Università Roma Tre

Complex numbers; holomorphic functions and the Cauchy_Riemann formula. Some examples of holomorphic functions; the Riemann sphere and e the point at infinity. The linear fractional transformations. Integral of a complex function along a curve; index of a point with respect to a curve. Cauchy theorem; Cauchy formula. The Liouville theorem. The mean value thorem, the maximum principle and the principle of identity of holomorphic functions. The almost-uniform limit of holomorphic functions is holomorphic. Shcwarz lemma and the automorphisms of the disc. The metric of Poincare' on the disc and its geodesics. Laurent series; the general form of Cauchy theorem. Removable singularities; poles and essential singularities; the Casorati-Weierstrass theorem. Euler's product for the sine. Meromorphic functions. The argument principle and the theorem of Rouche'. Holomorphic maps are open; the almost uniform limit of univalent functions is eithe univalent or constant; Lagrange inversion formula. Harmonic functions; the mean value property, the maximum principle and Dirichlet problem; Poisson kernel; continuous functions with the mean value property are harmonic. Schwarz reflection principle. Analytic extension. Jensen's formula for the zeroes of a holomorphic function. Normal families and compactness for the almost-uniform topology. The Riemann mapping theorem. When two rings are conformally equivalent. The small theorem of Picard. Holomorphic functions and fluidodynamics.

W. Rudin, Real and complex Analysis, McGraw-Hill.