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.

Assignments

• 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.