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