of the Coursera platform. It is only the launching point for my study of and fascination with complex analysis. I'd say the course covers about three-fifths of the content for an engineering mathematics treatment of functions of complex variables. I'm a student of fluid mechanics and my ignorance of advanced concepts in complex analysis has kept me from mastering some of the advanced topics in FM. I'm well on my way to go beyond those barriers, now.

By Afonso M B C

•Sep 22, 2020

The way the contents are presented make them easy to follow along with even if you haven't worked with complex analysis at all before. The assignments (quizzes) and peer-graded assignments are a great way to apply and practice what we learned throughout the lectures. The "bonus" content (like the Riemann Hypothesis and interesting applications of theorems) make the content much more interesting because we can see how it's applied to arrive at important results.

By Sam G

•Jun 25, 2017

This course was incredible. The teacher was made the experience understandable and fun giving light onto a new world of mathematics that I had only dreamed of investigating.

The lectures each week were insightful and interesting for all skill levels. (Although it may have been easier had I not had external exams to do)

I would highly recommend this course which spans from A-level to Year 3 university studies as it is presented in a clever and accessible way.

By Peter K

•Jan 12, 2019

Excellent pacing of materials. The assignments strike that rare balance of enforcing key concepts that were discussed during that week's lectures and not being too difficult. And more than that, in my opinion, you can certainly tell that the instructor really cares about teaching in general. I took this class on a whim because I saw it was available and thought "eh, why not?", and I am delighted to say that this feels stumbling on some hidden treasure.

By Michael L

•Apr 12, 2021

Fantastic course. After my lifelong fascination with real analysis and the real numbers, it's been literally and figuratively expansive (even mind expanding) to move into the complex plane. Dr. Bonfert-Taylor motivates and explains everything clearly. The examples are simple enough to understand yet nontrivial enough to be interesting, and are complemented with intuitive drawings. The lectures are invariably fascinating throughout. I thank her.

By Jerry H

•Aug 3, 2018

This is an excellent and challenging course! A great introduction to complex analysis. I feel some materials covered in the course might lack rigorous derivations, but the proofs are really intuitive. I hope Professor can make more courses like this one on Coursera about other topics in mathematics like linear algebra, number theory, abstract algebra, topology, differential geometry and functional analysis etc.

By Sandeep N

•Jan 16, 2017

Very nice course. It was very informative. Even though I did it in Audit mode, I am going to recommend my PhD students to do the course for improving themselves as physicists and computer science by knowing what actually goes behind the scene mathematically. I will recommend the course for all students of electronics (with strong desire to understand the mathematical basis) in general.

By Raymond M

•Aug 21, 2019

Enjoyable. There seemed to be a few math rendering errors in some of the assignments. The homework assignments involving peer review were good.

You've got to appreciate how the originally recorded class needed more, but then the author added more! The lesson 7, Calculus of Residues was an appropriate addition, I didn't mind it being last, but it did not go into Branch Cuts.

By Ayush T

•Jan 26, 2019

This is really good course for complex analysis. Things have discussed in details and assignents are also very good. It covers most of the topics which I felt were relevent for my work and studies. I wish it had Reimannian Optimization, but even without it the course is really well designed as well as well taught.

By Brahadeesh S

•Sep 5, 2017

The course content as well as the presentation of the material by the instructor are both wonderful. The quizzes and peer-graded assignments are optimally designed. Although there are few proofs of theorems, the applications and examples are in sufficient number that the understanding of the theorems is obtained.

By Henry P k

•Sep 22, 2017

This was a well thought out and taught course in complex analysis. A good amount of detail and examples were nicely done. A particularly attractive feature of the course is that all slides were available in a downloadable .pdf format. The instructor was quite good and presented well prepared lectures.

By JUAN F P S

•Jan 20, 2021

A tight course that adequately covers all the introductory topics, where they take you by the hand properly for each topic. There aren't proofs of every thing seen, but there is always an attempt to promote or give a hint. It is recommended to use a book and additional content to consolidate concepts.

By Phillip M

•Jul 13, 2020

Prof Bonfert-Taylor is an excellent teacher. The pacing and examples she used help to build my knowledge and confidence in tackling the material.

I found the course demanding but very interesting.Don't underestimate the amount of time you'll need to finish the weekly assignments; they're not that easy.

By Charles K

•Jan 16, 2017

While there are a plethora of MOOCs around aimed at first-year maths, Prof Bonfert-Taylor should be congratulated for being one of the few to bring a course on higher mathematics to the masses. This is a particularly valuable effort that's accessible to anyone with a grasp of first-year calculus.

By Nora S I

•Nov 10, 2016

Pedagogically insurmountable! I recommend it! I am a PhD student writing the thesis, and need to refresh old subjects. Well, this was better than any math course I took in the University. The professor serves the aim to teach, and not to feed her/his ego, really next level education!! Loved it!

By Matt

•Feb 28, 2021

I found this to be an excellent course. The teaching style used by Prof. Bonfert-Taylor is very effective in my opinion. Examples and proofs were worked in great detail. Exercises were challenging and represented a blend of theory and application. Thank you so much for providing this course!

By George B

•Aug 16, 2016

Very clear and well paced class. I like the early introduction to Julia sets to introduce recent mysteries of in the subject before teaching the classical results. The professor is a very clear speaker and was generous enough with her time to answer some of the forum questions.

By Sushma S

•May 19, 2020

This course is absolutely amazing. I am very happy to complete it and gain so much knowledge during this course. Highly recommended to all the students and teachers to attempt the course. Its very beneficial for future also.

Thank you so much to whole team of Cousera !!!

By Fathima S

•Oct 28, 2018

This is a great course for anyone who wants to learn the fundamentals of Complex Analysis.

The instructor has also included a few applications that us go "Wow!!"

Assignments are well-thought and they make sure that the key-ideas are comprehensively gathered by the learner.

By Shri H P

•Jul 30, 2020

This course was really awesome. Especially Course instructor’s way of teaching was nice. She explained everything so clearly. Even the assignments are interesting and makes us to think a lot. Thank you ma’am and the coursera team for making this learning amazing.

By Ken L

•Feb 4, 2021

This is a tough course, but extremely rewarding. I feel the need of more practice and will continue to study complex analysis. Everything very clearly presented, good slides, demanding tests and designed to stimulate mathematical thinking, not just rote learning.

By Jorge P L

•Aug 3, 2017

Very good Course... Given the complexity and the abstraction of complex analysis I was surprised to see I could follow perfectly well the lectures and actually learn..

Petra did a very good job explaining every step in the learning path.

Contratulations Petra!

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