Learn the basic results and techniques of the theory of ordinary differential equations and dynamical systems.Acquire skill in manipulation and transforms of complex numbers and understand the first but deep results of complex function theory.

Lectures and exercise sessions.

Written and oral test.

Differential and integral calculus for scalar and vector functions, matrices and linear transformations, sequences and series, power series, complex numbers, polar coordinates.

First part: Ordinary differential equations. General results on initial value problems (existence and uniqueness, extension of solutions, comparison theorem,
continuous dependence on data). Integration techniques for some special classes of equations. Linear differential equations and systems (general results
and computation of the exponential matrix). Cauchy-Peano's theorem. Qualitative study of solutions. Asymptotic (long-time) behavior and stability. Omega-limits,
attractors. Use of ordinary differential equations for mathematical modelling.

Second part: functions of one complex variable. Complex differentiability and analyticity. Power series. Integration along paths. Holomorphic functions and complex
primitives. Cauchy's integral theorem. Meromorphic functions and singularities. Logarithms in the complex field. Winding number, residue theorem and applications
to integrals. Further properties of holomorphic functions (argument principle, analytic extension, Rouché's theorem, sequences of holomorphic functions).
Geometrical properties: open mapping theorem, conformal mappings.

M. W. Hirsch, S. Smale, R. L. Devaney: Differential equations, dynamical systems, and an introduction to chaos. Pure and Applied Mathematics, Vol. 60. Elsevier/Academic Press, Amsterdam, 2004.

A. Ambrosetti: Appunti sulle equazioni differenziali ordinarie. Springer Verlag, 2011.

H. Amann: Ordinary differential equations. An introduction to nonlinear analysis. de Gruyter Studies in Mathematics, Vol. 13. Walter de Gruyter & Co., Berlin, 1990.

V. I. Arnold: Ordinary differential equations. Universitext, Springer-Verlag, 2006. Second printing of the 1992 edition.

S. Salsa, A. Squellati: Esercizi di analisi matematica 2. Masson, 1994.

E. M. Stein - R. Shakarchi: Complex analysis, Princeton Lectures in Analysis II, Princeton University Press (2003)

T. Needham: Visual Complex Analysis. Oxford University Press, 1997.

S.G. Krantz: A guide to complex variables. Mathematical Association of America, 2008. Lecture notes on ordinary differential equations will also be provided.