Università Roma Tre

Complex numbers; complex differentiability and Cauchy-Riemann formula; the Riemann sphere; linear fractionary maps; the line integral; Cauchy's theorem and Cauchy's formula; Lioville's theorem; the fundamental theorem of algebra; the identity principle; the Euler product for the sine; the mean value property, the maximum principle and Schwarz's lemma; the Moebius maps are the automorphisms of the disc; the hyperbolic metric on the disc and its geodesics; the contractions of the hyperbolic metric; Laurent series; theorem of the removable singularity; poles and essential singularities; the Casorati-Weierstrass theorem; the residues theorem; the argument principle; the theorem of Ropuche'; normal families and quasi-uniform convergence; harmonic functions are the real part of holomorphic functions; uniqueness for the Dirichlet problem; Poisson's kernel; functions which have the mean value property are harmonic; the Schwarz refelction principle; the Riemann mapping theorem.

W. Rudin, Real and complex analysis, Tata McGraw Hill.

J. B. Conway, Functions of one complex variable, Springer.