Physics

ALFREDO LEONARDO URBANO
ANDREA CAPUTO

PART I
General goals:
The goal of the course is the study and the comprehension of advanced mathematical techniques. The student will acquire the basic concepts of complex
analysis and functional analysis and she/he will be able to apply them to the study of problems in classical (electromagnetism and continuum media) and
quantum physics. The know-how that the student acquires attending this course is indispensable for the Physics courses of the third year.

Specific goals:
Introduction to the fundamental concepts of complex analysis, i) to provide the student with a deep knowledge and understanding of these concepts, and ii) to
allow him/her to successfully apply them in various physical contexts. In particular, the student must be able to use techniques of integration in the complex
domain in all the physical
contexts in which they have applications. In order to achieve these goals, and to help the student to develop the capability i) to communicate the acquired
knowledges, and
ii) to continue the studies autonomously, we plan to involve him (her), during the theoretical lectures and exercises, through general and specific questions
related to the
subject; or through the presentation in depth of some specific subject agreed with the teacher

PART II
General goals:
The goal of the course is the study and the comprehension of advanced mathematical techniques. The student will acquire the basic concepts of complex
analysis and functional analysis and she/he will be able to apply them to the study of problems in classical (electromagnetism and continuum media) and
quantum physics. The know-how that the student acquires attending this course is indispensable for the Physics courses of the third year.

Specific goals:
Introduction to the fundamental concepts of functional analysis, i) to provide the student with a deep knowledge and understanding of these concepts, and ii) to
allow him/her to successfully apply them in various physical contexts. In particular, the student must be able to use the Fourier series, the Fourier and Laplace
transforms, and the distributions in all the physical contexts in which they have applications; in addition the student must be able to work with Hilbert spaces and
with operators on functional spaces of physical interest. In order to achieve these goals, and to help the student to develop the capability i) to communicate the
acquired knowledges, and ii) to continue the studies autonomously, we plan to involve him (her), during the theoretical lectures and exercises, through general
and specific questions related to the subject; or through the presentation in depth of some specific subject agreed with the teacher

The fundamental prerequisite is the basic knowledge of the courses in mathematics of the first year and of the first semester of the second year. In particular, specific knowledge in the following fields is required:
1. Basic concepts of Calculus: real analysis of functions of one or many variables; derivatives, integrals, series.
2. Basic concepts of Linear Algebra.
It is important that the student has knowledge of classical Physics, in particular mechanics and thermodynamics.

Complex numbers and their properties. Analytic functions. Multivalued functions. Complex integrals.
Cauchy theorem and Cauchy integral formula.
Liouville and Morera theorems. The fundamental theorem of algebra.
Maximum modulus theorem. Singularities and their classification.
Taylor and Laurent series. Residues theorem and applications.

Banach spaces. Hilbert spaces. Linear functionals and distributions.
Linear operators in Hilbert spaces, self-adjoint operators, unitary operators, and spectrum.
Lp spaces. Fourier series.
Orthogonal polynomial sequences. Fourier and Laplace transform.
Applications: linear ordinary and partial differential equations relevant in physics.
Green's function.

C. Bernardini, O. Ragnisco, P. M. Santini "Metodi Matematici della Fisica", Carocci, 2014.
M. W. Hirsch, S. Smale and R. L. Devaney, "Differential Equations, Dynamical Systems, and an Introduction to Chaos", Academic Press, 2012.
M. Petrini, G. Pradisi, A. Zaffaroni, "A Guide to Mathematical Methods for Physicists", World Scientific.
F. Calogero, "Metodi Matematici della Fisica", Dispense Istituto di Fisica, Universita' di Roma, 1975.
F. Cesi "Rudimenti di analisi infinito dimensionale", dispense.
N. Kolmogorov, S. V. Fomin, "Elementi di teoria delle funzioni e di analisi funzionale", Editori Riuniti.

C. Bernardini, O. Ragnisco, P. M. Santini "Metodi Matematici della Fisica", Carocci, 2014.
M. W. Hirsch, S. Smale and R. L. Devaney, "Differential Equations, Dynamical Systems, and an Introduction to Chaos", Academic Press, 2012.
L. V. Ahlfors, "Complex Analysis", Mc Graw-Hill 1979.
M. Petrini, G. Pradisi, A. Zaffaroni, "A Guide to Mathematical Methods for Physicists", World Scientific.
C. Presilla, "Elementi di Analisi Complessa" (2a edizione), Springer, UNITEXT 2014.
F. Calogero, "Metodi Matematici della Fisica", Dispense Istituto di Fisica, Universita' di Roma, 1975.
F. Cesi "Rudimenti di analisi infinito dimensionale", dispense.

The format of the course consists of lectures at the blackboard.

The course is based on the in-class frequency of traditional chalkboard lectures.

The final grading will be based on a written and an oral exams.

The written exam consists of a test on the topics covered during the course. To pass the written test, the student should be able to carry out exercises on arguments explained during
the course and apply the method that he/she learned to examples similar to the one discussed. For the evaluation the following points will be considered:

- accuracy of the concepts laid out
- clarity and accuracy of the exposition
- ability to analytically develop the theory
- ability in problem-solving (method and results)

The oral exam consists of a discussion on the topics covered during the course. To pass the oral exam the student should be able to present an argument or repeat a calculation discussed during the course and apply the method that he/she learned to examples similar to the one discussed. For the evaluation the following points will be considered:

- accuracy of the concepts laid out
- clarity and accuracy of the exposition
- ability to analytically develop the theory