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) |
Instructor: Dr. Gantumur Tsogtgerel
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.
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%