2018/2019

2019/2020

DM270

MAT/05 (MATHEMATICAL ANALYSIS)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

2°

1st semester (30/09/2019 - 10/01/2020)

9

84 lesson hours

Italian

WRITTEN AND ORAL TEST

SCHIMPERNA GIULIO FERNANDO (titolare) - 9 ECTS

In order to follow this course with the due preparation it is necessary to have acquired the basic competences resulting from the courses of the first year, in particular: differential and integral calculus for scalar and vector-valued functions, matrices and linear transformations, sequences and series, power series, complex numbers, differential forms, polar coordinates.

Learn the basic results and techniques of the theory of ordinary differential equations and dynamical systems. Be able to apply the methods of the ODE theory to study simple real-world applications and physical models. Understand the first but deep results of complex function theory and become familiar with transformations in the complex plane. Learn the main integration techniques based on the Residue theorem.

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.

Extended summary

Models and examples of ODE's. General results concerning existence, uniqueness, comparison and stability for Cauchy problems. Elementary techniques for solving simple differential equations. Cauchy-Peano's theorem (existence without uniqueness). Linear systems of ODE's: general results and structure of the space of solutions, exponential matrix. Asymptotic behaviour of dynamical systems, stability (linearisation and Lyapunov method).

Function series and power series. Examples of complex functions. Differentiability. Contour integrals. Holomorphic functions. Cauchy's theorem. Singularities, Laurent expansion and residues. Cauchy's theorem. Application to the evaluation of integrals. Analytic extension. Argument principle. Rouché's theorem. Sequences of holomorphic functions. Further geometrical properties of holomorphic functions.

Lectures and exercise sessions.
The two parts of the course (1 - ordinary differential equationa and 2 - complex analysis) will proceed in parallel (generally 2 or 3 hours per week will be devoted to complex anlysis and the remaining ones to differential equations).
The exercise sessions will be scheduled after finishing the corresponding part of the theory.

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.

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

G. Gilardi, Analisi Matematica 3, McGraw- Hill Italia.

Lecture notes will be also provided.

Written and oral test.
The written test will be constituted by a number of exercises. Some of the exercises will have a theoretical character (proofs of simple properties or construction of examples or counterexamples); other ones will be aimed at verifying the acquisition of the basic computational techniques developed in the course (explicit resolution of differential equations or calculus of some integral by complex variable methods).
The oral exam will be aimed at verifying the comprehension of the main results of the theory and the capacity to illustrate them by means of concrete examples. The evaluation will be focused mainly on the comprehension of the main results and of their implications, rather than on the "technical" aspects of the proofs.