Complex Analysis I

Complex differentiability and Cauchy-Riemann equations. Holomorphic functions and primitives. Local Cauchy theorems, Morera theorem, power series representations, and fundamental properties of holomorphic functions. Logarithms, winding numbers, and global Cauchy theorems. Singularities, Laurent series, and residue theorem. Argument principle and open mapping theorem. Conformal mapping, Schwarz-Pick lemma, and Mobius transformations. Normal families and Riemann mapping theorem. Harmonic and subharmonic functions, Poisson integrals, and Schwarz reflection principle. Simply connected plane domains.