Mathematical Analysis 3
- Professors:
- Savaré Giuseppe
- Year:
- 2015/2016
- Course code:
- 502210
- ECTS:
- 9
- SSD:
- MAT/05
- DM:
- 270/04
- Lessons:
- 84
- Period:
- I semester
- Language:
- Italian
Objectives
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.
Teaching methods
Lectures and exercise sessions.
Examination
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.
Syllabus
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.
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.
Bibliography
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)
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)
Dipartimento di Matematica ''F. Casorati''
Università degli Studi di Pavia -
Via Ferrata, 5 - 27100 Pavia
Tel +39.0382.985600 - Fax +39.0382.985602