Teaching > MAT290H1F
MAT290H1F - Advanced Engineering Mathematics (Fall 2026)
Last updated: October 13, 2026
Calendar Description
Credit value: 0.50
Hours: 36.6L / 24.4T
An introduction to complex variables and ordinary differential equations. Topics include: Laplace transforms, ordinary higher-order linear differential equations with constant coefficients; transform methods; complex numbers and the complex plane; complex functions; limits and continuity; derivatives and integrals; analytic functions and the Cauchy-Riemann equations; power series as analytic functions; the logarithmic and exponential functions; Cauchy's integral theorem, Laurent series, residues, Cauchy's integral formula, and the Laplace transform as an analytic function. Examples are drawn from electrical systems.
Learning Objectives
By the end of the course, students should be able to:
- work confidently with complex numbers and elementary complex functions;
- understand ordinary differential equations, their solutions, and what they tell us about dynamical systems;
- use the Laplace transform to study differential equations and linear systems;
- understand complex functions and why differentiability in the complex plane has much more structure than in ordinary calculus;
- understand integration in the complex plane, discover the remarkable properties that make it so powerful, and see how it connects with ideas from vector calculus;
- use complex series and residue theory to evaluate integrals that seem out of reach by ordinary calculus.
Instructors
| Instructor | Office Hours | |
|---|---|---|
| Adrian Nachman | adrian.nachman (at) utoronto.ca | Posted on Quercus |
| Erfan Meskar | e.meskar (at) utoronto.ca | Posted on Quercus |
| Manfredi Maggiore | maggiore (at) ece.utoronto.ca | Posted on Quercus |
If you email an instructor about the course, please include MAT290 in the subject line.
Lectures
| Section | Instructor | Day, Time and Room |
|---|---|---|
| LEC101 | Erfan Meskar | Mon 5-6PM in SF1101 |
| Wed 3-4PM in SF1101 | ||
| Thu 5-6PM in SF1101 | ||
| LEC102 | Manfredi Maggiore | Wed 3-4PM in BA1180 |
| Thu 1-2PM in ES1050 (exception on Oct 22: AH400) | ||
| Fri 2-3PM in BA1130 | ||
| LEC103 | Adrian Nachman | Mon 9-10AM in BA1170 |
| Tue 9-10AM in BA1170 | ||
| Fri 9-10AM in BA1170 | ||
| LEC104 | Adrian Nachman | Mon 10-11AM in BA1170 |
| Tue 10-11AM in UC140 | ||
| Fri 10-11AM in RW110 |
The lectures present the central ideas and the most important or difficult parts of the course. The course notes and assigned reading provide the fuller development, additional examples, proofs, and supporting material.
Lecture recordings. Lecture recordings will not be provided. Students who miss a lecture are responsible for catching up using the course notes, assigned readings, and other materials posted on Quercus.
Quizzes
There are five quizzes during the term. Quizzes will be written during the Thursday 2-3PM tutorial period, in each student's regular Thursday tutorial room.
- One problem on each quiz will be taken from the set of assigned homework problems listed in the Course Schedule.
- The only aid allowed is a non-programmable calculator. Any other resources that you may require (e.g., a Table of Laplace Transforms) will be provided with the quiz.
- Before each quiz, we will post an announcement on Quercus specifying the material covered.
Policy on missed quizzes. There will be no make-up quizzes. For the first missed quiz covered by an approved petition, the quiz grade will be the average of the grades on the other quizzes completed. For each subsequent missed quiz covered by an approved petition, the weight of that quiz will be transferred to the final examination.
| Date | Time | Location |
|---|---|---|
| Oct 1 | 2-3PM | Tutorial room |
| Oct 15 | 2-3PM | Tutorial room |
| Nov 5 | 2-3PM | Tutorial room |
| Nov 19 | 2-3PM | Tutorial room |
| Dec 3 | 2-3PM | Tutorial room |
Quiz Regrade Requests
Regrade requests are intended for clear marking errors. Requests for additional partial credit based on disagreement with the marking scheme will not be considered. A regrade may result in your mark increasing, decreasing, or remaining unchanged.
If you submit a regrade request, provide a clear and detailed justification explaining why you believe a marking error occurred. Do not request a regrade merely in the hope of receiving a higher mark.
Regrade requests must be submitted through Quercus within three working days after quiz grades are released. Detailed instructions for submitting a request will be posted in a Quercus announcement when the time comes.
Tutorials
Students have two tutorial hours per week. Tutorials start on the week of September 21. Five of the Thursday tutorial periods will be used for quizzes, on the dates listed above.
| Section | Thursday tutorial | Monday/Tuesday tutorial |
|---|---|---|
| TUT101 | Thu, 2-3PM in WB119 | Tue, 10-11AM in ES B142 (except Tue, Sept 22, in MY330) |
| TUT102 | Thu, 2-3PM in GB303 | Tue, 10-11AM in BA1200 |
| TUT103 | Thu, 2-3PM in GB304 | Mon, 9-10AM in GB244 |
| TUT104 | Thu, 2-3PM in SU B120 | Mon, 9-10AM in GB303 |
| TUT105 | Thu, 2-3PM in HA403 | Mon, 1-2PM in GB244 |
| TUT106 | Thu, 2-3PM in HS610 | Mon, 1-2PM in BA1220 |
| TUT107 | Thu, 2-3PM in WB116 (except Thu, Sept 24 and Nov 5, in WB130) | Tue, 11AM-12PM in BA1210 |
| TUT108 | Thu, 2-3PM in GB248 | Tue, 11AM-12PM in BA1200 |
Tutorials are an essential part of the course. They are intended for problem solving, discussion of the assigned problems, and preparation for the quizzes. Students should work through the assigned problems before attending tutorial.
Course Materials
Textbook (required). Dennis G. Zill, Advanced Engineering Mathematics, 7th ed., Jones & Bartlett Learning.
Course notes. Unit 2, on ordinary differential equations, will follow a dedicated set of lecture notes posted on Quercus.
The lectures are not intended to reproduce every detail of the written material. Students are responsible for the assigned readings and problems, including material that may be treated only briefly in lecture.
Homework Assignments
Problem sets will be assigned each week but will not be graded. Working through the assigned problems before attending tutorial is essential for success in the course and for getting the most out of the tutorials. The problems are designed to develop fluency with the material and to prepare students for quizzes and the final examination.
Composition of Final Mark
| Assessment | Weight |
|---|---|
| 5 Quizzes | 50% |
| Final Exam | 50% |
Mandatory Final Examination: This course includes a mandatory final examination. All students are required to complete the final exam; there is no provision for an assessed grade in place of the examination. Students who are unable to write the exam for valid reasons must submit a petition, and if approved will be granted a deferred exam.
Minimum Final Exam Requirement: To earn a passing grade in this course, students must achieve a minimum grade of 40% on the final examination, regardless of their performance on other course components. Students who do not achieve at least 40% on the final exam will not pass the course, and a grade of 49% will be assigned, regardless of their overall course average.
Course Outline
The precise pace may vary during the term. The principal topics are:
- UNIT 1: Complex numbers and elementary complex functions
- Cartesian and polar representations; magnitude and argument
- Euler's formula; powers and roots
- Complex exponential, logarithm, trigonometric and related functions
- UNIT 2: Ordinary differential equations
- ODEs as dynamical models; order and classification
- State-space representation, states, outputs, and vector fields
- Initial value problems; state trajectories and orbits
- Autonomous ODEs, translation invariance, and equilibria
- Existence and uniqueness of solutions and their geometric interpretation
- Stability and asymptotic stability of equilibria; stability theorem for scalar ODEs
- Linear ODEs: existence and uniqueness; fundamental sets of solutions and structure of the general solution
- Constant-coefficient ODEs and the characteristic polynomial; construction of a fundamental set of solutions; stability theorem for constant-coefficient ODEs
- UNIT 3: The Laplace transform
- Definition, existence, and transform properties
- Transforms of derivatives and solution of initial value problems
- Partial-fraction methods and inverse transforms
- Time shifts, unit-step functions, convolution, and impulses
- Transfer functions and impulse responses of LTI systems
- UNIT 4: Complex functions and analyticity
- Topology of the complex plane; limits and continuity
- Complex differentiability, analytic functions, and the Cauchy-Riemann equations
- Harmonic functions and geometric consequences of analyticity
- UNIT 5: Contour integration
- Contours and contour integrals; basic properties and estimates
- Cauchy-Goursat theorem and deformation of contours
- Antiderivatives, path independence, and the Cauchy integral formula
- Consequences of the Cauchy integral formula
- UNIT 6: Complex series
- Complex series and power series
- Taylor and Laurent expansions
- UNIT 7: Residue theory
- Isolated singularities and residues
- Cauchy's residue theorem
- Evaluation of real integrals using residues
- The Laplace transform as an analytic function and inverse Laplace transformation
Academic Integrity
Students are expected to know and follow the University of Toronto's standards of academic integrity. Work submitted for assessment must comply with the course instructions and the University's Code of Behaviour on Academic Matters.
Accessibility and Accommodations
Students who require academic accommodations should register with Accessibility Services and follow the Faculty's procedures for requesting accommodations. Students are encouraged to make arrangements as early as possible.