Final exam

  • The final exam is scheduled on Tuesday April 21, starting at 2pm, in the Currie Gym.
  • It is a closed book, closed note exam. No calculators will be allowed or needed.
  • There will be 6 questions, with 1–2 questions corresponding to the midterm material, 1 question on computing integrals, and 3–4 questions on the rest.
  • Here are some practice problems.
  • Please also review the past assignments: Solutions posted below are provided by Dylan Cant. Thank you Dylan!
  • Scroll down to find practice problems and other material related to the midterm.
  • Instructor's office hours for the week of Apr 20 are: M 10:00–12:00, or by appointment.


  • Assignment 1 [tex] due Wednesday February 18 (Solutions)
  • Assignment 2 [tex] due Wednesday March 18 (Solutions)
  • Assignment 3 [tex] due Wednesday April 1 (Solutions)
  • Assignment 4 [tex] not to be handed in (Solutions)

    Lecture notes

  • Complex numbers
  • Complex differentiability
  • Power series
  • Elementary functions
  • Fundamental theorems (last updated on Mar 30)
  • Isolated singularities (last updated on Apr 16)
  • Class schedule

  • WF 11:35–12:55, Arts building 145 (+ lecture on Apr 14)

    Date Topics
    W 1/7 Historical introduction to complex numbers
    F 1/9 Axioms for complex numbers. Vector model.
    W 1/14 Matrix model. Algebra of complex numbers.
    F 1/16 Geometry of C. The extension problem.
    W 1/21 Topology of C. Limits and continuity. Complex differentiability.
    F 1/23 Holomorphy. Real differentiability.
    W 1/28 The Cauchy-Riemann equations.
    F 1/30 Uniform convergence.
    W 2/4 Absolutely uniform convergence.
    F 2/6 Double series.
    W 2/11 Power series.
    F 2/13 Constant functions. The exponential. Surjectivity onto the multiplicative group.
    W 2/18 The real exponential. The kernel and periodicity of the exponential.
    F 2/20 Multivalued functions. The argument. Logarithms.
    W 2/25 More on logarithms. Powers. Roots. Circular functions.
    F 2/27 Midterm exam
    W 3/11 Contour integration. Goursat's theorem.
    F 3/13 Local integrability. Cauchy's theorem for homotopic loops.
    W 3/18 Relative homotopy. Evaluation of real definite integrals.
    F 3/20 Cauchy integral formula. Cauchy-Taylor theorem. Analytic continuation.
    W 3/25 Cauchy estimates. Liouville's threorem. Morera's theorem. Weierstrass convergence theorem.
    F 3/27 Identity theorem. Open mapping theorem. Maximum principle.
    W 4/1 Laurent decomposition and Laurent series.
    F 4/3 Good Friday
    W 4/8 Isolated singularities.
    F 4/10 Residues and indices. The residue theorem.
    Tu 4/14 The argument principle. Rouché's theorem.
    Tu 4/21 Final exam (2pm, Currie Gym)

    Reference books

  • Elias Stein and Rami Shakarchi, Complex analysis. Princeton 2003.
  • Theodore Gamelin, Complex analysis. Springer 2001.

    Online resources

  • Lecture notes by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka
  • Complex variables modules by John Mathews and Russell Howell

    Course outline

    Instructor: Dr. Gantumur Tsogtgerel

  • Office hours: T 13:30–14:30, R 10:00–11:00, or by appointment
  • Office: Burnside Hall 1123

    Prerequisite: MATH 248 (Honours Advanced Calculus)

    Restriction: Intended for Honours Physics and Engineering students. Not open to students who have taken or are taking MATH 316.

    Topics: Very standard topics, except vector valued functions, which we will cover if time permits.

  • Power series, analytic functions, elementary functions
  • Holomorphic functions, contour integration, Cauchy's function theory
  • Isolated singularities and the residue
  • Conformal maps, harmonic functions
  • Vector valued analytic functions, spectral theory of matrices (if time permits)

    Calendar description: Functions of a complex variable; Cauchy-Riemann equations; Cauchy's theorem and consequences. Taylor and Laurent expansions. Residue calculus; evaluation of real integrals; integral representation of special functions; the complex inversion integral. Conformal mapping; Schwarz-Christoffel transformation; Poisson's integral formulas; applications.

    Homework: 3-4 written assignments.

    Exams: A midterm and final.

    Grading: Homework 20% + Midterm 20% + Final 60%

    Midterm exam

  • The grades are on MyCourses!
  • Solutions to the midterm problems.
  • Here are some practice problems and hints to their solution.
  • Solutions to 3 selected problems.