D1: Basic knowledge in complex analysis, Fourier theory, and Hilbert spaces; D2: Ability to apply knowledge to the computation of integrals, to the solution of differential equations using Fourier series, to basic operations with operators on Hilbert spaces; D3: Ability to choose which of the techniques studied, even possibly with appropriate modifications, are best suited to solving problems in physics; D4: Ability to communicate, in writing and / or orally, the logical steps that lead to the solution of a problem with the methods studied; D5: Ability to consult textbooks to deepen the topics covered.

* Mathematical analysis 1 & 2;
* Geometry.

1) Functions of a complex variable:
* analitic functions
* Cauchy's theorem
* applications to the calculation of integrals
2) Fourier Transform:
* Fourier transform in L^1
* Fourier transform in L^2
* transform of tempered distributions
3) Hilbert space
* complete orthonormal systems
* operators
* basics on spectrum

G. Cicogna "Metodi matematici della Fisica", 2015 Springer ;

G. Cicogna "Exercises and Problems in Mathematical Methods of Physics", 2018 Springer ;

M. Petrini, G. Pradisi, A. Zaffaroni "A Guide to Mathematical Methods for Physicists with Problems and Solutions", 2018 World Scientific Publishing Europe ;

"Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics (3rd Edition)", 2002 Prentice Hall.

1) Functions of a complex variable:

* analytic functions: review of complex numbers and real analytic functions, power series on C, examples of functions on C, notion of limit on C, derivability in C, definition of analytic function, Cauchy-Riemann conditions, two-sided series as examples of analytic functions;

* Cauchy's theorem: path integral for vector fields and functions on C, Green's theorem and applications to path deformations, Cauchy's theorem, Cauchy's integral formula, infinite derivability of analytic functions, Taylor-Laurent series, analytic extension, the example of Euler's Gamma, types of isolated singularities, Liouville's theorem, fundamental theorem of algebra;

* application to the computation of integrals: meromorphic functions, residue at an isolated singularity, formula for the residue at a pole of order n, residue theorem, application to the computation of real integrals, singularity and residue at infinity, external residue theorem, Jordan's lemma and applications, multi-valued functions, branches, cuts and branch points, use of multi-valued functions for the computation of integrals;

2) Fourier Transform:

* Fourier transform in L^1: introduction on Fourier series, applications, from the series to the transform, elements on Lebesgue integral, dominated convergence theorem, Fubini-Tonelli theorem, definition of L^1 and L^2, definition of transform in L^1, properties, continuity, boundedness, convolution and product, derivatives and multiplication by variable, Riemann-Lebesgue theorem, examples of transforms in L^1, transform of the Gaussian;

* Fourier transform in L^2: motivations, scalar product on L^2, Cauchy-Schwartz inequality, approximants in L^1, Parseval identity for approximants in L^1, definition by means of the limit of the transform on L^2, Parseval identity, antitransform, properties of the transform in L^2, product and convolution, derivatives and multiplication by variable, interpretation of the transform as decomposition into harmonics at a fixed frequency, uncertainty principle;

* transform of tempered distributions: orthogonality of the basis of functions for the Fourier series, generalization to the case of the transform, notion of Dirac's delta, definition of Schwarz space and tempered distribution, limit in distributional sense, operations on distributions, derivative, multiplication by a polynomial, Fourier transform, examples of distributions and their Fourier transforms, Heaviside theta, sign, principal part of 1 / x, Dirac delta and its derivatives, constant function, application to Green's functions for non-homogeneous linear equations;

3) Hilbert spaces

* complete orthonormal systems: infinite dimensional vector spaces, basic concept, series of vectors, Banach spaces and notion of convergence, examples of L^p and l^p, problem of determining coefficients in the expansion, notion of scalar product and induced norm, definition of Hilbert space, independent system, orthonormal system, Gram-Schmidt, complete orthonormal system, example of the Fourier series, characterization of complete orthonormal systems, complete non-orthonormal systems;

* operators: linear operators in infinite dim, domain, continuity, extension by continuity, norm and bounded operators, equivalence between continuity and boundedness, linear functionals, Riesz theorem, adjoint of a bounded operator and its properties, more general case of dense domain, symmetric and self-adjoint operator, unitary operator, example of the Fourier transform;

* basics on spectrum: eigenvalues ​​and eigenvectors in infinite dim, spectrum, example of derivative and multiplication by x.

Lectures and tutorials.

The Moodle of the course will be used to assign home exercises each week, to collect exams from past years, and to keep a record of the topics covered in each lesson.

The exam is intended to assess the ability to solve exercises and problems using the methods studied during the course. This is evaluated through a written exam consisting of three problems: one on complex analysis, one on the Fourier transform, and one on Hilbert spaces, to be completed within 3 hours. Each problem is divided into parts of varying difficulty, with each part assigned a clearly indicated score in the exam text. The total score is 33. Honors ("cum laude") are awarded for a perfect score of 33. The written exam is passed with a minimum score of 18. Students who pass the written exam may choose to take an oral exam as well. The oral exam is a one-hour interview during which the student is asked to solve two short exercises on topics covered in the course. The oral exam is graded on a scale of 30, and the final grade is the arithmetic average of the written and oral exam grades. The exam text and its solution are promptly published on the course's Moodle website after the exam.