# MATHEMATICAL METHODS FOR PHYSICS

MATHEMATICAL METHODS FOR PHYSICS (MOD. 1) (6 credits) | MATHEMATICAL METHODS FOR PHYSICS (MOD. 2) (6 credits) |

## Learning outcomes of the course unit

At the end of the course the student will have better knowledge and understanding as she/he will: master the basic results in analytic functions theory, being in particular able to understand how many outcomes stem from the holomorphic nature of a function; master the main topics of linear spaces theory, being in particular able to understand how many outcomes stem from spectral theory; master basic techniques in Fourier analysis and orthogonal polynomials.

The student will be able to apply knowledge and understanding and in particular she/he will be able to: apply results of analytic functions wherever they are relevant for subsequent studies in physics, and in particular compute integrals by means of the residue theorem, integrate systems of linear ordinary differential equations, compute Taylor and Laurent series, gaining understanding on possible singularities; compute the spectrum of linear operators (in particular in the context of quantum mechanics) and compute functions of operators; solve linear partial differential equations by means of separation of variables, being eventually able to apply Fourier series to comply to initial/boundary conditions; make use of Fourier transforms, gaining understanding that a few problems are actually simpler to solve in momentum space.

The student will be able to make judgements and in particular she/he will be able to: make her/his choice of a suitable technique to solve a given problem, recognising that often one has more than one choice; having put forward a solution to a problem, prove the correctness; doubt the correctness of vague arguments, looking instead for formal neatness.

The student will also have acquired communication skills as she/he will be able to: present her/his results in a clean, precise and concise way; present her/his results both synthetically as for the overall picture and analytically as for the most delicate points; argue her/his thesis in public, in particular acting in a team.

Finally, the student will have acquired learning skills as she/he will be able to: progress in the study of more advanced subjects in physics, recognising the overall mathematical structures; progress in the study of subjects in mathematics beyond what she/he has already learnt, whenever this is due.

## Prerequisites

Basics in calculus, geometry and linear algebra.

## Course contents summary

The course aims at providing a reasonable completion of the mathematical training of the student on her/his way to the degree in Physics. It is quite a large range of contents, actually subjected to have the necessary precision and a clearly visible overall picture going together well. A clear emphasis will be on calculation skills.

Basic training will be integrated with complex analysis and analytic functions theory (holomorphy, singularities, Taylor and Laurent series, analytic continuation, computation of integrals by means of residue theorem).

The most salient content will be the theory of linear operators in finite-dimensional spaces, to which end subjects in linear algebra will be summarised and extended. The final goal is a reasonable comprehensive treatment of spectral theory and of functions of operators. In this context an introduction to the solution of systems of linear ordinary differential equations will be provided.

Basics on topology and metric spaces will pave the way to infinite-dimensional spaces, and in particular function spaces. Spectral theory in Hilbert spaces will be discussed without aiming at comprehensiveness.

Many subjects will be stressed to be bridging the two main sections (analytic functions and linear operators), in particular Fourier transform and Fourier series, orthogonal polynomials.

All the subjects will be discussed in the perspective of problems in mathematical physics and a first approach to quantum mechanics.

## Course contents

Number fields and the complex numbers field. Basics on funtions of complex variables; limit and continuity in complex analysis. Derivatives and holomorphic nature.

(Line) integrals in the complex plane. Cauchy theorem and formula. Zeros and singularities of analytic functions; polydromy and branch points.

Recap on functions and power series for real functions. Power series, Taylor and Laurent series for analytic functions and relationships to singularities. Analytic continuation. Residue theorem and computation of integrals by means of it. Jordan lemma. Basics on principal value of an integral.

Real and complex linear spaces. Linear dependence and independence, dimension of a space and bases. Isomorphic spaces. Scalar product and unitary linear spaces. Orthonormal bases.

Linear functionals. Riesz theorem. Dirac notation. Completeness and projectors.

Linear operators and matrix representations. Change of basis. Linear spaces of operators, inverse operators and norms. Adjoint operator. Hermitian, unitary and normal operators. Connections to group theories (basics of).

Resolvent operator and spectrum of an operator; eigenvalues and eigenvectors. Diagonalizability and spectral decomposition; projectors and nilpotents operators. Calculation of functions of operators by means of series expansions and Riesz-Dunford formula.

Introduction to infinite-dimensional linear spaces. Metric and topological spaces. Hilbert spaces and separability. Bases, Fourier coefficients and completeness. Function spaces; L1 and L2 spaces.

Linear functionals in Hilbert spaces; Riesz theorem. Weak and strong convergence. Basics on distributions; Dirac delta fuction.

Linear operators in Hilbert spaces; adjoint operator; symmetric and hermitian operators; hermitian extension of a symmetric operator; unitary operators. Basics in spectral theory in Hilbert spaces. Point, continuos and residual spectrum.

Solution of basic linear partial differential equations in physics by means of separation of variables.

Legendre equation and Legendre polynomials. Orthogonal polynomials.

Fourier series and Fourier transform.

## Recommended readings

Lecture notes (available on ELLY) will be the main (self-consistent) reference.

However, students will bear in mind that there are many excellent books on the subjects the course will be dealing with, none of which will be the only reference. An incomplete list includes:

- V. Smirnov, Corso di Matematica superiore, vol.III,2 (MIR)

- E.Kolmogorov, S.Fomin, Elementi di teoria delle funzioni e dell'analisi funzionale (ER)

- F.G.Tricomi, Metodi Matematici della Fisica (Cedam)

- M.Spiegel, Variabili Complesse (Schaum, Etas)

- M.Spiegel, Analisi di Fourier (Schaum, Etas)

All these textbooks are available in the University libraries.

We will have a systematic reference for the subjects in linear spaces

- E. Onofri, Teoria degli Operatori lineari, http://www.fis.unipr.it/home/enrico.onofri/#Lezioni

## Teaching methods

Lectures and exercises (with students involved). Problems will be assigned to be worked out at home, mostly using the COMPITO function on ELLY. The lecture notes will be available on ELLY and they will be uploaded step by step, following the progress of the lectures. Students are strongly advised to check the uploaded lecture notes systematically, since those are the main reference for their study.

The two modules will be run in parallel, to let the students appreciate the logical and contents deep connections, in particular when the contents naturally merge (function spaces, Fourier analysis, orthogonal polynomials).

## Assessment methods and criteria

There will be an intermediate written test. That is mostly intended as self-evaluation, but if it is well done (and only in that case) it will be taken into account in the final grade (2 points added as a bonus).

Written and oral exams. The written exam is made of exercises (usually 3, covering both parts of the course and equally contributing to the final grade), intended to prove calculation ability; exercises are variants of the ones worked out during the course. The student will not be allowed to have books or notes with her/him. Written exam duration is 3 hours. Students are admitted to the oral exam with a grade of 16 or more. In case one exercise is badly failed, but a passing grade is got, an exercise on the same subject should be solved during the oral exam. Students will be invited to join a session in which the written exam exercises will be worked out in detail. Students will be be in any case contacted (via e-mail) individually for an overall account of the results of their written exam.

The oral exam will cover fundamental subjects, aiming at proving that the students master them. As a starting point, two subjects will be discussed: one will be assigned by the lecturer a few days before the exam takes place (this is usually connected to the outcome of the written exam), while the other one will be chosen by the student. The two subject must cover both analytic functions and linear spaces.

Both the written grade and the oral exam will contribute to the final grade, the written exam grade not setting a limit to the final grade.