Fall 2026
Web page: https://www.math.utoronto.ca/ilia/Teaching/MAT354.2026/
Class Location & Time: Tue, 1 PM - 3 PM, IB220; Thu, 2 PM - 3 PM, IB379.
Tutorials: Wednesdays 12 PM - 1 PM, IB395. The first tutorial will be on Wednesday, September 16.
Instructor: Ilia Binder (ilia@math.toronto.edu).
Office Hours: Thursdays, 12 PM -1 PM, DH3040 and by appointment.
Announcement: There will be no office hours on Thursday, October 15.
Teaching Assistant: Tymofiy Sompura(tim.sompura@mail.utoronto.ca).
Office Hours: Wednesdays, 11 AM - 12 PM, DH2034. The first TA Office Hour will be on Wednesday, September 23.
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: MAT257Y5 or [(MAT137Y5 or MAT139H5 or MAT157Y5 or MAT159H5) and (MAT202H5 or MAT240H5 or MAT337H5) and (MAT232H5 or MAT233H5)]
Exclusion: MAT334H5 or MAT334H1 or MAT354H1 or MATC34H3 or MATD34H3
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 many dramatic consequences: the Cauchy representation formula, the Fundamental Theorem of Algebra, the 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.
September 8: An informal introduction. Complex numbers as vectors; conjugation, absolute value, the field axioms, and the matrix form. Ahlfors, pp. 1-14.
September 10: Polar form, rotation as multiplication, De Moivre's formula, and integer powers. Ahlfors, pp. 12-15.
September 15: Roots of unity and nth roots of complex numbers. Stereographic projection. Limits and continuity. Ahlfors, pp. 15-24.
September 17: Analytic functions: complex vs. real differentiability, and the Cauchy-Riemann equations. Ahlfors, pp. 24-26.
September 22: Differentiability from continuous partials, and the Looman-Menchoff theorem. Harmonic functions. Polynomial functions: roots, multiplicity, the Gauss-Lucas theorem, and Taylor polynomials. Ahlfors, pp. 26-29.
September 24: Proof of the Gauss-Lucas theorem. Rational functions: poles and zeros, behavior at infinity, and the singular part at infinity. Ahlfors, pp. 26-32.
September 29: Rational functions: the partial fraction decomposition. Convergence of sequences and series of complex numbers. Power series: the radius of convergence and the Cauchy-Hadamard formula. Ahlfors, pp. 33-39.
October 1: Analyticity of power series: termwise differentiation and Taylor coefficients. (An alternative proof via the Weierstrass double-series rearrangement.) The complex exponential, Euler's formula, and the logarithm: the multivalued logarithm, the principal value, and branches. Ahlfors, pp. 39-44.
October 6: Continuing the logarithm; trigonometric, hyperbolic, and inverse functions. Planar topology: open and closed sets, connectedness, and path-connectedness, through the topologist's sine curve example. Ahlfors, pp. 43-58.
October 8: Planar topology continued: path-connectedness of open connected sets, regions, compactness, and locally uniform convergence. Ahlfors, pp. 56-66.
October 15: Planar topology: discrete sets. Conformality. Fractional linear transformations: definition and derivative. Ahlfors, pp. 53, 58, 67-76.
October 20: Fractional linear transformations: the complex projective line, the matrix representation, and preservation of circles and lines. The cross ratio, symmetry, and automorphisms of the disc and half-plane. Classification of Moebius maps through elliptic maps; Apollonius and Steiner circles. Ahlfors, pp. 76-86.
October 22: Classification of Moebius maps continued: hyperbolic and loxodromic maps, reflections, and parabolic maps. Integrals of complex-valued functions: definition, basic properties, and the integral inequality. Ahlfors, pp. 86-88, 101-102.
November 3: Line and contour integrals: parametrization, orientation, and integrals around circles. Length, path independence, primitives, the rectangle criterion, and Green's theorem. The Cauchy-Goursat theorem, including finitely many exceptional points. Ahlfors, pp. 101-114.
November 5: The Cauchy integral: power-series expansion, remainder estimates, and derivatives. The winding number: definition and proof that it is an integer. Ahlfors, pp. 115, 121-122.
Homework.
The homework assignments will be posted here on Tuesdays. The assignments will be due at noon on the second Thursday after posting. The assignments should be submitted through Crowdmark. 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 September 24.
Assignment #4, due October 22.
Assignment #5, due November 12.
Midterm test. The in-person Midterm test will be held during the regular class meeting time on Tuesday, October 13. There will be four problems, covering the material of Chapters I and II of the textbook. During the test, you can use the course textbook and course notes. The ACORN Absence Declaration Tool cannot be used for the midterm exam.
The 2023 midterm test, posted for students to practice.
A compact handout containing the material of the course notes from September 8 to October 1 (figures omitted), for use during the midterm. Students will be provided with a printout of this handout at the test.
Aids allowed during the midterm. Students will be provided with printed copies of the first two chapters of Ahlfors and the relevant lecture notes. Students may also bring any additional printed or handwritten materials. The use of electronic devices of any kind is strictly prohibited.
Final exam. The oral final exam will be held on Tuesday, December 15, 2026, from 1 to 4 p.m. The location will be announced later. You will receive supporting material containing a list of problems on Tuesday, December 8, 2026, at 1 p.m., one week before the exam. You will have 96 hours to upload your solutions, which are due on Saturday, December 12, 2026, at 1 p.m. The exam will be conducted as a series of 10-minute in-person interviews, where each of you will present some of your solutions and answer additional questions related to the course.
Grading. Grades will be based on eight homework assignments (2% each), Midterm test (34%), and Final exam (50%). 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. Submissions made on the due day after the noon deadline are considered one day late. Submit requests for special consideration for late assignments or missed exams via e-mail within a week of the original due date. No make-up midterm tests or finals will be offered. 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. 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.
Academic Integrity.
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, and Code of Student Conduct.
Generative AI policy.
Students may use generative AI for learning, ideas, inspiration, or grammar checks, but must properly think and write all content themselves and not copy text from AI. No form of AI may be used as a reference. All writing and presentations should be properly referenced and logically or mathematically justified.