Fall 2023
Web page: http://www.math.toronto.edu/ilia/MAT354.2023/.
Class Location & Time: Tue, 11 AM - 1 PM, IB395; Thu, 12 PM - 1 PM, IB379.
Tutorials: Wednesdays 7 PM - 8 PM, IB379. The first tutorial will be on Wednesday, September 20.
Instructor: Ilia Binder (ilia@math.toronto.edu).
Office Hours: Thursdays, 10 AM -11 AM, DH3016 and by appointment.
Teaching Assistant: Ahmed Ellithy (ahmed.ellithy@mail.utoronto.ca).
Office Hours: Wednesday, 8 PM - 9PM, on Zoom. The first TA Office Hour will be on Wednesday, September 20.
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: MAT334H1 or MAT334H5 or MAT354H1 or MATC34H3 or MATD34H3
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 mterial, 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 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 7: An informal introduction.
September 12: Complex numbers: geometric and algebraic properties. Ahlfors, pp. 1-17.
September 14: Stereographic projection. Complex analysis: limits and continuity. Ahlfors, pp. 18-24.
September 19: Analytic functions. Cauchy-Riemann equation. Ahlfors, pp. 24-27.
September 21: Harmonic functions. Polynomial functions. Ahlfors, pp. 27-29.
September 26: Rational functions. Convergence of sequences and series of complex numbers and functions. Ahlfors, pp. 29-37.
September 28: Power series. Ahlfors, pp. 38-41.
October 3: Exponent, logarithm, and trigonometric functions. Ahlfors, pp. 42-48.
October 5: Planar Topology. Ahlfors, pp. 49-66.
October 6: Conformality. Ahlfors, pp. 67-75.
October 19: Fractional Linear transformations. The Cross Ratio. Ahlfors, pp. 76-80.
October 24: Classification of Fractional Linear transformations. The Line Integrals. Ahlfors, pp. 80-88, 101-104.
October 26: Length of a curve. Green Theorem. Path independence and the existence of a primitive. Ahlfors, pp. 105-108.
October 31: Cauchy-Goursat Theorem. Ahlfors, pp. 109-114.
November 2: Cauchy Integral. The winding number. Ahlfors, pp. 115-118 and a note on Cauchy Integral.
November 3: The Cauchy Integral formula and Taylor series representation. Consequences of the Cauchy Integral formula. Ahlfors, pp. 118-123 and p. 179.
November 7: Isolated singularities. Ahlfors, pp. 124-129.
November 9: Logarithmic derivative. Local Argument Principle. Rouche Theorem. Ahlfors, pp. 130-131 and p. 153.
November 10: The local mapping. The Maximum Modulus Theorem. Ahlfors, pp. 131 - 135.
November 14: Schwarz Lemma and non-Euclidean Geometry. Chains and Cycles Ahlfors, pp. 135 - 138.
November 16: More general form of Cauchy Theorem. Ahlfors, pp. 130-148.
November 17: The Laurent Series. Calculus of residues. Ahlfors, pp. 149 - 152 and pp.184-186.
November 23: Calculating residues. Hurwitz Theorem. Ahlfors, pp. 152 - 154 and p.178.
November 24: Applications of Residue Calculus. Ahlfors, pp. 154 - 159.
November 28: More applications of Residue Calculus. Normal families. Ahlfors, pp. 159 - 161 and pp. 219 - 227.
November 30: The Riemann mapping theorem. Ahlfors, pp. 229 - 232.
December 5: The course review.
Homework.
The homework assignments are posted here on Thursdays. The first assignment is posted on September 14. The assignments will be due on the following Thursday, at noon. 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 21.
Assignment #2, due September 28.
Assignment #3, due October 5. A misprint corrected on October 3.
Assignment #4, due November 2.
Assignment #5, due November 9. Misprints corrected on November 7 and 8.
Assignment #6, due November 16.
Assignment #7, due November 23. Misprint corrected on November 22.
Assignment #8, due November 30.
Midterm test.
The in-person Midterm test was held during the regular class meeting time on Tuesday, October 17. There will be four problems, covering the material of Chapters I and II of the testbook. During the test, you can use the course textbook and course notes. The ACORN Absence Declaration Tool cannot be used for the midtem exam. Final exam.
The oral exam will be conducted on December 14, 9 am - 12 pm, at IB380. You will receive supporting material containing a list of problems on December 7, at 9 am. You will need to upload your solutions by 9 am on December 11. No late submissions will be accepted. 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 (3% each), Midterm test (28%), and Final exam (48%). 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.
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.
Suggested preparation: all homework and midterm problems.
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.
Generative AI policy.
The knowing use of generative artificial intelligence tools, including ChatGPT and other AI writing and coding assistants, for the completion of, or to support the completion of, an examination, term test, assignment, or any other form of academic assessment, may be considered an academic offence in this course.