ID:
501257
Duration (hours):
56
CFU:
6
SSD:
ANALISI MATEMATICA
Year:
2025
Overview
Date/time interval
Primo Semestre (22/09/2025 - 09/01/2026)
Syllabus
Course Objectives
The expected learning outcomes are:
(first part)
- insight into the contents and significance of the basic theoretical results relating to initial-value problems for ordinary differential equations, both for scalar and for vector-valued functions, with particular attention to linear systems. Understanding the elementary concepts of equilibrium stability;
- ability in reproducing with awareness the main arguments of the theory;
- ability in framing and solving basic problems involving differential equations;
- ability in facing some differential problems (for example, the qualitative study of the solutions to a scalar equation) which require the adjustment of tools and arguments encountered and already used in the course.
(second part)
- insight into the contents and significance of the basic theoretical results relating to holomorphic functions.
- ability in reproducing with awareness the main arguments of the theory.
- ability in framing and solving some standard problems related to Complex Analysis at the elementary level.
- ability in facing some problems of Complex Analysis (for example, specific line integrals) which require the adjustment of tools and arguments encountered and already used in the course.
(first part)
- insight into the contents and significance of the basic theoretical results relating to initial-value problems for ordinary differential equations, both for scalar and for vector-valued functions, with particular attention to linear systems. Understanding the elementary concepts of equilibrium stability;
- ability in reproducing with awareness the main arguments of the theory;
- ability in framing and solving basic problems involving differential equations;
- ability in facing some differential problems (for example, the qualitative study of the solutions to a scalar equation) which require the adjustment of tools and arguments encountered and already used in the course.
(second part)
- insight into the contents and significance of the basic theoretical results relating to holomorphic functions.
- ability in reproducing with awareness the main arguments of the theory.
- ability in framing and solving some standard problems related to Complex Analysis at the elementary level.
- ability in facing some problems of Complex Analysis (for example, specific line integrals) which require the adjustment of tools and arguments encountered and already used in the course.
Course Prerequisites
It is necessary to master the main notions and tools of Mathematical Analysis and Linear Algebra dealt with in the courses of the first year, in particular: differential and integral calculus for scalar and vector-valued functions, differential forms, matrices and linear transformations.
Teaching Methods
Teaching will be mainly carried out through traditional classes. Supporting tutoring activities will be organized outside the official teaching hours and according to an agreed timetable.
Assessment Methods
The exam consists in a written test and in an oral part. The first one mainly aims to check the level of knowledge of the principal methods dealt with in the course for the study of ordinary differential equations or for the resolution of some types of problems in Complex Analysis. Exercises from previous written tests will be made available.
A threshold mark (16/30) is needed to pass to the oral part. This latter intends to verify the global understanding of the theoretical framework.
NOTE
The course "Equazioni differenziali e sistemi dinamici" (for Physicists) shares 6 CFU from the course "Analisi Matematica 3". Because of the different context of the two courses, for Physics students an alternative way, with respect to the standard one, will be available for the exam: this will consist only of the oral part, the student will be requested to master the statements of the main theoretical results, but, in addition, he will have to elaborate on a specific topic (to be agreed with the teacher) according to his interests. A list of possible topics will be made available; depending on the needs of the students, suitable tutoring activities will be organized.
A threshold mark (16/30) is needed to pass to the oral part. This latter intends to verify the global understanding of the theoretical framework.
NOTE
The course "Equazioni differenziali e sistemi dinamici" (for Physicists) shares 6 CFU from the course "Analisi Matematica 3". Because of the different context of the two courses, for Physics students an alternative way, with respect to the standard one, will be available for the exam: this will consist only of the oral part, the student will be requested to master the statements of the main theoretical results, but, in addition, he will have to elaborate on a specific topic (to be agreed with the teacher) according to his interests. A list of possible topics will be made available; depending on the needs of the students, suitable tutoring activities will be organized.
Texts
Some useful references.
(Differential equations)
- C.D. Pagani, S. Salsa: Analisi matematica 2. Zanichelli.
- S. Salsa, A. Squellati: Esercizi di analisi matematica Vol. 2. Zanichelli.
- 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.
(Complex Analysis)
- E. Stein, S. Shakarchi: Complex Analysis. Princeton University Press, 2003
- R. Narasimhan, Y. Nievergelt: Complex Analysis in One Variable, Birkhauser, 2000
- R. E. Rodríguez, I. Kra, J. P. Gilman: Complex Analysis: In the Spirit of Lipman Bers (Graduate Texts in Mathematics, 245), Springer
- G. Gilardi, Analisi Matematica 3. McGraw- Hill Italia.
Lecture notes will be also provided.
(Differential equations)
- C.D. Pagani, S. Salsa: Analisi matematica 2. Zanichelli.
- S. Salsa, A. Squellati: Esercizi di analisi matematica Vol. 2. Zanichelli.
- 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.
(Complex Analysis)
- E. Stein, S. Shakarchi: Complex Analysis. Princeton University Press, 2003
- R. Narasimhan, Y. Nievergelt: Complex Analysis in One Variable, Birkhauser, 2000
- R. E. Rodríguez, I. Kra, J. P. Gilman: Complex Analysis: In the Spirit of Lipman Bers (Graduate Texts in Mathematics, 245), Springer
- G. Gilardi, Analisi Matematica 3. McGraw- Hill Italia.
Lecture notes will be also provided.
Contents
The course consists of two parts: the first one is devoted to the theory of ordinary differential equations and systems; the second part presents the elementary notions of convergence for sequences of functions and introduces to the theory of functions of one complex variable. More extensively: First part. Models and examples of ODE's. General results concerning existence, uniqueness and continuation of a solution; comparison results; continuous dependence on initial data. Elementary techniques for solving simple differential equations. Linear systems of ODE's: structure of the solutions, exponential matrix. Asymptotic behaviour of dynamical systems, stability (linearisation and Lyapunov method). Second part. Sequences and series of functions. Power series. Complex differentiability and power expansion. Contour integrals. Holomorphic functions and primitives. Cauchy's Theorem. Meromorphic functions and singularities. Complex logarithm. Winding number. Residue theorem and application to the evaluation of integrals. Further elementary properties of holomorphic functions. NOTE The course "Equazioni differenziali e sistemi dinamici" (for Physicists) shares 6 CFU from the course "Analisi Matematica 3". Because of the different context of the two courses, for Physics students an alternative way, with respect to the standard one, will be available for the exam: this will consist only of the oral part, the student will be requested to master the statements of the main theoretical results, but, in addition, he will have to elaborate on a specific topic (to be agreed with the teacher) according to his interests. A list of possible topics will be made available; depending on the needs of the students, suitable tutoring activities will be organized.
Course Language
Italian
More information
Course details and materials will be provided through the Kiro page of the course.
Those who can certify (according to the rules of Pavia University) their impossibility to attend the classes of the course can directly contact the teacher to identify the best alternative activities.
Those who can certify (according to the rules of Pavia University) their impossibility to attend the classes of the course can directly contact the teacher to identify the best alternative activities.
Degrees
Degrees (3)
PHYSICS
Bachelor’s Degree
3 years
PHYSICAL SCIENCES
Master’s Degree
2 years
SCIENZE FISICHE
Master’s Degree
2 years
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