Università Roma Tre

The complex field. Holomorphic functions; Cauchy-Riemann equations.
Series and Abel's theorem. Exponential and logarithms.
Elementary conformal mappings. Complex integration; Cauchy's theorem; Cauchy's formula.
Local properties of holomorphic functions (singularities, zeroes and poles; local mapping theorem and
maximum principle).
Residues.
Harmonic functions.
Series expansions (Weierstrass' theorem, Taylor's series).
Partial fractions and infinite products.
Supplementary arguments (depending on time): entire functions and Hadamard's theorem. Riemann zeta function.
Riemann mapping theorem.

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