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.

The course is divided into two parts: the first one is devoted to the theory of ordinary differential equations and systems, with an introduction to the study of dynamical systems. The second part is an introduction to the theory of functions of one complex variable.



Models and examples of ODE's. General results concerning existence, uniqueness, comparison and stability for Cauchy problems. Elementary techniques for solving simple differential equations.

Linear systems of ODE's: general results and structure, exponential matrix. The method of Laplace transform.



Asymptotic behaviour of dynamical systems, stability (linearisation and Lyapunov method).



Example of complex functions. Differentiability.

Power series and contour integrals. Olomorphic functions. Cauchy theorem. Singularities, Laurent expansion, and residues. Cauchy theorem. Application to the evaluation of integrals. Analytic extension. Argument principle. Open mapping theorem. Further properties.

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 written by prof. Enrico Vitali (available on line)