Instructor: M. Kapranov, ph. 978-3998, email: kapranov@math.toronto.edu. Office hours: MW 3-4PM, F 1-2PM Book: S. Fisher, Complex variables. Numbers below refer to sections of this book.
Mon. Sep. 9: Reminder on complex numbers. Their arithmetic. Complex conjugation, division. Trigonometric representation. \S 1.1.
Wed. Sep. 11: Roots of complex numbers. Functions of a complex variable. Examples: multiplication by a complex number, polynomials, rational functions. Graphs. \S 1.2 (up to p. 18), \S 1.4.
Fri. Sep. 13: Open and closed sets. Domains. Continuity and limits in the complex domain. Infinite series. The exponential function. \S 1.4-5.
Mon. Sep. 16: Trigonometric, logarithmic and inverse trig. functions in the complex domain. \S 1.5.
Wed. Sep. 18: Analytic functions. Cauchy-Riemann equations. Conformal property of analytic functions. Analytic functions and harmonic functions. Analyticity of sums and products. Polynomials and rational functions in the complex domain. Their derivatives. \S 2.1.
Fri. Sep. 20 : Power series in the complex domain. The radius of convergence. Analiticity of the sum. Multiplication and division of power series. \S 2.2
Mon. Sep. 23: Exponential, trigonometric and binomial functions as power series. \S 2.2.
Wed. Sep. 25: Line integrals in the complex domain. Green's theorem. \S 1.6.
Fri. Sep. 27: The Cauchy theorem on integrals of analytic functions. \S 2.3.
Mon. Sep. 30: Cauchy's integral formula. \S 2.3 (up to p. 111).
Wed. Oct. 2: Power series representation of an analytic function. Integral representation of the higher derivarives of an analytic function. \S 2.4.
Fri. Oct. 4: MIDTERM
Mon. Oct. 7: Unique continuation of analytic functions. The fact that zeroes are isolated. The order of a zero. \S 2.4.
Wed. Oct. 9: Theorems of Liouville and Morera. \S 2.4.
Fri. Oct. 11: Classification of isolated singularities. \S 2.5.
Wed. Oct. 16: Essential singularities. Examples. Weierstrass' theorem on approaching any value.
Fri. Oct. 18: Residues and their computation. The partial fraction expansion of rational functions. \S 2.5.
Mon. Oct. 21: Laurent series. \S 2.5.
Wed. Oct. 23: The residue theorem and evaluation of integrals: case of rational functions. \S 2.6.
Fri. Oct. 25: Evaluation of integrals by residues: trigonometric case. \S 2.6.
Mon. Oct. 28: Evaluation of integrals by residues, pt. III. \S 2.6.
Wed. Oct. 30: The argument principle and the Rouch\'e theorem. The fundamental theorem of algebra. \S 3.1.
Fri. Nov. 1: The maximum principle and the mean value theorem. \S 3.2
Mon. Nov.4: Linear fractional transformations. Its behavior on lines and circles. The cross-ratio. \S 3.3.
Wed. Nov. 6: Analytic functions as conformal mappings. Representation of analytic functions by level curves. \S 3.4.
Fri. Nov. 8: Applications of analytic functions in hydrodynamics. \S 2.1.1, 3.4.1.
Mon. Nov. 11: The Riemann mapping theorem (without proof). Examples of conformal mappings. Domains bounded by two circle arcs. \S 3.5.
Wed. Nov. 13: The Schwartz-Christoffel transformations. \S 3.5.
Fri. Nov. 15: Harmonic functions: the harmonic conjugate, the maximum principle and the mean value. \S 4.1.
Mon. Nov. 18: The Dirichlet problem for harmonic functions. The Poisson integral representation. \S 4.3.
Wed. Nov. 20: Partial fraction expansions of analytic functions. The Mittag-Leffler theorem.
Fri. Nov. 22: Expansion of trigonometric functions into partial fractions. p. 169
Mon. Nov. 25: Expansion of trigonometric functions on partial fractions (cont.). Application to the sums of inverse powers and Bernoulli numbers.
Wed. Nov. 27: Infinite products. Their convergence. Weierstrass' canonical products.
Fri. Nov. 29: The Gamma function as a product. Its poles and residues.
This last week will be reserved for review or catchup.