Università Roma Tre

I. Elementary theory (Including: Complex numbers and the complex plane. Convergence. Sets in the complex plane. Functions on the complex plane. Continuous functions. Holomorphic functions. Power series. Integration along curves).
II. Cauchy's theorem and its applications (Including: Goursat's theorem; Cauchy's formula and calculation of residues. Analytical continuation. Morera's theorem. Schwarz's principle). Cauchy's theorem in simply connected domains.
III. Meromorphic functions and the logarithm (Including: zeros and poles; isolated singularities. Argument principle. Rouché's theorem).
IV. Conformal transformations (Including: elementary maps and linear fractional transformations); Riemann mapping theorem.
V. Laurent series; partial fractions and canonical products.

[S] Complex Analysis. Elias M. Stein, Rami Shakarchi Princeton University Press 2003, ISBN 10: 1400831156 / ISBN 13: 9781400831159

[A] Ahlfors, Lars V, Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978. xi+331 pp. ISBN 0-07-000657-1

[E] M. Evgrafov, Coll, Recueil de problèmes sur la théorie des fonctions analytiques, Traduction francaise, Editions Mir, 1974.