Università Roma Tre

Proof of the classification theorem of isolated singularities and proof of the residues theorem. The logarithmic derivative and the principle of the argument. Calculation of residues.
Classification of the connected open of C. The Riemann map theorem and the uniformization theorem (without proof). The Riemann sphere as a compactification of the complex plane. The group of linear transformations of the projective line and the linear transformations produced by them. The group of automorphisms of the complex plane.
The lemma of Schwarz and the group of automorphisms of the unitary disc. Elements of global analytical functions and function. The logarithm as a global analytical function.
The n-th rooty as a global analytical function. The bundle of germs of analytical functions and its properties. The Riemann surface associated with a global analytical function.
Examples and properties of Riemann surface. The Riemann surface associated with an algebraic function and properties. Summary and considerations on the course program.

L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.
E. Freitag, R. Busam: Complex Analysis, Springer.