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.

Adopted text:
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