COURSE PROGRAM Professor_ Andrei Muravnik
Introduction to Theory of Functions of one Complex Variable 1 algebra and analytic geometry of complex numbers, the Riemann sphere 2 power series and their convergence 3 differentiability and analyticity, differentiability of power series 4 elementary functions in the complex domain 5 conformality 6 Cauchy-Riemann equations, partial z and partial z bar 7 Mobius transformations 8 Path integrals, Cauchy's formula and theorem for a disc, analytic implies power series at each point, 9 Cauchy's estimate, Maximum Modulus Principle, Liouville's Theorem, Fundamental Theorem of Algebra, isolation and finite multiplicity of roots 10 homotopy of paths and Cauchy's theorem, motivation for winding numbers 11 winding numbers and Cauchy's integral formula, zero counting 12 Morera's Theorem, theorem on existence of primitives, isolated singularities 13 poles, essential singularities, Casorati-Weierstrass Theorem, Laurent expansion 14 The Residue Theorem, residue integrals |