Winter 2021
Web page: http://www.math.toronto.edu/ilia/MAT354.2021/.
Class Location & Time: Tue, 11 AM - 1 PM; Thu, 12 PM - 1 PM; online (the link and connection instructions are available on Quercus).
Tutorials: Mondays 11 AM - 12 PM, online (the link and connection instructions are available on Quercus). The first tutorial will be on Monday, January 18.
Instructor: Ilia Binder (ilia@math.toronto.edu).
Office Hours: Fridays, 10.30 AM -11.30 AM, and by appointment.
Teaching Assistant: Ilia Kirillov (ilia.kirillov@mail.utoronto.ca).
Office Hours: By appointment.
Required Text: Lars V. Ahlfors,Complex Analysis.
The book is out of print but the coursepack is available at the University of Toronto Bookstore.
Prerequisites: (MAT137Y5 or MAT157Y5),(MAT202H5 or MAT240H5), and (MAT232H5, MAT233H5, or MAT257Y5).
Exclusion: MAT334H1, MAT334H5, MAT354H1, or MATC34H3.
Prerequisites will be checked, and students not meeting them will be removed from the course by the end of the second week of classes. If a student believes that they do have the necessary background material, and are able to prove it (e.g., has a transfer credit from a different university), then they should submit a 'Prerequisite/Corequisite Waiver Request Form' by email.
Course outline.
The course is a rigorous introduction to Complex Analysis, one of the most exciting fields of modern Mathematics. We will begin with a review of Complex numbers and their Geometric and Algebraic properties. After that, we will start investigating holomorphic functions, including polynomials, rational functions, and trigonometric functions. We will carefully discuss the differences between Real and Complex differentiation. Following that, we will take a Complex Analysis approach to line integration and derive the fundamental theorem of Complex Analysis, the Cauchy Theorem. This theorem has a number of dramatic consequences: the Cauchy representation fomula, Fundamental Theorem of Algebra, Maximum Modulus Principle, and many others. Developing the theory, we will study Residual Calculus and Harmonic functions. The culmination of the course will be proof of the celebrated Rieman mapping theorem, which asserts that any simply connected planar domains (i.e. "a domain without holes") which is not the whole plane can be bijectively mapped by a holomorphic map to the unit disk.
Topics covered in class.
January 12: An informal introduction. Complex numbers: geometric and algebraic properties. Ahlfors, pp. 1-17.
January 14: Stereographic projection. Complex analysis: limits, continuity, differentiability. Ahlfors, pp. 18-24.
January 19: Analytic functions. Cauchy-Riemann equation. Polynomial and rational functions. Ahlfors, pp. 24-32.
January 21: Convergence of complex sequences and series. Uniform convergence. Complex Power series: radius of convergence. Ahlfors, pp. 33-38.
Some notes on limsup: 1, 2.
January 26: Analyticity of power series. Stolz Theorem. Exponential and trigonometric functions. Ahlfors, pp. 38-47.
January 28: Planar topology: a crash course. Ahlfors, pp. 49-67.
February 2: The concept of conformality. Fractional Linear transformations, the Cross Ratio, Classification of Fractional Linear Transformations: non-parabolic case. Ahlfors, pp. 67-86.
February 4: Classification of Fractional Linear Transformations: parabolic case. Line integrals: definition and basic properties. Exactness. Ahlfors, pp. 87-88, 101-107.
February 9: Line integrals: exactness and Green Theorem. Cauchy-Goursat Theorem. Cauchy Integral. Ahlfors, pp. 107-114 and a note on Cauchy Integral.
February 11: The winding number. The Cauchy Integral formula and Taylor series representation. Ahlfors, pp. 114-121 and 179.
February 25: Consequences of the Cauchy Integral formula. Ahlfors, pp. 122-123 and 176-177.
March 2: Isolated singularities. The local mapping. Ahlfors, pp. 124-132.
March 4: The local mapping and the Maximum modulus principle. Schwarz Lemma. Ahlfors, pp. 133-135.
March 9:
Schwarz-Pick Theorem and non-Euclidean geometry. The general form of Cauchy Theorem.
March 11:
Calculus of Residues. General Argument Principle. Rouchet Theorem. Ahlfors, pp. 148-153. March 16:
Hurwitz Theorem. Computing Integrals using Residue Calculus. Ahlfors, pp. 154-158 and p.178. March 18:
Computing Integrals using Residue Calculus. Harmonic functions. Ahlfors, p. 159 and pp.162-163. March 23:
Poisson formula. Schwarz's theorem. Subharmonic functions.Maximum Principle. Harnack inequality and Principle. Ahlfors, pp. 164-171 and pp.241-247. March 25:
The Reflection Principle. Laurent series. Ahlfors, pp. 172-174 and pp.184-186. March 30:
Mittag-Leffler's Theorem. Infinite products. Weierstrass factorization. Ahlfors, pp. 187-196. April 1:
Canonical products. The Gamma function: definition. Ahlfors, pp. 196-199. April 6:
The Gamma function. Normal families. The Riemann mapping theorem. Ahlfors, pp. 199-206, 219-225, and 229-230. April 8:
The Riemann mapping theorem. Course review. Ahlfors, pp. 231-233. Homework. The homework assignments are posted here on Thursdays. The first assignment will be posted on January 14. The assignments will be due on the following Thursday, at noon. The assignments should be submitted through Quercus. To submit, you can scan or take a photo of your work (or write your work electronically). Please make sure that the images are clear and easy to read before you submit them. Assignment #1, due January 21.
Assignment #2, due January 28.
Assignment #3, due February 4.
Assignment #4, due February 11.
Midterm test.
The Midterm test will be held during the regular class meeting time on Tuesday, February 23. There will be four problems, covering all the material discussed in class so far. During the test, you can use the course textbook and course notes. You will have twenty minutes after the end of the midterm to upload your solutions. During the midterm, you should be connected to our regular Zoom session. Your camera should be on. Any noncompliance will result in zero credit. Final exam.
The exam will be a take-home exam. You will receive a list of problems on April 12, at 9 am. You will need to upload your solutions by 9 am on April 16. No late submissions will be accepted. The exam itself will be conducted as a series of 10-minute breakout room interviews, where each of you will present some of your solutions and answer additional questions related to the course. The link to the final exam will be available on Quercus.
Grading. Grades will be based on eight homework assignments (3% each), Midterm test (25%), and Final exam (45%). The remaining 6% will come from class participation: taking part in online discussions, answering pop-up quizes, and such. To get the participation marks, you will have to have your camera on during the class Zoom call. I will also occasionally assign bonus problems. Late work. Extensions for homework deadlines will be considered only for medical reasons. Late assignments will lose 20% per day. Submission on the day the homework is due but after the noon deadline is considered to be one day late. Special consideration for late assignments or missed exams must be submitted via e-mail within a week of the original due date. There will be no make-up midterm tests or final. Justifiable absences must be declared on ROSI, undocumented absences will result in zero credit. In the case of a justifiable absence, the weight of the submitted work will be adjusted proportionally. E-mail policy. Notice of video recording and sharing. Academic Integrity.
For those of you in the different time zones, there will be a 7-9am sitting of the midterm on February 23. Please email the instructor by Thursday, February 18, if you want to take this version.
Suggested preparation: all homework problems and exercises 3, 6, page 108; exercises 1-3, page 120 of Ahlfors. You do not need to turn them in.
You can also look at the Warm-up questions.
Suggested preparation: all homework problems, midterm preparation problems, and exercises 2-3, page 178; exercise 5, page 184; exercise 3, page 186; exercise 4, page 190; exercise 4, page 193; exercises 1-2, page 232 of Ahlfors. You do not need to turn them in.
Office hours on Friday, April 9: 10am - 12 pm.
E-mails must originate from a utoronto.ca address and contain the course code MAT354 in the subject line. Please include your full name and student number in your e-mail.
The lectures for this course will not be recorded, for privacy reasons.
Honesty and fairness are fundamental to the University of Toronto’s mission. Plagiarism is a form of academic fraud and is treated
very seriously. The work that you submit must be your own and cannot contain anyone elses work or ideas without proper
attribution. You are expected to read the handout How not to plagiarize (http://www.writing.utoronto.ca/advice/using-sources/how-not-to-plagiarize) and to be familiar with the Code of behaviour on academic matters, which is linked from the UTM calendar under
the link Codes and policies.