Università Roma Tre

Complex numbers; some complex maps; the Riemann sphere; fractional-linear transformations; they preserve the cross ratio and the set of circles and straight lines. The apollonius circles. Integral of a complex function along a curve; length of a curve; properties of the integral; indicator function; formal sum of curves. Cauchy's theorem on rectangles and on all curves; Caucgy's formula. Liouville's principle. The identity principle for holomorphic functions; removable singularities; almost uniform convergence and its properties. Morera's lemma; mean value theorem and maximum principle; harmonic functions locally are the real part of holomorphic functions. Hadamard's three circles theorem. Euler's product for the sine. Harmonic functions and the electric potential; mean value theorem for harmonic functions, maximum principle and uniqueness for the Dirichlet problem. Dirichlet's kernel; the functions with the mean value property are harmonic. Schwarz's reflection principle. Laurent series; residues and residue calculus. The argument principle and Rouche's theorem. Holomorphic maps are open; a variant of the argument principle and Lagrange's inversion formula; the almost uniform limit of univalent functions is either univalent or constant. Schwarz's lemma; automorphisms of the disc; hyperbolic metric on the disc and its geodesics; the automorphisms of the disc preserve the hyperbolic metric. Analytic extension and monodromy theorem; some examples of Riemann surfaces. The Riemann mapping theorem; Picard's small theorem.

W. Rudin, Real and complex Analysis.

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