Programme

Section I: Complex variable functions

- Complex numbers
- Analytic functions
- Integration of complex functions
- Power series
- Integration using the Residue theorem
- Asymptotic expansions
- Distributions

Section II: Vector spaces and eigenvalue problems

- Vector spaces
- Euclidean spaces
- Eigenvalue problems in N and infinite-dimensional vector spaces
- Function of a matrix
- Fourier transform

Section III: Integral and differential operators

- Integral operators and integral equations
- Sturm-Liouville operators
- First order partial differential equations

Core Documentation

C. Bernardini, O. Ragnisco and P.M. Santini
Metodi matematici della Fisica, NIS 1993

B. Chabat, M. Lavrentiev
Methodes de la Theorie des fonctions d'une variable complexe, MIR 1972

A. I. Markusevic
Elementi di teoria delle funzioni analitiche, Editori Riuniti 1988

F. Bagarello
Fisica Matematica, Zanichelli, 2007

P. A. Grassi
Esercizi di metodi matematici, Casa Editrice Ambrosiana, 2018

Reference Bibliography

The reference text is the Bernardini-Ragnisco-Santini, in which all the topics covered in the course are exposed. The other texts offer interesting insights on complex variable functions (Chabat-Lavrentiev, Markusevic), on the theory of bounded operators and distributions (Bagarello). The Grassi workbook contains a wide selection of complex analysis problems with solutions.

Type of delivery of the course

In order to achieve the expected goals, lessons will take place through taught classes on the board which will also provide a suitable number of exercises to allow the student to apply what they have learned to problems that can be easily found in physics.

Attendance

Course attendance is optional; the study of the recommended texts is sufficient to achieve the goal of the course.

Type of evaluation

Learning verification occurs firstly by passing two written tests in November and January or, in the absence of a positive result for both proofs, by passing a written test. Both kinds of written exams last two hours and are organized in such a way for the student to solve three exercises chosen from a list of four. The written tests are aimed at verifying the level of effective understanding of the concepts and the students' ability to apply them in recurrent physics problems. All previous written examinations [from the 2017-2018 academic year] are available on the web-site of the course: http://dmf.matfis.uniroma3.it/fisica/triennale/scheda_corso.php?id=1179. The exam is approved if, in addition to the written assignments, the student will be able to pass an oral test based on questions extracted from the course program.