Università Roma Tre

I. Preliminaries: Complex numbers and the complex plane. Topology and convergence. Continuous functions. Holomorphic functions and Cauchy-Riemann equations. Power series (Cauchy-Hadamard formula). Integration along curves.
II. Cauchy's theorem and its applications: Goursat's theorem. Cauchy's theorem on starred sets. Cauchy's formula and residue calculus. Analytic continuation. Morera's theorem. Schwarz's principle.
III. Meromorphic functions and the logarithm: Zeros, poles, essential singularities. Meromorphic functions. Argument principle. Homotopy. The complex logarithm. Cauchy's theorem on simply connected regions.
IV. Canonical sums and products: Laurent series. Fourier-Laurent-Weierstrass theorem. Partial fractions; Mittag-Leffler theorem. Canonical Products and Weierstrass's Theorem.
V. Conformal Transformations: Elementary Conformal Maps and Fractional Linear Transformations (Möbius); Circle Automorphisms. Montel's Theorem and the Riemann Map Theorem.

Elias M. Stein, R. Shakarchi, Complex Analysis, Princeton University Press, 2003