|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.
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%