Enrollment year
2016/2017
Academic discipline
MAT/05 (MATHEMATICAL ANALYSIS)
Department
DEPARTMENT OF PHYSICS
Curriculum
Fisica teorica
Period
1st semester (02/10/2017 - 19/01/2018)
Lesson hours
56 lesson hours
Activity type
WRITTEN AND ORAL TEST
Prerequisites
Differential and integral calculus for scalar and vector functions, matrices and linear transformations, sequences and series, power series, complex numbers, polar coordinates.
Learning outcomes
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.
Course contents
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, exponential matrix.
Asymptotic behaviour of dynamical systems, stability (linearisation and Lyapunov method).
Examples of complex functions. Differentiability. Power series and contour integrals. Holomorphic functions. Cauchy's theorem. Singularities, Laurent expansion and residues. Cauchy theorem. Application to the evaluation of integrals. Analytic extension. Argument principle. Open mapping theorem. Further properties.
Teaching methods
Lectures and exercise sessions.
Reccomended or required readings
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.
E. M. Stein - R. Shakarchi: Complex analysis, Princeton Lectures in Analysis II, Princeton University Press (2003)
G. Gilardi, Analisi Matematica 3, McGraw- Hill Italia.
Lecture notes will be also provided.
Assessment methods
Written and oral test.
The written test will be constituted by a number of exercises. 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.
Further information
Written and oral test.
The written test will be constituted by a number of exercises. 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.
Sustainable development goals - Agenda 2030
Università di Pavia - Offerta formativa