We will use the book Elias M. Stein & Rami Shakarchi: Complex Analysis, Princeton Lectures in Analysis, Princeton University Press (2003). We will also cover some topics in greater detail than can be found in the textbook, as well as some topics not treated in the textbook. Material for these topics will be posted online as supplementary notes.
Chapter 1 (Preliminaries to Complex Analysis) + Supplementary notes (Complex linearity of Jacobian)
Chapter 2 (Cauchy's Theorem and Its Applications)
Chapter 3 (Meromorphic Functions and the Logarithm) + Supplementary notes (Laurent series, complex logarithms, harmonic functions, Poisson integral formula and Dirichlet problem on the disc)
Chapter 8 (Conformal Mappings): Sections 1-3 + Supplementary notes (Möbius transformations and automorphisms of the Riemann sphere)