MAT 334 – Complex Variables Summer 2021
Course Staff
Instructor:
Course Description
The complex numbers differ from the real numbers rather dramatically. The addition of solutions to the equation x2 + 1 = 0 to the real numbers seems like a small thing, but ends up having far reaching consequences.
To begin, we’ll introduce the complex numbers and various techniques for manipulating them alge- braically. This will not differ very much from working over R, except for the addition of a ”polar form” for complex numbers.
We’ll swiftly move to studying functions of complex variables. We’ll see that all polynomials have roots, that all differentiable functions are differentiable infinitely often, and that’s just a few ways in which complex functions are much nicer than real functions. On the other hand, we will see that complex logarithms are much more complicated, and introduce a “branching” phenomenon.
We’re going to look at: complex algebra, polynomials over C, the topology of C, functions and limits, sequences and series, exponents and logarithms, differentiation and holomorphic functions, the Cauchy-Riemann equations, power series and analytic functions, line integrals, Cauchy’s integral Theorem and formula, singularities and the Residue theorem, and contour integration.
Time permitting, we will also talk about the argument principle and max modulus principle, conformal mappings, and harmonic functions in depth.
Pre-requisites. MAT223H1/ MATA23H3/ MAT223H5/ MAT240H1/ MAT240H5 and MAT235Y1/ MAT235Y5/ (MATB41H3, MATB42H3)/ MAT237H1/ MAT237H5/ (MATB41H3, MATB42H3, MATB43H3)/ MAT257Y1.
Course Objectives. By the end of the course, we expect students to be able to:
Work comfortably with complex numbers and basic complex functions: polynomials, exponen-
tials, trigonometric functions, and logarithms Page 1
Name
Section
E-mail
Office Hours
Thad Janisse
LEC 0101
thad.janisse@mail.utoronto.ca
To be announced
Wenbo Li
LEC 5101
wenbomath.li@mail.utoronto.ca
To be announced
MAT 334 – Complex Variables Summer 2021
Be fluent in computing complex derivatives and line integrals
Know basic properties of complex functions and their consequences
Prove facts about complex numbers and complex functions
Contrast the differences between complex functions and real functions
Textbook. We do not have a required textbook for the course. We will work from two different sources:
Thad’s MAT334 Lecture Notes – these will be provided for free via Quercus. Complex Variables, 2nd Ed. by Fisher.
In addition, we will post weekly practice problems. These will include a number of problems and some further suggested problems from Fisher. It is strongly reccommended that you do enough of these practice problems to get comfortable with the material. There are going to be a lot of problems, so you are not expected to complete them all. However, you should try to choose problems to get good coverage of the material.
Course Website. The course will be run through Quercus. Both sections will share the same Quercus page.
Other Platforms. In addition to Quercus, we will have a Piazza for the class. You are encouraged to ask questions on Piazza if you need help, or to answer each others questions.
Our marking platform for this summer will be Crowdmark.
Contact – Lectures, Tutorials, Office Hours, etc.
Lectures. This course will have synchronous online lectures through Zoom. The Zoom links for each section will be posted on Quercus.
Lecture section 0101 will be recorded, and these recordings will typically be posted within 24 hours of the end of class.
Tutorials. We will not be using the scheduled tutorial times to run tutorials. Instead, we will schedule office hours with the TAs at times that are convenient for the class as a whole. At these office hours, you will be able to get help with the material and with the homework assessments.
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MAT 334 – Complex Variables Summer 2021
Technical Requirements. In order to participate in this course, students will be required to have: Reliable internet access. It is recommended that students have a high speed broadband connec-
tion (LAN, Cable, or DSL) with a minimum download speed of 5 Mbps. A computer satisfying the minimum technical requirements
Other recommended items include headphones, microphone, webcam, and a tablet or printer. (Please note if any of these are required for your class.)
If you are facing financial hardship and are unable to meet these requirements, you are encouraged to contact your college or divisional registrar (https://future.utoronto.ca/current-students/registrars/) to apply for an emergency bursary so that you can obtain the required items.
Office Hours. Please do not be hesitant to come ask us for help. The staff of MAT334 are available for extra help outside of class, during our scheduled office hours. These will be posted on Quercus.
Marking Scheme
Your final grade will be determined as follows:
Homework (Best 6 out of 8) Two Midterms:
Final Assesment
Course Grade
Homework
Starting in the third week of class, we will have weekly homework. These will be due Friday nights, at 11:59pm. The exact schedule of due dates is in the course schedule on the last page of this syllabus.
For the homeworks, you will be allowed to submit in groups of up to 3 people. It is expected that these homeworks will be completed collaboratively with your group, and that each group member contributes to and understands the submitted solutions.
30% 20% each 30% 100%
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MAT 334 – Complex Variables Summer 2021
Midterms
We will have two midterms. The date for the first midterm is during the midsummer exam break from June 17th to the 28th. The faculty will let us know the exact date and time in the first week or so of June. The second midterm will be held on Friday, July 30th. We will end out a survey early in the course to determine your availability, and will schedule the midterm with that in mind. More information regarding material covered will be posted to the course website closer to the date of the test.
Alternate sittings. We recognize that any choice of timeslot may conflict with other commitments you may have (such as sleep, for those in different timezones). We will offer two sittings of each midterm to account for this. They will not be the same test, but will be written to be comparable in coverage and difficulty.
Final Assessment
The final assessment will take place during the examination period in August, and will be 3 hours long, plus upload time. It will cover all the material presented in lectures (unless explicitly announced otherwise on Quercus). The date, time, and location of the exam will be arranged by the Faculty of Arts and Science, and posted once it has been set.
The final assessment period is August 18th – 30th. The Faculty will set the assessment date whenever they please, so make sure you’re available to write the final!
Missed Work
As flexibility for missed or late homework has been built into the marking scheme, late and missed homework will not be accepted for any reason.
Please note that Verification of Illness forms (also known as a “doctor’s note”) are temporarily not required. Students who are absent from class for any reason (e.g., COVID, cold, flu and other illness or injury, family situation) and who require consideration for missed academic work should report their absence through the online absence declaration. The declaration is available on ACORN under the Profile and Settings menu.
If you miss a midterm or the final assessment, then you must inform your course Instructor within 72
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MAT 334 – Complex Variables Summer 2021
hours of the test. No exceptions. If your request is approved, you may receive an accommodation in the form of an oral exam, written make-up test, or a re-weighting of your assessments.
Help
If you need help there are many resources available to you. Please come and ask us for help as soon as you need it. Try not to fall behind. We will have office hours, which are posted on the course page, and at the top of this handout. If we find any useful resources on the internet (such as video tutorials, problem banks, etc.), we will share them on the course Blackboard.
Email. Feel free to contact us by email if you have any questions about the course. We try to respond within 48 hours. That said, there are a couple of ground rules you should observe:
Any emails related to this course must have MAT334 in the subject line.
We will only correspond via utoronto email addresses. As such, if you send an email from your
gmail account, or other email provider, we will not respond.
If your question can easily be answered by refering to the syllabus, or to an announcement on Quercus, we will probably not respond.
As in all things, be respectful.
If you need to arrange something with us, such as an appointment if you can’t make our office hours,
please do so via email.
Accessibility Servies
The University provides academic accommodations for students with disabilities in accordance with the terms of the Ontario Human Rights Code. This occurs through a collaborative process that acknowledges a collective obligation to develop an accessible learning environment that both meets the needs of students and preserves the essential academic requirements of the University’s courses and programs.
Students with diverse learning styles and needs are welcome in this course. If you have a disability that may require accommodations, please feel free to approach your Course Instructor and/or the Accessibility Services office as soon as possible. The sooner you let us know your needs the quicker we can assist you in achieving your learning goals in this course.
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MAT 334 – Complex Variables Summer 2021
Link to Accessibility Services website.
Academic Integrity
All suspected cases of academic dishonesty will be investigated following procedures outlined in the Code of Behaviour on Academic Matters. If you have questions or concerns about what constitutes appropriate academic behaviour or appropriate research and citation methods, please reach out to your Course Instructor. Note that you are expected to seek out additional information on academic integrity from me or from other institutional resources (for example, the University of Toronto website on Academic Integrity http://academicintegrity.utoronto.ca/).
Below are some examples of academic misconduct:
Submitting an assignment for which you did not contribute
Communicating with anyone other than course staff about the content of a midterm or the final assessment during the assessment
Receiving aid of any kind in solving midterm or final assessment questions
Using any unauthorized aid on an assessment
Working with anyone outside your group on a homework assignment
This list is not exhaustive. It is your responsibility to understand what constitutes appropriate aca- demic behaviour.
A special note on group work: In submitting a group assignment, each member of the group certifies that the work submitted is their own. An offense by one member of the group in submitting homework is almost always an offense by the group as a whole.
Copyright
This course will be recorded on video and will be available to students in the course for viewing remotely and after each session.
Course videos and materials belong to your instructor, the University, and/or other sources depending on the specific facts of each situation and are protected by copyright. Do not download, copy, or share any course or student materials or videos without the explicit permission of the instructor.
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MAT 334 – Complex Variables Summer 2021
For questions about the recording and use of videos in which you appear, please contact your instructor.
Equity, Diversity, and Inclusion
The University of Toronto is committed to equity, human rights and respect for diversity. All members of the learning environment in this course should strive to create an atmosphere of mutual respect where all members of our community can express themselves, engage with each other, and respect one another’s differences. U of T does not condone discrimination or harassment against any persons or communities.
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MAT 334 – Complex Variables Summer 2021
Course Outline
The following is a rough outline of the material which will be covered.
Date
Notes
Fisher
Topics
Info
May 3 – 7
1.1 – 1.4
1.1
Complex Plane, Modulus and Argument, Conjugation, Vectors, Polar Form, Complex exponenial, de Moivre’s Theorem
May 10 – 14
1.5, 1.6, 1.7.1
1.2, 1.4
nth roots, geometry, functions of complex variables
Tutorials start
May 17 – 21
1.7.2, 1.7.3, 2.1
1.4, 1.5
Logarithms, Branches, Complex Powers, Trig functions, Limits
Homework 1
May 24 – 28
2.2 – 2.4
1.3, 2.1
Continuity, Topology and differentiation (Cauchy Riemann, analyticity)
Homework 2
May 31 – June 4
2.5, 3.1, 3.2.1
1.6, 2.1, 2.3
Harmonic Functions, Review of line integrals and Green’s theorem, algebra of curves, esti- mation of integrals, primitives, Cauchy’s In- tegral Theorem.
Homework 3
June 7 – 11
3.2.2, 3.3
1.6, 2.3 – 2.5
Path independence, an integation roadmap.
Homework 4
June 17 – 28
Study Break
Midterm 1
July 5 – 9
3.4.1, 3.4.2
2.3 – 2.4
Cauchy integral formula, Deformation of curves.
July 12 – 16
3.5, 4.1, 4.2
2.2, 2.5, 2.6
Consequences of CIF. Power series.
Homework 5
July 19 – 23
4.3 – 4.5
2.2, 2.5, 2.6
Zeroes and singularities in depth, Laurent se- ries.
Homework 6
July 26 – 30
4.5, 4.6
2.5, 2.6
Laurent series, Residue theorem.
Midterm 2
Aug 2 – 6
4.6, 4.7
2.5, 2.6, 3.3
Residue theorem continued. Contour inte- gration.
Homework 7
Aug 10 – 14
Catchup or more fun topics! Review.
Homework 8
Aug 18 – 30
Final assessment
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